Seminar on Geometric Analysis

21.06.2019

• 11:30-12:30 Jonas Hirsch (Universität Leipzig): Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

16.05.2019

• 11:30-13:00 Patrick Breuning: Kompaktheit von Immersionen ohne lokale C^1-Schranke

Abstract: Wir betrachten Immersionen, die lokal als Graph einer C^1-Funktion über dem affinen Tangentialraum über einer Kugel mit festem Radius r>0 geschrieben werden können. Ferner sei das Volumen gleichmäßig beschränkt. Ein klassisches Resultat von Joel Langer (1985) besagt: Ist die C^1-Norm der Graphenfunktionen gleichmäßig beschränkt und hinreichend klein, so ist die Menge an Immersionen kompakt. Wir zeigen, dass das Resultat ohne die Voraussetzung an die C^1-Norm noch immer in voller Stärke gilt. Insbesondere besitzt jede Folge derartiger Immersionen eine Teilfolge, die (nach Reparametrisierung) gleichmäßig in C^1 beschränkt ist - trotz fehlender Voraussetzung einer lokalen C^1-Schranke. Das Resultat gilt in beliebiger Dimension und Kodimension 1.

14.12.2017

• 09.45-11.15 Anna Siffert (Max-Planck-Institut für Mathematik, Bonn): Existence of metrics maximizing the first eigenvalue on closed surfaces

Abstract: We prove that for closed surfaces of fixed topological type, orientable or non-orientable, there exists a unit volume metric, smooth away from finitely many conical singularities, that maximizes the first eigenvalue of the Laplace operator among all unit volume metrics. The key ingredient are several monotonicity results, which have partially been conjectured to hold before. This is joint work with Henrik Matthiesen.

09.11.2017

• 11.30-12.30 Huy Nguyen (Queen Mary University of London): Mean curvature flow with Quadratic Curvature Bounds

Abstract: In this talk we will discuss mean curvature flow with quadratic curvature bounds. We firstly will discuss improvements to the Andrews-Baker result for higher co-dimension for surfaces of co-dimension two. We will then classify ancient solutions to high codimension mean curvature flow under a natural pinching condition and finally we will discuss singularity formation of the mean curvature flow in the sphere without positive mean curvature.

24.10.2017

• 14.00-15.00 Ben Sharp (University of Warwick): Minimal hypersurfaces in Riemannian manifolds

Abstract: Minimal hypersurfaces are critical points of the volume functional, and the Morse index tells us how many ways one can decrease their volume locally. We will present an overview of recent results which relate the Morse index to the geometry and topology of minimal hypersurfaces. These are joint works with Lucas Ambrozio-Alessandro Carlotto and Reto Buzano.

27.06.2017

• 14.00-15.00 Reto Buzano (Queen Mary University of London): The moduli space of two-convex embedded spheres and tori

Abstract: It is interesting to study the topology of the space of smoothly embedded n-spheres in . By Smale’s theorem, this space is contractible for and by Hatcher’s proof of the Smale conjecture, it is also contractible for . These results are of great importance, generalising in particular the Schoenflies theorem and Cerf’s theorem. In this talk, I will explain how mean curvature flow with surgery can be used to study a higher-dimensional variant of these results, proving in particular that the space of two-convex embedded spheres is path-connected in every dimension n. We then also look at the space of two-convex embedded tori where the question is more intriguing and the result in particular depends on the dimension n. This is all joint work with Robert Haslhofer and Or Hershkovits.

22.06.2017

• 11.30-12.30 Jan Metzger (Universität Potsdam): On the uniqueness of small surfaces minimizing the Willmore functional subject to a small area constraint

Abstract: We consider the Willmore functional for surfaces immersed in a compact Riemannian manifold M and study minimizers subject to a small area constratint. We show that if the scalar curvature of M has a non-degenerate maximum then for small enough area these minimizers are unique. This is joint work with Tobias Lamm and Felix Schulze.

16.06.2017

• 11.30-12.30 Ernst Kuwert (Albert-Ludwigs-Universität Freiburg): Willmore minimizers with prescribed isoperimetric ratio

Abstract: We discuss the existence of surfaces of type minimizing the Willmore functional with prescribed isoperimetric ratio, and some asymptotics as the ratio goes to zero.

13.06.2017

• 13.00-14.00 Paola Pozzi (Universität Duisburg-Essen): On the elastic flow of open curves in

Abstract: In this talk I will discuss the evolution of regular open curves in moving according to the -gradient flow of the elastic energy and subject to different sets of boundary conditions and constraints on the length of the curve.
The results presented are based on joint works with Anna Dall’Acqua and Chun-Chi Lin.

