Events before Winter 2023/24

The Geroch conjecture (proven by Schoen-Yau and Gromov-Lawson) says that the torus T^n does not admit a metric of positive scalar curvature. In this talk, I will explain how to generalize it to some non-compact settings using μ-bubbles. In particular, I will talk about why the connected sum of a Schoen-Yau-Schick n-manifold with an arbitrary n-manifold does not admit a complete metric of positive scalar curvature for n<8; this generalizes work of Chodosh and Li. I will also discuss about how to generalize Brendle-Hirsch-Johne’s non-existence result for metrics of positive m-intermediate curvature on N^n = M^{n-m} x T^m to to manifolds with arbitrary ends for n<8 and certain m. Here, m-intermediate curvature is a new notion of curvature interpolating between Ricci and scalar curvature.

July 4, 2023, 2:00pm-3:00pm. Florian Johne (Columbia University)

Intermediate curvature and a generalization of Geroch's conjecture

In this talk we explain a non-existence result for metrics of positive m-intermediate curvature (a notion of curvature reducing to positive Ricci curvature for m = 1, and positive scalar curvature for m = n-1) on closed orientable manifolds with topology N^n = M^{n-m} x T^m for n < 8. Our proof uses a slicing constructed by minimization of weighted areas, the associated stability inequality, and estimates on the gradients of the weights and the second fundamental form of the slices. This is joint work with Simon Brendle and Sven Hirsch.

June 14, 2023, 11:30am-1:00pm. Fabian Rupp (University of Vienna)

Around one and a half worlds in zero days - the minimal curvature of short world trips

The optimal path to circumnavigate the earth is to follow a closed geodesic. For nonround surfaces, the length of such closed geodesics may be arbitrarily small. We show that infinitesimally short geodesics produce a quantifiable amount of curvature of the surface, measured in terms of the Willmore energy. This is joint work with M. Müller (Freiburg) and C. Scharrer (Bonn).

May 10, 2023, 11:30am-1:00pm. Volker Branding (University of Vienna)

On the existence and stability of harmonic self-maps

Harmonic maps are a well-studied geometric variational problem for maps between Riemannian manifolds with many links to analysis, geometry and theoretical physics. The harmonic map equation is a semilinear elliptic partial differential equation of second order which, in its simplest form, reduces to the equation for geodesics. In the first part of the talk we will give a general introduction to the notion of harmonic maps and present a number of key results highlighting their structure. The second part of the talk will be concerned with harmonic self-maps which are harmonic maps for which domain and target manifold coincide. By imposing additional symmetry assumptions the equation for harmonic self-maps reduces to an ordinary differential equation allowing for the application of tools from dynamical systems. We will present several results on the stability of harmonic self-maps of cohomogeneity one manifolds. Moreover, we will report on the existence of an infinite family of harmonic self-maps of ellipsoids in all dimensions. This is joint work with Anna Siffert.

April 5, 2023, 11:30am-1:00pm. Thomas Koerber (University of Vienna)

Schoen's conjecture for limits of isoperimetric surfaces

R. Schoen has conjectured that an asymptotically flat Riemannian n-manifold (M,g) with non-negative scalar curvature is isometric to Euclidean space if it admits a non-compact area-minimizing hypersurface. This has been confirmed by O. Chodosh and M. Eichmair in the case where n=3. In this talk, I will present recent work with M. Eichmair where we confirm this conjecture in the case where 3<n<8 and the area-minimizing hypersurface arises as the limit of large isoperimetric hypersurfaces. By contrast, we show that a large part of spatial Schwarzschild of dimension 3<n<8 is foliated by non-compact area-minimizing hypersurfaces.

March 8, 2023, 11:30am-1:00pm. Katharina Brazda (University of Vienna)

Curvature varifolds and biomembrane shapes

Varifolds are Radon measures that generalize the notion of differentiable submanifolds of Euclidean spaces. In this talk, I will first give a gentle introduction to this branch of geometric measure theory, focusing on the class of curvature varifolds with orientation and boundary. In the second part, I will apply these concepts to describe configurations of biomembranes, for example the biconcave shapes of human red blood cells, which can be modeled as surfaces of minimal Canham-Helfrich energy. This quadratic curvature functional can be seen as a modification of the Willmore energy, including different bending rigidities and a spontaneous curvature parameter. I will present results on multiphase membranes with sharp phase-interfaces and on the gradient flow evolution of biomembranes, based on joint work with Martin Kruzik, Luca Lussardi, and Ulisse Stefanelli.

January 11, 2023, 11:30am-1:00pm. Jonas Knörr (Vienna University of Technology)

From valuations on sets to valuations on functions

Since Dehn's solution of Hilbert's third problem, valuations defined on families of sets have been a core part of convex and integral geometry. In recent years, the advances in Geometric Valuation Theory have fueled a corresponding theory for valuations on functions. I will talk about some of these developments and their relation to classical results about valuations on convex bodies, focusing in particular on some recent results on valuations on Lipschitz functions on the unit sphere. This is partly based on joint work with Andrea Colesanti and Daniele Pagnini.

December 14, 2022, 11:30am-1:00pm. Mattia Magnabosco (University of Bonn)

Failure of the CD condition in sub-Riemannian geometry

The Lott-Sturm-Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It was proved by Juillet that the CD(K,N) condition is not satisfied in a large class of sub-Riemannian manifolds, for every choice of the parameters K and N. In this talk I present a different strategy to disprove the CD condition in sub-Riemannian manifolds, extending Juillet's result to the setting of almost-Riemannian manifolds. In particular, I show that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the CD(K,N) condition, for any K and N.

November 9, 2022, 11:30am-1:00pm. Alice Marveggio (Institute of Science and Technology Austria)

Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

In its sharp-interface limit, the vectorial Allen-Cahn equation with a potential with N ≥ 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In this talk, we give a rigorous proof for (unconditional) convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow, assuming that a classical (smooth) solution to the latter exists. Our result is valid in two and three ambient dimensions, for a suitable class of multi-well potentials, and starting from well-prepared initial data. We even establish a rate of convergence. Our approach relies on a notion of relative entropy for the vectorial Allen-Cahn equation with multi-well potential and, in particular, on the recent concept of “gradient flow calibrations” for multiphase mean curvature flow. This is joint work with J. Fischer.