Comparing shapes of high genus surfaces

TL;DR

This paper introduces a new metric on the shape space of high genus surfaces using energies that measure area and angle distortions, with minimizers achieved by quasiconformal homeomorphisms.

Contribution

It provides a rigorous definition of surface shape and constructs a novel metric based on quasiconformal energy minimization for high genus surfaces.

Findings

Energy minimizers are quasiconformal homeomorphisms.

The metric is based on area and angle distortion energies.

Minimizers exist due to lower semicontinuity.

Abstract

In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer in a fixed homotopy class is achieved by a quasiconformal homeomorphism by the lower semicontinuity property of these two energies.