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For Zoom URL please contact Laura Keller

Autumn Semester 2013

Date / Time Speaker Title Location
17 September 2013
15:15-16:15 		Alexander Volkmann
Max-Planck Institut für Gravitationsphysik, Potsdam
Event Details

Analysis Seminar

Title A Li-Yau type inequality for free boundary surfaces with respect to the unit ball
Speaker, Affiliation Alexander Volkmann, Max-Planck Institut für Gravitationsphysik, Potsdam
Date, Time 17 September 2013, 15:15-16:15
Location HG G 43
Abstract A classical inequality due to Li and Yau states that for a closed immersed surface the Willmore energy can be bounded from below by $4 \pi$ times the maximum multiplicity of the surface. Subsequently, Leon Simon proved a monotonicity identity for closed immersed surfaces, which as a corollary lead to a new proof of the Li-Yau inequality. In this talk we consider compact free boundary surfaces with respect to the unit ball in $\mathbb R^n$, i.e. compact surfaces in $\mathbb R^n$, the boundaries of which meet the boundary of the unit ball orthogonally. Inspired by Simon's idea we prove a monotonicity identity in this setting. As a corollary we obtain a Li-Yau type inequality, which can be seen as a generalization of an inequality due to Fraser and Schoen to not necessarily minimal surfaces and to all codimensions. Using a similar idea Simon Brendle had already extended Fraser-Schoen's inequality to higher dimensional minimal surfaces in all codimensions.
A Li-Yau type inequality for free boundary surfaces with respect to the unit ballread_more
HG G 43
8 October 2013
15:15-16:15 		Pierre Germain
Courant Institute, NY
Event Details

Analysis Seminar

Title Weakly nonlinear, big box limit for the nonlinear Schroedinger equation on the torus
Speaker, Affiliation Pierre Germain, Courant Institute, NY
Date, Time 8 October 2013, 15:15-16:15
Location HG G 43
Abstract I will present the derivation of a new equation, obtained from the nonlinear Schroedinger equation on the 2 torus, in the limit of small data and as the size of the torus goes to infinity (which is related to the set up of the theory of weak turbulence). This new equation turns out to have very striking properties, which I will describe. This is joint work with Erwan Faou (Rennes) and Zaher Hani (NYU).
Weakly nonlinear, big box limit for the nonlinear Schroedinger equation on the torusread_more
HG G 43
15 October 2013
15:15-16:15 		Carlo Mariconda
Università di Padova
Event Details

Analysis Seminar

Title The non occurrence of the Lavrentiev gap for scalar multi-dimensional variational problems
Speaker, Affiliation Carlo Mariconda, Università di Padova
Date, Time 15 October 2013, 15:15-16:15
Location HG G 43
Assets https://math.ethz.ch/ndb/00006/03982/lavrentiev_abstract.pdffile_download
The non occurrence of the Lavrentiev gap for scalar multi-dimensional variational problemsread_more
HG G 43
29 October 2013
15:15-16:15 		Nishanth Gudapati
Albert-Einstein-Institut, Potsdam/Golm
Event Details

Analysis Seminar

Title Critical Self-Gravitating Wave Maps
Speaker, Affiliation Nishanth Gudapati, Albert-Einstein-Institut, Potsdam/Golm
Date, Time 29 October 2013, 15:15-16:15
Location HG G 43
Abstract Wave maps are maps from a Lorentzian manifold to a Riemannian manifold which are critical points of a Lagrangian which is a natural geometrical generalization of the free wave Lagrangian. Self-gravitating wave maps are those from an asymptotically flat Lorentzian manifold which evolves according to Einstein's equations of general relativity with the wave map itself as the source. If the domain manifold is 2+1 dimensional, the energy of wave map is scale invariant, hence it is referred to as critical. Apart from a purely geometrical interest, the motivation to study critical self-gravitating wave maps is that they occur naturally in 3+1 Einstein's equations of general relativity. Therefore, studying critical self-gravitating wave maps could be a fruitful way of understanding the ever elusive global behaviour of Einstein's equations. In this talk, after a brief discussion on the background and formulation of the Cauchy problem of critical self-gravitating wave maps, we shall present a recent proof of the non-concentration of energy of critical equivariant self-gravitating wave maps before pointing out potential generalizations and applicable methods therein.
Critical Self-Gravitating Wave Mapsread_more
HG G 43
5 November 2013
15:15-16:15 		Alessandro Carlotto
University of Stanford
Event Details

