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# Seminars winter 2013/14

There are several research seminars in the Potsdam-Berlin area which are closely related to Geometric Analysis:

• Usually biweekly during term, on Thursdays from 15:15 till 17:45, the seminar "Topics in Geometric Analysis" at the Free University Berlin (FU) or the Albert Einstein Institute (AEI) run by Klaus Ecker, Theodora Bourni, and Ulrich Menne. The dates are October 24 (FU), November 7 (AEI) and 21 (FU), December 5 (AEI) and 12 (FU), January 9 (AEI) and 23 (FU) and February 6 (AEI).
• Usually, twice a term, on Thursdays from 15:00 till 17:30, the "Seminar Differential Geometry and Analysis" at the Free University Berlin, the Leibniz University Hannover and the Univerisity of Magdeburg run by Roger Bielawski (Hannover), Klaus Ecker (FU Berlin), Hans-Christoph Grunau (Magdeburg), Miles Simon (Magdeburg) and Knut Smoczyk (Hannover).
• In the weeks where there is no "Topics in Geometric Analysis" seminar there will usually be a reading seminar on "Geometric Measure Theory" at the Albert Einstein Institute (AEI) or the Freie Universität Berlin (FU). It will take place on October 17 (AEI), November 1 (FU), 13 (FU), 28 (AEI), and December 11 (FU). Details see below.
Timetable of seminars
DatesTypePlaceSpeakerTopic
October 17th, 2013, 14:15-15:45 GMT reading seminar

Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam

Theodora Bourni (FU Berlin)

Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.

October 17th, 2013, 16:15-17:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin)

Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.

October 24th, 2013, 15:15-16:15 Topics in Geometric Analysis FU Berlin, Room 005, Arnimallee 3, 14195 Berlin
Giuseppe Tinaglia (King's College London)
The topology of limits of embedded minimal disks

The work of Colding-Minicozzi gives that a sequence of embedded minimal disks converges, up to a subsequence, to a minimal lamination away from a closed set of singular points. In several examples of Colding-Minicozzi and others, the leaves of such lamination are disks, while Hoffman-White recently produced examples where some of the leaves are annuli. In this talk I will describe several results on the topology of the leaves of such lamination in a manifold that admits an isoperimetric inequality for minimal surfaces. For instance, each leaf has genus zero. This is joint work with Jacob Bernstein.

October 24th, 2013, 16:45-17:45 Topics in Geometric Analysis FU Berlin, Room 005, Arnimallee 3, 14195 Berlin Spencer T. Hughes (Cambridge University)
Minimal, two-valued graphs

The use of multi-valued functions in analysing the singularities of minimal submanifolds is well-established. For example, they were used by Almgren in his work on the size of the singular set of an area-minimizing current and by Schoen-Simon and Wickramasekera in various works on stable, minimal hypersurfaces. Despite progress in these contexts, gaining precise descriptions of the singularities of just minimal (i.e. `stationary', but not necessarily stable or area-minimizing) submanifolds is still considered difficult and many fundamental questions are open.

In this talk I will describe some recent results on the regularity and singularity theory of minimal, two-valued Lipschitz graphs in arbitrary codimension. In codimension one, a two-valued Lipschitz function whose graph is minimal must automatically be $C^{1,1/2}$ (as a two-valued function) and every tangent cone to the graph is a pair of hyperplanes. In higher codimension, things are naturally more complicated and I will present results that give detailed descriptions of some of the unavoidable singularities.

October 31st, 2013, 14:15-15:45 GMT reading seminar FU Berlin, Room 130, Arnimallee 3, 14195 Berlin Theodora Bourni (FU Berlin)

Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.

October 31st, 2013, 16:15-17:45 GMT reading seminar FU Berlin, Room 130, Arnimallee 3, 14195 Berlin Theodora Bourni (FU Berlin)

Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.

