Scholarlink

<p>In this paper we provide a new interpretation of the Willmore energy as the curvature of a quaternionic connection that has a clear geometric interpretation in terms of mean curvature spheres rolling over the surface. Building on this interpretation, we show that the Möbius invariant discretization of the Willmore energy introduced by Bobenko can be interpreted as the curvature of a discrete connection defined by rolling the edge circumspheres. We describe how a choice of discrete mean curvature spheres produces a Möbius invariant discrete Willmore energy. In this way, we define a new discrete Willmore energy for surfaces built out of spherical pieces. To facilitate the computations we describe a new quaternionic description of the conformal three-sphere, along with realizations of the spaces of circles, spheres, and point pairs in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^3\)</EquationSource> </InlineEquation>. It is obtained by specializing the quaternionic projective model of the conformal four-sphere.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Rolling Spheres and the Willmore Energy

  • 期刊论文

  • 发表日期：

    2025年11月9日

查看全文
  • Felix Knöppel,
  • Ulrich Pinkall,
  • Peter Schröder,
  • Yousuf Soliman

摘要

In this paper we provide a new interpretation of the Willmore energy as the curvature of a quaternionic connection that has a clear geometric interpretation in terms of mean curvature spheres rolling over the surface. Building on this interpretation, we show that the Möbius invariant discretization of the Willmore energy introduced by Bobenko can be interpreted as the curvature of a discrete connection defined by rolling the edge circumspheres. We describe how a choice of discrete mean curvature spheres produces a Möbius invariant discrete Willmore energy. In this way, we define a new discrete Willmore energy for surfaces built out of spherical pieces. To facilitate the computations we describe a new quaternionic description of the conformal three-sphere, along with realizations of the spaces of circles, spheres, and point pairs in \(\mathbb {R}^3\) . It is obtained by specializing the quaternionic projective model of the conformal four-sphere.