Geometric Analysis, Free Boundary Problems and Measure Theory

Abstracts for the talks

Costante Bellettini:
Compactness questions for triholomorphic maps
A triholomorphic map u between hyperKahler manifolds solves the "quaternion del-bar" equation du=I du i + J du j + K du k. Such a map turns out, under suitable assumptions, to be stationary harmonic. We focus on compactness issues regarding the quantization of the Dirichlet energy and the structure of the blow-up set. We can relax the assumptions on the manifolds, in particular we can take the domain to be merely "quaternion Kahler": this leads to the weaker notion of "almost-stationarity", without however affecting the compacntess results. This is a joint work with G. Tian (Princeton).

Esther Cabezas-Rivas:
Almost non-negative curvature, what's new?
We will review some classical problems in Differential Geometry, which lead us to work with manifolds with almost non-negative curvature. In particular, we will explain during the talk why it is natural to wonder weather for these manifolds a topological invariant called Â-genus vanishes (this question was proposed by John Lott in 1997). We will provide a positive answer by investigating sequences of spin manifolds with lower sectional curvature bound, upper diameter bound and the property that the Dirac operator is not invertible. As a key ingredient of the proof we prove a generalization (under weaker curvature assumptions) of the renowned theorem by Gromov about almost flat manifolds. This is joint work with Burkhard Wilking.

Guido De Philippis:
BV estimates in optimal transportation and applications
I will study BV regularity for solutions of variational problems in Optimal Transportation and present some applications. In particular I will focus on BV bounds for the Wasserstein projection on the set of measure with density bounded by a prescribed BV loc function f. I will also show how to recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. This is a joint work with A. Meszaros, F. Santambrogio and B. Velichkov.

Alessio Figalli:
Local and non-local minimal surfaces
Nonlocal minimal surfaces naturally appear when studying the structure of interphases that arise in classical phase field models with very long space correlations. These surfaces are boundaries of sets whose characteristic functions minimize a fractional Sobolev norm, and they generalize the classical notion of minimal surfaces in geometric measure theory. In this talk we’ll explain and compare the general regularity theory for both local and non-local minimal surfaces, and discuss several recent developments and open problems.

Matteo Focardi:
Endpoint regularity of 2d Mumford-Shah minimizers
We discuss an epsilon-regularity result at the endpoint of connected arcs for 2-dimensional Mumford-Shah minimizers obtained in a joint work with C. De Lellis (U. Zuerich). As an outcome of our analysis, if in a ball Br(x) the jump set of a given Mumford-Shah minimizer is sufficiently close in the Hausdorff distance to a radius of Br(x), then in a smaller ball the jump set is a connected arc terminating at some interior point and C1 up to the tip.

Robert Haslhofer:
Weak solutions for the Ricci flow
We introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions of the Ricci flow in the nonsmooth setting. Given a family (M,g_t) of Riemannian manifolds, we consider the path space of its space time. Our first characterization says that (M,g_t) evolves by Ricci flow if and only if a sharp infinite dimensional gradient estimate holds for all functions on path space. We prove additional characterizations in terms of the regularity of martingales on path space, as well as characterizations in terms of log-Sobolev and spectral gap inequalities for a family of Ornstein-Uhlenbeck type operators. Our estimates are infinite dimensional generalizations of much more elementary estimates for the linear heat equation on (M,g_t), which themselves generalize the Bakry-Emery-Ledoux estimates for spaces with lower Ricci curvature bounds. Based on our characterizations we can define a notion of weak solutions for the Ricci flow. This is joint work with Aaron Naber.

Stanislav Hencl:
Sobolev homeomorphism that cannot be approximated by diffeomorphisms in Wˆ1,1
We show that it is possible to construct a homeomorphism f in the Sobolev space Wˆ1,1([0,1]ˆ4,Rˆ4) such that there are no smooth (or piecewise affine) homeomorphisms f˙k that converge to f in Wˆ1,1 norm. This is joint result with Benjamin Vejnar.

Brian Krummel:
Fine structure and higher regularity of the branch sets
I will discuss work on the structure of the branch set of two-valued solutions to the Laplaces equation and the minimal surface system. Previously, the dimension of the branch set was known; we consider the fine structure of the branch set. In joint work with Neshan Wickramasekera, we show that the branch set is countably (n 2)-rectifiable. Moreover, I have independently shown that the branch set is locally real analytic on a relatively open dense subset of the branch set. Essential ingredients for both results include the monotonicity formula for frequency functions due to F. J. Almgren and a blow-up method, which was originally applied by Leon Simon to multiplicity one classes of minimal submanifolds. The analyticity result requires applying the blow-up method in a new setting where tangent maps are not necessarily translation invariant along (n 2)-dimensional subspaces.

Francesco Maggi:
Local and nonlocal almost-constant mean curvature hypersurfaces
Alexandrov's theorem asserts that a (bounded, embedded) constant mean curvature (cmc) hypersurface must be a sphere. It is well-known that if this condition is relaxed and the mean curvature is just assumed to be close to a constant, then the corresponding hypersurfaces does not need to be close to a sphere. Indeed any family of nearby spheres with equal radii connected by short catenoidal necks can be slightly perturbed to obtain examples of almost-cmc hypersurfaces (Kapouleas, Butscher, Mazzeo).

