42. Österreichisches und Süddeutsches
Kolloquium zur Differentialgeometrie

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).

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.

The classical notion of isotropic 3-space refers to a 3-dimensional real vector space with degenerate inner product. This space sits, on many occasions, between Euclidean and Minkowski geometry as a boundary case. In this talk, we will describe isotropic space as well as its spherical and hyperbolic analogue (the latter one also known as half-pipe geometry). In particular, we will show how the sphere geometric setup usually used to describe Euclidean geometry is well suited to describe the extrinsic geometry of surfaces in these spaces. As an application, we will prove a Weierstrass-type representation for constant mean curvature surfaces in isotropic geometries.

In pursuit of finding nicely shaped curves and surfaces, several examples of self-repulsive functionals have been introduced since the 1990s which raised challenging questions regarding analysis and simulation. In particular they can be employed to model topological constraints such as impermeability. The talk will provide a survey that includes some recent results by the speaker.

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

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.

We study existence, uniqueness, and regularity of minimizers for a manifold-constrained version of the Rudin-Osher-Fatemi model for image denoising, which appears in multiple references of applied literature, but lacks analytical foundations. This leads to study a system of elliptic PDEs with Neumann boundary conditions.

Our outcomes can be regarded as the extension to the harder situation of \(p=1\) of the regularity theory for p-harmonic maps, started by classical works of Eells-Sampson and Schoen-Uhlenbeck. In fact, we generalize the optimal regularity results for the classical Euclidean scalar model, without further requirements on the convexity of the boundary, in three different directions: vector-valued functions, manifold-constrained and curved domain. To achieve the results, it is crucial on a strong interplay between geometric and analytical techniques within the proofs.

Additionally, we provide variants of the regularity statement of independent interest: for 1-dimensional domains (related to signal denoising), local Lipschitz regularity (meaningful for image processing) and Lipschitz regularity for a perturbed model coming from fluid mechanics.

This is joint work with Salvador Moll and Vicent Pallardó-Julià.

Isothermic surfaces form a class of integrable surfaces that includes minimal and constant mean curvature surfaces. The question of controlling their global behaviour is challenging, which can be simplified by restricting to planar or spherical curvature lines. For example, this strategy has led to the famous Wente torus, the first non-round, compact, closed CMC-surface.

In this talk, we will discuss the novel method of "lifted-folding" (joint work with F. Burstall, J. Cho, T. Hoffmann and M.Pember). It reduces the construction of isothermic surfaces foliated by a family of spherical curvature lines to lower-dimensional data: namely, to holomorphic maps built from elastic curves and folding functions.

The presentation will concentrate on discrete quadrilateral surfaces, which subsequently result in kinematically snapping structures composed of spatial snapping 4-bars.

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.

William Thurston in his Geometrization Program suggested a new approach to locally homogeneous Riemannian structures and highlighted their significance for 3-dimensional topology. It was soon recognized that Thurston's approach is also very fruitful in non-Riemannian settings, particularly in Lorentzian, as it was demonstrated by the pioneering work of Geoffrey Mess on (2+1)-spacetimes of constant curvature. Geometries of constant curvature, whether Riemannian or Lorentzian, can be considered as subgeometries of projective geometry, which in particular allows to perform interesting geometric transitions between different geometries. In my talk I will describe how this viewpoint allows to deduce rigidity results on anti-de Sitter (2+1)-spacetimes, using the resolution of analogous problems in the setting of Minkowski spacetimes. The talk is partially based on joint works with François Fillastre and Jean-Marc Schlenker.

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.