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# Seminars summer 2016

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), Klaus Ecker (FU), Ulrich Menne (AEI), and Jan Metzger (UP). The dates are Apr 28 (FU), May 12 (AEI), May 26 (UP), Jun 9 (FU), Jun 23 (AEI), and Jul 7 (UP).
• There is also a reading seminar on "Geometric Measure Theory" at the Albert Einstein Institute (AEI).

Timetable of seminars
DatesTypePlaceSpeakerTopic
Apr 21th, 2016, 09:15-10:45, 11:15-12:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ulrich Menne (AEI)

Topological vector spaces (Ulrich Menne, Weakly differentiable functions on varifolds, 2015; § 2)

Distributions, regularizations (Herbert Federer, Geometric Measure Theory, 1969; 4.1.1-4.1.4)

Apr 21th, 2016, 14:15-15:45, 16:15-17:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Sławomir Kolasiński (AEI) Distributions representable by integration, differential forms and m-vectorfields (Herbert Federer, Geometric Measure Theory, 1969; 4.1.5-4.1.7)
Apr 28th, 2016, 16:15-17:15/td> Topics in Geometric Analysis< FU Berlin, Room 119, Arnimallee 3 (Hinterhaus), 14195 Berlin Joaquín Pérez (Universidad de Granada)

Minimal laminations in Euclidean three-space

The classical Hoffman-Meeks conjecture states that if M is a minimal surface with finite total curvature in \mathbb{R}^3 with genus g and k ends, then k\leq g+2. This open problem motivates the study of the possible limits of a sequence of embedded minimal surfaces M_n\subset \mathbb{R}^3 with fixed genus. Typically, minimal laminations with singularities appear as such limits. By using Colding-Minicozzi theory, we will give a convergence result for (a subsequence of) the M_n if we assume a uniform bound for the injectivity radius of the M_n outside a closed countable set of \mathbb{R}^3. We will also show how one can use this convergence result to obtain a (non-explicit) bound k\leq C(g) only depending on the genus, for the Hoffman-Meeks conjecture.

Apr 28th, 2016, 17:45-18:45 Topics in Geometric Analysis FU Berlin, Room 119, Arnimallee 3 (Hinterhaus), 14195 Berlin Carlo Sinistrari (Università di Roma "Tor Vergata")

Convex ancient solutions of the mean curvature flow

We consider compact convex hypersurfaces evolving by mean curvature flow which are ancient, that is, defined for all negative times. Solutions with these properties occur as the limit of rescalings near a singularity of a general mean convex solution of the flow. The easiest example is a shrinking sphere, but other examples are known having an oval shape which becomes more and more eccentric for negative times.

In this talk we consider various sufficient conditions which ensure that our solution is a shrinking sphere. Examples are: a uniform pinching on the principal curvatures, a growth rate assumption on the diameter, a bound on the ratio of outer and inner radius, a bound on the isoperimetric ratio. These results are in collaboration with G. Huisken.

May 12th, 2016, 16:15-17:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Thomas Schmidt (Universität Hamburg)

Measure data and obstacle problems for the total variation and the area functional

The talk is concerned with minimization problems for the total variation and the area functional, in which either an obstacle constraint is imposed or a lower-order term with a measure datum is present. These problems are naturally set in the space BV of functions of bounded variation, and the focus is on existence of BV minimizers, convex duality, and connections with BV supersolutions to nonlinear PDE. A crucial technical tool is a new Anzellotti type pairing between divergence-measure fields and gradient measures.

Most of these results have been obtained in collaboration with Christoph Scheven (Duisburg-Essen).

May 12th, 2016, 17:45-18:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Marcus Khuri (Stony Brook University)

Inequalities Involving Angular Momentum and Charge

We establish mass-angular momentum-charge inequalities in higher dimensions, including within the context of minimal supergravity. These in particular give variational characterizations of some well known stationary and static black holes. We also exhibit a special case of the Penrose inequality with angular momentum in the classical 4D setting.

