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Remark about scalar curvature on certain noncompact manifolds

John Lott

TL;DR

The paper addresses when the interior of a genus $g$ handlebody can support a complete metric with nonnegative scalar curvature outside a compact set, identifying an end-obstruction governed by end-area growth via $A(r)$. It combines $\mu$-bubble techniques to derive curvature bounds that force the end to have controlled topology (genus $\le 1$) under a sharp area-growth condition $\liminf_{r\to\infty} r^{-2} A(r) < 12/\pi$, and proves a higher-dimensional result: if $N$ is compact with nonnegative scalar curvature and $X$ is an $S^1$-bundle over $N$, then $[0,\infty)\times X$ admits a complete metric with positive scalar curvature. Additionally, the paper provides explicit warped-product constructions yielding PSC ends in various dimensions and Thurston geometries (notably Nil) and discusses the relationship between end obstructions, index theory, and Thurston-type cross-sections. These results clarify when noncompact ends can carry PSC structures and present concrete obstructions and constructions across dimensions.

Abstract

We give a sufficient condition to rule out complete Riemannian metrics with nonnegative scalar curvature on the interior of handlebodies. In higher dimensions, we give examples of ends of manifolds with positive scalar curvature metrics.

Remark about scalar curvature on certain noncompact manifolds

TL;DR

The paper addresses when the interior of a genus handlebody can support a complete metric with nonnegative scalar curvature outside a compact set, identifying an end-obstruction governed by end-area growth via . It combines -bubble techniques to derive curvature bounds that force the end to have controlled topology (genus ) under a sharp area-growth condition , and proves a higher-dimensional result: if is compact with nonnegative scalar curvature and is an -bundle over , then admits a complete metric with positive scalar curvature. Additionally, the paper provides explicit warped-product constructions yielding PSC ends in various dimensions and Thurston geometries (notably Nil) and discusses the relationship between end obstructions, index theory, and Thurston-type cross-sections. These results clarify when noncompact ends can carry PSC structures and present concrete obstructions and constructions across dimensions.

Abstract

We give a sufficient condition to rule out complete Riemannian metrics with nonnegative scalar curvature on the interior of handlebodies. In higher dimensions, we give examples of ends of manifolds with positive scalar curvature metrics.
Paper Structure (4 sections, 2 theorems, 26 equations)

This paper contains 4 sections, 2 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']}
  3. Proof of Theorem \ref{['thm2']}
  4. Discussion

Key Result

Theorem 1

Let $\overline{M}$ be an orientable three dimensional handlebody. Let $M$ be the interior of $\overline{M}$. Suppose that $M$ has a complete Riemannian metric with nonnegative scalar curvature outside of a compact subset. Choose a basepoint $m_0 \in M$. Given $r > 0$, let $A(r)$ be the infimum of th

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2