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arXiv · 2103.15693

Discrete Yamabe problem for polyhedral surfaces

Hana Dal Poz Kouřimská
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Abstract

We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.

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Hana Dal Poz Kouřimská. 2023-01-11. Discrete Yamabe problem for polyhedral surfaces. https://doi.org/10.1007/s00454-023-00484-2

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