Comparing shapes of high genus surfaces

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.

Keywords

differential geometry, metric geometry

2010 Mathematics Subject Classification

49Q10, 53A05

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The author was supported in part by NSF Grant DMS-1719582.

Received 22 November 2019

Accepted 19 January 2021

Published 25 April 2022