WorkshopsAn extremal eigenvalue problem and minimal surfaces in the ball
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We consider the spectrum of the Dirichlet-Neumann map. This is the spectrum of the operator which sends a function on the boundary of a manifold to the normal derivative of its harmonic extension. We show how the question of finding surfaces with fixed boundary length and largest first eigenvalue is intimately connected to the study of minimal surfaces in the ball which meet the boundary orthogonally (free boundary solutions). We describe recent results on the characterization of optimal surfaces for this problem. This approach also leads to the solution of problems on the geometry of this class of minimal submanifolds such as optimal isoperimetric inequalities. This is joint work with Ailana Fraser.
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Richard Schoen
2011-07-11
14:30:00 - 15:15:00
國際會議廳 , Astronomy and Mathematics Building
We consider the spectrum of the Dirichlet-Neumann map. This is the spectrum of the operator which sends a function on the boundary of a manifold to the normal derivative of its harmonic extension. We show how the question of finding surfaces with fixed boundary length and largest first eigenvalue is intimately connected to the study of minimal surfaces in the ball which meet the boundary orthogonally (free boundary solutions). We describe recent results on the characterization of optimal surfaces for this problem. This approach also leads to the solution of problems on the geometry of this class of minimal submanifolds such as optimal isoperimetric inequalities. This is joint work with Ailana Fraser.
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