### Geometric Analysis Seminar

### Oberseminar Geometrische Analysis

Dienstag 16 Uhr c.t., **Raum 903, **RMayer-Straße 6-8

Prof. Dr. E. Cabezas Rivas

Prof. Dr. Th. Mettler

Aktuelle Vorträge:

**Wintersemester 2017/18 (16.10.17 - 09.02.18)17.10.2017 Gabriel Paternain (University of Cambridge) 31.10.2017 Andy Sanders (Universität Heidelberg)21.11.2017 Sebastian Heller (Universität Hannover)**

**28.11.2017 Nawal Sadawi (Frankfurt) Titel: **Das Chern-Gauss-Bonnet Theorem (Bachelorarbeit)

05.12.2017 Friederike Dittberner (Universität Kontanz)

09.01.2018 Maciej Dunajski (University of Cambridge)

05.12.2017 Friederike Dittberner (Universität Kontanz)

09.01.2018 Maciej Dunajski (University of Cambridge)

**30.01.2018 Anna Wienhard (Universität Heidelberg) **

ARCHIV

**Sommersemester 2017 (10. April bis 14. Juli 2017)**

**25.04.2017 Franziska Borer (ETH Zurich)**

Titel: Uniqueness of Weak Solutions for the Normalised Ricci Flow on Closed Surfaces

Abstract: "We show uniqueness of classical solutions of the normalised two-dimensional Hamilton–Ricci flow on closed,

smooth manifolds for H^{^2}-data among solutions satisfying (essentially) only a uniform bound for the Liouville energy

and a natural space-time L^{^2}-bound for the time derivative of the solution. The result is surprising when compared

with results for the harmonic map heat flow, where non-uniqueness through reverse bubbling may occur."

**02.05. 2017 Katharina Neusser (Charles University Prag**)**Titel**: C-projective structures of degree of mobility at least two **Abstract**: In recent years there has been renewed interest in c-projective geometry, which is a natural analogue

of real projective geometry in the setting of complex manifolds, and in its applications in Kähler geometry.

While a projective structure on a manifold is given by a class of affine connections that have the same

(unparametrised) geodesics, a c-projective structure on a complex manifold is given by a class of affine complex

connections that have the same ``J-planar'' curves. In this talk we will be mainly concerned with c-projective

structures admitting compatible Kähler metrics (i.e. their Levi-Civita connections induce the c-projective structure),

and will present some work on the geometric and topological consequences of having at least two compatible

Kähler metrics. An application of these considerations is a proof of the Yano--Obata conjecture for complete

Kähler manifolds---a metric c-projective analogue of the conformal Lichnerowicz conjecture. This talk is based

on joint work with D. Calderbank, M. Eastwood and V. Matveev.

16.05.2017**Karin Melnick (University of Maryland)Titel: **Topology of automorphism groups of parabolic geometries

**Abstract**: It is well known that the automorphism group of a rigid geometric structure is a Lie group. In fact, as

there are multiple notions of rigid geometric structures, the property that the local automorphisms form a Lie

pseudogroup is sometimes taken as an informal definition of rigidity for a geometric structure. In which topology

is this the case? The classical theorems of Myers and Steenrod say that C

^{^0}convergence of isometries of a smooth

Riemannian metric implies C^\infty convergence; in particular, the compact-open and C^\infty topologies coincide

on the isometry group. I will present joint results with C. Frances in which we prove the same result for local

automorphisms of smooth parabolic geometries, a rich class of geometric structures including conformal

and projective structures. As a consequence, the automorphism group admits the structure of a Lie group

in the compact-open topology.

23.05.2017

23.05.2017

**Charles Frances (Université de Strasbourg)**

**Title:**Dynamics and topology for 3-dimensional Lorentz manifolds

**It is a well known phenomenon that in contrast to what happens for Riemannian manifolds, compact Lorentz structures might have a noncompact group of isometries. Such a property of the isometry group generally has strong consequences both on the geometry, and on the topology of the manifold. The aim of the talk is to present new results and methods on the subject, with an emphasis on closed $3$-dimensional manifolds.**

Abstract:

Abstract:

**13.06.2017 Sophia Jahns (Universität Tuebingen)**

**Titel:** Trapped Light in Stationary Spacetimes

**Abstract:** Light can circle a massive object (like a black hole or a neutron star) at a „fixed distance“, or, more generally, circle the object without falling in or escaping to infinity. This phenomenon is called trapping of light and well understood in static, asymptotically flat (AF) spacetimes. If we drop the requirement of staticity, similar behavior of light is known, but there is no definiton of trapping available.

After a short introduction to General Relativity, I present some known results about trapping of light in static AF spacetimes. Using the Kerr spacetime as a model, I then show how trapping can be better understood in the framework of phase space and work towards a definition for photon regions in stationary AF spacetimes. **20.06.2017 Joel Kübler (Frankfurt)Titel: **Travelling-Wave-Lösungen nichtlinearer Klein-Gordon-Gleichungen auf der Sphäre (Vortrag Master-Arbeit)

**Wir betrachten spezielle Lösungen einer nichtlinearen Klein-Gordon-Gleichung auf kompakten Mannigfaltigkeiten, die eine Verallgemeinerung von euklidischen Traveling-Waves darstellen. Mithilfe variationeller Methoden zeigen wir die Existenz solcher Lösungen und erläutern deren Eigenschaften. Insbesondere konzentrieren wir uns auf die 2-Sphäre und untersuchen dort unter abgeschwächten Voraussetzungen Lösungen, die anschaulich um eine Achse rotieren. Dies führt schließlich auf Einbettungsresultate für geeignete Hilberträume, die in enger Beziehung zu fraktionalen Sobolev-Räumen stehen.**

Abstract:

Abstract:

27.06.2017 Ruth Kellerhals (Université de Fribourg)

27.06.2017 Ruth Kellerhals (Université de Fribourg)

**Titel:** "Higher logarithmic integrals and non-euclidean volume" **Abstract**: Starting with a simple concrete integral expression I shall explain its connection to spherical volume and the respective computational difficulties. Then, I shall pass to related higher logarithmic integrals and present recent work about hyperbolic volume in 5 dimensions.

