Existence and geometric structure of metrics on surfaces which extremize eigenvalues



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].


spectral theory minimal surface free boundary condition Steklov eigenvalue 

Mathematical subject classification

53A10 58C40 


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Copyright information

© Sociedade Brasileira de Matemática 2013

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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