Surgery and total mean curvature

TL;DR

This paper proves Gromov's conjecture on the total mean curvature of fill-ins using surgery techniques and positive mass theorems, covering various cases including spin and non-spin manifolds with different curvature conditions.

Contribution

It introduces new surgery methods and extends positive mass theorems to handle fill-ins with negative or non-negative mean curvature, proving Gromov's conjecture in multiple cases.

Findings

Proved Gromov's conjecture for fill-ins of spheres.

Developed a quantitative surgery process for non-spin fill-ins.

Extended positive mass theorem techniques to cases with negative mean curvature.

Abstract

We prove Gromov's conjecture on the total 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 classical surgery result of Lawson-Michelsohn and 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.