November 7th, 2013, 15:15-16:15
Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Benoît Daniel (Lorraine Univ.-Institut Élie Cartan) Minimal isometric immersions into S^2 x R and H^2 x R

It is a classical result that any simply connected minimal surface in Euclidean space R^3 admits a one-parameter family of minimal isometric deformations, called the associate family. Conversely, two minimal isometric immersions of the same Riemannian surface into R^3 are associate. We will investigate extensions of these results for minimal surfaces in S^2 x R and H^2 x R, where S^2 is the constant curvature 2-sphere and H^2 the hyperbolic plane.

November 7th, 2013, 16:45-17:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam

Felix Schulze (UCL Departement of Mathematics)

A local regularity theorem for the network flow

The network flow is the evolution of a network of curves under curve shortening flow in the plane, where it is allowed that at triple points three curves meet under a 120 degree condition. We present here a local regularity theorem for the network flow, which is similar to the result of B. White for smooth mean curvature flow, and discuss applications.

November 13th, 2013, 09:15-10:45 (Wednesday) GMT reading seminar FU Berlin, Room 137, Takustraße 9, 14195 Berlin Theodora Bourni (FU Berlin) Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.
November 13th, 2013, 11:15-12:45 (Wednesday) GMT reading seminar FU Berlin, Room 137, Takustraße 9, 14195 Berlin Theodora Bourni (FU Berlin) Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.
November 14th, 2013, 15:00-16:00 Seminar Differential Geometry and Analysis University of Magdeburg, Universitätsplatz 2, Building 03, Room 21 Matthias Röger (Dortmund)
Minimization of bending energies under constraints
November 14th, 2013, 16:30-17:30 Seminar Differential Geometry and Analysis University of Magdeburg, Universitätsplatz 2, Building 03, Room 21 Mohameden Ould Ahmedou (Gießen) Conformal metrics of prescribed Q-curvature on 4-manifolds
November 21th, 2013, 15:15-16:15 Topics in Geometric Analysis FU Berlin, Room 130, Arnimallee 3, 14195 Berlin Mathew Langford (Australian National University)
Curvature estimates for fully non-linear curvature flows

We study the parabolic evolution of surfaces and hypersurfaces in which the normal speed is prescribed by some function of the Weingarten curvature. The speed functions we consider are homogeneous of degree one in the curvature, and therefore cause the hypersurfaces to contract. For compact hypersurfaces, singularities will form in finite time; however, except in the setting of surface flows, additional conditions are needed to ensure that only curvature singularities occur (convexity or concavity of the speed function is sufficient). We will show that surface flows and flows by convex speed functions have good asymptotic behaviour: Wherever the speed is becoming large, the Weingarten curvature approaches the smallest convex cone containing the Weingarten curvatures of the self-similarly shrinking cylinders admitted by the initial curvature condition. On the other hand, for concave speeds additional conditions are necessary. This work is joint with Ben Andrews and James McCoy.

November 21th, 2013, 16:45-17:45 Topics in Geometric Analysis FU Berlin, Room 130, Arnimallee 3, 14195 Berlin Francisco Martin (University of Granada)
The bridge principle at infinity for minimal surfaces in $H^3$

In this talk I will try to give an idea for the difficulties in the proof of the bridge principle for minimal surfaces in hyperbolic three-space: Two properly embedded minimal surfaces $M_1$ and $M_2$, which are together strictly stable, can be connected by a sufficiently thin minimal rectangle at the ideal boundary of $\mathbb{H}^3$, called the bridge. Additionally, the embeddedness of the resulting surface is preserved.

This theorem allows a rigorous proof of a conjecture by Antonio Ros about the existence of properly embedded minimal surfaces in $\mathbb{H}^3$ with arbitrary topology.

This is a joint work with Brian White.

December 4th 28th, 2013, 09:15-10:45 (Wednesday)
GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin) Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.
December 4th, 2013, 11:15-12:45 (Wednesday) GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin) Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.
December 5th, 2013, 15:15-16:15
Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Thomas Schmidt (University of Erlangen-Nürnberg)
Plateau's problem in infinite-dimensional spaces

The classical Plateau problem consists in finding a surface of minimal area among all 2-dimensional surfaces in Euclidean 3-space with a prescribed boundary. However, Plateau's problem makes sense in a much more general setup, and its oriented version can be formulated – with the metric currents of Ambrosio & Kirchheim – for surfaces of any dimension in an arbitrary metric space. The talk will report on some recent existence and regularity results for this generalized Plateau problem, mostly in the case that the ambient space is a possibly infinite-dimensional Banach or Hilbert space. These results have been obtained in collaboration with Luigi Ambrosio and Camillo De Lellis.

