Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields

TL;DR

This paper extends the positive mass theorem to spin initial data sets with multiple ends and dominant energy shields, using a modified Witten approach with an extra timelike dimension in the spinor bundle.

Contribution

It introduces a new proof of the positive mass theorem for complex initial data sets with arbitrary ends, incorporating dominant energy shields and a modified Witten method.

Findings

Proves positive mass theorem for initial data with dominant energy shields.

Shows existence of a neighborhood where violations of the theorem must occur.

Establishes a rigidity statement for the theorem.

Abstract

We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and a shield of dominant energy (a subset of $M$ on which the dominant energy scalar $\mu-|J|$ has a positive lower bound). In a similar vein, we show that for an asymptotically flat end $\mathcal{E}$ that violates the positive mass theorem (i.e. $\mathrm{E} < |\mathrm{P}|$), there exists a constant $R>0$, depending only on $\mathcal{E}$, such that any initial data set containing $\mathcal{E}$ must violate the hypotheses of Witten's proof of the positive mass theorem in an $R$-neighborhood of $\mathcal{E}$. This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten's approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.