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Symmetry and isoperimetry for Riemannian surfaces

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Abstract

For a domain \(\Omega \) in a geodesically convex surface, we introduce a scattering energy \(\mathcal {E}(\Omega )\), which measures the asymmetry of \(\Omega \) by quantifying its incompatibility with an isometric circle action. We prove several sharp quantitative isoperimetric inequalities involving \(\mathcal {E}(\Omega )\) and characterize the domains with vanishing scattering energy by their convexity and rotational symmetry. We also give a new of the sharp Sobolev inequality for Riemannian surfaces.

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Acknowledgements

We are very happy to thank Christopher Croke, Joseph H.G. Fu and Robert Kusner for their interest in this work and several helpful discussions. We thank Peter Topping for sharing with us a complex analytic proof of the sharp Euclidean Sobolev inequality (see Sect. 3).

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Joseph Ansel Hoisington

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Peter McGrath

Authors
  1. Joseph Ansel Hoisington
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  2. Peter McGrath
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Correspondence to Peter McGrath.

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Communicated by P. Topping.

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Hoisington, J.A., McGrath, P. Symmetry and isoperimetry for Riemannian surfaces. Calc. Var. 61, 6 (2022). https://doi.org/10.1007/s00526-021-02117-z

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