Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Urysohn width of hypersurfaces and positive macroscopic scalar curvature

Teo Gil Moreno de Mora Sardà

TL;DR

This work develops a macroscopic analogue of lower scalar curvature bounds by linking positive macroscopic scalar curvature at scale $R$ to the existence of a nontrivial codimension-1 hypersurface with small $Urysohn$ width, namely $UW_{n-2}(\Sigma) \le \frac{n-1}{n}R$. The approach combines Guth's macroscopic Schoen–Yau descent with a Stability Lemma for almost minimising hypersurfaces to produce width bounds, under assumptions on $H_{n-1}(M;G)$ and $\mathrm{sys}_1(M)$. It further furnishes counterexamples showing macroscopic curvature bounds do not control all topological invariants: prolate metrics on $\mathbb{S}^k\times\mathbb{S}^{n-k}$ and Berger-type metrics on $\mathbb{R}P^3$ illustrate the limitations. Altogether, the paper extends Bray–Brendle–Neves’s area-type results to a macroscopic setting and clarifies the relationship between macroscopic curvature and width/topology in noncompact or non-simply connected contexts.

Abstract

We prove that if a complete Riemannian $n$-manifold with non-trivial codimension 1 homology with $\mathbb{Z}_2$-coefficients or $\mathbb{Z}$-coefficients has positive macroscopic scalar curvature large enough, then it contains a non-nullhomologous hypersurface of small Urysohn $(n-2)$-width. This constitutes a macroscopic analogue of a theorem by Bray--Brendle--Neves on the area of non-contractible 2-spheres in a closed Riemannian 3-manifold with positive scalar curvature. Our proof is based on an adaptation of Guth's macroscopic version of the Schoen-Yau descent argument.

Urysohn width of hypersurfaces and positive macroscopic scalar curvature

TL;DR

This work develops a macroscopic analogue of lower scalar curvature bounds by linking positive macroscopic scalar curvature at scale to the existence of a nontrivial codimension-1 hypersurface with small width, namely . The approach combines Guth's macroscopic Schoen–Yau descent with a Stability Lemma for almost minimising hypersurfaces to produce width bounds, under assumptions on and . It further furnishes counterexamples showing macroscopic curvature bounds do not control all topological invariants: prolate metrics on and Berger-type metrics on illustrate the limitations. Altogether, the paper extends Bray–Brendle–Neves’s area-type results to a macroscopic setting and clarifies the relationship between macroscopic curvature and width/topology in noncompact or non-simply connected contexts.

Abstract

We prove that if a complete Riemannian -manifold with non-trivial codimension 1 homology with -coefficients or -coefficients has positive macroscopic scalar curvature large enough, then it contains a non-nullhomologous hypersurface of small Urysohn -width. This constitutes a macroscopic analogue of a theorem by Bray--Brendle--Neves on the area of non-contractible 2-spheres in a closed Riemannian 3-manifold with positive scalar curvature. Our proof is based on an adaptation of Guth's macroscopic version of the Schoen-Yau descent argument.

Paper Structure

This paper contains 6 sections, 13 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The function $V^n_s(R)$
  3. The Stability Lemma
  4. Proof of the main theorem
  5. Prolate product metrics on $\mathbb{S}^k \times \mathbb{S}^{n-k}$
  6. Berger metrics on $\mathbb{R}\mathrm{P}^3$

Key Result

Theorem 1.2

Let $M$ be a closed Riemannian 3-manifold with $\pi_2(M) \neq 0$. Suppose that $\mathop{\mathrm{scal}}\nolimits \geq s > 0$. Then Moreover, equality holds if and only if the universal cover of $M$ is isometric to the standard Riemannian cylinder $\mathbb{S}^2(1) \times \mathbb{R}$ up to scaling.

Figures (3)

  • Figure 1: The function $s \mapsto V^n_s(R)/b_n$ for $n=3$ and $R = 0.1 , 0.2, \dots , 2.5$.
  • Figure 2: Subdivision of the ball $B(x,t)$ into levels $L_1, \dots, L_N$ separated by the dividers $D_1, \dots, D_N$.
  • Figure 3: Modification of the surface $\Sigma$ by replacing each $D_i$ by the corresponding $D'_i$.

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2: Bray_Brendle_Neves_2010
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Definition 1.6
  • Theorem 1.7: Guth_2017
  • Corollary 1.8: Guth_2017
  • Theorem 1.9
  • Proposition 1.10
  • ...and 13 more