[Paper Review] Steklov eigenvalues on annulus

Xu-Qian Fan, Luen-Fai Tam+1 more|arXiv (Cornell University)|Oct 29, 2013

Geometric Analysis and Curvature Flows8 references3 citations

TL;DR

This paper computes the supremum of the $k$-th normalized Steklov eigenvalue over rotationally symmetric conformal metrics on the annulus $[0,T] \times \mathbb{S}^1$ for $k > 1$, showing that the supremum is achieved by metrics corresponding to embedded or immersed minimal surfaces that meet the boundary of a ball orthogonally. For $k=2$, the supremum is $4\pi$, but it is not achieved by any finite $T$, consistent with prior results on non-achievable second eigenvalues.

ABSTRACT

We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We also obtain some partial results on the comparison of the normalized Stekov eigenvalues of rotationally symmetric metrics and general conformal metrics on the cylinder. A counter example is constructed to show that for that the first normalized Steklov eigenvalue of rotationally symmetric metric may not be larger.

Motivation & Objective

• To determine the supremum of the $k$-th normalized Steklov eigenvalue over all rotationally symmetric conformal metrics on the annulus $[0,T] \times \mathbb{S}^1$ for $k > 1$.
• To characterize the geometric structure of metrics that achieve the supremum, particularly in terms of minimal surfaces in the ball.
• To compare normalized Steklov eigenvalues of rotationally symmetric metrics with those of general conformal metrics on the annulus.
• To investigate whether the supremum of the second normalized Steklov eigenvalue can be achieved, especially in light of known non-achievable results.

Proposed method

• Parameterize the annulus with a rotationally symmetric conformal metric $g = f(t)^2(dt^2 + d\theta^2)$ and express eigenvalues in terms of $f(0)/f(T)$ and $T$.
• Use spectral analysis and separation of variables to derive eigenvalue equations for Steklov eigenfunctions on the cylinder.
• Apply variational methods and symmetry reduction to compute the supremum of the $k$-th normalized eigenvalue $\tilde{\sigma}_k(g) = \sigma_k(g) \cdot L(\partial M)$.
• Relate extremal metrics to minimal surfaces via the construction of isometric embeddings into $\mathbb{R}^3$ that meet the boundary of a ball orthogonally.
• Construct a counterexample to show that for fixed $T$, the first normalized Steklov eigenvalue of a rotationally symmetric metric may not exceed that of a general conformal metric.
• Use asymptotic and comparison arguments involving Bessel functions and hyperbolic trigonometric identities to solve transcendental equations for critical $T$ values.

Experimental results

Research questions

• RQ1 What is the supremum of the $k$-th normalized Steklov eigenvalue among all rotationally symmetric conformal metrics on the annulus $[0,T] \times \mathbb{S}^1$ for $k > 1$?
• RQ2 Which geometric structures—specifically, minimal surfaces—correspond to metrics that achieve the supremum of the normalized Steklov eigenvalues?
• RQ3 Can the supremum of the second normalized Steklov eigenvalue be achieved by any rotationally symmetric metric on a finite annulus?
• RQ4 How do the normalized Steklov eigenvalues of rotationally symmetric metrics compare to those of general conformal metrics on the annulus?

Key findings

• The supremum of the $(2k-1)$-th normalized Steklov eigenvalue is $M_{2k-1} = \frac{4k\pi}{T_{2,0}(1)}$, achieved when $f(1) = f(T)$ and $T = \frac{2}{k}T_{2,0}(1)$, where $T_{2,0}(1)$ is the unique positive root of $s = \coth s$.
• The supremum of the second normalized Steklov eigenvalue is $M_2 = 4\pi$, but it is not achieved for any finite $T$, consistent with prior results on non-achievable eigenvalues.
• For even indices, $M_{2k} = 4k\pi \tanh\left(\frac{k T_{k,1}(1)}{2}\right)$, where $T_{k,1}(1)$ is the unique positive root of $k \tanh(ks/2) = \coth(s/2)$, and the supremum is achieved when $f(1) = f(T)$ and $T = T_{k,1}(1)$.
• All extremal metrics for odd $k$ correspond to embedded or immersed minimal surfaces that meet the boundary of a ball orthogonally, except for $k=2$, where the supremum is not achieved.
• A counterexample is constructed showing that for fixed $T$, the first normalized Steklov eigenvalue of a rotationally symmetric metric may be smaller than that of a general conformal metric.
• The comparison between normalized eigenvalues of rotationally symmetric and general metrics holds for a large class of metrics, but fails in general, as demonstrated by Theorem 5.1.