42. Österreichisches und Süddeutsches
Kolloquium zur Differentialgeometrie

Biharmonic and conformal biharmonic maps between Riemannian manifolds are two fourth order generalizations of the well-studied harmonic map equation. Since biharmonic and conformal biharmonic maps arise as critical points of a non-coercive energy functional and due to the large number of derivatives it is a challenging task to obtain existence theorems for the latter.

In the first part of the talk we will provide an introduction to both biharmonic and conformal biharmonic maps and highlight some of the key results that have been established up to now. Finally, we will focus on maps to the Euclidean sphere and will present a number of recent existence results.

I present an approach to Lorentzian geometry and General Relativity that does neither rely on smoothness nor on manifolds, thereby leaving the framework of classical differential geometry. This opens up the possibility to study curvature (bounds) for spacetimes of low regularity or even more general spaces. An analogous shift in perspective proved extremely fruitful in the Riemannian case (Alexandrov-, CAT(k)- and CD-spaces).

After reviewing this approach I will report on recent progress: a first-order calculus for causal functions, its implications for the Lorentzian splitting theorems and a Lorentzian version of Gromov-Hausdorff convergence, including a precompactness theorem.

The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this talk, I will present recent joint work with M. Eichmair that resolves this conjecture: The exterior mass of an asymptotically flat support surface S with nonnegative mean curvature is bounded in terms of the area of the outermost free boundary minimal surface supported on S.

If equality holds, then the exterior surface of S is a half-catenoid. In particular, we obtain a new characterization of the catenoid among all complete embedded minimal surfaces with finite total curvature. To prove this result, we study minimal capillary surfaces supported on S that minimize the free energy and discover a quantity associated with these surfaces that is nondecreasing as the contact angle increases.

We study the length-preserving elastic flow in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions.

We discuss how a suitable non-flatness assumption ensures global existence and subconvergence to critical points. This is joint work with Manuel Schlierf.

A hypersurface of a Lorentzian manifold is called null, if its induced metric is degenerate. In such geometries, some classical geometric concepts are unavailable and for this reason there is little knowledge about curvature flows in such spaces.

Building upon a definition of mean curvature flow I developed with Henri Roesch a few years ago, we extend and improve the analysis to constrained mean curvature flows, which will then enable us to foliate large classes of null hypersurfaces by surfaces of constant spacetime mean curvature. This is joint work with Wilhelm Klingenberg (Durham) and Ben Lambert (Leeds).

Inspired by the work of Michor et al. on the manifold of immersed curves, we propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of \(H^s\) regularity, \(s\in(3/2,2)\). The construction is motivated by the concept of tangent-point energies, a family of self-avoiding functionals on curves and surfaces of arbitrary dimension. Their notion of self-repulsion allowed us to capture the topological property "embeddedness" in a continuous way.

This talk illustrates the impact of this metric and highlights its most important features, namely metric and geodesic completeness, relative compactness of bounded sets with respect to the weak \(H^s\) topology, and existence of length-minimizing geodesics between every pair of curves in the same knot class. This is joint work with Philipp Reiter and Henrik Schumacher.

https://arxiv.org/abs/2501.16647