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Scalar curvature deformations with non-compact boundaries

Helge Frerichs

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, including progress on a generalised Geroch conjecture with boundary and some investigation of the Whitehead manifold.

Scalar curvature deformations with non-compact boundaries

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, including progress on a generalised Geroch conjecture with boundary and some investigation of the Whitehead manifold.
Paper Structure (11 sections, 25 theorems, 132 equations, 5 figures)

This paper contains 11 sections, 25 theorems, 132 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Completeness of doubling metrics
  3. The deformation principle
  4. Uniformisation for geodesic collar neighbourhoods
  5. A two-stage deformation scheme
  6. Applications
  7. Non-existence results for half-spaces
  8. Adding topology
  9. The Whitehead manifold
  10. Construction
  11. Creating non-compact boundaries

Key Result

Theorem 2.2

If $g$ is a complete doubling metric on $M$, then $G:=g\cup g$ is a complete metric on $\mathsf{D}M$.

Figures (5)

  • Figure 1: Tubular and collar neighbourhoods of $\partial M$
  • Figure 2: Collar neighbourhoods in $M=\mathbb{R}_{x_1\geq 0}^2\setminus(\{0\}\times\mathbb{R}_{\leq 0})$ -- $A$ and $B$ are closed, whereas the collar in the left picture is not
  • Figure 3: Construction for $f^\delta$ by boxes
  • Figure 4: $P_2$ in $\mathbb{R}^3$
  • Figure 5: Tube $R$ around the ray $[0,\infty)$ (above); Tori $W_0$ (blue), $W_1$ (light grey) and $W_2$ (dark grey) together with the initial part of $R$ (below)

Theorems & Definitions (65)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8
  • proof
  • ...and 55 more