Analysis Seminar

Title Complete minimal surfaces in asymptotically flat spaces and localized solutions of the Einstein constraint equations
Speaker, Affiliation Alessandro Carlotto, University of Stanford
Date, Time 5 November 2013, 15:15-16:15
Location HG G 43
Abstract To what extent can the global theory of complete minimal surfaces in the Euclidean space can be extended to asymptotically flat 3-manifolds? Starting with the most basic question, we might ask whether complete stable minimal surfaces actually exist in presence of a positive ADM mass, thus considering a generalized Bernstein problem. The answer to this question turns out to be NO if the ambient metric has a nice expansion at infinity (namely if it is asymptotically Schwarschild, in a weak sense), while it is YES in full generality. The former rigidity result implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. The proof of this theorem is based on a characterization of finite index minimal surfaces via improved curvature estimates, on classical infinitesimal rigidity results by Fischer-Colbrie and Schoen and on the positive mass theorem by Schoen-Yau. More specifically, we also show that a minimal surface inside an asymptotically flat 3-manifold has finitely many ends and each of these is a graph of a function that has a suitable expansion at infinity, in analogy with a classical result by Schoen for Euclidean spaces. On othe other hand, the latter result (which is joint with Richard Schoen) follows from the construction of a new class of solutions of the Einstein constraint equations that have positive mass but are (exactly) Euclidean on the complement of a cone of arbitrarily small opening angle. This flexibility theorem sharply contrasts various recent scalar curvature rigidity results both in the compact and in the free-boundary setting. Our main theorems can be extended to analyze marginally outer-trapped surfaces (MOTS) in non time-symmetric initial data sets: we first prove that a non-compact stable MOTS in an initial data set (M,g,k) is conformally diffeomorphic to either the plane C or to the cylinder A and in the latter case infinitesimal rigidity holds. If the data have harmonic asymptotics, the former case is proven to be globally rigid in the sense that the presence of a stable MOTS forces an isometric embedding of (M,g,k) in the Minkowski space-time as a space-like slice.
Complete minimal surfaces in asymptotically flat spaces and localized solutions of the Einstein constraint equations read_more
HG G 43
* 11 November 2013
16:15-17:15 		Prof. Dr. Gerhard Huisken
Event Details

Analysis Seminar

Title Mean curvature flow with surgery for embedded mean convex surfaces in R^3
Speaker, Affiliation Prof. Dr. Gerhard Huisken,
Date, Time 11 November 2013, 16:15-17:15
Location HG E 22
Mean curvature flow with surgery for embedded mean convex surfaces in R^3
HG E 22
12 November 2013
15:15-16:15 		Dr. Melanie Rupflin
University of Leipzig
Event Details

Analysis Seminar

Title Global solutions and Asymptotics of Teichmüller harmonic map flow
Speaker, Affiliation Dr. Melanie Rupflin, University of Leipzig
Date, Time 12 November 2013, 15:15-16:15
Location HG G 43
Abstract Teichmüller harmonic map flow is designed to evolve maps from a closed surface to a general target manifold towards branched minimal immersions. Defined as gradient flow of energy considered as a function of both a map and a metric on the domain, the flow enjoys the strong regularity properties known from harmonic map heat flow for as long as there is no degeneration in Teichmüller space. In this talk we will discuss the asymptotic behaviour of global solutions of the flow, guaranteed to exist in certain settings, and prove that any global solution changes (or decomposes) the initial data into (a union of) branched minimal immersions, possibly parametrized over surfaces of lower genus. This is joint work with Peter Topping.
Global solutions and Asymptotics of Teichmüller harmonic map flowread_more
HG G 43
3 December 2013
15:15-16:15 		Dr. Hui Nguyen
The University of Queensland
Event Details

Analysis Seminar

Title A Classification of Singular Willmore Spheres
Speaker, Affiliation Dr. Hui Nguyen, The University of Queensland
Date, Time 3 December 2013, 15:15-16:15
Location HG G 43
Abstract In this talk, we will discuss a recent classification of singular Willmore two-spheres immersed in Euclidean three-space and four-space with a bound on the possible singularities. Singular Willmore spheres are surfaces with finite total curvature that satisfy the Euler-Lagrange equation of the Willmore functional except at a finite number of points. These objects appear naturally as singularity models of the Willmore flow and as bubbles in energy identities. We show that if the singular points have the bound |D| < 4, then in Euclidean three space they must be the Möbius transformation of a complete non-compact genus zero minimal surface with finite total curvature and in Euclidean four space they must be the Möbius transformation of a complete non-compact genus zero minimal surface and/or the image of a holomorphic curve in the twistor space of R^4 under the Penrose twistor fibration.
A Classification of Singular Willmore Spheresread_more
HG G 43

Notes: red marked events are important, events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location and if you want you can subscribe to the iCal/ics Calender.

Organizers: Francesca Da Lio, Tom Ilmanen, Thomas Kappeler, Tristan Rivière, Michael Struwe

Archive: SS 24 AS 23 SS 23 AS 22 SS 22 AS 21 SS 21 AS 20 SS 20 AS 19 SS 19 AS 18 SS 18 AS 17 SS 17 AS 16 SS 16 AS 15 SS 15 AS 14 SS 14 AS 13 SS 13 AS 12 SS 12 AS 11 SS 11 AS 10 SS 10 AS 09

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