**04.07.2017 Farid Madani (Frankfurt)Titel: ** Lokal konform Kählersche Geometrie und konforme Kählersche Metriken.

**Abstract: **Nach einer kurzen Einführung in die lokal konform Kählersche Geometrie, werden konforme nichthomothetische

Kählersche Metriken auf einer kompakten Mannigfaltigkeit klassifiziert. Der Vortrag basiert auf einer gemeinsamen Arbeit mit

A. Moroianu und M. Pilca.

Sondertermin am Mittwoch, den 26.07.2017 um 16 Uhr, Raum 110

Bachelorvortrag von **Nikolai Krasnosselski**

Titel: Krümmungsfluss unter Potenzen der Gausskrümmung

**Sondertermin 17.8.2017, Raum 110Tristan Daus (Frankfurt), BachelorvortragTitel:** Inverse flow in AdS-Schwarzschild manifold

(proof of new Minkowski inequalities)

**Wintersemester 2016/1715.11.2016 Franz Schuster (TU Wien/Österreich)Titel**: Affine vs. Euclidean isoperimetric inequalities

**Abstract**: In this talk we explain how every even, zonal measure on the Euclidean unit

sphere gives rise to an isoperimetric inequality for sets of finite perimeter which

directly implies the classical Euclidean isoperimetric inequality. The strongest

member of this large family of inequalities is shown to be the only affine invariant

one among them – the Petty projection inequality. As application, a family of

sharp Sobolev inequalities for functions of bounded variation is obtained, each of

which is stronger than the classical Sobolev inequality.

(joint work with Christoph Haberl)

**6.12.2016**

**Kasri Khani-Alemouti (Frankfurt)**

**Titel**: Symmetrische Räume

**13.12.2016 Thomas Mettler (Frankfurt)**

Titel:Minimal Lagrangian connections

Titel:

**A connection on the tangent bundle of a smooth manifold M can be understood**

Abstract:

Abstract:

as a map into an affine bundle over M, whose total space carries a pseudo-Riemannian metric

as well as a symplectic form, both of which can be constructed in a canonical fashion from the

projective equivalence class of the connection. This viewpoint gives rise to the notion of a

minimal Lagrangian connection. I will discuss the classification of minimal Lagrangian

connections on compact oriented surfaces of non-vanishing Euler characteristic and show

how minimal Lagrangian connections naturally generalise the notion of an Einstein metric.

10.01.2017 François Fillastre (Université de Cergy-Pontoise)

10.01.2017 François Fillastre (Université de Cergy-Pontoise)

**Titel:**A remark about the space of flat metrics with conical singularities on a compact surface

**Abstract:**W.P. Thurston showed that the area form naturally endows the space of

flat metrics on the sphere with prescribed n cone singularities of

positive curvature with a structure complex hyperbolic structure of

dimension n-3. Using classical polyhedral geometry in Euclidean space,

we note that this space is decomposed by (real) hyperbolic convex

polyhedra of dimensions (n-3) and between 0 and (n-1)/2.

By a result of W.~Veech, there is a fibration of the space of flat

metrics on a compact surface with prescribed cone singularities of

negative curvature, and the area form naturally endows each leaves with

a structure of (complex) pseudo-sphere. Here the signature (that may be

degenerated) depends on the choice of the angles.

Using polyhedral surfaces in Minkowski space, we show that this space is

decomposed by spherical convex polyhedra.

**17.01.2017 Luca Martinazzi (Universität Basel)** **Title**: The fractional Liouville equation in dimension 1 - Geometry, compactness and quantization

**Abstract**: I will introduce the fractional Liouville equation on the circle S^1 and its geometric

interpretation in terms of conformal immersions of the unit disk into the complex plane. Using

this interpretation we can show that the solutions of the fractional Liouville equation have very

precise compactness properties (including quantization and half-quantization) with a clear

geometric counterpart. I will also compare these result to analogue ones for the classical Liouville

equation in dimension 2, used to prescribe the Gaussian and Q-curvature. This is a joint work

with Francesca Da Lio and Tristan Riviere.

**Sommersemester 2016**

12.04.2016 **Ignace Aristide Minlend (AIMS Mbour/Senegal)**Titel: Existence of self-cheeger sets on Riemannian Manifolds

Abstract

19.04.2016

**Micha Wasem (ETH Zürich)**

Titel: Convex Integration, Isometric Extensions and Approximations of Curves

Abstract: In this talk, I will present some applications of convex integration — the tool used in order

to prove the celebrated Nash-Kuiper theorem. I will show how convex integration can be used to

describe parallel parking explicitly, what it has to do with an optimality question related to a coin

trick and how it leads to the construction of knots with prescribed curvature.

26.04.2016

**Roland Hildebrand (WIAS Berlin)**

Titel: Canonical barriers on regular convex cones

Abstract: Conic optimization is concerned with the minimization of linear objective functions over

affine sections of regular convex cones. One type of solution methods are the so-called interior point

methods, which need a barrier function defined on the interior of the cone. The barrier function is a

smooth, locally strongly convex, logarithmically homogeneous, self-concordant function

which tends to infinity if the argument tends to the boundary of the cone. The speed of the interior

point algorithm depends on a scalar parameter of the barrier. In this talk we present a universal barrier,

i.e., one which is defined for every regular convex cone, which has a parameter at most equal to the

dimension of the cone. The construction is closely linked to the Calabi conjecture on hyperbolic affine hyperspheres.

