Surveys in Differential Geometry

Volume 24 (2019)

Positive scalar curvature and minimal hypersurface singularities

Pages: 441 – 480

DOI:  https://dx.doi.org/10.4310/SDG.2019.v24.n1.a10

Authors

Richard Schoen (Department of Mathematics, University of California, Irvine, Calif., U.S.A.)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of [SY1] to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension $1$ or $2$ the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.

The first author was partially supported by NSF grant DMS-2005431.

Published 29 December 2021