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Homological $n$-systole in $(n+1)$-manifolds and bi-Ricci curvature

Jianchun Chu, Man-Chun Lee, Jintian Zhu

Abstract

In this paper, we prove an optimal systolic inequality and the corresponding rigidity in the equality case on closed manifolds with positive bi-Ricci curvature, which generalizes the work of Bray-Brendle-Neves. The proof is given in all dimensions based on the method of minimal surfaces under the Generic Regularity Hypothesis, which is known to be true up to dimension ten.

Homological $n$-systole in $(n+1)$-manifolds and bi-Ricci curvature

Abstract

In this paper, we prove an optimal systolic inequality and the corresponding rigidity in the equality case on closed manifolds with positive bi-Ricci curvature, which generalizes the work of Bray-Brendle-Neves. The proof is given in all dimensions based on the method of minimal surfaces under the Generic Regularity Hypothesis, which is known to be true up to dimension ten.

Paper Structure

This paper contains 8 sections, 14 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Thm: main']} when $n+1\leq 7$
  3. Systolic inequality
  4. Rigidity
  5. Preliminaries
  6. Proof of Proposition \ref{['Prop: totally geodesic approximation']}
  7. Proof of Theorem \ref{['Thm: main']} using generic regularity hypothesis
  8. Curvature change under conformal deformation

Key Result

Theorem 1.1

Let $(M^{n+1},g)$ be an orientable $(n+1)$-dimensional closed Riemannian manifold with $H_n(M)\neq 0$ and $\mathop{\mathrm{biRic}}\nolimits\geq n-1$. Suppose additionally that the Generic Regularity Hypothesis holds. Then we have where the equality yields that the universal cover of $(M,g)$ splits isometrically as $\mathbb S^n\times \mathbb R$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2: Theorem 1 of AntonelliXu2024
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Lemma 2.7
  • ...and 20 more