We show that these examples actually capture the only possible behavior of almost-cmc hypersurfaces, by proving various quantitative bounds on the distance between an almost-cmc hypersurface and a collection of tangent
spheres of equal radii in terms of their mean curvature oscillation. This is a joint work with G. Ciraolo (U Palermo).

We next discuss these issues for the nonlocal mean curvature introduced by
Caffarelli and Souganidis, showing in particular a remarkable rigidity
property of the nonlocal problem which prevents bubbling phenomena, in
other words, every nonlocal almost-cmc hypersurface must be close to a
single sphere. This is a joint work with G. Ciraolo, A. Figalli (UT Austin)
and M. Novaga (U Pisa).

Ulrich Menne:
Weakly differentiable functions on varifolds
In this talk a theory of weakly differentiable functions on
rectifiable varifolds with locally bounded first variation will be presented. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. Results include a variety of Sobolev Poincaré type embeddings, embeddings into spaces of continuous and sometimes Hölder continuous functions, pointwise differentiability results both of
approximate and integral type as well as coarea formulae.

As applications the finiteness of the geodesic distance associated to varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.

Andrea Mondino:
Some properties of non-smooth spaces with Ricci curvature lower bounds
The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the ’80ies and was pushed by Cheeger and Colding in the ’90ies who investigated the structure of the spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds.
A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can a give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors (let me quote the fundamental work of Ambrosio-Gigli-Savaré among others) and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, etc.), permitting to give further insights in the theory. In the seminar I will give an overview of the topic.

Mircea Petrache:
Asymptotics of particles minimizing power-law interactions: results and questions
I will describe a result obtained with Sylvia Serfaty about asymptotic limits of particle systems in d with pairwise interactions modeled by Riesz kernels |x|s for s [max{0,d 2},d[. Motivations for such choices of s arise in several fields, including Coulomb gases, eigenvalues for some random matrix ensembles, Fekete sets and spherical designs from approximation theory and the physics of seminconductors immersed in strong magnetic fields. I will briefly recall how the first order term in the asymptotic expansion of the equilibrium energy (the mean field limit) can be obtained. Then I will show how to study and control a next order ”renormalized energy” that governs microscopic patterns of points and is an energy of infinite configurations of points. In the second half of the talk I will explain some open problems, namely 1) The long-stanting open questions about crystallization at the microscopic scale. 2) The case of interactions corresponding to the more non-local ”fat tailed” interaction energies with s [0,d 2[. 3) Some GMT open questions in the case of integer s under the constraint that the points belong to a fixed s-rectifiable set.

Arshak Petrosyan:
An epiperimetric inequality approach to the regularity of the free boundary in the thin and fractional obstacle problems
We will discuss generalizations of Weiss’s homogeneity improvement approach to the thin and fractional obstacle problems. The main ingredients are an epiperimetric inequality and a monotonicity formula, which give a powerful combination in the analysis of free boundaries and establish the C1 regularity of the regular set. The advantage of this method is that it is purely energy based and allows generalization to the case of thin obstacles living on codimension one C1,1 manifolds, or more generally, the thin obstacle problem for the divergence form operators with Lipschitz coefficients. The method can also be used in the study of the obstacle problem for the fractional Laplacian with drift, when the fractional order is greater that one half.

Based on joint works with Nicola Garofalo, Camelia Pop, and Mariana Smit Vega Garcia.

Peter Topping:
The harmonic map flow revisited
The harmonic map flow is a gradient flow of the harmonic map energy, introduced by Eells and Sampson in 1964. I will describe an alternative gradient flow that tries to find minimal immersions rather than harmonic maps, and give an overview of the theory so far, including forthcoming work.

Joint work with Melanie Rupflin.

Georg Weiss:
Equilibrium points of a singular cooperative system with free boundary
We study maps minimising the energy

<center class="math-display"> <img src="/fileadmin/conf_img/tex_2779c0x.png" alt="∫ (|∇u|2 + 2|u|) dx, D " class="math-display"></center> which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations <center class="math-display"> <img src="/fileadmin/conf_img/tex_2779c1x.png" alt="Δu = u-χ , u = (u ,⋅⋅⋅,u ) . |u | {|u|&amp;#x003E;0} 1 m " class="math-display"></center>

Our results here concern regularity of the solution as well as that of the free boundary. The main ingredient consists in an epiperimetric inequality. (The result is a joint work with J. Andersson, H. Shahgholian and N. Uraltseva, accepted for publication in Advances in Mathematics)

Stefan Wenger:
Area minimizing discs in metric spaces
In this talk, I will discuss a solution to the classical problem of Plateau in the setting of proper metric spaces. Precisely, I will show that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves then I will show that such a solution is locally Hoelder continuous in the interior and continuous up to the boundary. These results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces. This is joint work with Alexander Lytchak.


Date and Location

June 15 - 17, 2015
Max Planck Institute for Mathematics in the Sciences
Inselstraße 22
04103 Leipzig
see travel instructions

Scientific Organizers

Bernd Kirchheim
Universität Leipzig

Stephan Luckhaus
Universität Leipzig

Emanuele Spadaro
MPI für Mathematik in den Naturwissenschaften

Administrative Contact

Katja Heid
Contact by Email
05.04.2017, 12:42