03.05. 2016

**Jonas Knörr (Frankfurt)**

Titel: The hard Lefschetz theorem

10.05. 2016

**Daniele Alessandrini (Universität Heidelberg)**

Titel: Geometric structures on manifolds and Higgs bundles

Abstract: Higgs bundles can be used to construct geometric structures

on manifolds. I will explain how to use them to construct the closed

Anti-de Sitter 3-manifolds, and some real and complex projective

structures with holonomy in the Hitchin components. The construction

uses the solutions of Hitchin equations, a system of global elliptic

PDEs on a Riemann surface. This is joint work with Qiongling Li.

17.05.2016

**Wojciech Kryński (IMPAN Warschau)**

Titel: On geometry of GL(2,R)-structures. Abstract: The GL(2,R)-structures appear as natural generalizations of 3-dimensional

conformal geometry and are immanently connected to geometric theory of ODEs and certain integrable PDEs. We shall present

the basic concepts in the GL(2,R)-geometry and its applications as higher-dimensional counterparts of the Einstein-Weyl structures.

We shall also present new results on connections between the GL(2,R)-structures and complex geometry.

24.05.2016

**Dmitry Faifman (University of Toronto/Kanada)**

Titel: Some kinematic formulas for O(p,q).

Abstract: A central object in integral geometry are the kinematic formulas. Given a group G of motions of the plane, one wishes to write a closed expression for the average of the values of a certain G-invariant valuation - an integro-geometric quantity - of the intersection of two convex bodies (or manifolds) over their various relative positions under the affine action of G. I will first recall the theory behind such formulas for compact groups G, then present an attempt at obtaining kinematic formulas when G is the non-compact O(p,q).

07.06.2016

**Saikat Mazumdar (Nancy)**

Titel: Higher order Elliptic problems with Critical Sobolev Growth on a compact Riemannian Manifold:

Best constants and existence.

Abstract: We investigate the existence of solutions to a nonlinear elliptic problem involving the critical Sobolev

exponent for a Polyharmomic operator on a Riemannian manifold M. We first show that the best constant of the

Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a

consequence derive the existence of minimizers when the energy functional goes below a quantified threshold.

Next, higher energy solutions are obtained by Coron's topological method, provided that the minimizing solution

does not exist and the manifold satisfies a certain topological assumption. To perform the topological argument,

we obtain a decomposition of Palais-Smale sequences as a sum of bubbles and adapt Lions's concentration-compactness lemma.

21.06.2016 **Lukas Poerschke (Frankfurt)Titel: **Komplexe Raumformen

**Wintersemester 2015/16**13.10.2015

**Elhadji Abdoulaye Thiam**(AIMS Senegal, z. Zt. Frankfurt)

**Titel**: Hardy-Sobolev inequality with cylindrical weight on Riemannian manifolds

Abstract

03.11.2015

**Ignace Aristide Minlend**(AIMS Senegal, z. Zt. Frankfurt)

**Titel**: Construction of solutions to Serrin's overdetermined problem on the 2-sphere

Abstract

10.11.2015

**Bernardo González Merino**(TU München)

**Titel**: On the Minkowski measure of symmetry"

Abstract: The Minkowski measure of symmetry s(K) of a convex body K, is the smallest positive dilatation

of K containing a translate of -K. In this talk we will explain some of its basic properties in detail.

Afterwards, we will show how s(.) can be used to strengthen, smoothen, and join different geometric

inequalities, as well as its connections to other concepts such as diametrical completeness, Jung's

inequality, or Banach-Mazur distance

24.11.2015 ** Alexander Lytchak** (Universität Köln)**Beginn 16.30 Uhr****Titel**: Classical Plateau problem in non-classical spaces

19.01.2016 **Nicolas Tholozan** (University of Luxembourg)**Titel**: Entropy of Hilbert geometries**Abstract**

02.02.2016 **Ana Peón-Nieto** (Universität Heidelberg)**Titel**: SU (p, p + 1)-HIGGS BUNDLES AND THE HITCHIN MAP

Abstract

09.02.2016 **Stefan Rosemann** (Universität Jena)**Titel**: Complex projective transformations on (pseudo-)Kähler manifolds **Abstract:** The complex projective transformations of a (pseudo-)Kähler metric are defined by the property that they preserve the set of so-called J-planar curves. These curves satisfy that the acceleration is complex proportional to the velocity and can be viewed as natural generalizations of geodesics to the complex setting. In may talk I will discuss the proof of the following statement: for a closed (pseudo-)Kähler manifold all infinitesimal complex-projective transformations preserve the canonical connection of the metric unless the metric is the Fubini-Study metric. This statement is due to recent joint work with V. Matveev and A. Bolsinov and generalizes previously obtained results in the positive definite case.

Sommersemester 2015**26.05.2015 Dr. Farid Madani (GU Frankfurt am Main)****Titel**: S^1-Yamabe invariant on 3-manifolds.

**Abstract**: After a short overview on the (non-equivariant) Yamabe invariant, we introduce the equivariant one.

We show that the S^1-Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the

(non-equivariant) Yamabe invariant of the 3-sphere. Moreover, we give a topological upper bound for the

S^1-Yamabe invariant of any closed oriented 3-manifold endowed with a circle action. This is joint work

with Bernd Ammann and Mihaela Pilca.**23.06.2015 Olaf Müller (Universität Regensburg)Titel: **Conformal techniques, bounded geometry and the Yamabe flow

**Abstract:**After giving an overview over some recently developed conformal methods in Riemannian and Lorentzian geometry,

we focus on a result of a joint work with Marc Nardmann (Dortmund) stating that every conformal class contains a metric of

bounded geometry. Finally, we sketch implications of the result in the theory of the Yamabe flow on noncompact manifolds.

