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On the spectrum of bounded immersions. (English) Zbl 1327.53076

In the paper under review, the authors investigate relations between the spectrum of a non-compact, extrinsically bounded submanifold \(\varphi: M^m\to N^n\) and the Hausdorff dimension of its limit lim \(\varphi\).
They address Yau’s question whether the spectrum of bounded minimal surfaces in \(\mathbb{R}^3\) is discrete or not. The first result (Theorem 2.4) gives sufficient conditions on the size of the limit set of a bounded submanifold for its spectrum to be discrete. It applies to a number of examples recently constructed by many authors and gives a complete answer to the question of S. T. Yau [Asian J. Math. 4, No. 1, 235–278 (2000; Zbl 1031.53004)].
A counterpart to Theorem 2.4 would be a set of geometric conditions implying that the essential spectrum of a Riemannian manifold is not empty. A criterion, called the ball property, that does not involve curvatures, is presented. It can be used to study the spectrum of the complete minimal surfaces. It guarantees the existence of elements in the essential spectrum. As applications the essential spectrum of some examples of L. P. de M. Jorge and F. Xavier [Ann. Math. (2) 112, 203–206 (1980; Zbl 0455.53004)] and H. Rosenberg and E. Toubiana [Bull. Sci. Math., II. Sér. 111, 241–245 (1987; Zbl 0631.53012)] are determined.
Finally some open problems are stated.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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