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Eigenvalue problems and free boundary minimal surfaces in spherical caps
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Given a compact surface with boundary, we introduce a family of functionals on the space of its Riemannian metrics, defined via eigenvalues of a Steklov-type problem. We prove that each such functional is uniformly bounded from above, and we characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions.
Forward citations
Cited by 2 Pith papers
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A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere
The radius map of de Oliveira's free-boundary minimal annuli in S³ folds above the hemisphere, forcing non-uniqueness and isolated spectral degenerations detected by a general Robin defect identity.
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Free boundary minimal M\"obius band in spherical caps
For each r in (0,π/2), a free boundary minimal Möbius band immersed by first Steklov eigenfunctions exists in the four-dimensional spherical cap, and any such immersion is intrinsically rotationally symmetric.
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