Projektseminar Geometrische Analysis
Sommersemester 2026
Vorträge
Zeit und Ort
Dienstag, 14.7.26, 16:15–17:45, Seminarraum 125
Zusammenfassung
A surface of least area that intersects a given support surface at a constant angle is called a minimal capillary surface. Minimal capillary surfaces arise naturally in the context of fluid dynamics as the idealized interface between two fluids in a container bounded by the support surface. In recent years, minimal capillary surfaces have been shown to capture aspects of the geometry of the support surface that appear to be beyond the reach of established tools in geometric analysis. In this talk, I will describe some of these developments, including my contributions in joint work with M. Eichmair such as the resolution of the Penrose inequality in extrinsic geometry conjectured by G. Huisken and the characterization of stable minimal capillary surfaces in the so-called tangential limit.
Zeit und Ort
Dienstag, 30.6.26, 16:15–17:45, Seminarraum 125
Zusammenfassung
I will discuss some recently developed dimension descent methods in connection with the high-dimensional positive mass theorem. The focus will be on weighted scalar curvature and effective dimension, stable minimal hypersurface descent, and conformal blow-up near singular sets.
Zeit und Ort
Dienstag, 23.6.26, 16:15–17:45, Seminarraum 125
Zusammenfassung
We establish general monotone quantities and sharp mass-capacity inequalities related to p-capacitary functions in 3-dimensional asymptotically flat half-spaces of simple topology with nonnegative scalar curvature and nonnegative boundary mean curvature. These inequalities attain equality on a Schwarzschild half-space outside a rotationally symmetric half sphere. This is based on a joint work with Prof. Chao Xia and Prof. Jiabin Yin.
Zeit und Ort
Freitag, 12.6.26, 10:15–12:00, Seminarraum 125
Zusammenfassung
In this talk, I will explain our idea to establish positive mass theorem (PMT) up to dimension 19, which was later developed by Brendle and Wang to show PMT in all dimensions. This is a joint work with Yuchen Bi, Tianze Hao, Shihang He and Yuguang Shi.
Zeit und Ort
Dienstag, 9.6.26, 16:15–17:45, Seminarraum 125
Zusammenfassung
In this talk I will present some recent results about the quantitative stability of Sobolev-type inequalities on the \(d\)-dimensional round sphere, each of which can be viewed as a generalization of the seminal stability result by Bianchi and Egnell. The first part of my talk concerns the quantitative stability of the reverse fractional Sobolev inequality. Implementing the classical proof strategy by Bianchi and Egnell is non-trivial here because the underlying operator \(A_{2s}\) is not positive definite when \(s > d/2\). Remarkably, the case \(s - d/2 \in (1,2)\) constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer. The second part concerns a fully nonlinear functional inequality for the \(\sigma_2\)-curvature. We prove its stability in a reverse setting occurring in three dimensions. As a geometric application, this implies a quantitative refinement of the almost-Schur lemma of De Lellis and Topping in the special case of the round 3-sphere. This second result is joint work with Jonas Peteranderl (LMU München).
Zeit und Ort
Dienstag, 2.6.26, 17:00–18:00, Seminarraum 125
Zusammenfassung
In this talk, we firstly introduce a capillary John ellipsoid theorem for capillary convex bodies in the Euclidean half-space \(\overline{\mathbb{R}^{n+1}_{+}}\). This theorem yields a non-collapsing estimate for capillary hypersurfaces, which provides a new approach to obtaining \(C^{0}\) estimates for solutions to some capillary curvature problems (including the capillary \(L_{p}\) Christoffel–Minkowski problem and the capillary \(L_{p}\) curvature problem). Then, as a further application, we study the capillary \(L_{p}\) dual Minkowski problem, establish existence for \(1<p\leq q\leq 3\), and improve existing results for the case \(p>q\) in \(\overline{\mathbb{R}^{3}_{+}}\).
Zeit und Ort
Dienstag, 2.6.26, 16:00–17:00, Seminarraum 125
Zusammenfassung
In this talk, I will introduce a new notion of convexity in the unit sphere called horo-convexity, inspired by its analogue in hyperbolic space. For horo-convex hypersurfaces, we prove the smooth convergence of the Guan/Li inverse curvature flow and, as a consequence, establish the full set of quermassintegral inequalities on the sphere. If time permits, I will also discuss parallel results for the hypersurface with capillary boundary lying in the Euclidean unit ball. This talk is based on joint works with Julian Scheuer.
Zeit und Ort
Dienstag, 19.5.26, 16:15–17:45, Seminarraum 125
Zusammenfassung
We investigate the validity of the optimal higher-order Sobolev inequality \(H_k^2(M^n)=\{u\in L^2\hbox{ s.t. }\nabla^iu\in L^2\hbox{ for all }i\leq k\}\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)\) on a closed Riemannian manifold when the remainder term is the L^2-norm. Unlike the case k=1, the optimal inequality does not hold in general for k>1. We prove conditions for the validity and non-validity that depend on the geometry of the manifold. Our conditions are sharp when k=2 and in small dimensions. This is joint work with Lorenzo Carletti (Bruxelles).
Zeit und Ort
Dienstag, 12.5.26, 16:15–17:45, Seminarraum 125
Zusammenfassung
In this talk, I revisit the k-Yamabe problem. We prove that the 2-Yamabe constant defined for metrics with positive scalar curvature is same as the the 2-Yamabe constant defined for metrics in positive \(\Gamma_2\) cone. From the variational aspect, the corresponding functional can not be considered in more general space. This is a joint work with Yuxin Ge and Guofang Wang.
Zeit und Ort
Dienstag, 5.5.26, 16:15–17:45, Seminarraum 125
Zusammenfassung
Harmonic maps are critical points of the Dirichlet energy, while few of them are stable. The Morse index measures how unstable a harmonic map is. It is known that every harmonic map from S^n has index at least n+1. When S^3 to S^2, Revière showed index 4 implies Hopf fibration. In this talk, we give a new perspective and prove the rigidity for arbitrary dimension. This is a joint work with Qun Chen and Guofang Wang.