# Mean curvature versus diameter and energy quantization

## TL;DR

This paper extends a theorem relating mean curvature and diameter to broader settings and uses it to prove energy quantization for pseudo-holomorphic curves in certain symplectic manifolds.

## Contribution

It generalizes Topping's theorem to almost everywhere immersed submanifolds in compact Riemannian manifolds and applies this to establish energy quantization results.

## Key findings

- Extended Topping's theorem to broader class of submanifolds
- Proved energy quantization for pseudo-holomorphic curves of all genus
- Demonstrated applications in symplectic geometry

## Abstract

We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of $\mathbb{R} ^{n} $ to almost everywhere immersed, closed submanifolds of a compact Riemannian manifold. We use this to prove quantization of energy for pseudo-holomorphic closed curves, of all genus, in a compact locally conformally symplectic manifold.