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# Seminars summer 2015

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

• Usually biweekly during term, on Thursdays, the joint seminar "Topics in Geometric Analysis" taking place in turns at the University of Potsdam (UP), the Albert Einstein Institute (AEI) and the Free University Berlin (FU). It is organised by Theodora Bourni (FU), Ulrich Menne (AEI), and Jan Metzger (UP). The dates are Apr 23 (UP), May 7 (FU), May 21 (AEI), Jun 4 (UP), Jun 18 (FU) and Jul 2 (AEI).
• 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). It will take place on Apr 16, Apr 30, May 28, Jun 11, Jun 25, Jul 7, and Jul 16. Details see below.
Timetable of seminars
DatesTypePlaceSpeakerTopic
Apr 16th, 2015, 09:45-11:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Apr 16th, 2015, 12:30-13:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Apr 23th, 2015, 16:15-17:15 Topics in Geometric Analysis University of Potsdam, Room 1.15, Building 9, Am Neues Palais 10, 14469 Potsdam Manuel Ritoré (Universidad de Granada)

Isoperimetric inequalities in unbounded convex bodies

I shall consider the problem of minimizing the relative perimeter under a volume constraint in the interior of an unbounded convex body with arbitrary boundary. I shall give an example of a convex body whose isoperimetric profile is identically zero and give a characterization of the convex bodies with positive isoperimetric profile in terms of their asymptotic cylinders. I shall also show existence of isoperimetric regions in a generalized sense, and prove the concavity of the function I^{(n+1)/n}, where I is the isoperimetric profile and R^{n+1} is the ambient Euclidean space.

This is joint work in progress with Gian Paolo Leonardi and Efstratios Vernadakis.

Apr 23th, 2015, 17:45-18:45 Topics in Geometric Analysis University of Potsdam, Room 1.15, Building 9, Am Neues Palais 10, 14469 Potsdam Julian Scheuer (Universität Freiburg) Explicit rigidity for almost-umbilical hypersurfaces
Apr 30th, 2015, 09:45-11:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Apr 30th, 2015, 12:30-13:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
May 7th, 2015, 16:15-17:15 Topics in Geometric Analysis FU Berlin, Room 007/008, Arnimallee 6, 14195 Berlin Tristan Rivière (ETH Zürich)

Some results on the calculus of variations of Riemann surfaces

We shall present var­i­ous as­pects of the minimization of func­tion­als for immersion into the spheres of sur­faces under con­strained con­for­mal class. We will focus in par­tic­u­lar on the vari­a­tions of the Will­more and the area func­tional in the class of weak con­for­mal im­mer­sions of given Rie­mann sur­faces. We will in par­tic­u­lar give a char­ac­ter­i­za­tion of rie­mann tori into S3 min­i­miz­ing lo­cally the con­for­mal vol­ume.

May 7th, 2015, 17:45-18:45 Topics in Geometric Analysis FU Berlin, Room 007/008, Arnimallee 6, 14195 Berlin Benjamin Sharp (Imperial College London)

Compactness theorems for minimal hypersurfaces with bounded index

I will present a new compactness theorem for minimal hypersurfaces embedded in a closed Riemannian manifold N^{n+1} with n<7. When n=2 and N has positive Ricci curvature, Choi and Schoen proved that a sequence of minimal hypersurfaces with bounded genus converges smoothly and graphically to some minimal limit. A corollary of our main theorem recovers the result of Choi-Schoen and extends this appropriately for all n<7.

May 21st, 2015, 16:15-17:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Lu Wang (Imperial College London)

A Topological Property of Asymptotically Conical Self-shrinkers with Small Entropy

For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the round sphere uniquely minimizes the entropy among all non-flat two-dimensional self-shrinkers. This confirms a conjecture of Colding-Ilmanen-Minicozzi-White in dimension two.

May 21st, 2015, 17:45-18:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Martin Meurer (RWTH Aachen)

Integral Menger Curvature and Rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space

In 1999 J.C. Léger proved that a one-dimensional set with finite total Menger curvature is 1-rectifiable. In this talk we will present a generalisation of this result to sets of arbitrary dimension and co-dimension. We will give a short sketch of the proof and discuss different higher dimensional versions of Menger curvature known from the literature.