02.05.2017

• 14.00-15.00 Alix Deruelle (Université Pierre et Marie Curie): Expanders of the harmonic map flow

Abstract: Expanding self-similarities of a given evolution equation create an ambiguity in the
continuation of the flow after it reached a first singularity. In this talk, we investigate the
possibility of smoothing out any map from the n-sphere, , to another sphere, that is homotopic
to a constant by a self-similarity of the harmonic map flow. To do so, in the spirit of Chen-Struwe,
we introduce a one-parameter family of Ginzburg-Landau equations that exhibit the same homogeneity
and once the existence of expanders for this family is granted, we pass to the limit. We also study
the singular set of such solutions as well as the uniqueness issue.
The talk is based on joint work with Tobias Lamm.

10.02.2015

• 14:00-15:00 Daniele Valtorta (EPFL): Quantitative stratification and applications to p-harmonic maps

Abstract: The quantitative stratification technique can be used to study the singularities of different problems in mathematics. Based on blow-up analysis, it studies the interactions of infinitesimal symmetries and almost symmetries at different points in order to give bounds on the singularities, which in this context should be thought of as points with few symmetries. The aim of the talk is to describe this technique through the example of the recent results obtained on the regularity of p-harmonic maps. This is a joint work with Prof. Cheeger, Naber and Veronelli.

20.01.2015

• 14:00-15:00 Huy Nguyen (University of Queensland): The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifolds

18.11.2014

• 14:00-15:00 Glen Wheeler (University of Wollongong): Unstable Willmore Surfaces

Abstract: In this talk I will describe some recent work with Anna Dall'Acqua
(Ulm) and Klaus Deckelnick (OvGU Magdeburg) that established the existence
of Willmore surfaces with boundary that are unstable. The natural approach
of using Palais-Smale and Mountain Pass doesn't (really) work. I'll explain
why this is the case. We overcame this problem by using a completely
different (and new) approach. I will finish by describing some open
questions arising from the work.

14.10.2014

• 14:00-15:00 Benjamin Sharp (Imperial College London): Compactness theorems for minimal hypersurfaces of bounded index

25.07.2014

• 11:00-12:00 Nathalie Tassotti (Universität Wien): A positive mass theorem for low-regularity Riemannian metrics

04.07.2014

• 14:00-15:30 Roberta Alessandroni (Universität Freiburg): Local solutions to a free boundary problem for the Willmore functional

29.04.2014

• 13:00-13:45 Qi Ding (MPI Leipzig): Minimal hypersurfaces in manifolds with nonnegative Ricci curvature

• 14:30-15:15 Elena Mäder-Baumdicker (Universität Freiburg): Der area preserving curve shortening flow mit freien Neumann-Randwerten

04.02.2014

• 13:00-14:00 Jan Metzger (Universität Potsdam): Jenkins-Serrin-Spruck-Theorie für Graphen mit konstanter mittlerer Krümmung

Abstract: Ziel der klassischen Jenkins-Serrin-Spruck-Theorie ist die Konstruktion
von zweidimensionalen Graphen mit konstanter mittlerer Krümmung, deren
Randwerte unendlich sein können. Dies führt auf
Kompatibilitätsbedingungen an das zugrunde liegende Gebiet, die
sogenannten Jenkins-Serrin-Spruck Bedingungen. Bisherige Arbeiten
erforderten zusätzlich gewisse Symmetriebedingungen sowie die Existenz
einer Sublösung um die Existenz der Graphen zu garantieren. In diesem
Vortrag möchte ich erläutern, wie man ohne diese zusätzlichen
Voraussetzungen auskommt. Dies ist eine gemeinsame Arbeit mit Michael
Eichmair.

• 14:30-15:30 Carla Cederbaum (Universität Tübingen): Zur Definition von Masse und Schwerpunkt isolierter Systeme in der Newtonschen Gravitationslehre und der Allgemeinen Relativitätstheorie

Abstract: Isolierte Systeme wie beispielsweise Sterne, schwarze Löcher oder Galaxien spielen eine wichtige Rolle sowohl in der Newtonschen Gravitationslehre (NG) als auch in der Allgemeinen Relativitätstheorie (ART). Während in der NG die Definition von Masse und Schwerpunkt mittels der Massendichte naheliegt, gibt es in der ART mehrere vielversprechende Ansätze. Die wichtigsten Ansätze zur Definition des Schwerpunkts in der ART gehen auf Arnowitt, Deser und Misner (ADM) sowie auf Huisken und Yau (HY) zurück. Unter gewissen Annahmen an die Asymptotik stimmen ADM- und HY-Schwerpunkt überein (Huang, Metzger-Eichmair, Nerz). Beide Begriffe hängen jedoch auf empfindliche Weise von den gewählten asymptotischen Koordinaten ab; wir werden insbesondere ein explizites Beispiel vorstellen, in denen beide Schwerpunkte divergieren -- und ein analoges Beispiel in der NG (C-Nerz). Außerdem werden wir (im Spezialfall unbewegter Systeme) den sogenannten Newtonschen Limes der ART untersuchen und so den relativistischen Schwerpunkt mit dem Newtonschen direkt in Verbindung setzen.

