Antoine Song, Caltech

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Abstract: Consider a closed surface of genus at least 2 endowed with a Riemannian metric g, and let (S,g) be its universal cover. There are two important invariants for (S,g): the first eigenvalue λ of the Laplacian and the volume entropy h, which measures the exponential growth rate of the volume of geodesic balls. We can normalize g so that h=1. Then a classical inequality states that λ is at most 1/4. When g is a hyperbolic metric, equality holds. We will discuss a stability property for the hyperbolic plane: if λ is close to the upper bound 1/4, then (S,g) is close to the hyperbolic plane in a Benjamini-Schramm topology.

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