Oberseminar Geometrische Analysis

Dienstag 14 Uhr c.t.,  Raum 404

 

Prof. Dr. A. Bernig

Prof. Dr. T. Weth

 

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Aktuelle Vorträge:

 

 

Sommersemester 2012

 

03.07.2012   Susanna Dann (Missouri)

 

Titel: tba

 

 

 

 

 

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

 

Abstract

 

 

 

 

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).

 

Bitte beachten: zusätzlicher Termin am Freitag, den 14.01.2011, 12:00 Uhr, Raum 404

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

Dieser Vortrag findet um 16 ct im Raum 903 statt!

 

 

 

Archiv

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.

 

 

 

 

geändert am 23. Mai 2012  E-Mail: Bürobuero@math.uni-frankfurt.de

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