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Every closed surface of genus at least 18 is Loewner

Qiongling Li, Weixu Su

Abstract

In this paper, we obtain an improved upper bound involving the systole and area for the volume entropy of a Riemannian surface. As a result, we show that every orientable and closed Riemannian surface of genus $g\geq 18$ satisfies Loewner's systolic ratio inequality. We also show that every closed orientable and nonpositively curved Riemannnian surface of genus $g\geq 11$ satisfies Loewner's systolic ratio inequality.

Every closed surface of genus at least 18 is Loewner

Abstract

In this paper, we obtain an improved upper bound involving the systole and area for the volume entropy of a Riemannian surface. As a result, we show that every orientable and closed Riemannian surface of genus satisfies Loewner's systolic ratio inequality. We also show that every closed orientable and nonpositively curved Riemannnian surface of genus satisfies Loewner's systolic ratio inequality.
Paper Structure (6 sections, 3 theorems, 49 equations, 1 figure)

This paper contains 6 sections, 3 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An upper bound for the entropy
  3. Entropy
  4. Upper bound
  5. Proof of Theorem \ref{['thm:loewner']}
  6. Proof of Theorem \ref{['thm:nonpositive']}

Key Result

Corollary 2.1

Every surface $M$ of genus $g\geq 1$ satisfies whenever $0<\eta<\frac{1}{9}$.

Figures (1)

  • Figure 1: Picture in Case 1. The disks $B(\hat{q}_k)$'s cover the segment connecting $c((\gamma_k+\beta)s)$ and $c((\gamma_k+\beta+t\alpha)s)$. There is a smallest $l_0$ such that $d(q_k,\hat{q}_{l_0})\geq (\beta-t\alpha)s.$

Theorems & Definitions (10)

  • proof
  • Corollary 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • proof
  • proof
  • proof