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On Gromov's compactness question regarding positive scalar curvature

Shmuel Weinberger, Zhizhang Xie and Guoliang Yu

Vol. 9 (2024), No. 4, 583–612
Abstract

We give both positive and negative answers to Gromov’s compactness question regarding positive scalar curvature metrics on noncompact manifolds. First we construct examples that give a negative answer to Gromov’s compactness question. These examples are based on the nonvanishing of certain index-theoretic invariants that arise at the infinity of the given underlying manifold. This is a lim 1 phenomenon and naturally leads one to conjecture that Gromov’s compactness question has a positive answer provided that these lim 1 invariants also vanish. We prove this is indeed the case for a class of 1-tame manifolds.

Keywords
Gromov's compactness conjecture on scalar curvature, index invariants at infinity
Mathematical Subject Classification
Primary: 58B34
Milestones
Received: 1 May 2024
Accepted: 20 September 2024
Published: 22 October 2024
Authors
Shmuel Weinberger
Department of Mathematics
University of Chicago
Chicago, IL
United States
Zhizhang Xie
Department of Mathematics
Texas A&M University
College Station, TX
United States
Guoliang Yu
Department of Mathematics
Texas A&M University
College Station, TX
United States