Wintersemester 2014/15

28.10.2014 Gil Solanes (UAB)

Titel:** Invariant valuations in complex and quaternionic spaces. **

11.11.2014 Florian Besau (TU Wien)

Titel: The spherical convex floating body

Abstract

**Donnerstag, 04.12.2014, Raum 404, 15.15 Uhr Prof. Dr. Franz Schuster (TU Wien)**

Titel: Crofton formulas for Minkowski valuations and the Christoffel problem

Abstract: The classical Crofton formula for rigid motion invariant valuations is one of the starting points for many developments in modern integral geometry.

In this talk we present a new Crofton formula for translation invariant and SO(n) equivariant Minkowski valuations which leads to a surprising connection to C. Berg's solution of the Christoffel problem for area measures of order one of convex bodies. **09.12.2014 Dr. Thomas Mettler(ETH Zürich)**

Title: Projective surfaces, holomorphic curves and the SL(3,R)-Hitchin component

Abstract: A projective structure P on a surface M is an equivalence class of affine torsion-free connections on M where two connections are called projectively equivalent if they share the same geodesics up to parametrisation. An oriented projective surface (M,P) defines a complex surface Z together with a projection to M whose fibres are holomorphically embedded disks. Moreover, a conformal connection in the projective equivalence class corresponds to a section whose image is a holomorphic curve in Z. Findig a section of Z->M whose image is “as close as possible" to a holomorphic curve turns out to be related to the parametrisation of the SL(3,R)-Hitchin component in terms of holomorphic cubic differentials.

Sommersemester 2014

**29.04.2014 Prof. Uwe Semmelmann (Univ. Stuttgart)**Title: Almost complex structures on quaternion-Kähler manifolds and homogeneous spaces

**Abstract: In meinem Vortrag möchte ich zeigen, wie man die Nicht-Existenz von**

fast-komplexen Strukturen auf verschiedenen Klassen von Mannigfaltigkeiten

zeigen kann. Die Beweise beruhen auf einer Anwendung des Atiyah Singer

Indexsatzes für gewisse getwistete Dirac Operatoren.

13.05.2014 Manuel Ritoré (Univ. Granada)

13.05.2014 Manuel Ritoré (Univ. Granada)

Title: Large isoperimetric regions in the product of a compact

manifold with Euclidean space

Abstrakt: Given a compact Riemannian manifold $M$ without boundary, we

show that large isoperimetric regions in the Riemannian product $M\times

R^k$ of $M$ with the $k$-dimensional Euclidean space $R^k$ are tubular

neighborhoods of $M\times\{x\}$, $x\in R^k$.

**Title: On minimal surfaces in Finsler spaces (joint work with P. Overath)**

03.06.2014 Prof. Dr. Heiko von der Mosel (RWTH Aachen)

03.06.2014 Prof. Dr. Heiko von der Mosel (RWTH Aachen)

Abstract: In contrast to classic minimal surface theory relatively little seems to be

known about minimal surfaces in Finsler manifolds. We explore a connection

between the Busemann-Hausdorff volume in Finsler spaces and Cartan functionals

to prove new results in that direction, such as Bernstein theorems, a

uniqueness result, and removability of singularities for Finsler-minimal

graphs, isoperimetric inequalities and enclosure theorems for minimal

immersions in Finsler space, and we treat the Plateau problem in Finsler

$3$-space.

**Title: Valuations on Riemannian manifolds**

24. Juni 2014 Prof. Joseph Fu (University of Georgia, Athens)

24. Juni 2014 Prof. Joseph Fu (University of Georgia, Athens)

Abstract: A smooth valuation on a general manifold M is a finitely additive set function m, defined on a restricted class of “nice” subsets A, determined by a pair of differential forms, one living on M and the other on its cosphere bundle. The value m(A) is then expressed as the sum of the integral over A of the first and the integral of the second over the manifold of outward conormals to A. S. Alesker has shown that smooth valuations admit a natural multiplication.

Historically the first such expression predated this formal definition, in the form of Chern’s generalized Gauss-Bonnet theorem giving the Euler characteristic of a Riemannian manifold with boundary in terms of integrals of differential forms arising from the curvature tensor. We introduce a universal family of valuations that arise in similar fashion, and note that under Alesker multiplication it is naturally a module over the polynomial algebra R[t]. We determine the multiplication table for this module and give an application to the integral geometry of complex space forms.

**8. Juli 2014 Prof. Dmitry Faifman (Tel Aviv)**

Title: Concentration of measure and the flat torus.

Abstract. The phenomenon of concentration of measure goes back to Paul

Levy, and was used extensively by Gromov, Milman, Pisier and many others

in the study of the local theory of Banach spaces, also known as

asymptotic geometry. We will survey some theorems describing the source

of concentration of measure, and give examples. Then we will present a

result from our recent short note (joint with Klartag and Milman), where

no concentration of measure is present, but some consequences of

concentration appear nevertheless.

**29. Juli 2014 Christian Beck (Frankfurt**)

Titel: Freies Randwertproblem fur Niveaus mit mehreren Phasen

Abstrakt

**Wintersemester 2013/14**

**29.10.2013 Lukas Parapatits, TU Wien**

TITLE: Minkowski Valuations and the Special Linear Group

ABSTRACT: The systematic study of the space of valuations, i.e. finitely additive maps on convex bodies, that are compatible with some subgroup of linear transformations has its origins in the work of Hadwiger. Through the seminal work of Ludwig, convex-body-valued valuations that intertwine the special linear group have become the focus of recent research. In this talk, I will give a survey on characterizations of these Minkowski valuations.

**17.12.2013 Sven Jarohs, Frankfurt**

Titel: Overdetermined problems involving the fractional Laplacian.

