Lectures on Mean Curvature Flow and Related Equations

Draft Version, recompiled August 1998

Tom Ilmanen

Former title: “Some Techniques in Geometric Heat Flow.” These notes were prepared in AMS-TEX for the Conference on Partial Differential Equations & Applications to Geometry, 21 August - 1 September, 1995, ICTP, Trieste. The author was supported in part by an NSF Postdoctoral Fellowship. ii

Lecture 1

Our aim in these lectures is to study singularity formation, nonuniqueness, and topological change in motion by mean curvature. The outline is:

Lecture 1 - Introduction Lecture 2 - Flows in R^{3} Lecture 3 - Flows in R^{3}, continued Lecture 4 - Nonuniqueness in Geometric Heat Flows

Let {M_{t}}_{t∈R} be an evolving family of hypersurfaces in R^{n}. We say that M_{t} is moving by mean curvature if it satisfies the nonlinear parabolic equation

(MCF)  \frac{∂}{∂t}x = H⃗(x),  x ∈ M_{t}, t ∈ R.

Here H⃗(x) is the mean curvature vector of M_{t} at x, defined to be ∑_{1}^{n−1} λ_{i}ν, where λ_{i} are the principal curvatures, and ν is the unit normal. The expression ∂x/∂t stands for the normal velocity of the surface.

Motion by mean curvature is the gradient flow of the area functional (with respect to the L^{2} norm on the surface). We have the following formula for the decrease of the area of M_{t},

\frac{d}{dt}H^{n−1}(M_{t}) = − ∫_{M_{t}} H^{2} dH^{n−1}.

Mean curvature flow arises as a simplified model for various physical processes in which surface tension plays a role.

Since (MCF) is a parabolic equation, it will have a short-term smoothing effect on the surface. However, the surface can become singular later.

The simplest example is a sphere. The sphere shrinks without changing shape since the curvature is the same all around. We find that (MCF) becomes

\frac{dR}{dt} = −\frac{n − 1}{R}, R(0) = R_{0},

where R = R(t) is the radius of the sphere, which yields

R(t) = \sqrt{R_{0}^{2} − 2(n − 1)t},

so the sphere disappears at time R_{0}^{2}/2(n − 1).

Let me mention another early result. Huisken [Hu1] proved that if M_{0} is a bounded convex surface, then M_{t} becomes more and more nearly spherical as it shrinks, and at the instant it vanishes, it is asymptotic to the shrinking sphere given above. (See also Hamilton [H8].)

If we could control the curvature of M_{t}, we could keep it from becoming singular. But by differentiating (MCF), we have the following equation for the evolution of the norm of the extrinsic curvature (see [Hu1]):

\frac{∂}{∂t}|A|^{2} = Δ_{M_{t}}|A|^{2} − 2|D_{M_{t}}A|^{2} + 2|A|^{4}.

Here A = (A_{ij})_{i,j=1}^{n−1} is the second fundamental form of M_{t} in R^{n}, |A| is its length, and Δ_{M_{t}}, D_{M_{t}} are intrinsic derivatives on the surface M_{t}. Note that ∂|A|^{2}/∂t means the derivative with respect to a “meterial point” moving perpendicularly to the surface. This equation resembles

u_{t} = Δu + u^{2}

which has solutions that blow up to infinity. This suggests that it will be difficult to control the singularities. This is borne out by the example of the sphere, where the motion accelerates as the sphere gets smaller.

Generally speaking, the singularities of motion by mean curvature can be very complicated.

The mean curvature flow shares many characteristics with other socalled geometric heat flows, such as the harmonic map flow, the equation u_{t} = Δu + u^{p}, the Yang-Mills heat flow, and the Ricci flow. In particular, many of the ideas presented in these lectures will apply to these other geometric heat flows. I will have more to say about this in the last lecture.

Main Questions. Here are some major issues in mean curvature flow; they are at various stages of development by a number of mathematicians.

1. Nature of singularity formation, especially in R^{3} The basic mechanism is self-similar shrinking, as in the case of the convex bodies becoming spherical, but there are other, more subtle mechanisms, see Angenent-Velazquez [AV1, AV2], Velazquez [V].

2. Weak solutions past the onset of singularities Two definitions of weak solution have gained attention in recent years:

• the level-set flow, based on the idea of viscosity solutions (tangency with smooth test functions), see [ES1-ES4], [CGG],

• Brakke’s flow, using the varifolds of geometric measure theory (integration by parts with the surface measure), see [B].

3. Partial regularity The best result so far is : almost every evolution is smooth almost everywhere [I3]. It is based on Brakke’s local regularity theorem [B], together with the level-set flow. Recent work of White [W2-W4,W6] points towards a deeper theory of partial regularity.

4. Selection principle, in case of nonunique evolution As we shall see, the mean curvature flow supports nonunique evolution (past the first singularity). Which evolution is better? Various minimization principles are suggestive [Am], [I3, I5]. Perhaps the selected evolutions would decrease area more rapidly and be more regular.

5. Role of random processes Recently there has been progress in making rigorous the stochastic origins of mean curvature flow, that is, various interacting particle systems related to the Ising model, see Katsoulakis-Souganidis [ ], Griffeath [ ], and others. Could this yield clues about which flow to select?

Lecture 2: Flows in R^{3}

The outline of this lecture is A - Self-Shrinking Surfaces B - Examples of Self-Shrinkers in R^{3} C - Nonuniqueness in R^{3} D - Level-Set Flow E - Topological Changes Past a Singularity F - Evolution of Cones in R^{3}

We state the following principle, discovered by Huisken [Hu2]:

Singularity formation is modelled by self-shrinking surfaces. This is already exemplified by the above-mentioned evolution of convex hypersurfaces. We will accept this principle without proof for the moment; but let us explain what we mean by shrinking self-similarly.

A. Self-Shrinking Surfaces

By counting units on either side of the equation, the reader will agree that (MCF) is invariant under parabolic rescaling

x → λx, t → λ^{2}t.

We look for solutions that are invariant under the same scaling, that is, we impose the ansatz

N_{t} = \sqrt{−t} · N, t < 0.

where \sqrt{−t} · represents a homothety.

Another way to motivate this ansatz is to insert the general expression a(t) · N_{t} into (MCF) to obtain separate equations for a and N. We solve for a to obtain \sqrt{−t} (there are other interesting solutions, which we ignore for now). We also obtain the following elliptic parametric equation for N:

(SS)  H + \frac{x · ν}{2} = 0,  x ∈ N.

Here H = H⃗ · ν. This says:

If the surface N begins to move by its mean curvature vector, this produces the same normal velocity as moving by the vector −x/2.

That is to say, the surface is shrinking by homothety. We call N, and N_{t}, a self-shrinking surface.

It turns out that (SS) is the Euler-Lagrange equation of a functional, discovered by Huisken [Hu2]. To find this, let us try a functional of the general form

∫_{N} f(x) dH^{n−1}(x),

where f is to be determined. Let X be a C^{1} vectorfield of compact support. Vary N in the direction of X to yield a family N_{s} defined by N^{s} = Φ^{s}(N), where Φ^{s}(x) = x + sX(x), s ∈ (−ε, ε) is a C^{2} family of diffeomorphisms. Let Df denote ordinary derivation in the ambient space, let Df^{T} be the orthogonal projection of Df onto T_{x}N, and let Df^{⊥} = Df − Df^{T}. Write div_{N} X for the divergence of X along N, that is, ∑_{i=1}^{n−1} D_{e_{i}}X · e_{i} where e_{1} …, e_{n−1} is an orthonormal basis for T_{x}N. We calculate the first variation in the direction X:

(1)

\frac{d}{ds}|_{s=0} ∫_{N^{s}} f(x) = \frac{d}{ds}|_{s=0} ∫_{N} f(Φ^{s}(x))|J_{T_{x}N}Φ^{s}(x)|
= ∫_{N} Df · X + f div_{N} X
= ∫_{N} Df · X + f div_{N}(fX) − Df^{T} · X
= ∫_{N} −fX · H⃗ + Df^{⊥} · X

This is called the weighted first variation formula. In the last line we used the usual first variation formula

(2) ∫_{N} div_{N} X = ∫_{N} −X · H⃗  (= \frac{d}{ds}|_{s=0} H^{n−1}(N^{s}))

which the reader will recognize as an integration by parts formula. From (1) we obtain the Euler Lagrange equation

fH − Df · ν = 0,

which becomes (SS) if we choose f = e^{−|x|^{2}/4}. Accordingly, we define

J[N] = ∫_{N} e^{−|x|^{2}/4} dH^{n−1}(x).

B. Example of Self-Shrinkers in R^{3}.

We now present some further solutions of (SS).

A well known self-shrinker is the cylinder S^{1} × R in R^{3}.

Figure. Cylinder

There is also a rotationally symmetric torus discovered by Angenent [A5] by ODE methods.

Finding further solutions of (SS) has been difficult since they are always unstable as critical points of J. All the other examples are recent and are based on numerical computations, without proof. The first algorithms and most of the examples are due to Chopp [C].

The idea is to start with a surface in R^{3} such as the following, evolve it by mean curvature, and rescale the singularities to find interesting examples of self-shrinkers – we hope.

Figure. A Surface

Typically, the singularities that form will consist of little necks pinching off; these are aymptotic to shrinking cylinders. The reason for this is that one shrinking handle gets a little smaller than the others, and then, because of the high curvature around the handle, the shrinking accelerates and outpaces the other handles.

The more handles, the more ways the necks can get out of balance, and the more unstable the self-shrinking surface will be, if it is ever found.

The easiest way to control the instabilities is to impose a group symmetry, which forces certain handles to shrink off simultaneously. Then we expect to see a new type of singularity.

For example, consider the eight-element group G = D_{2} ⋊ Z_{2}, where D_{2} is the dihedral group of order 4, and ⋊ indicates a semidirect product. G acts on R^{3} via the rigid motions

(x, y, z) → (−x, y, z),  (x, y, z) → (x, −y, z);  (x, y, z) → (−y, x, −z).

Consider the initial surface depicted in the figure, which is invariant under G.

Figure. Initial Surface

This topology was suggested by Grayson and Ilmanen. The group forces the two interlocked handles to pinch off simultaneously, if they pinch off at all. Evolving this on the computer, Chopp [C] finds that they do pinch off, and rescaling the surface so that the handle size remains constant, it converges to the surface N shown in the figure.

Figure. Doubly Punctured Saddle

Note that N is asymptotic to a certain cone (i.e. homothety invariant set) at spatial infinity. Equivalently, N_{t} ≡ \sqrt{−t} · N converges to a cone N_{0} as t → 0^{−}. We can think of the flow as pulling the surface toward the origin so fast that it streaks out and becomes a cone at the singular time.^{1}

We can look for analogous surfaces based on the group D_{m} ⋊ Z_{2} for any m ≥ 2. In the figure is a self-shrinker with m = 3. I found it using Brakke’s Evolver at the GANG lab at the University of Massachusetts, with technical advice from K. Brakke and J. Sullivan.

I also tried m = 9. The resulting surface is depicted in the figure. It looks like the union of a sphere and a plane (each a self-shrinker in its own right) desingularized along the curve of intersection by a high-frequency “zipper” of perforations that looks like the second Scherk surface (the socalled Scherk tower).

These surfaces are the only ones known (besides the sphere and cylinder) where the group symmetry by itself is enough to control the instabilities.

Figure. Scherk Surface

Now we pass to examples which require a mountain-pass argument in addition to the group symmetry. We give the example from Angenent-Chopp-Ilmanen [ACI], using the algorithm from Chopp’s paper [C]. Consider the initial surface depicted below, which is a cylinder with holes of a smaller radius bored through it. It is symmetric under the action of the group G = D_{4} × Z_{2}, where the eight-element dihedral group D_{4} acts on the y-z plane (without affecting x) and Z_{2} acts by (x, y, z) → (−x, y, z).

Because of the group symmetry, the holes pinch off symmetrically, but this time there are two topologically distinct extremes.

1It can be proven in full generality that as t → 0^{−}, any self-shrinker converges to a (unique) cone locally in the Hausdorff metric on closed sets. The proof uses the monotonicity formula of Lecture 3, section G; the proof is beyond the scope of the lectures.

Figure. Triply Punctured Saddle

(a) If the bored holes are very thin, they pinch off in four necks radiating from the origin like spokes of a wheel, leaving a pointy sphere surrounded by a dented cylinder. (After the sphere disappears, the cylinder develops a single neckpinch at the origin.)

(b) If the bored holes are very broad, then the center of the cylinder breaks in four longitudinal necks, with axes parallel to the axis of the cylinder, leaving two pointy stumps, which move away from each other.

Between these two extremes, some third behavior must occur. To find it, we foliated R^{3} with the level sets of a G-invariant function u whose zero-set is the above surface, and evolved these level-sets simultaneously.

Figure. Nonuply Punctured Saddle

Figure. Initial Surface

To recognize the level-set that develops the interesting singularity, define R(z, t) to be the radius of the radial necks divided by the radius of the longitudinal necks, for the level set {u(·, t) = z}.

At one extreme, R(z, t) → 0 as the radial necks pinch off. At the other extreme, R(z, t) → ∞ as the longitudinal necks pinch off. At each time t, select the level-set z for which R is changing the least, and make it the new zero-set. On this level-set, the two sets of necks are shrinking at around the same rate. Also, rescale the whole picture by a homothety so that the two radii remain roughly constant in size.

