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Existence and geometric structure of metrics on surfaces which extremize eigenvalues. (English) Zbl 1336.53070

The author has published numerous results regarding sharp upper bounds on the first eigenvalue of a surface (with or without boundary) in terms of area or boundary length and the surface topology. This article contains a result on a new coarse upper bound for non-orientable surfaces with boundary. It is mostly a survey article presenting recent progress made by the author, in the field of differential and spectral geometry.
The problem that may be of interest to most researchers in this field, regardless of their particular area, may be the very first one that is exposed: “To determine a metric \(g\) on a given closed smooth surface \(M^2\), normalized so that \(A(g)=1\), with the largest first eigenvalue”. As the author himself expresses it, “another way to pose the question is the problem of finding a sharp upper bound on the eigenvalue \(\lambda_1\) in terms of the area and topology of a surface \(M\)”. The article should be of interest to all researchers in the field, and is written in a manner that makes it accessible to junior researchers and advanced graduate students.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C43 Differential geometric aspects of harmonic maps
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
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[1] G. Besson. Sur la multiplicité de la première valeur propre des surfaces riemanniennes. Ann. Inst. Fourier (Grenoble), 30(1) (1980), x, 109-128. · Zbl 0417.30033 · doi:10.5802/aif.777
[2] R. Brooks and E. Makover. Riemann surfaces with large first eigenvalue. J. Anal. Math., 83 (2001), 243-258. · Zbl 0981.30031 · doi:10.1007/BF02790263
[3] Buser, P.; Burger, M.; Dodziuk, J., Riemann surfaces of large genus and large λ1, No. 1339, 54-63 (1988), Berlin · doi:10.1007/BFb0083046
[4] S.Y. Cheng. Eigenfunctions and nodal sets. Comment. Math. Helv., 51(1) (1976), 43-55. · Zbl 0334.35022 · doi:10.1007/BF02568142
[5] F. Da Lio and T. Riviére. Three-term commutator estimates and the regularity of \[\tfrac{1} {2} \]-harmonic maps into spheres. Anal. PDE, 4 (2011), 149-190. · Zbl 1241.35035 · doi:10.2140/apde.2011.4.149
[6] A. El Soufi, H. Giacomini and M. Jazar. A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle. Duke Math. J., 135 (2006), 181-202. · Zbl 1109.58029 · doi:10.1215/S0012-7094-06-13514-7
[7] A. El Soufi and S. Ilias. Immersionsminimales, première valeur propre du laplacien et volume conforme. Math. Ann., 275(2) (1986), 257-267. · Zbl 0675.53045 · doi:10.1007/BF01458460
[8] A. El Soufi and S. Ilias. Riemannian manifolds admitting isometric immersions by their first eigenfunctions. Pacific J. Math., 195(1) (2000), 91-99. · Zbl 1030.53043 · doi:10.2140/pjm.2000.195.91
[9] A. Fraser and R. Schoen. The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math., 226(5) (2011), 4011-4030. · Zbl 1215.53052 · doi:10.1016/j.aim.2010.11.007
[10] A. Fraser and R. Schoen. Sharp eigenvalue bounds and minimal surfaces in the ball, arXiv:1209.3789 [math.DG] (2012).
[11] A. Girouard. Fundamental tone, concentration of density, and conformal degeneration on surfaces. Canad. J. Math., 61 (2009), 548-565. · Zbl 1198.53039 · doi:10.4153/CJM-2009-029-1
[12] A. Girouard and I. Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electron. Res. Announc. Math. Sci., 19 (2012), 77-85. · Zbl 1257.58019
[13] J. Hersch. Quatre propriétés isopérimétriqes de membranes sphériques homogènes. C.R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1645-A1648.
[14] D. Jakobson, M. Levitin, N. Nadirashvili, N. Nigram and I. Polterovich. How large can the first eigenvalue be on a surface of genus two? IMRN, 63 (2005), 3967-3985. · Zbl 1114.58026 · doi:10.1155/IMRN.2005.3967
[15] D. Jakobson, N. Nadirashvili and I. Polterovich. Extremal metric for the first eigenvalue on a Klein bottle. Cand. J. Math., 58 (2006), 381-400. · Zbl 1104.58008 · doi:10.4153/CJM-2006-016-0
[16] P. Jammes. Prescription du spectre de Steklov dans une classe conforme, arXiv:1209.4571 [math.DG] (2012).
[17] M. Karpukhin, G. Kokarev and I. Polterovich. Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces, arXiv:1209.4869v2 [math.DG] (2013).
[18] G. Kokarev. Variational aspects of Laplace eigenvalues on Riemannian surfaces, arXiv:1103.2448 [math.SP] (2011).
[19] Kokarev, G.; Nadirashvili, N., On first Neumann eigenvalue bounds for conformal metrics, No. 12, 229-238 (2010), New York · Zbl 1187.58030 · doi:10.1007/978-1-4419-1343-2_10
[20] N. Korevaar. Upper bounds for eigenvalues of conformal surfaces. J. Diff. Geom., 37 (1993), 73-93. · Zbl 0794.58045
[21] P. Li and S.-T. Yau. A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math., 69(2) (1982), 269-291. · Zbl 0503.53042 · doi:10.1007/BF01399507
[22] S. Montiel and A. Ros. Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math., 83(1) (1985), 153-166. · Zbl 0584.53026 · doi:10.1007/BF01388756
[23] N. Nadirashvili. Multiple eigenvalues of the Laplace operator, (Russian) Mat. Sb. (N.S.), 133(175) (1987), 223-237; translation in Math. USSR-Sb., 61 (1988), 225-238.
[24] N. Nadirashvili. Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal., 6(5) (1996), 877-897. · Zbl 0868.58079 · doi:10.1007/BF02246788
[25] N. Nadirashvili and Y. Sire. Conformal spectrum and harmonic maps, arXiv:1007.3104v3 [math.DG], (2011).
[26] J.C.C. Nitsche. Stationary partitioning of convex bodies. Arch. Rational Mech. Anal., 89(1) (1985), 1-19. · Zbl 0572.52005 · doi:10.1007/BF00281743
[27] Vignéras, M-F, Quelques remarques sur la conjecture \[\lambda_1 \geqslant \tfrac{1} {4}, No. 38, 321-343 (1983)\], Boston, MA
[28] R. Weinstock. Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal., 3 (1954), 745-753. · Zbl 0056.09801
[29] P. Yang and S.-T. Yau. Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7(1) (1980), 55-63. · Zbl 0446.58017
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