In this talk, we minimize the energy \begin{equation*} \mathcal{E}(u) = \int_\Omega (\Delta u)^2 \; \mathrm{d}x + |\{x \in \Omega: u(x) \neq 0 \}|, \end{equation*} among functions that attain prescribed positive boundary values. Minimizers must perform a balancing act. The second term requires a large flatness region at the zero level. However, the first term (a bending energy) prevents minimizers from reaching the zero level close to the boundary. Our question of interest is the global regularity of minimizers, which might break down at the boundary of the zero level. While $C^{1,\alpha}$-regularity is readily retrieved from previous literature, it turns out that $C^2$-regularity fails in general. For radial minimizers, this loss of regularity can be computed explicitly. Yet, it is not clear whether one can obtain radial minimizers at all, given the lack of a maximum principle for the biharmonic equation. I hope to convince you that this obstruction can be overcome --- radiality can be obtained by means of a nonstandard annular rearrangement procedure! This is a joint work with Hans-Christoph Grunau (Magdeburg).

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 prove a new Lojasiewicz-Simon gradient inequality for energies defined on Banach manifolds. This abstract framework is well suited for studying non-linear boundary problems. Indeed, as an application, we deduce a Lojasiewicz-Simon gradient inequality for the Willmore energy of surfaces with free orthogonal boundary conditions and prove asymptotic stability of local minimizers for the associated $L^2$-gradient flow.

Classical Euler elastic curves are obtained as critical values of the bending energy while constraining the length. Wente himself observed that on his famous Wente tori, the first non-round closed surfaces of constant mean curvature, some curves are planar elastic eights. Recent research shows that these tori are in fact foliated by a family of planar constrained elastic curves in space forms. These are critical points of the bending energy with constrained length and area. In this talk, we shall present a construction method for isothermic surfaces that are foliated by a family of planar or spherical curvature lines. This integrable surface class includes, for example, minimal and constant mean curvature surfaces in Euclidean, hyperbolic and spherical space. Our approach starts with 2-dimensional data, specific holomorphic maps built from constrained elastic curves, which are appropriately lifted-folded to 3-space. The results presented are based on several recent works with F.Burstall, J.Cho, T.Hoffmann, M.Pember and J.Steinmeier.

Consider a connected surface of finite area without boundary, properly embedded in Euclidean space. By a geometric inequality of Leon Simon from 1993, such a surface must be compact, provided its mean curvature has bounded Lebesgue 2-norm. In 2008, Simon’s inequality was improved by Peter Topping who showed that the diameter of such a surface is in fact cotrolled by the Lebesgue 1-norm of its mean curvature. Over the past decade, Topping’s diameter bound has inspired various generalizations in differential geometry and geometric measure theory. In this talk, I will present an overview of these developments, pointing out some applications and related open problems.

In my talk I will present the geometric and physical properties of so-called higher-power harmonic maps, a generalization of harmonic maps described by C. Wood and A. Ramachandran in 2023. Due to the algebraic structure of their equations, they have strong similarities with Yang-Mills theory and are therefore particularly suitable for use in physical field theory. After a short introduction, I will explain the coupling of these maps to a gravitational field and the resulting coupled partial differential equations. I will show how explicit solutions of the system can be constructed on Lie groups under symmetry assumptions. If there is enough time, I will present an instanton theory for higher-power harmonic maps: This yields an easier first-order differential equation whose solutions coincide with those of the original equation. In addition to the simpler construction of solutions, this theory also allows an estimation of the energy of these maps via a topological invariant.

In the calculus of variations, Lavrentiev's phenomenon refers to the situation when the minimum value of a certain functional varies significantly depending on the space of functions considered. In this talk, we present a new example of Lavrentiev's phenomenon in the context of nonlinear elasticity. This example is based on an interplay of the elastic energy’s resistance to infinite compression and the Ciarlet–Necas condition, a constraint preventing global interpenetration of matter on sets of full measure.

While knot energies for curves have been extensively studied over the past three decades, there remains a lack of well-behaved geometric functionals for higher-dimensional knotted objects—functionals that effectively penalize self-intersections and could be minimized within ambient isotopy classes. In this talk, we present both classical and recent families of such energies, exploring their respective advantages and limitations.