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Existence and geometric structure of metrics on surfaces which extremize eigenvalues

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Abstract

This is an exposition of the research area around our lecture at the 60th anniversary conference of IMPA which was held in October of 2012. It is a survey of results which have been obtained over many years concerning sharp upper bounds on the first eigenvalue of a surface, either with or without boundary, in terms or area or boundary length and the surface topology. It is mostly expository, but contains a new coarse upper bound for non-orientable surfaces with boundary. It also contains a classical reformulation of recent results in [10].

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Richard Schoen

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  1. Richard Schoen
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Correspondence to Richard Schoen.

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The author was partially supported by NSF grant DMS-1105323.

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Schoen, R. Existence and geometric structure of metrics on surfaces which extremize eigenvalues. Bull Braz Math Soc, New Series 44, 777–807 (2013). https://doi.org/10.1007/s00574-013-0034-6

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  • DOI: https://doi.org/10.1007/s00574-013-0034-6

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