Abstract: In 1971 Serrin proved that if there is a positive solution to the Poissonproblem with constant nonnegative right-hand side and Dirichlet boundary conditions in a domain Omega, such that the outernormal derivative of u along the boundary of Omega is constant, then Omega must be a ball. Several works since then have been devoted to the study of this kind of overdetermined problems. I will show how this result and its generalization can be extended to problems involving the fractional Laplacian. After a short introduction to the fractional Laplacian, I will present a version of Hopf's Lemma for weak continuous solutions for linear problems involving that operator. This especially induces the strong maximum principle. Based on this I will give the main geometric ideas of the proof and reason why the nonlocal structure enables us to a more general result than in the local case. My talk is based on a joint work with Moustapha Fall.

**28.01.2014 Dr. Astrid Berg (TU Wien)**

Title: Log-Concavity Properties of Minkowski Valuations (joint work with Lukas Parapatits, Franz E. Schuster, Manuel Weberndorfer)

Abstract: The famous Brunn-Minkowski inequality expresses the log-concavity of the volume functional. We use a recent result of Wannerer and Schuster to extend this classical inequality and prove log-concavity properties of intrinsic volumes of rigid motion compatible Minkowski valuations. We also obtain new Orlicz-Brunn-Minkowski inequalities for such valuations. These inequalities extend previous results by Alesker, Bernig, Lutwak, Parapatits and Schuster.

**Sommersemester 2013**

**14.05.2013 Prof. Dr. Arnd Scheel**, University of Minnesota **Titel:** Pattern selection in the wake of fronts **Abstract:** Motivated by the formation of complex patterns in the wake of growth

processes in bacterial colonies, we'll study more generally patterns

formed in the wake of moving fronts. We discuss a number of mathematical

problems that arise when one tries to predict which wavenumbers and what

type of pattern is created by the growth process. We therefore describe

in some detail how to derive linear predictions and analyze nonlinear

mechanisms that lead to failure of these predictions. This is joint work

with Matt Holzer and Ryan Goh.

**11.06.2013 Prof. Dr. Elisabeth Werne**r (Western Reserve University, Cleveland)

**Title**: Divergence for s-concave and log concave functions **Abstrac**t: In information theory, probability theory and statistics, an

f-divergence is a function that measures the difference between two (probability)

distributions. It is a generalization of commonly used divergences such as relative

entropy (Kullback-Leibler divergence), Renyi divergences, total variation

distance etc. Here, we introduce f-divergence for s-concave and for log-concave functions.

**25.06.2013 Prof. Dmitry Faifman** (University of Tel Aviv, Israel)

**Title**: Generalized translation-invariant valuations and the Lorentz group. **Abstract**:

The field of valuations on convex bodies has been developing rapidly in the

last decade, stemming from the solution of McMullen's conjecture by

Alesker, and the rich algebraic structure that was subsequently discovered

in the space of smooth valuations. Among other things, it allowed the

complete classification of valuations invariant under various compact

groups, notably the unitary group, in several works by Alesker, Bernig and

Fu, thus extending the classical orthogonal-invariant classification due to

Hadwiger. For non-compact groups, however, the situation is more obscure.

In particular, some technical difficulties of analytic nature appear, as

much of the algebraic structure only exists for smooth valuations, while

valuations arising in the non-compact setting are typically non-smooth. In

this talk, I will describe the space of generalized translation-invariant

valuations, and extend some of the now-standard operations on valuations to

the space of generalized valuations. Then I will present a complete

classification of the generalized valuations that are invariant under the

Lorentz group. The talk is based on a joint work with Semyon Alesker.

**02.07.2013 Prof. Eva Vedel Jensen **(Aarhus Universität, Dänemark)

**Title: **Recent developments in rotational integral geometry **Abstract: **In this talk, I will give an overview of recent developments

in rotational integral geometry. The focus will be on rotational integral

formulae for Minkowski tensors. The overview will include a rotational

Crofton formula and a principal rotational formula for Minkowski tensors.

Applications to local stereology will shortly be mentioned. Finally, I will

present and discuss some of the open problems in rotational integral geometry.

**09.07.2013 Tolga Yesil **(Goethe-Universität Frankfurt am Main)

**Titel:** Der Zerlegungssatz von Gerard **Abstract:** Der Zerlegungssatz von Gerard besagt, dass sich jede beschränkte

Folge des homogenen fraktionalen Sobolev-Hilbertraums als asymptotisch

orthogonale Superposition translatierender und dilatierender Elemente

dieses Raums approximieren lässt, wobei diese Approximation im Sinne der

Norm des Bildraums unter der korrospondierenden Sobolev-Einbettung, also

in einen gewissen Lebesgueraum, zu verstehen ist.

In diesem Vortrag soll die Konstruktion dieser "Profilzerlegung"

erläutert werden, indem, die wesentlichen Beweisschritte tangierend,

die Extraktion des ersten Profils demonstriert wird.

Dazu wir es nötig sein, ein kleines fourieranalytisches Repertoire -

genauer die sogenannte Littlewood-Paley Zerlegung - sowie eine auf

dieser Zerlegung fußende verbesserte Sobolev-Ungleichung

vorauszuschicken. Damit gewappnet werden wir die im Beweis verwendete(n)

Ausschöpfungsmethode(n) beschreiben und anwenden können.