May 19th, 2016, 09:15-10:45, 11:15-12:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ananda Lahiri (AEI)

Cartesian products, homotopies, joins, oriented simplexes, flat chains (Herbert Federer, Geometric Measure Theory, 1969; 4.1.8-4.1.13)

May 19th, 2016, 14:15-15:45, 16:15-17:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam

Ananda Lahiri (AEI) and

Ulrich Menne (AEI)

Cartesian products, homotopies, joins, oriented simplexes, flat chains (Herbert Federer, Geometric Measure Theory, 1969; 4.1.8-4.1.13) continued.

Flat chains, relation to integralgeometry measure, polyhedral chains and flat approximation (Herbert Federer, Geometric Measure Theory, 1969; 4.1.14-4.1.23)

May 26th, 2016, 16:15-17:15 Topics in Geometric Analysis University of Potsdam, Campus Golm, Building 9, Room 2.22, Karl-Liebknecht-Straße 24-25, 14476 Potsdam Martin Reiris

A classification theorem for static solutions of the vacuum Einstein equations

I will discuss a classification theorem for metrically complete solutions of the static vacuum Einstein equations with a compact but non-necessarily connected horizon. It is stated that any such solution is either: i) a Boost, ii) a Schwarzschild black hole, or iii) is of Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Korotkin-Nicolai black holes.

May 26th, 2016, 17:45-18:45 Topics in Geometric Analysis University of Potsdam, Campus Golm, Building 9, Room 2.22, Karl-Liebknecht-Straße 24-25, 14476 Potsdam Laurent Hauswirth (Université Paris-Est Marne-la-Vallée)

Surface theory in homogenous 3-space and harmonic maps

I will give an introduction to the differential geometry of surfaces in Thurston's homogeneous 3-manifolds with a particular scope on the role of harmonic map in the theory of constant mean curvature surfaces and minimal surfaces in HxR (H the hyperbolic plane) and the Heisenberg Riemannian space

June 09th, 2016, 16:15-17:15 Topics in Geometric Analysis FU Berlin, Room 119, Arnimallee 3 (Hinterhaus), 14195 Berlin Ulrich Menne (AEI)

Weakly differentiable functions and Sobolev functions on varifolds

In geometric analysis both Sobolev functions on smooth Riemannian manifolds and models of possibly singular surfaces, such as varifolds and currents which generalise the concept of submanifold, are tools of basic importance. In this talk a theory of Sobolev functions on varifolds is presented which allows to combine these two tools.

June 09th, 2016, 17:45-18:45 Topics in Geometric Analysis FU Berlin, Room 119, Arnimallee 3 (Hinterhaus), 14195 Berlin Elena Mäder-Baumdicker (Karlsruher Institut für Technologie)

Existence of minimizing Willmore Klein bottles in Euclidean four-space

We consider immersed Klein bottles in Euclidean four-space with low Willmore energy. It turns out that there are three distinct homotopy classes of immersions that are regularly homotopic to an embedding. One is characterized by the property that the immersions have Euler normal number zero. This class contains embedded Klein bottles with Willmore energy strictly less than $8\pi$. We prove that the infimum of the Willmore energy among all immersed Klein bottles in euclidean four-space is attained by a smooth embedding that is in this first homotopy class. In the other two homotopy classes we have that the Willmore energy is bounded from below by $8\pi$. We classify all immersed Klein bottles with Willmore energy $8\pi$ and Euler normal number $+4$ or $-4$. These surfaces are minimizers of the second or the third homotopy class.

June 23th, 2016, 16:15-17:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Klaus Kröncke (Universität Hamburg)

Stable and unstable Einstein warped products

In this talk, we systematically investigate the stability properties of certain warped product Einstein manifolds. We characterize stability of these metrics in terms of an eigenvalue condition of the Einstein operator on the base manifold. In particular, we prove that all complete manifolds carrying imaginary Killing spinors are strictly stable. Moreover, we show that Ricci-flat and hyperbolic cones over Kähler-Einstein Fano manifolds and over nonnegatively curved Einstein manifolds are stable if the cone has dimension n>=10.

June 23th, 2016, 17:45-18:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI)

Sets with Lusin property and second order rectifiability

Alberti proved that if $A$ is a $m$ dimensional closed set in $\mathbf{R}^{m}$, if $k = 1, \ldots m-1$ and $\Sigma^{k}(A) = \{ x \in \partial\,A : \mathscr{H}^{m-k}(\mathscr{N}(A,x) > 0 \}$ then $\Sigma^{k}(A)$ is countably $(\mathscr{H}^{k},k)$ rectifiable of class $2$. In this talk we introduce a new notion of unit normal bundle that it allows to suitably generalize the Alberti theorem to any closed set. Assuming a Lusin type property for the unit normal bundle we study the second order rectifiability properties of $k$ dimensional closed sets in $\mathbf{R}^{m}$.