This creates a sequence of surfaces which, according to our computation, converges to the following self-shrinking surface N. It represents a singularity where all eight necks pinch off simultaneously. Note that N is asymptotic to a double-lobed cone C at infinity, and N_{t} ≡ \sqrt{−t} → N_{0} ≡ C as t → 0^{−}.

Figure. Four-Handle Surface

Figure. Cone at t = 0

Examples with similar topology should exist for any group G = D_{m}×Z_{2}, m ≥ 2. The case m = 2 was done in the original paper of Chopp [C].

Open Problem. Prove that the m-punctured saddle and the m-handle surface exist for m ≥ 2.

C. Nonuniqueness in R^{3}

We now attempt to continue the evolution of the four-handle surface past t = 0. We obtain

Result (Angenent-Chopp-Ilmanen). The four-handle surface evolves nonuniquely after t = 0.

Here is how we establish this. Observe that the cone C = N_{0} is the union of two subcones C^{−} = C ∩ {x ≤ 0} and C^{+} = C ∩ {x ≥ 0}, each of which is a graph over the y-z plane. By standard techniques for the evolution of Lipschitz graphs (see [EH1, EH2]), C^{−} and C^{+} have smooth forward evolutions P_{t}^{−}, P_{t}^{+} for t > 0, which are smooth graphs over the y-z-plane. Observe that by the strong maximum principle, P_{t}^{−} and P_{t}^{+} are disjoint after the first instant. Then the union

P′′_{t} = P_{t}^{+} ∪ P_{t}^{−}

is a forward evolution of the double cone N_{0} that continues the four-handle evolution. We call P′′_{t} the two-sheeted evolution.

Since P_{t}^{+} is a graph, it is the unique evolution for its initial condition N_{0}. It follows that P_{t}^{+} inherits the symmetry of the equation (MCF), namely

P_{t}^{+} = \sqrt{t} · P^{+}, t > 0,

for some smooth surface P^{+}. (The argument is this: the rescaled flow λ^{−1} · P_{λ^{2}t}^{+} solves (MCF) with the same initial condition N_{0}, so it must be equal to P_{t}^{+}.) The same argument works for P_{t}^{−}.

Therefore P′′_{t} = \sqrt{t} · P′′, t > 0. We call such an evolution self-similarly expanding. (This represents a second solution a(t) of the ODE refered to in section A.)

But there is also a second possible evolution: the two lobes of C can join up near 0 to form a surface of one sheet. Intuitively, this occurs because area decreases sharply by tranching a hole between the cones.

We will construct this one-sheeted evolution by a variational argument which shows that in a certain sense, the two-sheeted evolution is not “minimizing”. Inspired by P′′_{t}, we impose the self-expander ansatz

P_{t} = \sqrt{t} · P, t > 0.

We also require that P is asymptotic to C at infinity, so that P_{t} → C as t → 0^{+}.

By separation of variables as before, this leads to the elliptic equation

(SE)  H − \frac{x · ν}{2} = 0,  x ∈ P,

which says that P expands by homothety under (MCF). This is the Euler-Lagrange equation of the functional

K[P] := ∫_{P} e^{|x|^{2}/4} dH^{n−1}(x).

This functional, in contrast to the self-shrinking functional J, easily yields critical points by minimization; this comes from the convexity of e^{|x|^{2}/4}. In fact we have

Theorem (Ilmanen [I5]). For any closed, homothety invariant set C and any component E of the complement of C, there exists a surface P minimizing K with respect to compact replacements, such that P separates the set

\{x ∈ E : dist(x, ∂E) ≥ \sqrt{2(n − 1)}\}

from the set

\{x ∈ R^{n} \ Ē : dist(x, ∂E) ≥ \sqrt{2(n − 1)}\}.

This theorem is proven by minimizing K subject to the boundary Q_{R} ≡ ∂E ∩ ∂B_{R} to obtain a surface P_{R}, then passing a subsequence P_{R_{i}} → P as R_{i} → ∞. This is possible in the space of locally integral (n − 1)-currents (or locally finite perimeters) using the lower semincontinuity of K and the compactness theorem of DeGiorgi.

The “separation” condition implies in particular that the limit P is nontrivial, that is, the approximators P_{R} do not chase off to infinity as R → ∞. This is proven by suitable barriers. It corresponds to the fact that P_{t} cannot move farther than distance \sqrt{2(n − 1)} from ∂E before t = 1, because by the maximum principle, P_{t} cannot collide with the shrinking sphere ∂B_{R(t)}(x) before it disappears at t = 1, where R(t) = \sqrt{2(n − 1) − 2(n − 1)t}.

Because of the separation condition, P_{t} → ∂E as t → 0^{+} in the sense of Hausdorff distance of closed sets, so P_{t} is a forward evolution of ∂E.

Furthermore (by the regularity theory of DeGiorgi, Federer, Almgren, Simons) the singularities of P have codimension at least 7, so in particular P_{t} is smooth in R^{3}.

Next we will consider the evolution of the double cone D_{α} consisting of all points making an angle of α with the x-axis, 0 < α < π. We have

Proposition. For each n ≥ 3, there is a critical angle α_{crit}(n) ∈ (0, π/2) such that for α ≥ α_{crit}, D_{α} has both a two-sheeted and a one-sheeted evolution.

Proof Sketch. We have already constructed the 2-sheeted evolution P′′_{t}(α) of D_{α}. Write P′′(α) = P′′_{1}(α).

Suppose that α is very close to π/2. By applying the maximum principle with very large shrinking balls, we can prove that for every R > 0, ε > 0, there exists α(R, ε) < π/2 such that for all α ∈ (α(R, ε), π/2 we have

P′′(α) ∩ B_{R} lies within ε of the plane {x = 0}

Then by taking ε very small and α very close to π/2, we see that the two-sheeted P′′(α) cannot be a minimizer of K, because we can replace the two disks P′′(α) ∩ B_{R} by a narrow strip in ∂B_{R} connecting the edge of the disks, thereby greatly reducing K.

This shows P′′(α) ≠ P′(α), where P′(α) is the minimizer of K provided by the Theorem above.

Therefore D_{α} evolves nonuniquely. By further analysis, we can show that P′(α) really has only one sheet; for details see [ACI]. This completes the proof sketch.

To complete the analysis of the four-handle surface, we need a more quantitative result. Since D_{α} is rotationally symmetric, we may assume that P′(α) is rotationally symmetric, and reduce (SE) to an ODE. Using Mathematica, we plotted some solutions of this ODE for R^{3}, and obtained the following graphs. To obtain the self-expanding evolutions, rotate the graphs around the x-axis. (Notice that above the critical aperture, there are actually two one-sheeted evolutions of the same topology, but different geometry.)

Figure. Self Expanders Starting From D_{α} As a result of this numerical investigation, we found that

α_{crit}(3) ≈ 66.04^{∘}.

Returning to the computed four-handle surface, we find by inspection that the cone N_{0} has “aperture” around 72^{∘}, that is, the cone D_{72^{∘}} fits between N_{0} and the x-axis.

Employing the one-sheeted evolution for D_{α} as a barrier, we can then prove that there exists a one-sheeted evolution

P′_{t} = \sqrt{t} · P′, t > 0,

with initial condition N_{0}. The details of this argument are in [ACI]. This establishes (numerically) that the four-handle surface evolves nonuniquely after t = 0.

We ask: which is the preferred evolution? A good candidate is the minimizer of K, since it seems to decrease area fastest, in some weighted sense (this is a suggestion of K. Brakke). But how can we generalize this idea to arbitrary mean curvature flows?

One might ask: what is the probability of various forward evolutions? (Katsoulakis) One way to approach this is to randomize the initial condition or the equation and look at the probability of various outcomes. This seems susceptible to a Monte Carlo simulation.

D. Level-Set Flow.

Our purpose in sections D-F is to investigate the topological changes that can occur in the flow after the onset of singularities.

To do that, we need another tool, called the level-set flow, which is a sort of envelope for all the possible nonunique flows. The level-set flow is due to Chen-Giga-Goto [CGG] and Evans-Spruck [ES1-ES4], based on the idea of viscosity solutions [CIL]. The geometric formulation that I will present in this section appears in Ilmanen [I2], see also Soner [So].

The level-set flow will allow us to distinguish so-called “outermost” flows among all the possible evolutions of a given surface. They seem to be the simplest topologically.

We observe first that smooth flows M_{t} and M′_{t} obey the maximum principle, that is, if M_{t} and M′_{t} are disjoint, they remain disjoint, and in fact the distance between them is nondecreasing. Reason: at the point of closest approach, the curvature vectors cause the surfaces to pull slightly further apart.

We will now make this into a definition of a weak solution, by using smooth flows as test surfaces for a nonsmooth flow. The basic idea is this : the level-set flow is the largest flowing set that does not violate the maximum principle with smooth test flows.

Definition. A family {Γ_{t}}_{t≥0} of closed sets is a “subsolution” of the mean curvature flow provided that for every compact, smooth mean curvature flow {Q_{t}}_{t∈[a,b]},

if Γ_{a} ∩ Q_{a} = ∅, then Γ_{t} ∩ Q_{t} = ∅ for a ≤ t ≤ b.

It follows immediately, by using the translation invariance of (MCF), that dist(Γ_{t}, Q_{t}) is nondecreasing.

Proposition. For any closed set Γ_{0} there exists a unique maximal subsolution {Γ_{t}}_{t≥0} with initial condition Γ_{0}, that is,

(a) {Γ_{t}}_{t≥0} is a subsolution, and

(b) If {Δ_{t}}_{t≥0} is any subsolution with Δ_{0} = Γ_{0}, then Δ_{t} ⊆ Γ_{t} for all t ≥ 0.

This is proven by taking the closure of the union of all subsolutions. The maximal subsolution is called the level-set flow. It agrees with the definition given in [ES1] and [CGG]. We should remark that the level-set flow has the semigroup property.

The definition would be rather empty without verifying at least some properties in common with the classical mean curvature flow. Here is a basic theorem (for proof, see [I2] and [ES1]).

Theorem. (a) The level-set flow is equal to the smooth flow for as long as the latter exists.

(b) (Maximum Principle) Two level-set flows disjoint at time t = a remain disjoint for all times t ≥ a. (True by definition if one of them is smooth.)

(c) (Filling Property) Let u(·, 0) be a Lipschitz function. Then there is a Lipschitz function u(·, ·) such that for each z, the set Γ_{t}^{z} ≡ {u(·, t) = z} is moving by the level-set flow.

Property (c) explains the term “level-set flow”, and establishes the relationship with the work of Evans-Spruck and Chan-Giga-Goto. The main point of (c) is that no gaps develop between evolving sets that “foliate” space.

By definition, the level-set flow contains all possible mean curvature flows, so: Nonuniqueness of the mean curvature flow corresponds to “fattening” of the level-set flow.

For example, the flows P′_{t} and P′′_{t} for the four-handle surface are both subsolutions, so they lie within the corresponding level-set flow Γ_{t}.

By further analysis [ACI], we establish that for t > 0, Γ_{t} consists precisely of the region bounded by P′_{t} and P′′_{t}. (Γ_{t} is equal to N_{t} the four handle surface for t ≤ 0.)

More generally, let Γ_{0} be the zero-set of a Lipschitz function u(·, 0), and let u(x, t) be the evolution provided by the “filling” property. Then the two sets

M_{t}^{+} = ∂{u(·, t) > 0}, M_{t}^{−} = ∂{u(·, t) < 0},

are called “outermost” flows.^{2} 2In what sense do they satisfy the equation? Using the existence and compactness theorems of Ilmanen [I3], one can prove that each of M_{t}^{±} is the support of a Brakke flow.

In the case of fattening, the outermost flows are distinct; the two sides of the surface peel apart after the onset of fattening.

In the next section, we will see that outermost flows have some special topological properties.

Figure. Level-Set Flow of D_{α}

E. Topological Changes Past a Singularity

As background for the next section, we will say something about the possible topological changes that can arise during the flow. The results in this section are due to B. White.

Let {Γ_{t}}_{t≥0} be a level-set flow, and let E_{t} = R^{n} \ Γ_{t}. For a ≤ b, define

E[a, b] = ∪_{a≤t≤b}E_{t} × {t}.

The set E[a, b] is open in R^{n}×[a, b]. (Reason: E[a, b] = {u ≠ 0}∩R^{n}×[a, b]).

Theorem (White [W2]). The inclusions E_{a} ⊆ E[a, b], E[a, b] ⊇ E_{b} induce the following maps, which are injective, surjective and isomorphic as indicated.

(a) π_{0}(E_{a}) ≅ π_{0}(E[a, b]),

(b) π_{1}(E_{a}) ↠ π_{1}(E[a, b]),

(c) H_{n−2}(E[a, b]) ↩ H_{n−2}(E_{b}),

(d) H_{n−1}(E[a, b]) ≅ H_{n−1}(E_{b}).

The proof of this theorem uses the maximum principle very cleverly, together with technical tools from [I3]. See [W4] for the proof.

If n = 3, then combining (b) and (c) we obtain

H_{1}(E_{a}) ↠ H_{1}(E[a, b]) ↩ H_{1}(E_{b}), a ≤ b,

so we have

Corollary (White). In R^{3}, dim H_{1}(E_{t}) is nonincreasing.

