Construction of surfaces with large systolic ratio

TL;DR

This paper constructs surfaces with large systolic ratios using cutting and pasting techniques, deriving inequalities relating the systolic ratios across different genera, and extends these ideas to product manifolds.

Contribution

It introduces new methods to construct surfaces with large systolic ratios and establishes inequalities linking ratios across different genera, expanding known bounds.

Findings

Established a subadditive inequality for systolic ratios across genera.

Extended bounds on homological systolic ratios for many genera.

Constructed product manifolds with large systolic ratios from lower-dimensional cases.

Abstract

Let $(M,g)$ be a closed, oriented, Riemannian manifold of dimension $m$. We call a systole a shortest non-contractible loop in $(M,g)$ and denote by $sys(M,g)$ its length. Let $SR(M,g)=\frac{{sys(M,g)}^m}{vol(M,g)}$ be the systolic ratio of $(M,g)$. Denote by $SR(k)$ the supremum of $SR(S,g)$ among the surfaces of fixed genus $k \neq 0$. In Section 2 we construct surfaces with large systolic ratio from surfaces with systolic ratio close to the optimal value $SR(k)$ using cutting and pasting techniques. For all $k_i \geq 1$, this enables us to prove: $$\frac{1}{SR(k_1 + k_2)} \leq \frac{1}{SR(k_1)} + \frac{1}{SR(k_2)}.$$ We furthermore derive the equivalent intersystolic inequality for $SR_h(k)$, the supremum of the homological systolic ratio. As a consequence we greatly enlarge the number of genera $k$ for which the bound $SR_h(k) \geq SR(k) \gtrsim \frac{4}{9\pi} \frac{\log(k)^2}{k}$ is valid and show that that $SR_h(k) \leq \frac{(\log(195k)+8)^2}{\pi(k-1)}$ for all $k \geq 76$. In Section 3 we expand on this idea. There we construct product manifolds with large systolic ratio from lower dimensional manifolds.