June 30th, 2016, 09:15-10:45, 11:15-12:45 GMT reading seminar Albert Einstein Institute, Main Lecture Hall, Am Mühlenberg 1, 14476 Potsdam

Ulrich Menne (AEI)

Yangqin Fang (AEI)

Flat chains, relation to integralgeometry measure, polyhedral chains and flat approximation (Herbert Federer, Geometric Measure Theory, 1969; 4.1.14-4.1.23)

Rectifiable currents, Lipschitz neighbourhood retracts, tranformation formula (Herbert Federer, Geometric Measure Theory, 1969; 4.1.24-4.1.30)

June 30th, 2016, 14:15-15:45, 16:15-17:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Yangqin Fang (AEI) Rectifiable currents, Lipschitz neighbourhood retracts, tranformation formula (Herbert Federer, Geometric Measure Theory, 1969; 4.1.24-4.1.30)
Jul 7th, 2016, 16:15-17:15 Topics in Geometric Analysis University of Potsdam, Campus Golm, Building 9, Room 2.22, Karl-Liebknecht-Straße 24-25, 14476 Potsdam Julien Cortier (Institut Fourier, Grenoble)

Mass-like invariants for asymptotically hyperbolic manifolds

We focus on Riemannian manifolds with one end asymptotic to the hyperbolic space geometry. Such objects arise in general relativity as slices of asymptotically anti-de Sitter (adS) spacetimes. Under an assumption on the decay rate of the metric, a number of authors (Abbott-Deser, Chrusciel-Herzlich, Wang...) have defined global quantities (mass and center of mass) which enjoy “asymptotic invariance” properties that we will review, and for which the group PO(n,1) of isometries of the hyperbolic space plays a central role. We will then see how to construct and classify other such asymptotic invariants when we relax the assumption on the decay rate. They are attached to finite dimensional representations of PO(n,1). We shall finally see how every such invariant is naturally linked to a curvature operator (e.g. the scalar curvature for the classical mass). This is based on a joint work with Mattias Dahl and Romain Gicquaud.

Jul 7th, 2016, 17:45-18:45 Topics in Geometric Analysis University of Potsdam, Campus Golm, Building 9, Room 2.22, Karl-Liebknecht-Straße 24-25, 14476 Potsdam Enrico Valdinoci (Weierstraß-Institut für Angewandte Analysis und Stochastik)

A notion of fractional perimeter and nonlocal minimal surfaces

We give an introductory exposition about some surfaces which minimize a nonlocal perimeter functional. These objects naturally arise in the study of the interfaces of phase transitions, when the particles exhibit long-range interactions or when boundary effects are present. In addition, these nonlocal minimal surfaces have concrete applications in several areas, such as image processing and mathematical biology. We will present some results concerning interior regularity and rigidity of nonlocal minimal surfaces, with some quantitative estimates and some qualitative descriptions in several examples. We will describe also the (rather unusual) boundary behavior of these object and their connection with the fractional Allen-Cahn equation.

Jul 21st, 2016, 09:15-10:45, 11:15-12:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) Oriented submanifolds, projective maps and polyhedral chains, duality formulae, Lip product of vectorfields; Slicing normal currents by real valued functions, maps with singularities, (part of) cubical subdivisions (Herbert Federer, Geometric Measure Theory, 1969; 4.1.31-4.2.3)
July 21st, 2016, 14:15-15:45, 16:15-17:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam

Mario Santilli (AEI)

Sławomir Kolasiński (AEI)

Oriented submanifolds, projective maps and polyhedral chains, duality formulae, Lip product of vectorfields; Slicing normal currents by real valued functions, maps with singularities (Herbert Federer, Geometric Measure Theory, 1969; 4.1.31-4.2.2)

Cubical subdivisions (part of) (Herbert Federer, Geometric Measure Theory, 1969; 4.2.3-4.2.5)