Now let M_{t}^{+} be a compact outermost flow in R^{3}. Then M_{t}^{+} divides R^{3} into two open regions, U_{t}^{+} = {u(·, t) > 0} and U_{t}^{−} = R^{3} \ U_{t}^{+} \ M_{t}^{+}. If M_{t}^{+} happens to be smooth, then by the Mayer-Vietoris sequence and Poincare duality,

2 genus(M_{t}) = dim H_{1}(M_{t}^{+}) = 2 dim H_{1}(U_{t}^{+}),

which is nonincreasing since {R^{3} \ U_{t}^{+}}_{t≥0} is a level-set flow, as can be seen by applying the definition directly (or see [I1] for a proof). This shows:

The genus of a compact outermost flow is nonincreasing in time, whenever it is defined.

F. Evolution of Cones in R^{3}

Suppose that C is a cone in R^{3}, and consider its evolution. In this section we ask two questions:

(i) What is the topology of an outermost flow of C? (ii) Do all evolutions of C have to be self-similar?

(i) Topology. Let Γ_{t} be the level-set flow of C. Since it is unique, we deduce using the parabolic scaling of (MCF) and of C that Γ_{t} itself is self-expanding:

Γ_{t} = \sqrt{t} · Γ_{1}, t > 0.

Then the “outermost” flows will also be self-expanding, and so ∂Γ_{1} will solve the elliptic equation (SE), in some sense. We have the following result.

Proposition (Ilmanen [I5]). The boundary of Γ_{t} is smooth for t > 0.

Proof Sketch. We will show that we can approximate ∂Γ_{1} from outside by hypersurfaces that are stationary and stable for K.

Fix R > 0, ε > 0. Let S_{ε} = B_{R} ∩ ∂{x : dist(x, Γ_{1}) < ε}. Consider the minimization problem

(3)  min{K[P] : P ∈ I_{2}(R^{3}), ∂P = ∂S_{ε}, spt P ⊂⊂ R^{3} \ Γ_{1}},

where I_{2}(R^{3}) is the space of integral 2-currents in R^{3}.

Since {Γ_{t}}_{t≥0} solves (MCF) in the level-set sense and is self-expanding, it follows that Γ_{1} solves (SE) in a “viscosity” sense. Therefore, at least formally, Γ_{1} should be a barrier for equation (SE), and by the strong maximum principle, there should exist a minimizer of (3) that is disjoint from Γ_{1}.

In [ISZ, I3, I5], this is made precise, yielding a hypersurface P_{ε} ⊆ R^{3}\Γ_{1} with ∂P_{ε} = ∂S_{ε}, which is stationary and stable for K. It is smooth by the estimates of Schoen-Simon [ScS]. Passing ε → 0, the compactness theorem of [ScS] yields a smooth surface P_{R} ⊆ \overline{R^{3} \ Γ_{1}} with ∂P_{R} ⊆ ∂Γ_{1}.

Now \{Γ_{t}\}_{t≥0} is a level-set flow, and \{\sqrt{t} · P_{R}\}_{t≥0} is a smooth mean curvature flow with boundary lying in Γ_{t}. It follows that \{Γ_{t} ∪ (\sqrt{t} · P_{R})\}_{t≥0} is a “subsolution” as defined in section D. Then, since {Γ_{t}}_{t≥0} is maximal,

P_{R} ⊆ Γ_{1}.

Then ∂Γ_{1} ∩ B_{r} = P_{R} (it requires a short argument to see that every point of ∂Γ_{1} is approached by P_{ε}). Since R was arbitrary, ∂Γ_{1} is a smooth surface solving (SE), which completes the proof of the proposition.

We now turn to as simple result about the topology of ∂Γ_{t}. It is in the same vein as the above theorem of White (see also [W1]); we will be able to give a complete proof here.

Let M be a surface that is a boundary component of a three manifold K. We call M incompressible in K if every embedded curve in M that bounds an embedded disk in K, also bounds an embedded disk in M. By the Loop Theorem (Hempel [H]), it is equivalent to require that π_{1}(M) injects into π_{1}(K).

If M is not a closed surface, we define the genus of M to be the smallest genus of a closed surface containing M.

Proposition (Ilmanen-White). Let C ⊆ R^{3} be a cone, and let Γ_{t} be its level-set flow. For a.e. R > 0, each component of Γ_{t} ∩ B_{R} is a 2-disk or a 3-ball, and ∂Γ_{t} has genus zero and is incompressible in R^{3} \ Γ_{t}.

The import of the proposition is that the outermost flow of a cone is as simple as possible, topologically speaking. Since cones arise as spatial blowups of a singularity, this helps explain (infinitesimally) why the genus of the outermost flow is nonincreasing.

Example. Recall the double cone D_{α} of section C. If α is small, then topologically, Γ_{t} consists of two disks. If α is large, then Γ_{t} consists of two disks and an annulus. The annulus is incompressible in the complement of Γ_{t}.

Proof. Let γ be a loop in Γ_{1} ∩B_{R}. By classical results for minimal surfaces, γ bounds a smoothly immersed, K-minimizing disk D in B_{R} (using also the convexity of B_{R} with respect to K). Note that although it is merely immersed, \sqrt{t} · D is nevertheless a “subsolution”, except along its boundary. By the same maximality argument as above, D ⊆ Γ_{1}∩B_{R}. Therefore π_{1}(Γ_{1}∩ B_{R}) = 0.

By the previous proposition, for a.e. R > 0, each component K of Γ_{1}∩B_{R} is a 2-manifold with boundary or a topological 3-manifold with boundary. In the first case, K is a 2-disk. In the second case, K is a 3-ball with a finite number of interior balls removed. By maximality again, K is a 3-ball.

2. Thus ∂(Γ_{1} ∩ B_{R}) consists of spheres, so ∂Γ_{1} ∩ B_{R} has genus zero and ∂Γ_{1} has genus zero.

3. Now suppose that γ ⊆ ∂Γ_{1} bounds an embedded disk D′ ⊆ \overline{R^{3} \ Γ_{1}}. Then we may minimize K[D] in the space of integral 2-currents subject to

∂D = ∂D′, D ⊆ \overline{R^{3} \ Γ_{1}}.

Since ∂Γ_{1} is smooth and solves (SE), D is smooth and solves (SE), and either D ⊆ ∂Γ_{1} or D ∩ ∂Γ_{1} = ∂D ∩ ∂D_{1}.

By the maximality argument above, the former case holds, and ∂Γ_{1} is incompressible in \overline{R^{3} \ Γ_{1}}. This completes the proof.

By the nature of these methods, we can draw conclusions only about the outermost flows. What about the flows that lie strictly inside the level-set flow?

Genus Reduction Conjecture. Let M_{t} be any mean curvature flow in R^{3}. Then the genus of M_{t} strictly decreases at the moment of a singularity, unless the singularity is a neckpinch or shrinking sphere.

This covers genus reduction even for “interior” flows, and also says that, except for the cylinder and the sphere, a singularity consumes a certain amount of topology.

It includes the conjecture that every smooth self-shrinker of genus zero is the plane, cylinder, or sphere.

(ii) Self-Similarity. One might think that every evolution of a cone is self-similar. Actually, we expect non-self-similar evolutions of any cone that evolves nonuniquely. Here is a heuristic argument that can be made rigorous in some cases.

Suppose that P_{t}, P′_{t} are two distinct self-expanding evolutions of C. We may assume that either P_{1} or P′_{1}, say P_{1}, is unstable for the functional K, for otherwise we may construct an unstable critical point of K with initial condition C using the sweepout method of Pitts [Pi]; see also [ScS]. Then we construct a connecting orbit {M̃_{s}}_{s∈R} from P_{1} to some other critical point, say P′′_{1}, under the flow for K, namely

\frac{∂}{∂s}y = H⃗(y) + \frac{y}{2} , y ∈ M̃_{s}, s ∈ (−∞, ∞).

Under the change of coordinates t = e^{s}, x = \sqrt{t} y, M_{t} = \sqrt{t} · M̃_{s}, this equation becomes the ordinary mean curvature flow,

\frac{∂}{∂t}x = H⃗(x), x ∈ M_{t}, s ∈ (−∞, ∞).

Then M_{t} is close to P_{t} for t near 0 and close to P′_{t} for t near ∞, and is not self-similar.

This argument can be carried out rigorously for the nonunique examples in Lecture 4. In this way we obtain a huge number of non-self-similar evolutions of cones.

Here is a proposal for a rather spectacular example of the failure of selfsimilarity, due to B. White. Let M_{0} consist of the union of two coordinate planes, dividing R^{3} into four quadrants labeled I, II, III, IV in order around the axis. Consider a smooth approximator M_{0}^{ε} to M_{0} constructed as follows. Suppose the singular set of M_{0} is the z-axis. Along the line segments 2k < z < 2k + 1, make narrow slits that connect I to III, and along the line segments 2k + 1 < z < 2k + 2, make narrow slits that connect II to IV.

Figure. Starting Surface M_{0}^{ε}.

Consider the evolution M_{t}^{ε} of M_{0}^{ε}. It can be proven that it converges to the second Scherk surface as t → ∞. (See White [W5] for a similar argument.) This Scherk surface, known as the “Scherk tower”, is a minimal surface of the same topology and periodic symmetry as the above.

We speculate: as ε → 0, M_{t}^{ε} converges to a surface M_{t} such that M_{t} is smooth for t > 0 and approaches the Scherk surface as t → ∞.

This example leads to many questions: what if we vary the spacing of the holes? For example, what if the holes are invariant under a discrete group of homotheties, rather than translations? Or: if there are a finite number of slits, can we get a self-expander corresponding to a “finite Scherk tower” (Ilmanen-Kusner)? All of these questions are subject to computer investigation.

These examples leave open the possibility that the following conjecture might be true. It is motivated by the analogous result for backward blowups, presented in the next section, which follows from a monotonicity formula.

Conjecture. For any mean curvature flow M_{t} (including singular) and any point (x_{0}, t_{0}), the forward blowups defined by

M_{t}^{λ} = λ^{−1} · M_{t_{0}+λ^{2}t}, t > 0,

converge subsequentially to self-expanders.

Remark. There is a “forward” monotonicity formula involving e^{|x|^{2}/4}, valid on any surface with self-expanding boundary as follows,

∂M_{t} = Q_{t} where Q_{t} = \sqrt{t} · Q, t > 0.

However, this formula doesn’t seem to be much use for proving the above conjecture, since the integral is infinite on the forward evolution of any cone.

Lecture 3: Flows in R^{3}, continued

The outline of this lecture:

G - Monotonicity Formula H - Blowup Theorem in R^{3}

G. Monotonicity Formula

We will now discuss the precise meaning of Huisken’s principle: Singularity formation is modelled by self-shrinking surfaces.

That is, if we rescale a singularity parabolically, we obtain (subsequentially) a self-shrinking surface.

First we present the monotonicity formula of Huisken [Hu]. For motivation, we mention that if u solves the ordinary heat equation u_{t} = Δu and ϕ is a backwards heat kernel, then

\frac{d}{dt} ∫_{R^{n}} ϕ|Du|^{2} = ∫_{R^{n}} −2|Δu|^{2} ≤ 0

as the reader may easily calculate.

Let ρ_{x_{0},t_{0}}(x, t) be a backwards heat kernel defined on R^{n}, but with the scaling appropriate to R^{n−1}, that is,

ρ_{x_{0},t_{0}}(x, t) = \frac{1}{(4π(t_{0} − t))^{(n−1)/2}} e^{−|x−x_{0}|^{2}/4(t_{0}−t)}, t < T, x ∈ R^{n}.

Monotonicity Formula (Huisken). For any mean curvature flow M_{t}, 0 ≤ t < T, and any (x_{0}, t_{0}) ∈ R^{n} × [0, T], we have the decreasing quantity

\frac{d}{dt} ∫_{M_{t}} ρ_{x_{0},t_{0}}(x, t) H^{n−1}(x)

= ∫_{M_{t}} −ρ_{x_{0},t_{0}}(x, t) |H⃗(x, t) + \frac{(x − x_{0}) · ν(x, t)}{2(t_{0} − t)}|^{2} dH^{n−1}(x),

for 0 ≤ t < t_{0}.

There are well-known analogues for minimal surfaces (see Simon [S1]), for the harmonic map heat flow (Struwe [St]), the equation u_{t} = Δu + u^{p} (Giga-Kohn [GK]), and the Yang-Mills heat flow (Price [P]).

Proof. Assume that (x_{0}, t_{0}) = (0, 0) and write ρ = ρ_{0,0}(x, t). First we use the weighted first variation formula (1) with X = H⃗, then we use the regular first variation formula (in the “wrong” direction!) with X = Dρ to complete the square. Thus we derive for t < 0,

\frac{d}{dt} ∫_{M_{t}} ρ = ∫ −ρH^{2} + Dρ · H⃗ + ρ_{t}

= ∫ −ρ^{2} + 2Dρ · H − \frac{(Dρ · ν)^{2}}{ρ} + (div_{M} Dρ + \frac{(Dρ · ν)^{2}}{ρ} + ρ_{t})

= ∫ −ρ |H⃗ − \frac{x · ν}{2t} |^{2} + 0

because, by a miracle that may be verified by direct calculation, the quantity div_{M}(Dρ) + (Dρ · ν)^{2}/ρ + ρ_{t} vanishes identically when we choose ρ to be the backwards Gaussian above. (The reader might be interested in finding other functions ρ with this property.) This proves the monotonicity formula.

