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Index bounds and existence results for minimal surfaces and harmonic maps Sargent, Pam


In this work, we focus on three problems. First, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in ℝ³ tends to infinity as its genus or the number of boundary components tends to infinity. Second, we consider the relationship between the kth normalized eigenvalue of the Dirichlet-to-Neumann map (the kth Steklov eigenvalue) and the geometry of rotationally symmetric Möbius bands. More specifically, we look at the problem of finding a metric that maximizes the kth Steklov eigenvalue among all rotationally symmetric metrics on the Möbius band. We show that such a metric can always be found and that it is realized by the induced metric on a free boundary minimal Möbius band in B⁴. Third, we consider the existence problem for harmonic maps into CAT(1) spaces. If Σ is a compact Riemann surface, X is a compact locally CAT(1) space and φ : Σ → X is a continuous finite energy map, we use the technique of harmonic replacement to prove that either there exists a harmonic map u : Σ → X homotopic to φ or there exists a conformal harmonic map v : S² → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.

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