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<p>In this paper, we introduce a new discretization of the Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of some dual cell of a weighted triangulation at the conic singularity. This discrete Gaussian curvature has the same scaling properties as the classical Gaussian curvature and is an approximation of the classical Gaussian curvature. Then we establish a uniformization theorem for this discrete Gaussian curvature on surfaces with non-positive Euler number by variational principles with constraints. This can be taken as a discrete analogy of the classical uniformization theorem on surfaces.</p>

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A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces

  • 期刊论文

  • 发表日期：

    2026年5月28日

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  • Xu Xu,
  • Chao Zheng

摘要

In this paper, we introduce a new discretization of the Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of some dual cell of a weighted triangulation at the conic singularity. This discrete Gaussian curvature has the same scaling properties as the classical Gaussian curvature and is an approximation of the classical Gaussian curvature. Then we establish a uniformization theorem for this discrete Gaussian curvature on surfaces with non-positive Euler number by variational principles with constraints. This can be taken as a discrete analogy of the classical uniformization theorem on surfaces.