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Mathematics > Differential Geometry
Title: Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature
(Submitted on 24 Mar 2024)
Abstract: This paper deals with the positive mass theorem and the existence of isoperimetric sets on $3$-manifolds endowed with continuous complete metrics having nonnegative scalar curvature in a suitable weak sense.
We prove that if the manifold has an end that is $C^0$-locally asymptotically flat, and the metric is the local uniform limit of smooth metrics with vanishing lower bounds on the scalar curvature outside a compact set, then Huisken's isoperimetric mass is nonnegative. This addresses a version of a recent conjecture of Huisken about positive isoperimetric mass theorems for continuous metrics satisfying $R_g\geq 0$ in a weak sense. As a consequence, any fill-in of a truncation of a Schwarzschild space with negative ADM mass has nonnegative isoperimetric mass.
Moreover, in case the whole manifold is $C^0$-locally asymptotically flat and the metric is the local uniform limit of smooth metrics with vanishing lower bounds on the scalar curvature outside a compact set, we prove that, as a large scale effect, isoperimetric sets with arbitrarily large volume exist.
Submission history
From: Gioacchino Antonelli [view email][v1] Sun, 24 Mar 2024 00:43:35 GMT (71kb)
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