Green function rigidity and the mass of hypersurfaces under inversion

TL;DR

This paper proves the Green function rigidity conjecture for conformal Laplacian in dimensions n≥3 and for Paneitz operator in specific dimensions, using positive mass and energy theorems, and introduces a new ADM mass formula.

Contribution

It establishes the Green function rigidity conjecture for conformal Laplacian and Paneitz operator in certain dimensions, extending previous results and providing new insights into hypersurface mass.

Findings

Proved Green function rigidity for conformal Laplacian in n≥3.

Proved Green function rigidity for Paneitz operator in certain dimensions.

Derived a new formula for ADM mass of asymptotically flat hypersurfaces.

Abstract

This is a sequel to arXiv:2401.02087. We prove the Green function rigidity conjecture in arXiv:2401.02087 for conformal Laplacian in dimension $n\geq 3$. For the Paneitz operator, we prove the Green function rigidity conjecture when $n\neq 4k+2, k\geq 2$. Important ingredients in our proof are the positive mass theorem and the positive energy theorem for Paneitz operator. As a byproduct, we also obtain a new formula for the ADM mass of an asymptotically flat hypersurface that allows for a non-entire graph.