Asian Journal of Mathematics
Volume 25 (2021)
Comparing shapes of high genus surfaces
Pages: 579 – 596
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.
differential geometry, metric geometry
2010 Mathematics Subject Classification
The author was supported in part by NSF Grant DMS-1719582.
Received 22 November 2019
Accepted 19 January 2021
Published 25 April 2022