×

Berger’s isoperimetric problem and minimal immersions of surfaces. (English) Zbl 0868.58079

The author establishes inequalities between eigenvalues of the Laplacian of a compact surface and the geometrical characteristics of this surface, and he considers the case of an orientable surface of genus one, \(T^2\). It can be shown that in the class of flat tori of fixed area the first nonzero eigenvalue of the Beltrami-Laplace operator \(\lambda_1\) attains its supremum on the equilateral torus, i.e., on the torus \(\mathbb{R}^2/\Gamma\), where \(\Gamma\) is the lattice generated by (1,0) and \((1/2,\sqrt{3/2})\). The main result of this paper is that the flat equilateral torus is \(\lambda_1\)-maximal. It means that the author gives affirmative answers to Berger’s problem.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] M. Berger, Sur les premières valeurs propres des variétés Riemanniennes, Composito Math. 26 (1973), 129–149. · Zbl 0257.53048
[2] M. Berger, Systoles et applications selon Gromov, Asterisque 216 (1993), 279–10. · Zbl 0789.53040
[3] M. Berger, P. Ganduchon, E. Mazet, Le spectre d’une variété Riemannienne, Springer Lect. Notes Math. 194, 1971.
[4] G. Besson, Sur la multiplicité de la première valeur propre des surfaces Riemanniennes, Ann. Inst. Fourier 30 (1980), 109–128. · Zbl 0417.30033
[5] G. Besson, Properiété génériques des fonctions propres et multiplicité, Comment. Math. Helv. 64 (1989), 542–588. · Zbl 0697.58056
[6] S.Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), 43–55. · Zbl 0334.35022
[7] Y. Colin de Verdière, Sur la multiplicité de la première valeur propre non nulle du Laplacien, Comment. Math. Helv. 61 (1986), 254–270. · Zbl 0607.53028
[8] Y. Colin de Verdière, Sur une hypothèse de transversalité d’Arnold, Comment. Math. Helv. 63 (1988), 184–193. · Zbl 0672.58046
[9] R. Courant, D. Hilbert, Methoden der Mathematishen Physik, I, Springer Verlag, 1968. · Zbl 0161.29402
[10] A. El Soufi, S. Ilias, Imersions minimales, première valeur propre du Laplacien et volume conforme, Math. Ann. 275 (1986), 257–267. · Zbl 0675.53045
[11] H.M. Farkas, Special divisors and analytic subloci of Teichmueller space, Am. Journ. Math. 88 (1966), 881–901. · Zbl 0154.33101
[12] M. Gromov, Metric invariants of Kähler manifolds, Proceedings in Differential Geometry and Topology, Alghero, Italy, World Scientific (1992), 90–116. · Zbl 0888.53047
[13] J. Hersch, Quatre propriétés isopérimétriques de membranes sphérique homogènes, C.R.Acad. Paris 270 (1970), 1645–1648. · Zbl 0224.73083
[14] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, Local properties of solutions of Schrödinger equations, Comm. PDE 17 (1992), 491–522. · Zbl 0783.35054
[15] L. Hörmander, Linear Partial Differential Operators, Springer Verlag, 1963.
[16] T. Kato, Perturbation Theory for Linear Operators, Springer Verlag, 1976. · Zbl 0342.47009
[17] H.B. Lawson, Lectures on Minimal Submanifolds 1, Math. Lecture Series 9, Perish Inc., Berkeley, 1980. · Zbl 0434.53006
[18] P. Li, S.-T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), 269–291. · Zbl 0503.53042
[19] Th. Meis, Die minimale Blatterzahl der Konkretisirung einer kompakten Riemannischen Flache, Schriftenreihe des Mathematisches Institut der Universitat Munster, 1960.
[20] S. Montiel, A. Ros, Minimal immersion of surfaces by the first eigenfunctions and conformal area, Invent. Math. 83 (153–166), 1986. · Zbl 0584.53026
[21] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer Verlag, 1966. · Zbl 0142.38701
[22] N. Nadirashvili, Multiple eigenvalues of Laplace operator, Math. USSR Sbornik 61 (1988), 225–238. · Zbl 0672.35049
[23] B. Osgood, R.S. Phillips, P. Sarnak, Extremals of determinants of Laplacians, J. Funk. Anal. 80 (1988), 148–211. · Zbl 0653.53022
[24] P. Pu, Some inequalities in certain nonoreintable manifolds, Pacific J. Math. 2 (1952), 55–71. · Zbl 0046.39902
[25] R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, 1970. · Zbl 0193.18401
[26] G. Springer, Introduction to Riemann Surfaces, Addison-Wesley Publ. Comp., 1957. · Zbl 0078.06602
[27] S.K. Vodopianov, V.U. Goldstein, Yu.G. Resetnajak, On geometric properties of functions with generalized first derivatives, Russian Math. Surveys 34 (1979), 19–74. · Zbl 0429.30017
[28] P. Yang, S.-T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Sc. Sup. Pisa 7 (1980), 55–63. · Zbl 0446.58017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.