The sharp energy-capacity inequality

TL;DR

This paper proves a sharp inequality relating the -sensitive Hofer-Zehnder capacity and displacement energy of subsets in closed symplectic manifolds, extending non-squeezing results using spectral invariants.

Contribution

It establishes a sharp energy-capacity inequality for -sensitive Hofer-Zehnder capacity, extending non-squeezing theorems via spectral invariants and symplectic topology techniques.

Findings

-sensitive Hofer-Zehnder capacity is bounded by displacement energy.

The inequality is sharp, confirming optimal bounds.

New extensions of the Non-Squeezing Theorem are derived.

Abstract

Using the Oh-Schwarz spectral invariants and some arguments of Frauenfelder, Ginzburg, and Schlenk, we show that the \pi_1-sensitive Hofer-Zehnder capacity of any subset of a closed symplectic manifold is less than or equal to its displacement energy. This estimate is sharp, and implies some new extensions of the Non-Squeezing Theorem.