Define

Φ_{(x_{0},t_{0})}(t) = ∫_{M_{t}} ρ_{x_{0},t_{0}}(x, t) H^{n−1}(x).

Observe that Φ is a unitless quantity that measures the local area of M_{t} at time t and scale \sqrt{t_{0} − t}, smeared out by the Gaussian.

The monotonicity formula yields two consequences: first, Φ_{x_{0},t_{0}}(t) decreases to a well-defined limit as t → t_{0}; second, the quantity on the righthand side must approach zero in some sense. The reader will notice that this quantity is precisely the one that vanishes on self-shrinkers. Let us think how we can take advantage of this.

Assume that for some C > 0,

(4)  H^{n−1}(M_{0} ∩ B_{R}(x)) ≤ CR^{n−1},  R > 0, x ∈ R^{n}.

This local area bound is obviously satisfied for compact, smooth initial surfaces and all reasonable noncompact ones.

By the monotonicity formula, it follow that

(4′)  H^{n−1}(M_{t} ∩ B_{R}(x)) ≤ C,  R > 0, x ∈ R^{n}, t > 0,

(with a larger C) and

(5′)  ∫_{0}^{t_{0}} ∫_{M_{t}} ρ_{x_{0},t_{0}} |H − \frac{(x − x_{0}) · ν}{2(t_{0} − t)} |^{2} dH^{n−1}(x) dt < ∞.

Define the rescaled flow for any λ > 0,

M_{t}^{λ} = λ^{−1} · (M_{t_{0}+λ^{2}t} − x_{0}), −t_{0}/λ^{2} ≤ t < 0.

where − x_{0} represents translation and λ^{−1} · represents homothety. By parabolic invariance, {M_{t}^{λ}}_{t∈[−t_{0}/λ^{2},0)} is also a mean curvature flow. The quantities in (4′) and (5′) are scale invariant, so we obtain

(4′′) H^{n−1}(M_{t}^{λ} ∩ B_{R}(x)) ≤ C,  R > 0, x ∈ R^{n}, −t_{0}/λ^{2} ≤ t < 0,

and

(5′′)  ∫_{−100}^{0} ∫_{M_{t}^{λ}} ρ |H − \frac{x · ν}{−2t} |^{2} → 0  as λ → 0.

Therefore, in a formal sense, we get H + x · ν/2 = 0 in the limit, and M_{t}^{λ} converges (at least subsequentially) to a self-shrinker N_{t}. Formally, this justifies the slogan “developing singularities are modeled by self-shrinkiers”, which we used as motivation for the computer searches.

We call this limit surface, if it exists, a blowup of M_{t} at (x_{0}, t_{0}).

Example. The blowup of a shrinking convex surface at its unique point of disappearance is a self-shrinking sphere [Hu1].

Blowups always exist in the weak sense of Brakke (moving varifolds); see [I6] or White [W2].

Our aim in the next section is to prove that the blowups exist classically for surfaces in R^{3}.

H. Blowup Theorem in R^{3}.

In this section we will establish that in R^{3}, a blowup surface (at the first singular time) is smooth. This theorem might help to prove rigorously that the self-shrinkers of Chopp actually exist, and (if pushed a little further) could help in improving the partial regularity theory in R^{3}.

R^{3} Blowup Theorem. In R^{3}, if the evolving surface is embedded, satisfies (4), and has bounded genus, then the blowups are smooth (though not necessarily the convergence).

Remarks. 1. There is a version of this theorem, with a discrete set of singularities, for immersed surfaces in R^{3}, and for two dimensional surfaces in higher dimensions.

2. It is not know whether the full blowup sequence converges.

Sketch of Proof. The proof mostly involves known techniques in the compactness theory of 2-dimensional surfaces, with a few twists. For general background, see Sacks-Uhlenback [SaU], Schoen-Yau [SY], Choi-Schoen [CS], Gulliver [Gu1, Gu2], and many others; see also Ecker [E].

Let M_{t}^{λ} be the rescaled surface. Write M^{λ} = M_{−1}^{λ}. We will show that a subsequence of {M^{λ}}_{λ>0} passes to a smooth limit. Let us fix R > 0 and prove this in B_{R}. It may later be extended to R^{3} by diagonalization.

1. First, let accomplish what we can just using the monotonicity formula and general theorems of geometric measure theory; see [S1] for background.

By (4′′) and (5′′), there exists a sequence λ_{i} → 0 such that

(6) H^{n−1}(M^{λ_{i}} ∩ B_{R}) ≤ C,  H + x · ν/2 → 0  in L^{2}(M^{λ_{i}}).

After choosing a subsequence, there exists a Radon measure μ in B_{R} satisfying

μ = lim_{i→∞} H^{2}⌊M^{λ_{i}} ∩ B_{R}.

By Allard’s Compactness Theorem [S1] and the bound on ∫ H^{2}, the limit measure μ will be integer 2-rectifable, that is

dμ = θ(x) dH^{2}⌊X,

where X is an H^{2}-measurable, 2-rectifiable set and θ is an H^{2}⌊X-integrable, integer valued “multiplicity function”. The multiplicity function measures how the limit surfaces pile up. (Equivalently, μ is the mass measure of an integer rectifiable 2-varifold.)

In particular, μ has an approximate tangent plane almost everywhere, that is, for μ-a.e. x there exists a blowup measure ν defined by

ν(A) = lim_{σ→0} σ^{−2}μ(x + σ · A) for A ⊆ R^{2},

such that ν = kH^{2}⌊P for some 2-plane P and integer k ≥ 1. We will use T_{x}μ to denote the plane P whenever it exists. In particular, it is possible to express divergence-form equations weakly on μ.

Also by Allard’s Compactness Theorem, the convergence takes the stronger form

∫ ϕ(x, T_{x}μ) dμ(x) = lim_{i→∞} ∫_{M^{λ_{i}}} ϕ(x, T_{x}M^{λ_{i}}) dH^{2}(x)

for every test function ϕ : R^{3} × G_{2}(R^{3}) → R, where G_{2}(R^{3}) is the Grassmannian of 2-planes in R^{3}.

We define H for μ by taking the first variation formula (2) as a definition, and applying the Radon-Nykodym differentiation theorem. The quantity ∫ |H| is lower semicontinuous under weak convergence, and using the above displayed formula it is not hard to show that ∫ |H + x · ν|^{2} is lowersemicontinouous as well. Therefore μ solves (SS) weakly, that is,

∫ − div_{T_{x}μ} X + \frac{x · (T_{x}μ)^{⊥} · X}{2} dμ = 0

for all test vectorfields X ∈ C_{c}^{1}(B_{R}, R^{3}), where (T_{x}μ)^{⊥} represents projection onto the normal space of T_{x}μ. In particular H ∈ L^{∞}(μ).

We would then like to invoke Allard’s local regularity theorem in the form:

Allard’s Theorem (see [S1]). There are constants c > 0, ε_{0} > 0 depending on n with the following property. Let μ be an integer rectifiable Radon measure in B_{r}, and assume

|H(x)| ≤ M μ-a.e. x ∈ B_{r},

r ≤ c/M, 0 ∈ spt μ, and

μ(B_{r}) ≤ (1 + ε_{0})πr^{2}.

Then spt μ∩B_{r}/2 is a C^{1,α} manifold (in fact a graph over a domain in some plane).

By a bootstrap process, we would then obtain that spt μ∩B_{r}/2 is smooth and (SS) is satisfied classically.

2. Unfortunately, we cannot immediately apply Allard’s Theorem because the second hypothesis, which says that μ has roughly one layer, may not be satisfied.

For minimal surfaces, the archetypical counterexample is the catenoid. It satisfies H = 0, but on a large scale, there are roughly two layers. The catenoid is not a C^{1,α} graph over any plane because of the hole, which we can make as small as we wish by scaling.

Figure. Catenoid

To get around this difficulty we use special tricks that work only for two dimensional surfaces. The proof is completed by the two following lemmas. Let A = (A_{ij}) denote the second fundamental form.

Lemma 1. The ∫ H^{2} estimate yields an ∫ |A|^{2} estimate on a smaller ball.

Lemma 2. If M_{i} is a sequence of smooth, embedded surfaces in B_{R} with

H^{2}(M_{i} ∩ B_{R}) ≤ C,  H = f_{i} + g_{i},  ∫_{M_{i}∩B_{R}} |A|^{2} ≤ C,

where f_{i} is bounded in L^{∞} and g_{i} → 0 in L^{2}, then a subsequence converges to a Radon measure μ in B_{R} such that N = spt μ is a C^{1,α} manifold with H ∈ L^{∞}. The convergence may be badly behaved at a finite number of “concentration points” of |A|^{2}.

Lemma 1 is based on the Gauss-Bonnet formula with a cutoff function. Lemma 2 uses a well known concentrated-compactness argument in the

theory of surfaces, together with removal of point singularities. It amounts to the following: a singular point can only occur where topology concentrates; in any ball where this happens, ∫ |A|^{2} has a certain minimum value ε_{0}.

Our implementation runs as follows: away from concentration points of |A|^{2}, we use a Lemma of Simon [S2] to make each component of M_{i} ∩ B_{r}(x) very close to a plane, then pass to limits and apply Allard’s Regularity Theorem separately to the limit of each component.

An alternative is to use the weak |A|^{2} theory of Hutchinson [Hut1, Hut2]. One could also adapt the classical theory of conformally parametrized surfaces, or the blowup argument of Choi-Schoen [CS].

Proof of Lemma 1. We will prove the localized Gauss-Bonnet estimate: for any surface M immersed in R^{3} and any ε > 0,

(7)  (1 − ε) ∫_{M∩B_{1}} |A|^{2} ≤ ∫_{M∩B_{2}} H^{2} + 8πg(M ∩ B_{2}) + \frac{96πD}{ε} ,

where

D = sup_{r∈[1,2]} \frac{H^{2}(M ∩ B_{r})}{πr^{2}}.

Scaled to radius R, this will establish Lemma 1.

After proving this estimate by a convoluted argument involving curveshortening on M (!) and tube-counting, the author consulted with R. Gulliver, R. Kusner, T. Toro, B. White, and R. Ye, whose suggestions led to the following, much simpler proof.

The Gauss curvature of M is given by

2K = H^{2} − |A|^{2},

since K = λ_{1}λ_{2}, H = λ_{1} + λ_{2}, |A|^{2} = λ_{1}^{2} + λ_{2}^{2}. Then according to the Gauss-Bonnet formula with boundary, we have for a.e. r > 0,

∫_{M∩B_{r}} |A|^{2} = ∫_{M∩B_{r}} H^{2} − 4πχ(M ∩ B_{r}) + 2 ∫_{M∩∂B_{r}} k̃,

where k̃ is the geodesic curvature of the curve M ∩ B_{r} in M. The Euler characteristic is given by

χ(M ∩ B_{r}) = 2c(M ∩ B_{r}) − 2g(M ∩ B_{r}) − h(M ∩ B_{r}),

where c(M ∩ B_{r}) is the number of components of M ∩ B_{r}, g(M ∩ B_{r}) is the genus of M ∩ B_{r} (after capping off the boundary components by disks), and h(M ∩ B_{r}) is the number of components of M ∩ ∂B_{r}. Therefore

∫_{M∩B_{r}} |A|^{2} = ∫_{M∩B_{r}} H^{2} − 8πc(M ∩ B_{r}) + 8πg(M ∩ B_{r})
+ 4πh(M ∩ B_{r}) + 2 ∫_{M∩∂B_{r}} k̃.

The c term is negative and the g term is controlled by hypothesis. The two remaining terms on the right-hand side are essentially boundary terms. They can be unbounded when transversality is breaking down. We will control them by averaging over r ∈ [1, 2].

The following lower bound on curvature helps us to control the number of components.

Lemma 3 (Borsuk [Bo], Milnor [ ]). For any curve γ in R^{n},

∫_{γ} |k| ≥ 2πh(γ),

where k is the geodesic curvature of γ in R^{n}.

Next we calculate the curvature of γ.

Lemma 4. If γ is the transverse intersection of M^{2} and N^{n−1}, then

|k| ≤ \frac{|A_{M}(γ̇, γ̇)| + |A_{N}(γ̇, γ̇)|}{sin α},

where α is the angle between M, A_{M} and A_{N} are the second fundamental forms, and γ̇ is the unit tangent vector.

We leave the proof of Lemma 4 to the reader. Applying Lemma 4 we obtain for γ = M ∩ ∂B_{r},

|k| ≤ \frac{|A_{M}| + 1/r}{sin α} .

We continue the main estimate. Set ϕ = ϕ(|x|), a cutoff function to be specified below. By the above and Fubini’s Theorem,

∫_{M} ϕ|A|^{2} dH^{2} = ∫_{0}^{1} ∫_{M∩{ϕ>t}} |A|^{2} dH^{2} dt

= ∫_{0}^{1}( ∫_{M∩{ϕ>t}} H^{2} dH^{2} − 8πc(t) + 8πg(t) + 4πh(t)

+ 2 ∫_{M∩{ϕ=t}} |k̃| ds)dt

≤ ∫_{M} ϕH^{2} + 8πg(M ∩ B_{2}) + ∫_{0}^{1} ∫_{M∩{ϕ=t}} (2|k| + 2|k|) ds dt.