**18.09.2012 **Roland Hildebrand (Grenoble)

**Titel: **Affine hypersurface immersions with parallel cubic form **Abstract:**

**13.11.2012** Franz Schuster (Wien)

**Titel**: Umgekehrte Isoperimetrische Ungleichungen

** **

**08.01.2013 Dennis Amelunxen (Cornell University, Ithaca, USA)**

**Titel: **Spherical intrinsic volumes - Applications, asymptotics, inequalities

**Sommersemester 2012**

**19.06.2012 **Patrick Mentrup **Titel: **Konvexität der L_p-Schnittkörper

**03.07.2012** Susanna Dann (Missouri)

**Titel: **The Busemann-Petty Problem in Complex Hyperbolic Space **Abstract**: The Busemann-Petty problem asks whether origin-symmetric convex bodies in R^n

with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative

if n<5 and negative if n>=5. We study this problem in the complex hyperbolic n-space and prove

that the answer is affirmative for n<3 and negative for n>=3.

**10.07.2012 ** Mikhailo Saienko (Frankfurt) **Titel: **SU(n)-invariante Bewertungen

**Wintersemester 2011/12**

**13.12.2011** Dr. Mouhamed Mustafa Fall (Frankfurt)

**Titel: ** Sharp local upper bound for the first non-zero Neumann eigenvalue in

Riemanian manifolds. **Abstract:** We study the local Szegö-Weinberger isoperimetric profile in a

geodesic ball in a Riemannian manifold. This profile is obtained by maximizing

the first nontrivial Neumann eigenvalue of the Laplace-Beltrami

operator among subdomains of this geodesic ball with

fixed volume. We derive a sharp asymptotic upper bound of this profile

in terms of the scalar curvature. As a consequence, we deduce

an isoperimetric comparison principle relative to the

corresponding profile for 2-dimensional manifolds

**08.11.2011** Dr. Christoph Scheven (Universität Erlangen-Nürnberg)

**Titel**: *The flow for surfaces of prescribed mean curvature: Existence and asymptotics***Abstract**: The geometric significance of the H-surface equations lies in the fact

that conformal solutions parametrize a surface of prescribed mean

curvature H. Classical results guarantee the existence of solutions

under certain smallness assumptions on H, the most general of which is

an isoperimetric condition. The talk will present a new result on the

existence of solutions to the associated heat flow under the same

isoperimetric condition that implies existence in the elliptic case.

Moreover, it is shown that the solution to the flow subconverges to a

solution of the stationary problem as the time tends to infinity.

Moreover, some possible applications of the techniques to the heat flow

of m-harmonic maps are discussed.

**25.10.2011** Dr. Andzrej Muravnik

**Titel:** *Integral transforms of measures and properties of solutions of singular differential equations*

**Sommersemester 2011**

**12.07.2011** Dr. Christoph Haberl (Universität Salzburg)

**Titel:** * Valuations and affine Sobolev inequalities*

**Abstract**: Basic operators in convex geometry can be characterized as valuations

which are compatible with the special linear group. We present such

characterization results and show how these results lead to new affine

versions of classical Sobolev inequalities.

**12.07.2011 **Dr. Huy Nguyen (University of Warwick)**Titel**: * Geometric Rigidity of Surfaces with Bounded Total Curvature.***Abstract**: We consider surfaces conformally immersed in R^3 with L^2 bounds on the norm of the second fundamental form.

We will classify certain limit cases of these bounds, for example we will suitably generalize Osserman's classification

of complete non-compact minimal surfaces with total curvature equal to 8\pi to the case of complete non-compact surfaces

with total bounded curvature.

**05.07.2011 **Dr. Gil Solanes ((Universitat Autònoma de Barcelona)

**Titel: **

*Integral geometry in complex projective space.*

**Abstract**: The classical formulas of integral geometry by Blaschke and others have been recently extended to complex affine spaces by Bernig and Fu. This opens the door to the development of integral geometry in complex spaces of constant holomorphic curvature. I will present the first step in this project: the determination of Crofton formulas with respect to totally geodesic complex spaces. This is a joint work with J. Abardia and E.Gallego.

**05.07.2011** Dr. Yann Bernard (Universität Freiburg)

**Titel: ***Asymptotic analysis of branched Willmore surfaces***Abstract:** We consider Willmore surfaces in R^m (for m\ge3) with an isolated singularity of finite density at the origin.

The aim is to gather as much information as possible on the local behavior of the immersion and of the second fundamental

form around the singularity. In particular, we derive local asymptotitcs for the mean curvature vector near the singular point.

We also provide specific conditions under which the singularity is removable. This is joint-work with Tristan Rivière.

**28.06.2011 ** Professor Elisabeth Werner (Case Western Reserve University Cleveland)

**Titel**: *Non additivity of the Renyi entropy and Dvorezky's theorem*

**Abstract:** We show that the analysis of the minimum output $p$-Renyi entropy of a typical quantum channel essentially amounts to applying Dvoretzky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden--Winter and Hastings disproving the additivity conjecture for the minimal output $p$-Renyi entropy.

**14.06.2011 **Thomas Wannerer (TU Wien)

**Titel: **

*Rotation-equivariant Minkowski valuations*

**Abstract:**Valuations are a classical concept in convex geometry. They are

real-valued functions on the space of convex bodies satisfying an

additivity property. In recent years, the theory of valuations has been

generalized in various directions and in this talk we discuss

convex-body-valued valuations. We establish a new description Minkowski

valuations, discuss relations between this and other representations,

and give applications

**07.06.2011** PD Marco Kuehnel (Universität Freiburg)

**Titel**: *Krümmungserhaltende Deformationen hermitescher Mannigfaltigkeiten und Moduli von Vektorbuendeln auf nicht-Kähler-Flächen *

**Abstract**: Auf der Suche nach dem Deformationsverhalten offener

hermitescher Mannigfaltigkeiten trifft man auf einen Ansatz, der das

genannte Deformationsproblem für Kählermannigfaltigkeiten in einem

Spezialfall löst. Für kompakte nicht-Kähler-Flächen erscheint er aber besonders

interessant und führt schliesslich zu einer fast vollständigen

Klassifikation solcher Flächen mit $0$-dimensionalen Modulräumen von

Rang-2-Vektorbündeln. Hierbei ist dies im Rahmen der Klassifikation

von class VII-Flächen von gewissem Interesse.