December 5th, 2013, 16:45-17:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Klaus Kröncke (University of Potsdam) Einstein metrics and Ricci flow

We consider the Ricci flow close to compact Einstein metrics. We will prove stability and instability results of compact Einstein metrics with respect to the Ricci flow. These conditions are given in terms of the conformal Yamabe invariant and the Laplace spectrum.

These results generalize stability and instability results recently obtained by R. Haslhofer and R. Müller. They imply dynamical stability of some interesting classes of Einstein manifolds.

December 12th, 2013, 15:15-16:15
Topics in Geometric Analysis FU Berlin, Room 130, Arnimallee 3, 14195 Berlin Armin Schikorra (MIS Leipzig)
Fractional harmonic maps and applications

Fractional harmonic mappings are critical points of a generalized Dirichlet Energy where the gradient is replaced with a (non-local) differential operator.

I will present aspects of the regularity theory of (non-local) fractional harmonic maps into manifolds, which extends (and contains) the theory of (poly-)harmonic mappings.

I also will mention, how one can show regularity for critical points of the Moebius (Knot-) Energy, applying the techniques developed in this theory.

December 12th, 2013, 16:45-17:45 Topics in Geometric Analysis FU Berlin, Room 130, Arnimallee 3, 14195 Berlin Guido De Philippis (University of Bonn)
Spectral optimization problems with perimeter constraint

We prove existence and regularity of opens sets minimizing the k-th Dirichlet Laplacian eigenvalue under perimeter constraint. (joint work with Bozhidar Velichkov)

January 9th, 2014, 15:15-16:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Romain Gicquaud (University of Tours) Towards the proof of the Penrose inequality for asymptotically hyperbolic manifolds

The mass of an asymptotically hyperbolic manifold is an invariant associated to the behavior of the geometry of the manifold at infinity. It has been proven that if an asymptotically hyperbolic manifold has scalar curvature greater than the scalar curvature of the hyperbolic space, then its mass is non negative and is zero if and only if it is diffeomorphic to the hyperbolic space. However, understanding what mass measures is a challenging question. The Penrose inequality states that the mass can be bounded from below by some function of the area of the outermost minimal surface in the manifold. In this talk I will describe a strategy to prove this inequality together with partial results we obtained towards the complete proof.

This work in progress is join with Mattias Dahl and Anna Sakovich.

January 9th, 2014, 16:45-17:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Anda Degeratu (University of Freiburg) Witten spinors on nonspin manifolds

Unlike a 3-dimensional manifold, a higher dimensional manifold need not be spin. On an oriented Riemannian manifold the obstruction to having a spin structure is given by the second Stiefel-Whitney class. I will show that even when this obstruction does not vanish, it is still possible to define a notion of singular spin structure and associated singular Dirac operator. Then, modeling on Witten's proof of the Positive Mass Theorem, I will define the notion of Witten spinor on an asymptotically flat nonspin manifold, show their existence and describe their properties.

January 15th, 2014, 09:15-10:45 (Wednesday) GMT reading seminar FU Berlin, Hörsaal B (0.1.01), Arnimallee 14, 14195 Berlin
Mario Santilli (AEI) Michael Grüter and Jürgen Jost, "
January 15th, 2014, 11:15-12:45 (Wednesday) GMT reading seminar FU Berlin, Hörsaal B (0.1.01), Arnimallee 14, 14195 Berlin Mario Santilli (AEI) Michael Grüter and Jürgen Jost, "
January 16th, 2014, 15:15-16:15 Topics in Geometric Analysis Konrad-Zuse-Zentrum für Informationstechnik Berlin, Großer Hörsaal, Takustraße 7, 14195 Berlin Mariel Saez (Pontificia Universidad Católica de Chile) Mean curvature flow without singularities

We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow in one additional dimension.