Here c(t) is the number of components of M ∩ {ϕ > t}, etc. In the last line we applied Lemma 3, and used the fact that |k̃| ≤ |k|. Let D_{M}ϕ be the tangential derivative T_{x}M · Dϕ, and note that |D_{M}ϕ| = | sin α||Dϕ|. Using Lemma 4,

(last term) = ∫_{M∩B_{2}} 4|D_{M}ϕ||k|  by the co-area formula

≤ ∫_{M∩B_{2}} 4(|Dϕ| sin α) (\frac{|A_{M}| + 1}{sin α})

= ∫_{M∩B_{2}} 4|Dϕ|(|A_{M}| + 1)

≤ ∫_{M∩B_{2}} \frac{4|Dϕ|^{2}}{εϕ} + εϕ|A_{M}|^{2} + 4|Dϕ|

≤ ∫_{M∩B_{2}} C/ε + C + εϕ|A_{M}|^{2}, where we choose ϕ to be a cutoff function for B_{1} inside B_{2} with |Dϕ|^{2}/ϕ ≤ C, |Dϕ| ≤ C. Plugging back into the main formula, we get

(1 − ε) ∫ ϕ|A|^{2} ≤ ∫ ϕH^{2} + 8πg(M ∩ B_{2}) + \frac{C}{ε} H^{2}(M ∩ B_{2}).

Select the constants more carefully to get (7). This complete the proof of Lemma 1.

Proof of Lemma 2. Write μ_{i} = H^{2}⌊M_{i}. Using (6) and Lemma 1 to bound ∫_{M_{1}∩B_{R}} |A|^{2}, pick a subsequence such that |A|^{2} dμ_{i} converges to a Radon measure σ. Define

X = {x : σ({x}) > ε_{2}}

for ε_{2} > 0 to be chosen. These are called “concentration points”. Let N = spt μ. Then:

(a) X is a finite set.

(b) We will show that N \ X is smooth. For each x ∉ X, there is r = r(x) > 0 such that ∫_{M^{λ_{i}} ∩B_{r}(x)} |A|^{2} < ε_{2} for all sufficiently large i. We now invoke Simon’s Lemma [S2], which implies the following.

Lemma. For any C > 0, there is ε_{2} = ε_{2}(C, n) with the following property. Let M be a smooth 2-manifold properly immersed in B_{r} ⊆ R^{n}, and suppose

∫_{M∩B_{r}} |A|^{2} ≤ ε ≤ ε_{2}, H^{2}(M ∩ B_{r}) ≤ Cr^{2}.

Then for each y ∈ M ∩ B_{r/4}, there is a 2-plane H such that

M(y) ⊆ {x : dist(x, H) ≤ δ_{1}}, H^{2}(M(y)) ≤ (1 + δ_{2})π(r/2)^{2}

where M(y) is the component of M ∩ B_{r/2}(y) containing y, and δ_{1}, δ_{2} → 0 as ε → 0.

The original Lemma is more detailed: M ∩ B_{r′(y)} is a topological disk for suitable r′ ∈ [1/2, 3/4], and except for a small set, M(y) is a Lipschitz graph over a domain in H, with estimates.

The reader should think of this as a kind of W^{2,2} → W^{1,∞} Sobolev embedding theorem that just barely fails. That is,

∫ |A|^{2} ∼ ∫ |D^{2}u|^{2}, sup |ν − ν_{H}| ∼ ∥Du∥_{L^{∞}},

where ν_{H} is the normal to H. Now sup |ν − ν_{H}| just fails to be bounded, so M(y) just fails to be a graph.

Now let y_{i} ∈ M_{i} be any sequence converging to x, and let μ̃ be a subsequential limit of H^{2}⌊M(y_{i}). By previous considerations, μ̃ is a weak solution of (SS) in B_{r/2}(x). Using the Lemma, Allard’s Theorem applies to μ̃ (after making sure that r ≤ c/M) and so spt μ̃ ∩ B_{r/4}(x) is smooth. By the monotonicity formula for bounded |H| (see [S1]), μ̃(B_{r/2}(x)) ≥ c(r/2)^{2}. Therefore the number of components of M_{i} that approach x is bounded, and we may apply this process simultaneously to them all to conclude that N is a finite union of smooth surfaces near x.

Since M_{i} is embedded, by the strong maximum principle we see that these surfaces are disjoint or equal, so N is smooth near x. This shows that N \ X is smooth.

(c) Since |H| is bounded on N \ X, the concentration points are removable singularities by Gulliver [Gu1], [Gu2].

This completes the proof of Lemma 2 and of the Blowup Theorem.

The Blowup Theorem leaves open the possibility that M^{λ_{i}} converges to μ several layers deep, that is,

dμ = k dH^{2}⌊N

where N is smooth, but k > 1.

Even worse, near the concentration points there might be small connections between the layers: little catenoid-like necks, or more complicated structures.

Figure. Bad Convergence Conjecture. M^{λ_{i}} converges smoothly to N, with one layer.

The main difficulty is that the concentration points might skittle around on the surface of N in an uncontrolled way.

If we could prove this conjecture, it would follow that

Corollary. Every end of a self-shrinker is asymptotic either to a smooth cone or to a self-shrinking cylinder.

To be more precise: let N be a self-shrinker, then there exists R > 0 such that N \ B_{R} decomposes into a finite number of ends U_{j}, such that for each j, either

(a) As λ → 0, λ · U_{j} converges locally smoothly to a cone C_{j} such that C_{j} \ {0} is smooth.

(b) There is a vector v_{j} such that U_{j} − τv_{j} converges to the cylinder \{x : dist(x, span (v_{j})) = \sqrt{2}\} as τ → ∞.

Remark. It follows that N_{t} = \sqrt{−t} · N converges in the sense of Hausdorff distance to N_{0} = (∪C_{j}) ∪ (∪{τv_{j} : τ ≥ 0}.

The proof of this Corollary is beyond the present notes. The basic idea is to apply the Blowup Theorem (in its conjectured strong form) to the self-shrinker N_{t} at a point x = x_{0}, t = 0.

Conjecture (Rigidity of Cylinder). If one end of N is asymptotic to a cylinder, then N is isometric to a cylinder.^{3}

3Angenent has proposed a likely counterexample: the union of a plane and a cylinder, desingularized along the circle of intersection by tiny holes modelled on the Scherck surface.

Lecture 4: Nonuniqueness in Geometric Heat Flows

We wish to present some rigorous examples of nonuniqueness in higher dimensions,^{4} namely R^{4} through R^{7}. This work is due to Angenent-Velaz-quez-Ilmanen, to appear in [AIV]. The lecture has five parts:

A - Overview B - Cones C - Forward Time D - Past Time E - Reduction of Complexity

We would like to thank K. Brakke, D. Chopp, R. Kusner, R. Mazzeo, M. Paolini, R. Schoen, B. White, and others for valuable discussions.

A. Overview. Remarkably enough, the nonuniqueness phenomenon we are about to describe holds almost identically for several other so-called geometric heat flows, namely the harmonic map heat flow, the equation u_{t} = Δu + u^{p}, and (under investigation) the Yang-Mills heat flow. Let us present these.

1. The harmonic map heat flow is the equation

\frac{∂u}{∂t} = Δu + |Du|^{2}u, x ∈ R^{n}, t ∈ R,

where u takes values in the unit sphere S^{n} = {|u| = 1} in R^{n+1}. It is the gradient flow for the Dirichlet energy ∫ |Du|^{2} dx subject to this constraint. See [ESa], [Ch], [CS], [St].

2. The semilinear heat equation

\frac{∂u}{∂t} = Δu + u^{p}, x ∈ R^{n}, t ∈ R,

which is the gradient flow for the energy ∫ |Du|^{2}/2−u^{p+1}/(p+1). See [GK], etc. 4Of course, any example may be crossed with R^{k}.

3. The Yang-Mills heat flow is the parabolic equation

\frac{∂}{∂t}∇ = −2 div F_{∇}, x ∈ R^{n}, t ≥ 0.

Here ∇ is an affine connection, F_{∇} = (F_{ijβ}^{α}) = −∇_{i}∇_{j} + ∇_{j}∇_{i} is its curvature, and (div F)_{iβ}^{α} = ∇_{j}F_{ijβ}^{α}. This is the gradient flow for the Yang-Mills functional energy ∫ |F_{∇}|^{2}/2. See [JT].

Ricci Flow. These equations (except the u^{p} equation) were grouped together by Hamilton in his 1986 ICM lecture [H1] because of their geometric character and many deep analogies between them. This class also includes the Ricci flow

\frac{d}{dt}g_{ij} = −2R_{ij}, x ∈ M, t ≥ 0,

where g_{ij} is an evolving Riemannian metric on the manifold M and R_{ij} is its Ricci curvature. The Ricci flow has been the subject of intensive study by Hamilton [H2-H7] and others. A natural question is: can there be nonuniqueness for the Ricci flow, after the onset of singularities? This question does not yield to the method of analysis presented in this lcture. We will attempt to resolve these difficulties in future work.

The method (in modified form) also applies to nonlinear wave equations, such as the wave map equation u_{tt} + |u_{t}|^{2} = Δu + |Du|^{2}u; see Shatah-Tahvildar-Zadeh [ST].

Main Results. We start with a simple example, which appears in the back of Brakke’s book [B]. It is a network with triple junctions moving by mean curvature in the plane, which shrinks self-similiarly to a cross, and then expands self-similiarly – in more than one way. The angles at the junctions are maintained at 120^{∘} except when junctions merge.

Figure. Nonuniqueness in the Plane

However, we desire an example which is smooth, except at the point x = 0, t = 0. Inspired by Brakke’s example and the four-handle example of Lecture 2, we impose the following ansatz, which says that the flow is invariant under parabolic rescaling:

(a) For t < 0, the solution is self-shrinking.

(b) At t = 0, the solution becomes a cone C.

(c) For t > 0, the solution is self-expanding. To enable a rigorous analysis, we also impose:

(d) The solution is invariant under rotation by the group SO(p)×SO(q) acting on R^{n} = R^{p} × R^{q}, p, q ≥ 2.

We have expressed this ansatz for (MCF); for the other equations, “cone” should be replaced by “homothety invariant map” and SO(p)×SO(q) should be replaced by SO(n).

This ansatz has the effect of reducing (MCF) to an ODE that can be analyzed. Our principal result is the following.

Theorem [AIV]. The following equations have solutions satisfying (a)-(d), which evolve nonuniquely after an isolated singularity at x = 0, t = 0.

(i) The mean curvature flow, for hypersurfaces in R^{4} through R^{7}. (ii) The harmonic map heat flow, with domain R^{3} through R^{6}. (iii) The equation u_{t} = Δu + u^{p} in R^{n}, in the supercritical range

n ≥ 3, p_{crit} < p < p_{+},

where p_{crit} = (n + 2)/(n − 2) is critical for the Sobolev inequality, and p_{+} is given by

p_{+} = \{ \begin{matrix}∞ & n ≤ 10 \\ \frac{n^{2} − 8n + 4 + 8\sqrt{n − 1}}{(n − 10)(n − 2)} & n ≥ 11 \end{matrix}

Remarks. 1. The explanation of these mysterious ranges is the existence of a finite energy, unstable, static “cone” (i.e. homothety invariant solution). We shall say more about this below.

2. Nonuniqueness for motion by curvature with a time-varying driving term was discovered by Soner-Souganidis [SS]. This result was refined to the equation ∂x/∂t = H⃗ + ν by Bellettini-Paolini [BP].

3. Troy [T], Budd-Qi [BQ] and Lepin [L] studied the self-shrinking solutions for the u^{p} equation.

4. Coron [Co] found nonuniqueness for harmonic map heat flow (with singular initial data). Our examples satisfy the weak form of the harmonic map heat flow not only with respect to the usual range variations, but also with respect to domain variations. In particular they satisfy the monotonicity formula and usual energy inequality. See Chen-Li-Lin [CLL], Feldman [F], Freire [Fr]. Assuming these latter inequalities, Freire shows uniqueness in dimension n = 2.

For simplicity, we will restrict ourselves to discussing the mean curvature flow. We now start a gradual buildup to this theorem.

B. Cones. For each p, q ≥ 2 with p+q = n, there is a rotationally invariant stationary hypercone C_{p,q} defined by

C_{p,q} = {x = (y, z) ∈ R^{p} × R^{q} = R^{n} : |y|^{2}/(p − 1) = |z|^{2}/(q − 1)}.

The reader may verify by direct calculation that these cones satisfy H = 0. (In the case p = q this is clear by symmetry.)

It is well known (see Simons [Si], Bombieri-DeGiorgi-Giusti [BDG], Lawlor [L]), that

• 4 ≤ n ≤ 7 : C_{p,q} is unstable for the area functional (with respect to smooth perturbations), and therefore non-minimizing,

• {p, q} = {6, 2} : C_{p,q} is stable, and minimizing on one side, but nonminimizing on the other,

• n ≥ 8, {p, q} ≠ {6, 2} : C_{p,q} is a minimizer of area (with respect to compact replacements).

Remarks. 1. In the harmonic map case, the “cone” is the map x/|x|. In the u^{p} case, the “cone” is the function A_{p,n}|x|^{−2/(p−1)} for p > n/(n − 2), where A_{p,n} is a constant.

2. There is also an example of nonuniqueness involving the cone C_{6,2}. The mechanism is quite different, however. This was the first nonuniqueness example discovered, but unfortunately we will not have time to cover it.

