Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball
Tuesday, 17.10.23, 16:00-17:00, Raum 127, Ernst-Zermelo-Str. 1
Let \(\bSigma\) be a compact hypersurface with a capillary boundary in a unit ball, in this talk, I will discuss the relative isoperimetric problem for such kinds of hypersurfaces. We introduce the relative quermassintegrals for such hypersurfaces from the variational viewpoint. Then by introducing some constrained nonlinear curvature flows, which preserve one geometric quantity invariant and monotone increase another, we obtain the Alexandrov-Fenchel inequality for such hypersurfaces. The talk is based on joint work with Profs. Guofang Wang and Chao Xia.\n
FORMULATING CAPILLARY SURFACES IN THE CONTEXT OF VARIFOLDS
Tuesday, 5.12.23, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
Capillary surface is a fundamental geometric object, describing a particular boundary behavior of surface in a given container. In some recent works of Kagaya-Tonegawa (Hiroshima Math. J. 47(2): 139-153, 2017. https://doi.org/10.32917/hmj/1499392823), De Masi-De Philippis (https://doi.org/10.48550/arXiv.2111.09913), weak capillary surfaces are formulated, using the language of varifolds, and named ”pair of varifolds with fixed contact angle condition”.\n\nIn this talk, we will discuss some recent development in this direction, and prove a strong boundary maximum principle for a specific class of pairs of varifolds with fixed contact angle, which generalizes Li-Zhou’s boundary maximum principle for free boundary varifolds (Comm.\nAnal. Geom. (29): 1509–1521, 2021. https://doi.org/10.4310/CAG.2021.v29.n6.a7). The similar results in the context of rectifiable cones will be discussed as well. If time permits, we will also discuss some applications of the weak formulation, including the establishment of the\nSimon-type monotonicity identities as well as the Li-Yau-type inequalities.
Hyperbolicity and rigidity in moduli spaces of polarized manifolds
Tuesday, 6.2.24, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1