ABSTRACT

Scalar curvature exhibits a striking interplay between rigidity and flexibility under weak geometric control. In this talk, I will present two related results motivated by general relativity. The first concerns rigidity at spatial infinity inspired by the positive mass theorem: jointly with Jianchun Chu and Man-Chun Lee, we address a conjecture of Gromov asserting that nonnegative scalar curvature, together with purely zeroth-order metric decay faster than the Schwarzschild rate, forces a complete metric on Euclidean space to be flat. The second concerns flexibility under high-frequency limits motivated by Burnett’s conjecture for the Einstein equations: we solve and strengthen the Riemannian reverse-Burnett conjecture of Huneau and Luk, establishing that metrics with nonnegative scalar curvature arise as weak limits of scalar-flat metrics within the natural $W^{1,\infty}$ Burnett compactness class. More generally, this identifies scalar-curvature lower bounds as the precise relaxation of the corresponding constant scalar-curvature equations.