09.07.2013

• 14:00-15:00 Clemens Kienzler (Universität Bonn): Flat Fronts and Stability for the Porous Medium Equation

Abstract: Considering the Cauchy problem for the porous medium equation, we prove stability and partial regularity of the pressure of solutions close to flat fronts under conditions on the initial data that are optimal for the methods we use. In a perturbational setting we rely on the theory of singular integrals in spaces of homogeneous type to get linear estimates, and apply an analytic fixed point argument in special functions spaces for nonlinear results.

06.06.2013

• 16:00-17:00 Christine Breiner (Columbia University): Embedded constant mean curvature surfaces in Euclidean three space

Abstract: Constant mean curvature (CMC) surfaces are critical points to the area functional with an enclosed volume constraint. Classic examples include the round sphere and a one parameter family of rotationally invariant surfaces discovered by Delaunay. In this talk I outline a generalized gluing method we develop that produces infinitely many new examples of embedded CMC surfaces of finite topology. In particular, I explain how we solve the global linearized problem in the presence of possible obstructions and how we handle the remaining higher order terms. Finally, I will mention aspects of the proof we must alter to adapt the method to higher dimensions. This work is joint with Nicos Kapouleas.

17.04.2013

• 11:30-12:30 Felix Schulze (University College London): The half-space property and entire minimal graphs in MxR

07.03.2013

• 17:30-18:30 László Székelyhidi (Universität Leipzig): Isometric embeddings and turbulent energy cascades

Abstract: One of the cornerstones of the theory of 3-dimensional turbulence is the energy cascade: in a turbulent fluid the energy is transferred successively to smaller and smaller scales by some non-linear mechanism until at the smallest scale it is transferred to heat by viscous dissipation. A consequence is the so-called dissipation anomaly. Although as a heuristic explanation the energy cascade is widely accepted by the community, few rigorous results are known. This is also closely related to a famous conjecture of Onsager from 1949 and to Kolmogorov's theory of turbulence. In joint work with Camillo De Lellis we interpret this cascade using an old idea of Nash and obtain a method of construction, which can be seen as a "hard" PDE version of Gromov´s convex integration. In the talk I will recall the basic scheme of Nash, explain the similarities and differences with our scheme and report on current progress regarding Onsager's conjecture.

15.01.2013

• 14:00-15:00 Miles Simon (Universität Magdeburg): Local results for Ricci flow on regions with curvature bounded from below

06.12.2012

• 13:00-14:00 Reto Müller (Imperial College London): On finite time singularities of the Ricci flow

Abstract: In this talk, we study Perelman's W-entropy functional at finite time singularities of the Ricci flow. We first explain how this entropy can be used in the case of fast-forming singularities (the so-called Type I case) to prove that at any singular point there exists a sequence of rescaled flows that smoothly converges to a self-similar solution of the flow, a so-called gradient shrinking soliton. In the second part of the talk, we discuss a uniqueness result for these limit shrinking solitons.

20.11.2012

• 14:00-15:00 Benjamin Sharp (Imperial College London): Critical 'd-bar' problems in one complex dimension and some remarks on conformally invariant variational problems in two real dimensions

Abstract: We will consider a linear first order system, a connection 'd-bar' problem, on a vector bundle equipped with a connection, over a Riemann surface. We show optimal conditions on the connection forms which allow one to find a holomorphic frame, or in other words to prove the optimal regularity of our solution. The underlying geometric principle, due to Koszul-Malgrange, is classical and well known; it gives necessary and sufficient conditions for a connection to induce a holomorphic structure on a vector bundle over a complex manifold. We will explore the limits of this statement in one complex dimension, when the connection is not smooth and our findings lead to a very short proof of the regularity of harmonic maps in two dimensions as well as re-proving a recent estimate of Lamm and Lin concerning conformally invariant variational problems in two dimensions.

23.10.2012

• 14:45-15:45 Fernando Coda Marques (IMPA): Minimal surfaces and conformally invariant variational problems

Abstract: In this talk I will explain how to produce the Clifford torus as a min-max minimal surface in the three-sphere, and how to use this to prove the Willmore conjecture (joint work with André Neves).
Then I will explain another application: a sharp energy estimate for links in Euclidean space that had been conjectured by Freedman, He, and Wang (joint work with Ian Agol and A. Neves). I will try to emphasize what both solutions have in common.

• 16:15-17:15 Peter Hornung (MPI Leipzig): Some remarks about the Willmore functional with an isometry constraint

Abstract: In this talk we present a general approach to study the restriction of the Willmore functional to the class of isometric immersions of a given 2d Riemannian manifold into R^3. Our main focus will be on the derivation of the corresponding constrained Willmore equation.