Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

TL;DR

This paper proves the extension of boundary metrics to positive scalar curvature metrics inside manifolds, introduces a fill-in invariant, and explores its relation to positive mass theorems, providing insights into Gromov's conjectures.

Contribution

It solves an open problem on extending boundary metrics to PSC metrics and introduces a new fill-in invariant linked to positive mass theorems.

Findings

Extension of boundary metrics to PSC inside manifolds proven.

Introduces a fill-in invariant related to mass theorems.

Provides partial answers to Gromov's conjectures.

Abstract

In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, which completely solves an open problem due to Gromov (see Question \ref{extension1}). Then we introduce a fill-in invariant (see Definition \ref{fillininvariant}) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromov's conjectures formulated in \cite{Gro19} (see Conjecture \ref{conj0} and Conjecture \ref{conj1} below)