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Ruivo de Oliveira, Manuel

University of British Columbia

We consider free boundary minimal submanifolds in geodesic balls of simply connected space forms and relate them to the eigenvalues and eigenfunctions of the Dirichlet-to-Neumann map for the Helmholtz equation, generalizing a well-known connection from Euclidean space. This relation is applied to derive bounds on an isoperimetric ratio on such submanifolds as well as to study their balancing. We begin work towards an extremal eigenvalue problem by showing that some normalization of the eigenvalues is needed before considering a maximization problem analogous to that of Laplace eigenvalues on closed surfaces and of Steklov eigenvalues on compact surfaces with boundary. The first variation of the second eigenvalue of the Dirichlet-to-Neumann map for the Helmholtz equation along a family of metrics is calculated, extending earlier calculations of Berger \cite{berger_1973} and Fraser and Schoen \cite{fraser_schoen_2016}. We also compute the spectrum of the Dirichlet-to-Neumann map for the Helmholtz equation on Euclidean annuli. Finally, we study free boundary minimal surfaces of revolution in the sphere $\mathbb{S}^3$ and hyperbolic space $\mathbb{H}^3$. By looking at two distinct one-parameter families of complete minimal surfaces of revolution in $\mathbb{H}^3$, we prove that one contains a one-parameter family of free boundary minimal annuli and the other contains only the free boundary totally geodesic disks. In $\mathbb{S}^3$, we find that a one-parameter family of complete minimal surfaces of revolution contains many free boundary minimal annuli, which may be embedded or have self-intersections, contained in a geodesic ball of $\mathbb{S}^3$ or not. We conclude by showing that at least some of the free boundary minimal annuli we found are embedded and contained in a geodesic ball which may be smaller than, equal to or greater than a hemisphere.

Text

2024-04-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/88082

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/