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- W3110331085 abstract "We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Superscript m> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>S^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S squared> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>S^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the second nonzero eigenvalue becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. For higher dimensional spheres, the analogous upper bound was conjectured by Girouard, Nadirashvili and Polterovich. Our method to confirm the conjecture builds on Petrides’ work and recent developments on the hyperbolic center of mass and provides also a simpler proof for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S squared> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>S^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
- W3110331085 created "2020-12-07" @default.
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- W3110331085 date "2022-04-15" @default.
- W3110331085 modified "2023-09-24" @default.
- W3110331085 title "Maximization of the second Laplacian eigenvalue on the sphere" @default.
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- W3110331085 doi "https://doi.org/10.1090/proc/15908" @default.
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