3. There is no stationary cone in R^{3} (other than the plane), even though n = 3 lies within the range of dimensions 5 − 2\sqrt{2} < n < 5 + 2\sqrt{2} for which such a cone would be unstable. As a result, in R^{3}, the method presented here produces no embedded, rotationally symmetric examples of nonuniqueness, but there are likely to be immersed, rotationally symmetric examples (in addition to the non-rotationally symmetric example in Lecture 2).

As a warmup to our nonuniqueness analysis, we present the following general fact.

Proposition (Ilmanen, [I6]). Let C be a stationary cone of the form C = ∂E, where E is an open set of locally finite perimeter. If C is not area-minimizing, then C evolves nonuniquely.

Proof Sketch. Since C is stationary, one evolution is given by M_{t} ≡ C, t ≥ 0. On the other hand, by the Theorem quoted in Lecture 2, section C, there exists another evolution of the form P_{t} = \sqrt{t}·P, where P is a minimizer of the functional K[P] = ∫_{P} e^{|x|^{2}/4}. If P = C, then by blowing up at the origin, K converges to the area functional, and we find that C minimizes area. This establishes the Proposition.

In section C we will present a more precise statement: C_{p,q} has an infinite number of forward evolutions.

We give a heuristic explanation of this. The stationary cone C_{p,q} is unstable, which means that there exists a smooth, compactly supported perturbation X of C_{p,q} such that the second variation of area with respect to X is negative (see the figure). Since (MCF) is the gradient flow for area, such a perturbation will grow under the flow.

Figure. A Totally Unstable Cone

We may arrange that X is supported away from the origin. Then we scale X by a homothety until it is disjoint from its original position, and reverse its sign across C_{p,q}. (That is, X_{new}(x) = −λX(x/λ), where we assume that X is perpendicular to C_{p,q}). This creates a tendency for the cone to “break” on the opposite side. The relative effect of the new perturbation is the same as the original one, except speeded up by a factor λ^{−2}. By iterating this procedure, we may obtain conflicting tendencies of ever greater speed all the way down.

As we shall see in section C, the cone can indeed “break” on either side with any number of kinks.

A Nearby Minimal Surface. Before looking at dynamics, we will describe a smooth minimal surface near C_{p,q} which exhibits infinitely many kinks.

We pause to set up some notation. Suppose that a surface M is invariant under the natural action of SO(p)×SO(q) on R^{n} = R^{p}×R^{q}. Write x = (y, z) respecting this decomposition. Then M has the form

M = M(γ) ≡ {(y, z) : (|y|, |z| ∈ γ},

where γ is a curve in R^{2}. Write (r, u) = (|y|, |z|), and by abuse of notation, write x = (r, u). If γ is the graph of a function u = u(r), write M = M(u).

Let C_{α} be the cone M({u = r tan α}). Define

λ_{s} = tan α_{s} = \sqrt{\frac{q − 1)}{p − 1}},

so C_{p,q} = C_{α_{s}}.

We now prepare to reduce PDEs to ODEs via the rotational symmetry. The principal curvatures of M with respect to a unit normal ν are

k, \underbrace{\frac{− cos θ}{u} , …, \frac{− cos θ}{u}}_{q−1}, \underbrace{\frac{sin θ}{r} , …, \frac{sin θ}{r}}_{p−1}

where θ is defined by γ̇ = (cos θ, sin θ), ν = (− sin θ, cos θ), and k is the curvature of γ with respect to ν. Then we obtain:

(8)  H = k + \frac{(q − 1) sin θ}{u} − \frac{(p − 1) cos θ}{u} ,  x ∈ M.

Therefore the minimal surface equation H = 0 becomes the following ODE for γ,

k + \frac{(q − 1) sin θ}{u} − \frac{(p − 1) cos θ}{u} = 0, x ∈ γ.

Except at isolated vertical points, γ is locally the graph of a function u = u(r) solving the following equivalent ODE, using (cos θ, sin θ) = ±(1, Du)/ \sqrt{1 + |Du|^{2}}.

(9)  \frac{u_{rr}}{1 + u_{r}^{2}} + \frac{p − 1}{r}u_{r} − \frac{q − 1}{u} = 0,  r ≥ 0.

Now we turn to a smooth minimal surface that is a “companion” to C_{p,q}. An analogous example was found by Schoen-Uhlenbeck [ScU2] for harmonic maps.

Proposition. For 4 ≤ n ≤ 7, there exists a smooth, SO(p) × SO(q)- invariant minimal surface Q which is asymptotic to C_{p,q} at infinity, and crosses it infinitely many times.

In fact, Q = M(v), where v = v(r) is a function defined for all r ≥ 0, and (v/r, v_{r}) → (λ_{s}, λ_{s}) as r → ∞. Note that for any λ > 0, λ · Q is also a minimal surface, so we may assume v(0) = 1.

In the picture, the reader will observe that Q crosses C_{p,q} an infinite number of times as we pass to infinity.

Figure. A Wiggly Companion

Proof. Begin integrating (9) with the initial condition

u(0) = 1.

By analyzing the singular point at r = 0, we find that there is a unique smooth solution defined for small r, and that u_{r}(0) = 0 is forced. Furthermore, u_{r} > 0 for small r > 0. We now change coordinates by

η = log r,  X = u/r,  Y = u_{r},

rendering the equation automonous, and yielding

X_{η} = Y − X, Y_{η} = −\frac{(p − 1)(1 + Y^{2})(λ_{s}^{2} − XY)}{X}.

Examining the phase plane, it is easily seen that the solution remains in the first quadrant, and spirals in to the unique fixed point

(X, Y) = (λ_{s}, λ_{s}).

The details are left to the reader. This completes the proof.

Figure. Phase Plane

In the next two sections, we will look at the forward and backward evolution of cones. (The backward problem is more fragile.) We will ultimately produce

(1) a continuous family of self-expanding solutions (for t > 0),

(2) a discrete sequence of self-shrinking solutions (for t < 0), which we will match up at t = 0.

C. Forward Time. First we construct various expanders for t > 0. We recall the self-expanding equation

(SE)  H(x) − \frac{x · ν}{2} = 0,  x ∈ P,

which implies that P_{t} ≡ \sqrt{t} · P, t > 0, is a mean curvature flow.

Let P = M(γ). Using (8) and noting that

\frac{x · ν}{2} = \frac{(−r sin θ, u cos θ)}{2},

we find that γ solves the ODE

k + (\frac{p − 1}{r} + \frac{r}{2}) sin θ − (\frac{u}{2} + \frac{q − 1}{u}) cos θ = 0, x ∈ γ.

This is equivalent to the following equation wherever γ can be expressed as a graph u = u(r):

(10)  \frac{u_{rr}}{1 + u_{r}^{2}} + (\frac{p − 1}{r} + \frac{r}{2}) u_{r} − \frac{u}{2} − \frac{q − 1}{u} = 0,  r > 0.

Our next task is to describe the solutions of this ODE. One solution is u = λ_{s}r. That is, the static cone C_{p,q} is expanding by homothety, without anyone noticing.

Lemma. For each a > 0, there exists a minimal surface P^{a} = M(u^{a}), where u_{a} is the solution of (10) with initial condition

u_{a}(0) = a.

u_{a} satisfies (u_{a})_{r}(0) = 0, is defined for all r ≥ 0, and is asymptotic to some ray u = λ(a)r where 0 < λ(a) < ∞, in the sense that (u_{a}/r, (u_{a})_{r}) converges to (λ(a), λ(a)) as r → ∞.

Let γ_{a} denote graph (u_{a}). Define the angle α(a) by λ(a) = tan(α(a)). The lemma implies that the flow assumes a cone as initial condition:

P_{t}^{a} ≡ \sqrt{t} · P^{a} → C_{α(a)}, t → 0^{+}.

Proof Sketch. The proof is similar to that of the existence of the wiggly companion M(v). We perform the same change of variables and obtain

X_{η} = Y − X, Y_{η} = −(1 + Y^{2}) (\frac{(p − 1)(λ_{s}^{2} − XY)}{X} + \frac{e^{2η}}{2} (Y − X)) .

As before the condition 0 < X < ∞, 0 < Y < ∞ is maintained, and in fact (X, Y) never leaves any square with corners on the locus X = Y, XY = λ^{2}. The extra terms only help to maintain these conditions. However, there is so much friction that (X, Y) can slowly grind to a halt at some point (λ, λ) other than (λ_{s}, λ_{s}). The details will appear in [AIV].

Figure. Phase Plane

Next, we will investigate the behavior of α(a) as a varies. We will match asymptotics from two regions:

(i) the region |x| ≪ 1, where P^{a} is nearly a minimal surface.

(ii) the region |α(x) − α_{s}| ≪ 1, where we linearize (10) over the cone.

(i) Behavior near the origin.

Near the origin, we ignore the x/2 term in (SE) to obtain the minimal surface equation H = 0. We expect that u_{a} will be well approximated by the wiggly curve av(r/a) in the region r ≪ 1.

Let us examine how av(r/a) approaches the cone. First we will look at v. By analyzing the regular singular point of (9) at r = ∞ (or equivalently, studying the fixed point (λ_{s}, λ_{s})), we deduce

(11)  v(r) ≈ λ_{s}r + r^{−β}(A_{1} cos(μ log r) + A_{2} sin(μ log r)),  r ≫ 1,

where

(12)  β = \frac{n − 3}{2},  μ = \frac{\sqrt{8 − (n − 5)^{2}}}{2} ,

and A_{1}, A_{2} are constants that are uniquely determined. Then

u_{a} ≈ av(r/a) ≈ λ_{s}r + a(r/a)^{−β}(A_{1} cos(μ log(r/a)) + A_{2} sin(μ log(r/a)))

in the regime a ≪ r ≪ 1.

This analysis is valid precisely in the range 4 ≤ n ≤ 7; outside of this range the behavior is not oscillatory. This corresponds to the fact that the cone is unstable for area in these dimensions.

(ii) Behavior near the Static Cone.

We linearize equation (10) over C_{p,q}. Write u(r) = λ_{s}r +h(r). Then, up to an error that is quadratic in h and its derivatives, h solves the equation

(13)  \frac{h_{rr}}{1 + λ_{s}^{2}} + (\frac{p − 1}{r} + \frac{r}{2}) h_{r} + (−\frac{1}{2} + \frac{p − 1}{r^{2}}) h = 0.

Figure. Solutions of Linearized Equation

We find that h has the asymptotics given by (11) as r approaches the regular singular point r = 0. This is not surprising, since we have completed a “commutative diagram” of approximations. Therefore we can select a basis h_{1}, h_{2} for the solutions of (13) by imposing the normalization

h_{1} ≈ r^{−β} cos(μ log r),  h_{2} ≈ r^{−β} sin(μ log r),  r ≪ 1.

At the irregular singular point r = ∞, the solution h has the asymptotics

h ∼ C′_{1}r + C′_{2}e^{−(1+λ_{s}^{2})r^{2}/4}/r.

The decaying exponential stabilizes the solution, and explains why every initial condition is asymptotic to a cone at infinity.

Therefore there exist constants λ_{1}, λ_{2} such that

(14)  h_{1} ≈ λ_{1}r,  h_{2} ≈ λ_{2}r,  r ≫ 1.

(iii) Matching

Now we scale v and h to match in the intersection of regions (i) and (ii). Let a, C_{1}, C_{2} be undetermined constants and set

av(r/a) ≈ λ_{s}r + C_{1}h_{1}(r) + C_{2}h_{2}(r), a ≪ r ≪ 1.

This becomes upon substitution,

λ_{s}r + a(r/a)^{−β}(A_{1} cos(μ log(r/a)) + A_{2} sin(μ log(r/a))) ≈ λ_{s}r + C_{1}r^{−β} cos(μ log r) + C_{2}r^{−β} sin(μ log r), which yields

C_{1} ≈ a^{β+1}(A_{1} cos(μ log a) − A_{2} sin(μ log a)),
C_{2} ≈ a^{β+1}(A_{2} cos(μ log a) + A_{1} sin(μ log a)),

and using (14), we estimate the limiting slope

(15)

λ(a) ≈ λ_{s} + C_{1}λ_{1} + C_{2}λ_{2}
≈ λ_{s} + a^{β+1}((λ_{1}A_{1} + λ_{2}A_{2}) cos(μ log a)

+ (−λ_{1}A_{2} + λ_{2}A_{1}) sin(μ log a))

≡ λ_{s} + a^{β+1}(D_{1} cos(μ log a) + D_{2} sin(μ log a)).

Figure. Matching

Summarizing the above, we have Lemma 1 (Self-Expanders). For each a > 0, the self-expander P^{a} has the asymptotics given in (i) and (ii). The limiting angle α(a) is given asymptotically by

tan α(a) ≈ λ_{s} + a^{(n−1)/2}(D_{1} cos(μ log a) + D_{2} sin(μ log a)), a ≪ 1, where μ = \sqrt{8 − (n − 5)^{2}}/2. Also α(a) → π/2 as a → ∞.

The proof will appear in [AIV].

Figure. Graph of α(a)

By counting the set {α(a) = α}, we obtain the following. Corollary. The cone C_{p,q} has infinitely many self-expanding forward evolutions. The cone C_{α} has a very large L(α) number of forward evolutions, given by

L(α) = −\frac{2μ}{(n − 1)π} log(tan α − λ_{s}) + O(1).

