# Distortion of spheres and surfaces in space

## TL;DR

This paper investigates the minimal distortion embeddings of convex 2-spheres and establishes a sharp lower bound for the distortion of embedded surfaces with positive genus, revealing geometric constraints on surface embeddings.

## Contribution

It proves the existence of distortion minimizers among convex embedded 2-spheres with bounded eccentricity and establishes a sharp lower bound of π/2 for surfaces of positive genus.

## Key findings

- Distortion minimizers exist among convex embedded 2-spheres.
- Embedded surfaces of positive genus have a distortion lower bound of π/2.
- Convex 2-spheres can be embedded with uniformly bounded eccentricity.

## Abstract

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $\pi/2$ is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.