Asian Journal of Mathematics

Volume 25 (2021)

Number 4

Comparing shapes of high genus surfaces

Pages: 579 – 596



Yanwen Luo (Department of Mathematics, University of California, Davis, Calif., U.S.A.)


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

49Q10, 53A05

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

Received 22 November 2019

Accepted 19 January 2021

Published 25 April 2022