January 16th, 2014, 16:45-17:45 Topics in Geometric Analysis Konrad-Zuse-Zentrum für Informationstechnik Berlin, Großer Hörsaal, Takustraße 7, 14195 Berlin Nikolas D. Alikakos (University of Athens) Density estimates for the vector Allen-Cahn

The scalar Allen-Cahn equation is related to Minimal Surfaces and Minimal Graphs. The Vector Allen-Cahn is related to Plateau Complexes. These are non-orientable minimal objects with a hierarchical structure. We extend the Caffarelli-Cordoba estimates (central in the proof of the De Giorgi conjecture in the scalar case) to the vector case in a couple of ways, and give applications. This is joint work with Giorgio Fusco.

January 23th, 2014, 15:15-16:15 Topics in Geometric Analysis Konrad-Zuse-Zentrum für Informationstechnik Berlin, Großer Hörsaal, Takustraße 7, 14195 Berlin Gerhard Huisken (Mathematisches Forschungsinstitut Oberwolfach, University of Tübingen) Mean curvature flow with surgery for meanconvex embedded surfaces in R^3

The lecture reports joint work with S. Brendle on a surgery algorithm for mean curvature flow of 2-dimensional embedded meanconvex surfaces in Euclidean space. The algorithm makes use of new quantitative non-collapsing estimates for embedded surfaces and a pseudolocality result.

January 23th, 2014, 16:45-17:45 Topics in Geometric Analysis Konrad-Zuse-Zentrum für Informationstechnik Berlin, Großer Hörsaal, Takustraße 7, 14195 Berlin Juan J. L. Velázquez (University of Bonn) Singularities without horizon formation for the Einstein-Vlasov system

In this talk I will describe some solutions of the Einstein-Vlasov system with spherical symmetry which have been jointly obtained with Alan Rendall. Their main property is that they yield a geodesically incomplete metric, without the formation of a horizon. The main idea in the construction of these solutions is the proof of the existence of a family of self-similar solutions by means of a shooting argument for a four-dimensional dynamical system. However, the metric associated these self-similar solutions at large distances of the center is not asymptotically flat. Nevertheless, it is possible to modify the obtained solutions and to derive a family of solutions for which the corresponding metric behaves asymptotically as the Minkowsky metric far away from the center cutting the solutions at some positive radius and analyzing the properties of the whole Einstein-Vlasov system for larger values of the radius by means of a perturbative argument for a hyperbolic system.

January 30th, 2014, 10:30-12:00 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) Michael Grüter and Jürgen Jost, "
January 30th, 2014, 13:00-14:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) Michael Grüter and Jürgen Jost, "
January 30th, 2014, 15:00-16:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) Michael Grüter and Jürgen Jost, "
January 30th, 2014, 17:00-18:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) Michael Grüter and Jürgen Jost, "
February 6th, 2014, 15:15-16:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Panagiotis Gianniotis (University of Warwick) Boundary estimates for the Ricci flow

Shi's higher order estimates for the curvature and Hamilton's compactness theorem are essential tools in the study of the singularities of the Ricci flow on complete manifolds. In this talk I will consider the Ricci flow on manifolds with boundary and present some new higher order estimates valid near the boundary. Then, I will discuss a compactness result for sequences of Ricci flows, in which the mean curvature and conformal class of the boundary are appropriately controlled.

February 6th, 2014, 16:45-17:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Yangqin Fang (Université Paris-Sud) Existence of Minimizers for the Reifenberg Plateau Problem

We generalize an existence theorem of Reifenberg by using V. Feuvrier's method. That is, given $B$ a compact set in $\mathbb{R}^n$, and $G$ a commutative group, $L \subset G$ a subgroup, $1 < d < n$. Then we can find a compact set $E \subset \mathbb{R}^n$ such that $L$ is contained in the kernel of the homomorphism $\check{H}_{d-1} (B,G) \to \check{H}_{d-1} (B,G)$ induced by inclusion $B \to E$, and such that Hausdorff measure of $E \setminus B$ is minimal under these constraints.