**10.05.2011** Dr. Reto Müller (Scuola Normale Superiore di Pisa)

**Titel**: *A compactness theorem for complete Ricci shrinkers***Abstract:** We prove precompactness in an orbifold Cheeger-Gromov sense

of complete gradient Ricci shrinkers with a lower bound on their entropy and

a local integral Riemann bound. In particular, we do not need any pointwise

curvature assumptions, volume or diameter bounds. In dimension four, under

a technical assumption, we can replace the local integral Riemann bound by

an upper bound for the Euler characteristic. The proof relies on a Gauss-

Bonnet with cutoû argument. The results are joint work with Robert Haslhofer.

**15.02.2011** Professor Semyon Alesker (Tel Aviv University)

**Titel:** * A Radon type transform on valuations*

**Abstract:** Valuation is a classical notion of convex geometry with numerous applications to integral geometry. A valuation is a finitely additive measure on the class of all convex compact sets in R^n satisfying a continuity assumption. In recent years it turned out that the theory of valuations partly generalizes far beyond convexity, to all smooth manifolds. This extention has even broader relations to integral geometry, some of which will be discussed in the talk. We will describe the rich structures on valuations on manifolds assuming no preliminary knowledge of the classical theory. We introduce a Radon type transform on valuations and discuss its properties in various special cases.

**01.02.2011 **Miles Simon (Universität Freiburg)

**Titel**: *Ricci Fluss von Metriken mit Kegel Singularitäten*

**Abstract**: Wir zeigen, dass man positiv gekrümmte singuläre Kegel-Metriken mit dem Ricci Fluss evolieren kann. Die Metrik wird sofort glatt, und die Lösung ist ein expandierender Soliton.

**18.01.2011** Professor Mohameden Ould Ahmedou (Universität Giessen)

**Titel**: *A Poincare uniformization type-theorem on compact four-manifolds*

**Abstract**: In this talk we report on some progress for the study on the problem of existence of conformal metrics with constant $Q$-curvature on closed four-dimensional Riemannian manifolds. This problem amounts to solve a fourth-order nonlinear elliptic equation involving the Paneitz operator. The corresponding equation has a variational structure, however the associated Euler-Lagrange functional is not bounded from below nor from above in many situations. Furthermore, it does not satisfy the Palais-Smale condition in general. Using an algebraic topological argument, combined with a refined analysis of the loss of compactness, we solve the problem in many cases where blow-up does occur. Precisely, we prove that if the kernel of the Paneitz operator consists only of constant functions, then the above problem is solvable in all the cases (except one) left open after the celebrated works of Chang-Yang (Annals 1995) and Djadli-Malchiodi (Annals 2008).

Dr. Armin Schikorra (RWTH Aachen)

**Titel**: *Beispiele für Regularität elliptischer Systeme mit antisymmetrischen Potentialen*

**Abstract**: Es werden Beispiele aus einerseits der Theorie der fraktionalen polyharmonischen Abbildungen und andererseits aus der Theorie der degeneriert elliptischen $n$-harmonischen Abbildungen vorgestellt, welche als Fortsetzung von Riviere's berühmten Resultats von 2007 über die Regularität von kritischen Punkten von konform invarianten Variationsfunktionalen in zwei Dimensionen gesehen werden können.

**11.01.2011** Dr. Theodora Bourni (Max-Planck-Institut für Gravitationsphysik, Golm)

**Titel:** *Curvature estimates for surfaces with bounded mean curvature*

**Abstract:** In this talk I will discuss some recent results concerning estimates for the norm of the second fundamental form, |A|, for surfaces with bounded mean curvature. In particular I will show that for an embedded geodesic disk with bounded L^2 norm of |A|, |A| is bounded at interior points, provided that the W^{1,p} norm of its mean curvature is sufficiently small, p>2. This is joint work with Giuseppe Tinaglia.

**14.12.2010** Professor Jan Metzger (Universität Potsdam)

**Titel**: *Isoperimetric surfaces in asymptotically flat manifolds*

**Abstract**: In this talk I will present joint work with Michael Eichmair. We consider the isoperimetric problem in asymptotically flat manifolds which are close in C^0 to Schwarzschild. The main result is that for given large enough volume there exists a smooth connected isoperimetric surface enclosing this volume. We furthermore derive position estimates for this surface. A corollary of our analysis is that the constant mean curvature foliation constructed by Huisken and Yau consists of isoperimetric surfaces.

**7.12.2010 **Professor Oliver Schnürer (Universität Konstanz)

**Titel:** *Stability of entire graphs evolving under mean curvature flow and Gauss curvature flow*

**Abstract: **We consider the evolution of hypersurfaces described as graphs over R^n. The evolution is determined by the normal velocity. Here we consider the mean curvature and powers of the Gauss curvature as normal velocities. In both cases, we obtain stability results.

Long time existence for graphical solutions evolving under mean curvature flow is well known. Hence we focus on techniques to obtain stability results. For equations involving the Gauss curvature, we first have to establish a long time existence result. The proofs of our stability results are then similar to those for mean curvature flow.

In the first part, we investigate the qualitative behaviour of entire solutions to mean curvature flow. In the second part we mainly focus on an existence result for Gauss curvature flow.

**30.11.2010** Dr. Judit Abardia (Frankfurt)**Title**: *Projection bodies in complex vector spaces*

**Abstract**: The space of Minkowski valuations on an $n$-dimensional complex vector space which are continuous, translation invariant and contravariant under the complex special linear group is explicitly described. Each valuation with these properties is shown to satisfy geometric inequalities of Brunn-Minkowski, Alexandrov-Fenchel and Minkowski type.

