Geodesic disks uniquely maximize the first positive Neumann eigenvalue among simply connected fixed-area domains on the sphere.
Isoperimetric inequalities and sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces
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abstract
We consider the first eigenvalue of the magnetic Laplacian with zero magnetic field on simply connected compact surfaces and we establish isoperimetric inequalities and upper bounds in terms of a bound on the gaussian curvature. As a corollary, we prove that among all simply connected spherical domains of fixed area, the first eigenvalue is maximal for a geodesic disk with the pole of the magnetic potential at its center; also, for the sphere punctured at two points, the first eigenvalue is maximal when the punctures are antipodal.
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On the isoperimetric inequality for the first positive Neumann eigenvalue on the sphere
Geodesic disks uniquely maximize the first positive Neumann eigenvalue among simply connected fixed-area domains on the sphere.