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arXiv · 2601.10617

Surgery and total mean curvature

Georg Frenck·Bernhard Hanke·Sven Hirsch
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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.

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Georg Frenck, Bernhard Hanke, Sven Hirsch. 2026-01-15. Surgery and total mean curvature. https://arxiv.org/abs/2601.10617

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