Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space

Abstract

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: We prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is Steklov isospectral to the disk.

Appendix: Surfaces of Revolution with Connected Boundary are Steklov Isospectral

The goal of this appendix is to prove Proposition 1.10.

Let $M\subset \mathbb R^3$ be a surface of revolution with connected boundary $\partial M=\mathbb S^1\subset \mathbb R^2\times \{0\}$ and induced Riemannian metric g. We will construct a smooth function $\delta :M\rightarrow (0,\infty )$ such that $\delta \bigl |\bigr ._{\partial M}=1$ and $(M,\delta g)$ is isometric to the Euclidean unit disk $({\mathbb {D}},g_0)$.

The Riemannian metric on M can be written as $g=dr^2+h(r)^2d\theta ^2$ on $[0,L)\times \mathbb S^1$, where $d\theta$ is the usual metric on $\mathbb S^1$ and, as in Sect. 3.1, the function h satisfies $h(0)=1$ and $h(L)=0$. Define the diffeomorphism $\tau :[0,L)\rightarrow [0,\infty )$ by

$$\begin{aligned} \tau (r)=\int _0^r\frac{1}{h(s)}\,\mathrm{{d}}s. \end{aligned}$$

The map $\Phi :(0,L)\times \mathbb S^1\rightarrow (0,1)\times \mathbb S^1$ defined by

$$\begin{aligned} \Phi (r,\theta )=(1-e^{-\tau (r)},\theta ) \end{aligned}$$

induces a conformal diffeomorphism to the disk ${\mathbb {D}}$ represented by the metric $g_0=dr^2+(1-r)^2d\theta ^2$ on $[0,1)\times \mathbb S^1$. In particular, one has $\Phi ^\star (g_0)=\delta g$, where the conformal factor $\delta \in C^\infty (M)$ is given by

$$\begin{aligned} \delta (r,\theta )= \left( \frac{\mathrm{{d}}}{\mathrm{{d}}r}\left( 1-e^{-\tau (r)}\right) \right) ^2 =\frac{1}{h^2}e^{-2\tau (r)}. \end{aligned}$$

Moreover $\delta \bigl |\bigr ._{\partial M}=1$, since $\delta (0,\theta )=1$. This implies that (M, g) and $(M,\delta g)\cong ({\mathbb {D}},g_0)$ are Steklov isospectral.