Gradient flow dynamics for Willmore-type bending energies: global existence, convergence and analysis of singularities
Dienstag, 29.4.25, 16:00-17:45, Seminarraum 125
Geometric inequality related with \(\sigma_2\) curvature
Dienstag, 6.5.25, 16:15-17:45, Seminarraum 125
A note on the existence of nontrivial zero modes on Riemannian manifolds
Dienstag, 13.5.25, 16:15-17:45, Seminarraum 125
In this talk we consider nontrivial solutions, so called zero modes, to the Dirac equation on closed Riemannian manifolds. We will state and prove a necessary criterion for the existence of zero modes, which relates the norm of a certain vector field to the Yamabe constant of the manifold. In the end we will give some insight on a classification of manifolds on which this criterion is sharp.
Dirac-geodesics in surfaces
Dienstag, 20.5.25, 16:15-17:45, Seminarraum 125
In this talk, we consider the structure of Dirac-geodesics with curvature term in surfaces and give solutions on the 2-sphere and the hyperbolic plane, and then we give the structure of solutions in warped product spaces. Finally, we define the corresponding heat flow and prove the global existence and sub-convergence of the heat flow into any closed surfaces and space forms. This is a joint work with Prof. Q. Chen.
Stability of sharp spinorial Sobolev inequality on sphere
Dienstag, 27.5.25, 16:15-17:45, Seminarraum 125
In this talk we consider the sharp spinorial Sobolev inequality on S^n. From the variation point of view, this is a spinorial analogy of Yamabe problem. It is well known that the optimal Sobolev constant is the so-called Bär-Hijazi-Lott invariant which, as the Yamabe invariant, attains its maximum at round sphere. In this talk, we will prove on S^n that the Sobolev quotient being close to the optimal constant implies that spinor being close to an optimizer. Compared to the function case, the difficulty arises from the fact that the Dirac operator has unbounded spectrum both from above and blow. This is a joint work with Prof. Guofang Wang.
Bounding the area of submanifolds with prescribed boundary in terms of its curvature energy
Dienstag, 3.6.25, 16:15-17:45, Seminarraum 125
Given an (m-1)-dimensional, embedded, compact submanifold \(\Gamma\) in \(\mathbb{R}^n\), consider any compact, immersed m-dimensional submanifold whose boundary is exactly given by \(\Gamma\). In this talk, we show how the area of such an m-submanifold is controlled in terms of its curvature energy. The talk is based on joint work with Prof. Ernst Kuwert.
Sharp quantitative estimates of Struwe’s decomposition
Donnerstag, 5.6.25, 14:00-15:00, Seminarraum 125
Suppose \(u\in \dot{H}^1(\mathbb{R}^n)\). In a fundamental paper, Struwe proved that if \(u\geq 0\) and \(\|\Delta u+u^{\frac{n+2}{n-2}}\|_{H^{-1}}:=\Gamma(u)\to 0\) then \(dist(u,\mathcal{T})\to 0\), where \(dist(u,\mathcal{T})\) denotes the \(\dot{H}^1(\mathbb{R}^n)\)-distance of \(u\) from the manifold of sums of Talenti bubbles. In this talk, I will talk about a quantitative version of this Struwe’s decomposition. Precisely, we proved nonlinear quantitative estimates for dimension while Figalli-Glaudo proved a linear estimate for dimension . Furthermore, we showed that these estimates are sharp in the sense of the exponents are optimal. It is joint work with Liming Sun and Juncheng Wei.
Liouville type theorem for a class quasi-linear \(p\)-Laplace equation on the half space
Donnerstag, 5.6.25, 15:00-16:00, Seminarraum 125
In this talk, we will study the positive solutions of p-Laplacian with subcrtical exponent in the half space with the Neumann boundary condition. We deduce a Liouville Theorem via the method of vector field and integral by part motivated by Obata. This is a joint work with Xinan Ma and Yang Zhou.
TBA
Dienstag, 17.6.25, 16:15-17:45, Seminarraum 125
TBA
Dienstag, 1.7.25, 16:15-17:45, Seminarraum 125
TBA
Dienstag, 8.7.25, 16:15-17:45, Seminarraum 125
TBA
Dienstag, 15.7.25, 16:15-17:45, Seminarraum 125