Bernhard Hanke (Augsburg)
We prove Gromov’s conjecture on the mean of the mean curvature of fill-ins in various cases. Our methods are based on surgery to reduce the statement to fill-ins of spheres, which can be treated by instances of the positive mass theorem. For spin fill-ins, where we permit the mean curvature to take negative values, we build on a recent positive mass theorem with creases by Kazaras–Khuri–Lin. For non-spin fill-ins of spin manifolds, where we assume the mean curvature to be non-negative, we develop a novel quantitative surgery process to reduce the general situation to a result of Shi–Wang–Wei. We also treat the case of fill-ins of non-spin manifolds, provided there is a fixed positive lower bound on the mean curvature.
This is based on joint work with Georg Frenck and Sven Hirsch.
