March 6, 2024Open Access

Scalar curvature deformations with non-compact boundaries

Key Points

Non-existence results for Riemannian metrics with positive scalar curvature are established, indicating limitations in geometric configurations.
The principle shows that boundary conditions can be strengthened from mean convex to totally geodesic configurations effectively.
Application of the deformation principle ensures the preservation of geometric properties like completeness and quasi-isometry type during modifications of metrics and boundaries. The analysis involves spaces that may have non-compact boundaries, expanding conventional applicability of established theorems.

Abstract

We develop a general deformation principle for families of Riemannian metrics on smooth manifolds with possibly non-compact boundary, preserving lower scalar curvature bounds. The principle is used in order to strengthen boundary conditions, from mean convex to totally geodesic or doubling. The deformation principle preserves further geometric properties such as completeness and a given quasi-isometry type. As an application, we prove non-existence results for Riemannian metrics with (uniformly) positive scalar curvature and mean convex boundary.

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Helge Frerichs (Wed,) studied this question.

www.synapsesocial.com/papers/68e758b6b6db6435876d00a2 — DOI: https://doi.org/10.48550/arxiv.2403.03941

Scalar curvature deformations with non-compact boundaries

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