A dimension descent scheme for the positive mass theorem in arbitrary dimension

TL;DR

This paper extends the Schoen-Yau positive mass theorem proof to all dimensions using a new inductive scheme and advanced geometric techniques.

Contribution

It introduces a novel dimension descent scheme that overcomes singularity issues in higher-dimensional positive mass theorem proofs.

Findings

Successfully generalizes the positive mass theorem to arbitrary dimensions.

Develops an inductive approach combining shielding principles and conformal blow-up techniques.

Utilizes Cheeger-Naber bounds to control singular set dimensions.

Abstract

We describe how the Schoen-Yau proof of the positive mass theorem can be extended to arbitrary dimensions. To overcome the problem of singularities, we propose a new inductive scheme. To carry out the inductive step, we use a combination of several techniques, including the shielding principle of Lesourd-Unger-Yau, as well as a conformal blow-up argument in the spirit of Bi-Hao-He-Shi-Zhu. Our arguments also rely on the Cheeger-Naber bound for the Minkowski dimension of the singular set.