D. Past Time. Next, we try to concoct a self-shrinker for t < 0 that matches up with a cone at t = 0. We recall the self-shrinking equation

(SS)  H(x) + \frac{x · ν}{2} = 0,  x ∈ N.

If N = M(δ), then the curve δ solves the ODE

k + (\frac{p − 1}{r} − \frac{r}{2}) sin θ + (\frac{u}{2} − \frac{q − 1}{u}) cos θ = 0, x ∈ δ,

which is equivalent to the following ODE wherever γ is locally a graph,

(16)  \frac{u_{rr}}{1 + u_{r}^{2}} + (\frac{p − 1}{r} − \frac{r}{2}) u_{r} + \frac{u}{2} − \frac{q − 1}{u} = 0,  r > 0.

We must describe the solutions of this ODE. Define δ_{a} to be the curve solving (16) with initial condition

u(0) = a, a > 0.

In constrast to the self-expanding case, the solutions are wildly unstable for most values of a, and will not approach a cone at infinity (the transport terms have the wrong sign). To understand this behavior, we embark again on matched asymptotics in the regions

(i) near the origin (ii) near the cone.

(i) As before, we expect the solution δ_{a} of (16) to behave like the wiggly minimal surface M(av(r/a)) as long as it stays in the region |x| ≪ 1. In particular, for small a, δ_{a} will approach closely to the cone in the ocillatory manner given by (11). (ii) We linearize equation (16) over the cone. Write u(r) = λ_{s}r + g(r). This time we obtain the equation

(17)  \frac{g_{rr}}{1 + λ_{s}^{2}} + (\frac{p − 1}{r} − \frac{r}{2}) g_{r} + (\frac{1}{2} + \frac{p − 1}{r^{2}}) g = 0.

Again, the solution g has the asymptotics given by (11) as r approaches 0, so we can define a basis g_{1}, g_{2} for the solutions of (17) by requiring

g_{1} ≈ r^{−β} cos(μ log r),  g_{2} ≈ r^{−β} sin(μ log r),  forr ≪ 1.

As r → ∞, the solution g now has the asymptotics

g ∼ C′_{1}r + C′_{2}e^{(1+λ_{s}^{2})r^{2}/4}/r.

Figure. Solutions of Linearized Equation

The consequences for the nonlinear equation are the following. If we are fortunate enough that C′_{2} equals 0, then δ_{a} approaches a cone at infinity with slope λ_{s} + C′_{1}. Otherwise, δ_{a} very quickly leaves the cone, going either up or down. Once it gets a little away from the cone, the linearization loses its validity. Eventually γ_{a} “turns”, that is, the inclination θ passes π/2 or 0 and δ_{a} ceases to be an increasing graph. Subsequently, δ_{a} wanders all over R^{2}, seemingly at random. Pictures of such curves appear in Angenent [A5].

Accordingly, we define the solution g_{3} of (17) and constants B_{1}, B_{2} by requiring that

(18)  g_{3} ≈ r forr ≫ 1;  g_{3} = B_{1}g_{1} + B_{2}g_{2}.

(iii) Now we do the matching. Let a, C be undetermined constants and set

av(r/a) ≈ λ_{s}r + Cg_{3}(r), a ≪ r ≪ 1.

This becomes upon substitution,

λ_{s}r + a(r/a)^{−β}(A_{1} cos(μ log(r/a)) + A_{2} sin(μ log(r/a))) ≈ λ_{s}r + C(B_{1}r^{−β} cos(μ log r) + B_{2}r^{−β} sin(μ log r)),

which yields

CB_{1} ≈ a^{β+1}(A_{1} cos(μ log a) − A_{2} sin(μ log a)),
CB_{2} ≈ a^{β+1}(A_{2} cos(μ log a) + A_{1} sin(μ log a)),

or equivalently

C(B_{1} + iB_{2}) ≈ a^{β+1}e^{iμ log a}(A_{1} + iA_{2}).

This has a discrete sequence of solutions given by

μ log a_{k} ≈ E + πk,  C_{k} ≈ (−1)^{k}Da^{β+1},  k ∈ Z, k ≫ 1,

where D, E are defined by

B_{1} + iB_{2} = e^{iE}(A_{1} + iA_{2}) D = \begin{matrix}√ & \overline{A_{1}^{2} + A_{2}^{2}} \\ √ & \overline{B_{1}^{2} + B_{2}^{2}} \end{matrix} .

We now change E by a multiple of 2π so that k becomes the number of intersections of δ_{a} with the ray u = λ_{s}r (for large k).

We now interpret this result. As we gradually decrease a toward 0, because of (i) the solution δ_{a} will acquire more and more intersections with the cone u = λ_{s}r. As a passes a_{k}, the curve δ_{a} will acquire a new intersection as it switches from going “up” to going “down” or vice versa (that is, as the sign of C′_{2} changes). When a = a_{k}, δ_{a} goes neither up nor down, and δ_{a} is a complete graph asymptotic to a cone at infinity.

Solving for a_{k} and observing from (18) that λ_{k} = λ_{s} + C_{k}, we have the following result. Here F and G are given by F = e^{E}, G = DF^{β+1}.

Lemma 2 (Self-Shrinkers). For k sufficiently large, there exist numbers a_{k} > 0, α_{k} ∈ (0, π/2) with the asymptotic behavior

a_{k} ≈ F τ^{−k}, tan α_{k} ≈ λ_{s} + Gσ^{−k},

where

τ = e^{−π/μ}, σ = τ^{β+1} = e^{−π(n−1)/2μ},

and rotationally symmetric self-shrinkers N^{k} = M(δ_{a_{k}}), such that

(a) The curve δ_{a_{k}} is a complete graph that intersects u = λ_{s}r in k distinct points.

(b) N^{k} is asymptotic to the cone C_{α_{k}} at infinity.

The proof will appear in [AVI]. It is based on the asymptotics and shooting argument described above.

Figure. Typical Self-Shrinkers

It follows from (b) that

N_{t}^{k} ≡ \sqrt{−t} · N^{k} → C_{α_{k}} as t → 0^{−}.

Then we can conveniently look up the number of forward evolutions of C_{α_{k}} in Lemma 1 of the previous section. We substitute the value of α_{k}, and the various logarithms cancel to yield the following.

Corollary (Nonuniqueness) For large k, the self-shrinker N_{t}^{k} with k intersections has k + O(1) different continuations of the form P_{t}^{a}.

Remark. We have only considered continuations that are given by graphs u = u(r). There is another family of continuations given by graphs r = r(u).

Figure. Nonunique Evolution

E. Reduction of Complexity.

Proposition. The number of intersections with C_{p,q} drops by at least one after the singularity.

We prove this via the intersection theory of Angenent [A1], which says: if two evolving curves solve the same (smooth, nonlinear) parabolic equation, then

(a) the number of intersections of the two curves is finite after an instant,

(b) the number of intersections is nonincreasing, and strictly decreases whenever the curves are nontransverse.

Note that if M_{t} = M(γ_{t}) is a smooth mean curvature flow, then γ_{t} solves

(19)  \frac{∂}{∂t}x = k + (\frac{p − 1}{r} + \frac{r}{2}) sin θ − (\frac{u}{2} + \frac{q − 1}{u}) cos θ,  x ∈ γ_{t},

which is nonsingular away from the axes (see (8)).

Let σ be the ray u = r tan α_{s}, so C_{p,q} = M(σ).

Proof. Let {M_{t}}_{t∈R} be one of the shrinking-expanding solutions constructed above, with M_{t} = M(γ_{t}). We cannot apply Angenent’s Theorem directly, since γ_{0} meets σ at the origin, where both the curves and the equation are singular.

Recall the wiggly minimal surface Q = M(v) of section B, and write ε = graph(v). Then #(γ_{t} ∩ε) is nonincreasing for t ≠ 0. By passing to limits near t = 0 we get

#(γ_{s} ∩ ε) ≤ #(γ_{0} ∩ ε) ≤ #(γ_{t} ∩ ε), t < 0 < s.

In fact, we have

lim_{s→∞} #(γ_{s} ∩ ε) + 2 ≤ lim_{t→−∞} #(γ_{t} ∩ ε)

since γ_{t} is tangent to ε once coming down and once coming back up, at the point r = 0, u = v(0). (The strict decrease can be verified by hand, without having to extend Angenent’s theorem to the singular point r = 0.)

This proves the proposition, because

lim_{s→∞} #(γ_{s} ∩ ε) = #(γ_{1} ∩ σ),  lim_{t→−∞} #(γ_{t} ∩ ε) = #(γ_{−1} ∩ σ).

In fact, the number of intersections drops by at least two.

Remark. 1. As remarked above, there are also expanding evolutions of the form r = r(u); in this case, there is only one moment of tangency, so the number of intersections drops by at least one.

2. We would like a more exact estimate of the number of continuations. In light of the above results, one might conjecture (a) for each k ≥ 2, there is exactly one self-shrinker N_{t}^{k} of the form M({u = u(r)}) which is asymptotic to a cone at infinity and has k intersections with the cone, and (b) N_{t}^{k} has exactly 2k − 1 continuations P_{t}, with the following numbers of intersections with the cone:

0, 2, 2, …, k − 2, k − 2,  pulling one way,
1, 1, 3, 3, …, k − 1, k − 1,  pulling the other way.

Finally we show that the cone C_{p,q} itself can never arise as a blowup.

Proposition. Suppose {M_{t}}_{t∈[−b,0)} is a smooth, compact, rotationally symmetric mean curvature flow that becomes singular at t = 0. Then M_{t}^{λ} cannot converge to the cone C_{p,q}.

Proof. The basic idea, due to Grayson, is this: the self-shrinking solutions we have constructed “protect” the cone by having too many wiggles.

Let M_{t} = M(γ_{t}). For t < 0, γ_{t} meets the axes orthogonally (or not at all) and never contains (0, 0).

By Angenent [A1], γ_{t} ∩ σ is finite and and the intersection is transverse for almost every t ∈ (−b, 0). Suppose in particular that this holds at t = t_{0}.

For large k, δ_{k} is close to σ in C_{loc}^{1} and u/r is close to λ_{s} in C^{1}(R^{2} \B_{1}). It follows that for k sufficiently large,

#(γ_{t_{0}} ∩ δ_{k}) = #(γ_{t_{0}} ∩ σ) ≪ #(δ_{k} ∩ σ).

Since γ_{t} and \sqrt{−t} · δ_{k} both solve (19), by Angenent [A1], #(γ_{t} ∩ \sqrt{−t} · δ_{k}) is nonincreasing. On the other hand, if (−t)^{−1/2} · γ_{t} converges to σ, then #((−t)^{−1/2} · γ_{t} ∩ δ_{k}) → #(σ ∩ δ_{k}), which is a contradiction. This proves the proposition.

A similar proof, using the wiggly curve ε instead of δ_{k}, shows that if M_{t} (possibly noncompact) is smooth and rotationally symmetric for t < 0, then

M_{t} ↛ C_{p,q} as t → 0^{−}

provided that M_{t} originally has a finite number of intersections with the cone.

Example. Interestingly, we can attain C_{p,q} exactly if we allow triple junctions, in the case p = q and n = 4 or 6, with thanks to J. Sullivan and R. Kusner. Consider the self shrinkers

u\sqrt{n−2}(r) = \sqrt{n − 2}  (Cylinder)
u\sqrt{2(n−1)}(r) = \sqrt{2(n − 1) − r^{2}}  (Sphere)

which cross u = r at 135^{∘} and 90^{∘} respectively. Somewhere between, there is a curve u_{a} that hits u = r at 120^{∘}, at some point (b, b). Let γ_{1} denote the arc of u_{a} from (0, a) to (b, b), let γ_{2} be its reflection across u = r, and let γ_{3} be the part of u = r from (b, b) to infinity. Then

N_{t} = \sqrt{t} · M(γ_{1} ∩ γ_{2} ∩ γ_{3}), t < 0,

is a self-shrinker with triple junctions that meet at 120^{∘}, such that N_{t} → C_{p,q} as t → 0. A trapped bubble shrinks to nothing at the origin, and the cone explodes in infinitely many possibilities.

Tom Ilmanen Evanston August, 1995.

References

[AAG] Altschuler, S., Angenent, S., and Giga, Y., “Mean Curvature Flow Through Singularities for Surfaces of Rotation,” J. Geom. Analysis, to appear.

[AL] Abresch, U. and Langer, J., “The Normalized Curve Shortening Flow and Homothetic Solutions,” J. Diff. Geom. 23 (1986) 175-196.

[Am] Ambrosio, L., “Minimizing Movements,” lecture notes for a summer school course in Padova, Italy, June 20-24, 1994.

[A1] Angenent, S., “The Zero Set of a Solution of a Parabolic Equation,” J. für die Reine und Angew. der Math. 390 (1988) 79-96.

[A2] Angenent, S., “Parabolic Equations for Curves on Surfaces (I). Curves with p-Integrable Curvature,” Annals of Math. 132 (1990) 451-483.

[A3] Angenent, S., “Parabolic Equations for Curves on Surfaces (II). Intersections, Blow Up and Generalized Solutions,” Annals of Math. 133 (1991) 171-215.

[A4] Angenent, S., “On the Formation of Singularities in the Curve Shortening Flow,” J. Diff. Geom. 33 (1991) 601-633.

