Abstract
This is an exposition of the research area around our lecture at the 60th anniversary conference of IMPA which was held in October of 2012. It is a survey of results which have been obtained over many years concerning sharp upper bounds on the first eigenvalue of a surface, either with or without boundary, in terms or area or boundary length and the surface topology. It is mostly expository, but contains a new coarse upper bound for non-orientable surfaces with boundary. It also contains a classical reformulation of recent results in [10].
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References
G. Besson. Sur la multiplicité de la première valeur propre des surfaces riemanniennes. Ann. Inst. Fourier (Grenoble), 30(1) (1980), x, 109–128.
R. Brooks and E. Makover. Riemann surfaces with large first eigenvalue. J. Anal. Math., 83 (2001), 243–258.
P. Buser, M. Burger and J. Dodziuk. Riemann surfaces of large genus and large λ 1. Geometry and analysis on manifolds (Katata/Kyoto, 1987), 54–63, Lecture Notes in Math., 1339, Springer, Berlin (1988).
S.Y. Cheng. Eigenfunctions and nodal sets. Comment. Math. Helv., 51(1) (1976), 43–55.
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.
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.
A. El Soufi and S. Ilias. Immersionsminimales, première valeur propre du laplacien et volume conforme. Math. Ann., 275(2) (1986), 257–267.
A. El Soufi and S. Ilias. Riemannian manifolds admitting isometric immersions by their first eigenfunctions. Pacific J. Math., 195(1) (2000), 91–99.
A. Fraser and R. Schoen. The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math., 226(5) (2011), 4011–4030.
A. Fraser and R. Schoen. Sharp eigenvalue bounds and minimal surfaces in the ball, arXiv:1209.3789 [math.DG] (2012).
A. Girouard. Fundamental tone, concentration of density, and conformal degeneration on surfaces. Canad. J. Math., 61 (2009), 548–565.
A. Girouard and I. Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electron. Res. Announc. Math. Sci., 19 (2012), 77–85.
J. Hersch. Quatre propriétés isopérimétriqes de membranes sphériques homogènes. C.R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1645–A1648.
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.
D. Jakobson, N. Nadirashvili and I. Polterovich. Extremal metric for the first eigenvalue on a Klein bottle. Cand. J. Math., 58 (2006), 381–400.
P. Jammes. Prescription du spectre de Steklov dans une classe conforme, arXiv:1209.4571 [math.DG] (2012).
M. Karpukhin, G. Kokarev and I. Polterovich. Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces, arXiv:1209.4869v2 [math.DG] (2013).
G. Kokarev. Variational aspects of Laplace eigenvalues on Riemannian surfaces, arXiv:1103.2448 [math.SP] (2011).
G. Kokarev and N. Nadirashvili. On first Neumann eigenvalue bounds for conformal metrics. Around the research of Vladimir Maz’ya. II, 229–238, Int. Math. Ser. (N. Y.), 12, Springer, New York (2010).
N. Korevaar. Upper bounds for eigenvalues of conformal surfaces. J. Diff. Geom., 37 (1993), 73–93.
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.
S. Montiel and A. Ros. Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math., 83(1) (1985), 153–166.
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.
N. Nadirashvili. Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal., 6(5) (1996), 877–897.
N. Nadirashvili and Y. Sire. Conformal spectrum and harmonic maps, arXiv:1007.3104v3 [math.DG], (2011).
J.C.C. Nitsche. Stationary partitioning of convex bodies. Arch. Rational Mech. Anal., 89(1) (1985), 1–19.
M-F. Vignéras. Quelques remarques sur la conjecture \(\lambda _1 \geqslant \tfrac{1} {4}\). Seminar on Number Theory, Paris 1981–82, Progr. Math., vol. 38, Birkhauser, Boston, MA, (1983), 321–343.
R. Weinstock. Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal., 3 (1954), 745–753.
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.
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The author was partially supported by NSF grant DMS-1105323.
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Schoen, R. Existence and geometric structure of metrics on surfaces which extremize eigenvalues. Bull Braz Math Soc, New Series 44, 777–807 (2013). https://doi.org/10.1007/s00574-013-0034-6
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DOI: https://doi.org/10.1007/s00574-013-0034-6