Tuesday, August 31
The problem of finding upper bounds for eigenvalues of the Laplace operator has a rich history. In 1970, Joseph Hersch showed that for the sphere, the product of its surface area and lowest nonzero eigenvalue is at most 8π, regardless of the metric. Moreover, the bound is realized only by the usual constant curvature metric. In contrast, Bruno Colbois and Józef Dodziuk showed that for manifolds of dimension three and higher, the eigenvalues can be unbounded unless additional geometric constraints are imposed. We discuss upper bounds on eigenvalues in the setting of compact submanifolds of Euclidean space. No previous experience with eigenvalues on manifolds will be assumed.
This is joint work with Bruno Colbois and Ahmad El Soufi.
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