**23.11.2010** Dr. Patrick Breuning (Universität Freiburg)

**Title:*** Immersions with local Lipschitz representation*

**Abstract**: We consider immersions admitting uniform graph representations over the affine tangent space. Assume that any graph is defined on a ball of radius r and satisfies a specific property such as a Lipschitz bound or a C^0-bound. We show that such graph functions coming from immersions satisfy much better properties than a single function. Particularly we show that a sufficiently small C^0-norm of any graph implies Lipschitz continuity with small Lipschitz constant. This can be used e.g. for showing compactness of such immersions.

**16.11.2010 **Dr. Gil Solanes (Universitat Autònoma de Barcelona)

**Title:** * Total curvature of complete surfaces in hyperbolic space.*

**Abstract**: I will discuss on a Gauss-Bonnet formula for the integral of the extrinsic curvature of complete surfaces in hyperbolic space. The formula contains a contribution from infinity which is a Möbius invariant of the ideal boundary curve. It can be described as the renormalized volume of the set of spheres linked with the curve. This is related to Banchoff-Pohl's definition of the area enclosed by space curves. Also, connections with some known knot energies will be discussed.

**9.11.2010** Christian Beck (Frankfurt)

**Titel:** *Symmetrie von Minimierern von Variationsproblemen mit C1-Regularität*

**Abstract:** Es wird eine Methode vorgestellt, wie bei bestimmten Variationsproblemen, und zwar solchen, bei denen polarisierte Minimierer wieder Minimierer sind und ferner hinreichende Regularität aufweisen, bewiesen werden kann, dass eine gewisse Art von Symmetrie, nämlich lokal geblätterte Schwarz-Symmetrie, vorliegt. Anwenden werden wir die gewonnenen Fertigkeiten, um Aussagen über die Gestalt von Eigenfunktionen des p-Laplace-Operators unter Neumann-Randbedingungen zum ersten positiven Eigenwert auf einem beschränkten radialsymmetrischen Gebiet zu treffen.

**14.09.2010, 16.15 Uhr** Dr. Alvaro Guevera (Institut für Informatik)

**Title:** * Convergence Results for a Self-dual Regularization of Convex Problems*

**Abstract:** We study a one-parameter regularization technique for convex optimization problems, which has as its main feature its self-duality with respect to the usual convex conjugation. The technique, introduced by Goebel, can be defined for both convex and saddle functions. When applied to the latter, we show that if a saddle function has at least one saddle point, then the sequence of saddle points of the regularized saddle functions converges to the saddle point of minimal norm of the original one.

For convex problems with inequality and state constraints, we apply the regularization directly on the objective and constraint functions, and show that, under suitable conditions, the associated Lagrangians of the regularized problem hypo/epi-converge to the original Lagrangian, and that the associated value functions also epi-converge to the original one. Finally, we find explicit conditions ensuring that the regularized sequence satisfies Slater’s condition.

**23.02.2010.** Jean Van Schaftingen (Université catholique de Louvain )

**Titel: ***Existence of optimal functions for Poincaré-Sobolev inequalities*

**26.01.2010:** Thomas Mettler (Université de Fribourg)

**Titel: *** On projective surfaces with compatible Weyl structure*

**Abstract:** The existence problem for Riemannian metrics on a manifold with prescribed (unparametrized) geodesics is a natural problem in the theory of over determined systems of partial differential equations. Unfortunately even for surfaces its solution is somewhat unpleasant. However if one generalizes the problem and looks for Weyl structures on surfaces with prescribed geodesics, the problem becomes tractable with techniques from complex geometry as will be shown in this talk.

**12.01.2010:** Prof. Dr. Tobias Kaiser (Universität Passau)

**Titel: *** Zahme Geometrie und Analysis*

**Abstract: **O-minimale Strukturen stellen eine umfassende Verallgemeinerung der klassischen Geometrien algebraischer Prägung dar. Sie zeichnen sich durch exzellente Endlichkeits- und Zahmheitseigenschaften aus und erfassen wichtige Konzepte der Analysis. Nach einer kurzen Einführung in o-minimale Strukturen zeige ich deren Bezug zu und Anwendung bei dynamischen Systemen (Hilbert 16, Teil 2), komplexer Analysis (Riemannscher Abbildungssatz) und partiellen Differentialgleichungen (Dirichlet-Problem).

**3.11. 09:** Dr. Anna Dall' Acqua (Universität Magdeburg):

**Titel: ***The Dirichlet boundary value problem for Willmore surfaces of revolution*

**Abstract**: The Willmore functional is the integral of the square of the mean curvature over the unknown surface. We consider the minimisation problem among all surfaces which obey suitable boundary conditions. The Willmore equation as the corresponding Euler-Lagrange equation may be considered as frame invariant counterpart of the clamped plate equation. This equation is of interest not only in mechanics and membrane physics but also in differential geometry.

We consider the Willmore boundary value problem for surfaces of revolution with arbitrary symmetric Dirichlet boundary condition. Using direct methods of the calculus of variations, we prove existence and regularity of minimising solutions.

**27.10. 09:** Dr. Gautier Berck (Université de Fribourg):

**Titel: ***Different aspects of the intersection bodies*

**Abstract**: Classically, the intersection bodies where first introduced by Busemann in his theory of areas in normed and Finsler spaces. The notion was later successively extended by Lutwak, Zhang and Koldobsky leading to a complete solution of the first Busemann-Petty problem.

In the first, more descriptive, part of the talk, we will focus on Busemann's notion of area, the convexity of the intersection bodies and related convex geometric problems. In a second part, we will show how the theory of distributions and Fourier transforms may be used to address some of these problems.