Extensions and fill-ins with nonnegative scalar curvature

TL;DR

This paper investigates the fill-in problem for 2-sphere data as boundary of 3-manifolds with nonnegative scalar curvature, using Shi-Tam extensions, and establishes conditions for existence and nonexistence of such fill-ins along with a positive mass theorem.

Contribution

It introduces a new characterization of fill-in existence based on Shi-Tam extensions and relates it to the zero mass case, also proving a positive mass theorem in the non-fill-in scenario.

Findings

Characterizes the relationship between zero mass Shi-Tam extensions and fill-in nonexistence.

Establishes a positive mass theorem for non-fill-in cases.

Provides conditions distinguishing fill-in existence from nonexistence.

Abstract

Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown--York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of nonnegative scalar curvature. We characterize the relationship between two borderline cases: one in which the Shi--Tam extension has zero total mass, and another in which fill-ins of nonnegative scalar curvature fail to exist. Additionally, we prove a type of positive mass theorem in the latter case.