[A5] Angenent, S., “Shrinking Donuts,” Proc. of the Conf. on Ell. and Parab. Equ. held at Gregynog, Wales, Aug. 1989.

[ACI] Angenent, S., Chopp, D., and Ilmanen, T., “A Computed Example of Nonuniqueness of Mean Curvature Flow in R^{3},” Comm. Part. Diff. Eq., to appear.

[AIV] Angenent, S., Ilmanen, T., and Velazquez, J. J. L., “Nonuniqueness in Geometric Heat Flows,” in preparation.

[AV1] Angenent, S. and Velazquez, J. J. L., “Asymptotic Shape of Cusp Singularities in Curve Shortening,” Duke Math. J. (1995) 71-110.

[AV2] Angenent, S. and Velazquez, J. J. L., paper on asymptotic shape of degenerate neckpinches, in preparation.

[BSS] Barles, G., Soner, H. M., and Souganidis, P. E. “Front Propagation and Phase Field Theory,” SIAM J. Cont. Optim. 31 (1993) 439-469.

[BP] Bellettini, G. and Paolini, M., “Two Examples of Fattening for the Mean Curvature Flow with a Driving Force,” Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl. 5 (1994) 229–236.

[BDG] Bombieri, E., De Giorgi, E., and Giusti, E., “Minimal Cones and the

Bernstein Problem,” Invent. Math. 7 (1969) 243-268.

[Bo] Borsuk, K., “Sur la Courbure Total des Courbes Fermées,” Annales Soc. Polonaise 20 (1947) 251-265.

[B] Brakke, K., The Motion of a Surface by its Mean Curvature, Princeton Univ. Press, 1978.

[BQ] Budd, C. J. and Qi, Y. W., “The Existence of Bounded Solutions of a Semilinear Heat Equation,” J. Diff. Eq. 82 (1989) 207-218.

[Ch] Chen, Y., “The Weak Solutions to the Evolution Problem of Harmonic Maps,” Math. Z. 201 (1989) 69-74.

[CLL] Chen, Y., Li, and Lin, F.-H., preprint.

[CS] Chen, Y. and Struwe, M., “Existence and Partial Regularity Results for

the Heat Flow for Harmonic Maps,” Math. Z. 201 (1989) 83-103.

[CGG] Chen, Y.-G., Giga, Y., and Goto, S., “Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations,” J. Diff. Geom. 33 (1991) 749-786.

[CSc] Choi, H. I. and Schoen, R., “The Space of Minimal Embeddings of a Surface into a Three-Dimensional Manifold of Positive Ricci Curvature,” Invent. Math 81 (1985) 387-394.

[Co] Coron, J.-M., “Nonuniqueness for the Heat Flow of Harmonic Maps,” Ann. Inst. H. Poincare Non-Lineare 7 (1990) 335-344.

[C] Chopp, D. “Numerical Computation of Self-Similar Solutions for Mean

Curvature Flow,” Exper. Math. 3 (1994) 1-16.

[CIL] Crandall, M. G., Ishii, H., and Lions, P.-L., “User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations,” Bull. Amer. Math. Soc. 27 (1992) 1-67.

[D1] De Giorgi, E., “Some Conjectures on Flow by Mean Curvature,” white paper, 1990.

[D2] De Giorgi, E., “New Problems on Minimizing Movements,” Boundary Value Problems for PDE and Applications, C. Baiocchi and J.L. Lions, ed., Masson, 1992, 81-88.

[E] Ecker, K., “On Regularity for Mean Curvature Flow of Hypersurfaces,”

preprint.

[EH1] Ecker, K. and Huisken, G., “Mean Curvature Evolution of Entire Graphs,” Annals of Math. 130 (1989) 453-471.

[EH2] Ecker, K. and Huisken, G., “Interior Estimates for Hypersurfaces moving by Mean Curvature,” Invent. Math. 105 (1991) 635-681.

[ESa] Eells, J. and Sampson, J. H., “Harmonic Mappings of Riemannian Manifolds,” Ann. of Math. 86 (1964) 109-160.

[ES1] Evans, L. C. and Spruck, J., “Motion of Level Sets by Mean Curvature I,” J. Diff. Geom. 33 (1991) 635-681.

[ES2] Evans, L. C. and Spruck, J., “Motion of Level Sets by Mean Curvature II,” Trans. of the Amer. Math. Soc. 330 (1992) 321-332.

[ES3] Evans, L. C. and Spruck, J., “Motion of Level Sets by Mean Curvature III,” J. Geom. Anal. 2 (1992) 121-150.

[ES4] Evans, L. C. and Spruck, J., “Motion of Level Sets by Mean Curvature IV,” J. Geom. Anal. 5 (1995) 77-114.

[F] Feldman, M., “Partial Regularity for Harmonic Maps of Evolution into Sphere,” Comm. Part. Diff. Eq. 19 (1994) 761-790.

[Fr] Freire, A., “Uniqueness for the Harmonic Map Flow in Two Dimensions,” Calc. of Var. and Part. Diff. Eq., 1995.

[Ga] Gage, M., “Curve Shortening Makes Convex Curves Circular,” Invent. Math. 76 (1984) 357-364.

[GH] Gage, M. and Hamilton, R., “The Heat Equation Shrinking Convex Plane Curves,” J. Diff. Geom. 23 (1986) 69-96.

[GK] Giga, Y. and Kohn, R. V., “Asymptotically Self-Similar Blowup of Semilinear Heat Equations,” Comm. Pure Appl. Math. 38 (1985) 297-319.

[G1] Grayson, M. A., “The Heat Equation Shrinks Embedded Curves to Round Points,” J. Diff. Geom., 26 (1987) 285-314.

[G2] Grayson, M. A., “A Short Note on the Evolution of a Surface by its Mean Curvature,” Duke Math. J. 58 (1989) 555-558.

[Gu1] Gulliver, R., “Regularity of Minimizing Surfaces of Prescribed Mean Curvature,” Ann. of Math. 97 (1973) 275-305.

[Gu2] Gulliver, R., “Removability of Singular Points on Surfaces of Bounded Mean Curvature,” J. Diff. Geom. 11 (1976) 345-350.

[H1] Hamilton, R., hour talk at the International Congress of Mathematicians, 1986.

[H2] Hamilton, R., “Three-Manifolds with Positive Ricci Curvature,” J. Diff. Geom. 17 (1982) 255-306.

[H3] Hamilton, R., “Four-Manifolds with Positive Curvature Operator,” J. Diff. Geom. 24 (1986) 153-179.

[H4] Hamilton, R., “The Ricci Flow on Surfaces,” Contemp. Math. 71 (1988) 237-261.

[H5] Hamilton, R., “The Harnack Estimate for the Ricci Flow,” preprint.

[H6] Hamilton, R., “A Compactness Property for Solutions of the Ricci Flow,” preprint.

[H7] Hamilton, R., “Eternal Solutions to the Ricci Flow,” preprint.

[H8] Hamilton, R., “Convex Surfaces with Pinched Second Fundamental Form,” preprint.

[He] Hempel, J., 3-Manifolds, Annals of Math. Stud. #86, Princ. Univ. Press, 1976.

[Hu1] Huisken, G., “Flow by Mean Curvature of Convex Surfaces into Spheres,” J. Diff. Geom. 20 (1984) 237-266.

[Hu2] Huisken, G., “Asymptotic Behavior for Singularities of the Mean Curvature Flow,” J. Diff. Geom. 31 (1990) 285-299.

[Hut1] Hutchinson, J. E., “Second Fundamental Form for Varifolds and Existence of Surfaces Minimizing Curvature,” Indiana J. Math. 35 (1986) 45-71.

[Hut2] Hutchinson, J. E., “C^{1,α} Multi-Valued Function Regularity and Tangent Cone Behavior for Varifolds with Second Fundamental Form in L^{p},” Proc. Symp. Pure Math. 44 (1986) 281-306.

[I1] Ilmanen, T., “Generalized Motion of Sets by Mean Curvature on a Manifold,” Indiana J. Math. 41 (1992) 671-705.

[I2] Ilmanen, T., “The Level-Set Flow on a Manifold,” Proc. Symp. Pure Math. 54 Part 1 (1993) 193-204.

[I3] Ilmanen, T., “Elliptic Regularization and Partial Regularity for Motion by Mean Curvature,” Mem. Amer. Math. Soc. #520, 1994.

[I4] Ilmanen, T., “A Strong Maximum Principle for Singular Minimal Hypersurfaces,” Calc. of Var. and Part. Diff. Eq., submitted.

[I5] Ilmanen, T., “Dynamics of Stationary Cones,” in preparation.

[I6] Ilmanen, T., “Singularities of Mean Curvature Flow of Surfaces,” preprint.

[ISZ] Ilmanen, T., Sternberg, P., and Ziemer, W., “Equilibria of the Level-Set Flow,” J. Geom. Analysis, submitted.

[JT] Jaffe, A., and Taubes, C., Vortices and Monopoles, Progress in Physics, Vol. 2, Birkhauser, Boston, 1980.

[K] Kapouleas, N., “Complete Constant Mean Curvature Surfaces in Euclidean Three-Space,” Ann. Math. 131 (1990) 239-330.

[L] Lawlor, G., A Sufficient Criterion for a Cone to be Area-Minimizing, Mem. Amer. Math. Soc., May 1991.

[Le] Lepin, L. A., “Self Similar Solutions of a Semilinear Heat Equation,” Math. Mod. 2 (1990) 63-74 (in Russian).

[Mo] Morgan, F., Geometric Measure Theory: A Beginner’s Guide, Academic Press, 1988.

[OS] Osher, S. and Sethian, J. A., “Fronts Progagating with Curvature-De-pendent Speed: Algorithms Based on Hamilton-Jacobi Formulations,” J. Comp. Phys. 79 (1988) 12-49.

[Pi] Pitts, J. T., Existence and Regularity of Minimal Surfaces in Riemannian Manifolds, 1979.

[PR] Pitts, J. T. and Rubinstein, J. H., “Equivariant Minimax and Minimal Surfaces in Geometric Three-Manifolds,” Bull. AMS 19 (1988) 303-309.

[P] Price, P., “A Monotonicity Formula for Yang-Mills Fields,” Manu. Math.

43 (1983) 131-166.

[SaU] Sacks, J. and Uhlenbeck, K., “Minimal Immersions of Closed Riemann Surfaces,” Trans. Am. Math. Soc. 271 (1982) 639-652.

[ScS] Schoen, S., and Simon, L. M., “Regularity of Stable Minimal Hypersurfaces,” Comm. Pure Appl. Math. 34 (1981) 741-797.

[SSY] Schoen, S., Simon, L. M., and Yau, S.-T., “Curvature Estimates for Minimal Hypersurfaces,” Acta Math. 134 (1975) 275-288.

[ScU1] Schoen, R. and Uhlenbeck, K., “A Regularity Theorem for Harmonic Maps,” J. Diff. Geom. 17 (1982) 307-335.

[ScU2] Schoen, R. and Uhlenbeck, K., “Regularity of Minimizing Harmonic

Maps into the Sphere,” Maps,” Invent. Math. 78 (1984) 89-100.

[SY] Schoen, R., and Yau, S.-T., “Existence of Incompressible Minimal Surfaces and the Topology of Three-Dimensional Manifolds with Non-Negative Scalar Curvature,” Annals of Math. 110 (1979) 127-142.

[ST] Shatah, J. and Tahvildar-Zadeh, A. S., “On the Cauchy Problem for Equivariant Wave Maps,” Comm. Pure Appl. Math. 47 (1994) 719-754.

[S1] Simon, L. M., Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Analysis, Austr. Nat. Univ., Vol. 3, 1983.

[S2] Simon, L. M., “Existence of Surfaces Minimizing the Willmore Functional,” preprint.

[Si] Simons, J., “Minimal Varieties in Riemannian Manifolds,” Ann. of Math. 88 (1968) 62-105.

[So] Soner, H. M., “Motion of a Set by the Mean Curvature of its Boundary,” J. Diff. Eq. 101 (1993) 313-372.

[SoS] Soner, H. M. and Souganidis, P. E., “Singularities and Uniqueness of Cylindrically Symmetric Surfaces Moving by Mean Curvature,” Comm. Partial Diff. Eq. 18 (1993) 859-894.

[St] Struwe, M., “On the Evolution of Harmonic Maps in Higher Dimensions,” J. Diff. Geom. 28 (1988) 485-502.

[T] Troy, W. C., “The Existence of Bounded Solutions of a Semilinear Heat Equation,” SIAM J. Math. Anal. 18 (1987) 332-336.

[V] Velazquez, J. J.-L., “Curvature Blowup in Perturbations of Minimizing Cones Evolving by Mean Curvature Flow,” preprint.

[W1] White, B., “On the Topological Type of Minimal Submanifolds,” Topology 31 (1992) 445-448.

[W2] White, B., “Partial Regularity of Mean Convex Hypersurfaces Flowing by Mean Curvature,” Internat. Math. Res. Notices 4 (1994) 185-192.

[W3] White, B., “Stratification of Minimal Surfaces, Mean Curvature Flows, and Harmonic Maps,” preprint.

[W4] White, B., in preparation.

[W5] White, B., note on nonfattening in DeGiorgi’s proposed example.

[W6] White, B., “Brakke’s Regularity Theorem for Limits of Smooth Flows,” in preparation.