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.