An inverse spectral problem on the surfaces. (Un problème spectral inverse sur les surfaces.) (French) Zbl 1025.58005

Séminaire de théorie spectrale et géométrie. Année 2001-2002. St. Martin d’Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectr. Géom. 20, 139-142 (2002).
Minimal submanifolds are the solutions of a variational problem. They are critical points of the volume functional for all deformations with compact support. A minimal immersion is said to be stable if its stability operator is positive. For a minimal surface \(M\) in \({\mathbb R}^3\), the stability operator is \(S=\Delta + 2{\mathcal K}\), where \({\mathcal K}\) is the curvature of \(M\).
Let \((M,h)\) be a complete non-compact Riemannian surface. For any \(\lambda\in {\mathbb R}\), denote by \(L_{\lambda}=\Delta +\lambda {\mathcal K}\) and \(q_{\lambda}\) its associated quadratic form. It is known [see D. Fisher-Colbrie and R. Schoen, Commun. Pure Appl. Math. 33, 199-211 (1980; Zbl 0439.53060)] that the set \(I_h=\{\lambda\in {\mathbb R}\mid q_{\lambda}\) positive\(\}\) is an interval \(I_h=[a_h,b_h]\), with \(-\infty \leq a_h\leq 0\leq b_h\leq +\infty\).
In the paper under review, the author considers the unit complex disk \(D=\{z\in {\mathbb C}\mid |z|<1\}\) and proves that for any complete metric \(h\) conformal to the Euclidean metric on \(D\), one has \(b_h\leq\frac 14\). Moreover, for any \(\lambda\in [0,\frac 14]\), there exists a complete metric \(h\) conformal to the Euclidean metric on \(D\) such that \(b_h=\lambda\). The case \(b_h=\frac 14\) is investigated.
For the entire collection see [Zbl 1008.00009].


58E12 Variational problems concerning minimal surfaces (problems in two independent variables)


Zbl 0439.53060
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