Talks (titles and abstracts)

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Robert Balmer: tba

Costante Bellettini: tba

Alessandro Carlotto: Compactness analysis for free boundary minimal hypersurfaces

The study of free boundary minimal hypersurfaces, namely of those critical points for the area functional that meet the boundary of their ambient manifold orthogonally, dates back at least to Courant and yet has recently seen remarkable developments both in terms of existence results (via min-max and or gluing/desingularization methods) and in terms of a much deeper understanding of the connection with extremal metrics for the first Steklov eigenvalue. In this lecture, I will survey a number of results, of global character, centered around the following general questions: What conditions ensure geometric compactness (i.e. smooth, graphical convergence with multiplicity one) of a sequence of free boundary minimal hypersurfaces? What is the fine scale description of the geometric picture when multiply-sheeted convergence happens instead? As a byproduct, our analysis allows to obtain various consequences in terms of finiteness and topological control, which we can combine with a suitable bumpy metric theorem in this category to derive novel genericity theorems.
Based on joint work with Lucas Ambrozio and Benjamin Sharp.

Carla Cederbaum: A geometric boundary value problem related to the static vacuum equations in General Relativity

The Schwarzschild spacetime models the vacuum region outside a spherically symmetric static star or black hole in General Relativity. It possesses a very special, timelike hypersurface which is ruled by and “traps” null geodesics. This surface is called the “photon sphere”.

We will show that the Schwarzschild spacetime is the only static vacuum asymptotically flat general relativistic spacetime that possesses a suitably geometrically defined photon sphere. We will present two proofs, both extending classical static black hole uniqueness results and also discuss generalizations to higher dimensions. Part of this work is joint with Gregory J. Galloway.

Otis Chodosh: On the global uniqueness of large stable CMC regions in asymptotically flat 3-manifolds

Any asymptotically flat 3-manifold with positive mass admits a foliation at infinity by stable CMC spheres. I'll describe recent work establishing the global uniqueness of such spheres. This is joint work with M. Eichmair.

Fernando Coda Marques: The space of cycles, a Weyl's law for minimal hypersurfaces and Morse index estimates

The space of cycles in a compact Riemannian manifold has very rich topological structure. The space of hypercycles, for instance, taken
with coefficients modulo two, is weakly homotopically equivalent to the infinite dimensional real projective space. This reveals the existence of nontrivial k-parameter sweepouts for every k. We will discuss a proof of a Weyl's law conjectured by Gromov (joint work with Liokumovich and Neves) in which the eigenvalues of the Laplacian are replaced by the areas of minimal hypersurfaces constructed by minimax methods. We will also discuss current work with Neves about Morse index bounds in the min-max theory of minimal surfaces and the problem of multiplicity.

Olivier Druet: Existence of high energy solutions to elliptic PDEs with Moser-Trudinger growth

We prove existence of solutions of high energy of equations of the type \(-\Delta u = \lambda u e^{u^2}\) in a bounded domain of \(\rtwo\). This equation has a nonlinearity of critical Moser-Trudinger growth. Existence results are obtained through a careful analysis of the defect of compactness of sequences of solutions of this equation. This is a joint work with Pierre-Damien Thizy (Univ. of Cergy-Pontoise).

Ailana Fraser: tba

Robert Haslhofer: Ricci curvature and martingales

We generalize the classical Bochner formula for the heat flow on a manifold M to martingales on path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner as the classical Bochner formula on M is related to lower bounds on Ricci curvature. This establishes a new link between geometry and stochastic analysis, and provides a crucial new tool for the study of Einstein metrics and Ricci flow in the smooth and non-smooth setting. Joint work with Aaron Naber.

Paul Laurain: CMC surfaces in asymptotically flat manifolds

CMC surface plays a very important role in General Relativity. After remembering the state of art, I will discuss some improvement about the uniqueness of such objects. Then I will speak about some generalisations to Willmore surfaces and the notion of quasi-local mass.

Gang Liu: On Yau's uniformization conjecture

We confirm Yau's uniformization conjecture when the manifold has maximal volume growth. More precisely, we prove that if \(M^n\) is a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, then M is biholomorphic to \(C^n\).

Davi Maximo:  Minimal surfaces with bounded index

In this talk we will give a precise picture on how a sequence of minimal surfaces with bounded index might degenerate on given closed three-manifold. As an application, we prove several compactness results.

Aaron Naber: Structure theory for non collapsed spaces with lower Ricci curvature bounds.

Let (\(M_i,g_i->(X,d)\) be a Groom-Hausdorff limit of spaces with \(Rc>=-(n-1)\) and \(Vol(B_1)>v>0\).  It is classically understood one can stratify \(X= S^0(X)<=..<=S^{n-2}(X)<=X\) such that dim \(S^k<=k\).  The first main result of this talk is that \(S^k\) is k-rectifiable.  We will see this is sharp, as we will discuss examples such that \(Sing(X) = S^k\) is both k-rectifiable and a k-cantor set.  One can prove finite measure results for the quantitative stratifications.  Applications of the structure results include proving X is a manifold away from a n-2 rectifiable subset of finite n-2 measure, showing that the tangent cones of X are unique away from a set of n-2 measure zero, and giving a new proof of the n-4 finiteness conjecture for limits with bounded Ricci.  The structure results are joint with Cheeger/Jiang and the examples are joint with Li.

Melanie Rupflin: Eigenvalues of the Laplacian on degenerating surfaces

We consider the first non-zero eigenvalue \(\lambda_1\) of the Laplacian on hyperbolic surfaces for which one disconnecting geodesic collapses and establish that the \(L^2\) gradient of \(\lambda_1\) can be essentially explicitly described in terms of the dual of the degenerating length coordinate.

As a corollary we obtain that \(\lambda_1\) essentially only depends on the length of the collapsing geodesic \(\sigma\) and the topology of \(M\setminus \sigma\), with error estimates that are sharp for all surfaces of genus greater than 2.

Felix Schulze: Ricci flow from spaces with isolated conical singularities

Let \(M,g_0\) be a compact n-dimensional Riemannian manifold with a finite number of singular points, where at each singular point the metric is asymptotic to a cone over a compact \(n-1\)-dimensional manifold with curvature operator greater or equal to one. We show that there exists a smooth Ricci flow, possibly with isolated orbifold points,  starting from such a metric with curvature decaying like \(C/t\). The initial metric is attained in Gromov-Hausdorff distance and smoothly away from the singular points. To construct this solution, we desingularize the initial metric by glueing in expanding solitons with positive curvature operator, each asymptotic to the cone at the singular point, at a small scale s. Localizing a recent stability result of Deruelle-Lamm for such expanding solutions, we show that there exists a solution from the desingularized initial metric for a uniform time \(T>0\), independent of the glueing scale s. The solution is then obtained by letting \(s->0\). We also show that the so obtained limiting solution has the corresponding expanding soliton as a forward tangent flow. This is joint work with P. Gianniotis.

Neshan Wickramasekera: tba

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