$L^{\infty }$-metrics on tori and Schoen's conjecture

Abstract

We prove Schoen's conjecture on $L^{\infty }$-metrics for tori. More precisely, we show that any $L^{\infty }$-metric on a torus that is smooth and has non-negative scalar curvature away from an embedded submanifold with codimension at least three extends to a smooth flat metric. We also prove a LLarull-type theorem for $L^{\infty }$-metric. Our proof uses weighted scalar curvature and the relative index theorem.

37 pages, version 2: removed the extra condition on the fundamental group of the singular set and proved Schoen's conjecture for $L^{\infty }$ metrics on toris in full generality; also added a Llarull type rigidity theorem for $L^{\infty }$ metrics on spheres