May 28th, 2015, 09:45-11:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Anna Sakovich (AEI) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
May 28th, 2015, 12:30-13:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Anna Sakovich (AEI) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jun 4th, 2015, 16:15-17:15 Topics in Geometric Analysis University of Potsdam, Room 1.15, Building 9, Am Neues Palais 10, 14469 Potsdam Benjamin Lambert (Universität Konstanz)

Inverse Mean Curvature Flow inside the Sphere

Usually the inverse mean curvature flow (IMCF) of convex hypersurfaces exists for all time and flows outwards towards infinitely large spheres. In this talk we will consider IMCF on of convex disks inside the unit ball with a Neumann boundary condition on the sphere. We will see that conversely to the boundaryless case, global singularities occur in finite time, and the flow converges in C^{1, \beta} to a flat disk.

Jun 4th, 2015, 17:45-18:45 Topics in Geometric Analysis University of Potsdam, Room 1.15, Building 9, Am Neues Palais 10, 14469 Potsdam Simone Di Marino (Université Paris-Sud)

Sobolev spaces on metric measure spaces via derivations and integration by parts formula

I will illustrate recent developments of the theory of Sobolev spaces on metric measure spaces. In particular I will focus on the more recent viewpoint, inspired by Weaver’s derivations, which provides an integration by parts formula even in the abstract setting. I will show how this viewpoint is completely equivalent to the usual one constructed by approximation with Lipschitz functions under very mild assumptions on the metric measure structure. If there is time I will discuss also the theory for BV and $W^{1,1}$ spaces.

Jun 11th, 2015, 09:45-11:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Anna Sakovich (AEI) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jun 11th, 2015, 12:30-13:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Anna Sakovich (AEI) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jun 18th, 2015, 16:15-17:15 Topics in Geometric Analysis FU Berlin, Room 007/008, Arnimallee 6, 14195 Berlin Peter Topping (University of Warwick)

Refined asymptotics of the Teichmueller harmonic map flow

I will give an introduction to the Teichmueller harmonic map flow, which is related both to the harmonic map flow and the mean curvature flow. I will describe some work, completed earlier this year, establishing that the large time effect of the flow is to decompose the flow map into branched minimal immersions with no loss of energy.

Joint work with Tobias Huxol, Melanie Rupflin and Miaomiao Zhu.

Jun 18th, 2015, 17:45-18:45 Topics in Geometric Analysis FU Berlin, Room 007/008, Arnimallee 6, 14195 Berlin Debora Impera (Università degli Studi di Milano-Bicocca)

Rigidity results and topology at infinity of translating solitons of the mean curvature flow

In this talk I will discuss some rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. These results were recently obtained in collaboration with M. Rimoldi and our approach relies on the theory of f-minimal hypersurfaces.

Jun 25th, 2015, 09:15-11:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam

Anna Sakovich (AEI)

Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jun 25th, 2015, 12:30-13:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Anna Sakovich (AEI) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jul 2nd, 2015, 16:15-17:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ye Sle Cha (FU Berlin)

Geometric Inequalities in General Relativity for Non-Maximal Initial Data

The geometric inequalities (Penrose Inequalities) in general relativity which relate to the ADM mass, angular momentum and charge have been proven for a large class of the axially symmetric, asymptotically flat, maximal initial data of the Einstein-Maxwell equations. In this talk, we will introduce how to reduce the general formulation for the non-maximal initial data, to the known maximal case, whenever a system of elliptic equations admits a solution. Each equation in the system will be analyzed individually, and the solvability of the system in the near maximal case will be discussed. The talk is based on joint work with Marcus Khuri.

Jul 2nd, 2015, 17:45-18:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Melanie Rupflin (Universität Leipzig)

Horizontal curves of metrics and applications to geometric flows

The evolution of a family of metrics on a closed surface can always be described as a combination of a conformal change, a change obtained by pulling back with suitable diffeomorphisms and an evolution in "horizontal direction". As we shall discuss in this talk, curves that move only in horizontal direction turn out to be very well controlled even when the underlying conformal structure degenerates. As an application we will show that solutions of Teichmüller harmonic map flow can be continued canonically past any finite time singularity and consequently that this flow admits global weak solution for arbitrary initial data and target manifolds.

Joint work with Peter Topping.

Jul 9th, 2015, 09:45-11:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam

Ulrich Menne (AEI)

Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jul 9th, 2015, 12:30-13:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam

Ulrich Menne (AEI)

Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jul 16th, 2015, 09:45-11:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ulrich Menne (AEI) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jul 16th, 2015, 12:30-13:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ulrich Menne (AEI) Richard Schoen and Leon Simon, "Regularity of Stable Minimal Hypersurfaces", 1981.
Jul 23th, 2015, 09:45-11:30 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) Herbert Federer, "Geometric Measure Theory", 1969: 5.4.11-5.4.14.
Jul 23th, 2015, 12:30-13:45 GMT reading seminarsa Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) Herbert Federer, "Geometric Measure Theory", 1969: 5.4.11-5.4.14.