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@article{TRSPY_2019_305_a15, author = {Alexei V. Penskoi}, title = {Isoperimetric {Inequalities} for {Higher} {Eigenvalues} of the {Laplace--Beltrami} {Operator} on {Surfaces}}, journal = {Informatics and Automation}, pages = {291--308}, publisher = {mathdoc}, volume = {305}, year = {2019}, language = {ru}, url = {http://geodesic.mathdoc.fr/item/TRSPY_2019_305_a15/} }
TY - JOUR AU - Alexei V. Penskoi TI - Isoperimetric Inequalities for Higher Eigenvalues of the Laplace--Beltrami Operator on Surfaces JO - Informatics and Automation PY - 2019 SP - 291 EP - 308 VL - 305 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TRSPY_2019_305_a15/ LA - ru ID - TRSPY_2019_305_a15 ER -
Alexei V. Penskoi. Isoperimetric Inequalities for Higher Eigenvalues of the Laplace--Beltrami Operator on Surfaces. Informatics and Automation, Algebraic topology, combinatorics, and mathematical physics, Tome 305 (2019), pp. 291-308. http://geodesic.mathdoc.fr/item/TRSPY_2019_305_a15/
[1] Anné C., “Spectre du laplacien et écrasement d'anses”, Ann. sci. Éc. norm. supér. Sér. 4, 20:2 (1987), 271–280 | Zbl
[2] Bando S., Urakawa H., “Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds”, Tôhoku Math. J., 35:2 (1983), 155–172 | DOI | Zbl
[3] Barbosa J., “On minimal immersions of $S^2$ into $S^{2m}$”, Trans. Amer. Math. Soc., 210 (1975), 75–106 | Zbl
[4] Berger M., “Sur les premières valeurs propres des variétés riemanniennes”, Compos. math., 26 (1973), 129–149 | Zbl
[5] Bers L., “Local behavior of solutions of general linear elliptic equations”, Commun. Pure Appl. Math., 8 (1955), 473–496 | DOI | Zbl
[6] Besson G., “Sur la multiplicité de la premiere valeur propre des surfaces riemanniennes”, Ann. Inst. Fourier, 30:1 (1980), 109–128 | DOI | Zbl
[7] Bolton J., Woodward L.M., “Higher singularities and the twistor fibration $\pi :\mathbb CP^3\to S^4$”, Geom. dedicata, 80:1–3 (2000), 231–246 | DOI | Zbl
[8] Bolton J., Woodward L.M., “Linearly full harmonic 2-spheres in $S^4$ of area $20\pi $”, Int. J. Math., 12:5 (2001), 535–554 | DOI | Zbl
[9] Bryant R.L., “Conformal and minimal immersions of compact surfaces into the 4-sphere”, J. Diff. Geom., 17:3 (1982), 455–473 | DOI | Zbl
[10] Calabi E., “Minimal immersions of surfaces in Euclidean spheres”, J. Diff. Geom., 1:1–2 (1967), 111–125 | DOI | Zbl
[11] Cianci D., Karpukhin M., Medvedev V., On branched minimal immersions of surfaces by first eigenfunctions, E-print, 2017, arXiv: 1711.05916 [math.SP]
[12] Colbois B., El Soufi A., “Extremal eigenvalues of the Laplacian in a conformal class of metrics: The ‘conformal spectrum’”, Ann. Global Anal. Geom., 24:4 (2003), 337–349 | DOI | Zbl
[13] Courant R., Hilbert D., Methoden der mathematischen Physik, v. 1, 2, Grundl. Math. Wiss., 12, Springer, Berlin, 1931
[14] Eells J., Lemaire L., “A report on harmonic maps”, Bull. London Math. Soc., 10:1 (1978), 1–68 | DOI | Zbl
[15] Eells J., \textup {Jr.}, Sampson J.H., “Harmonic mappings of Riemannian manifolds”, Amer. J. Math., 86 (1964), 109–160 | DOI | Zbl
[16] Ejiri N., “The boundary of the space of full harmonic maps of $S^2$ into $S^{2m}(1)$ and extra eigenfunctions”, Jpn. J. Math, 24:1 (1998), 83–121 | DOI | Zbl
[17] El Soufi A., Giacomini H., Jazar M., “A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle”, Duke Math. J., 135:1 (2006), 181–202 ; arXiv: math/0701773 [math.MG] | DOI | Zbl
[18] El Soufi A., Ilias S., “Le volume conforme et ses applications d'après Li et Yau”, Séminaire de théorie spectrale et géométrie (Chambéry–Grenoble), 1983–1984, v. VII, Univ. Savoie, Fac. Sci., Serv. Math., Chambéry, 1984
[19] El Soufi A., Ilias S., “Immersions minimales, première valeur propre du laplacien et volume conforme”, Math. Ann., 275:2 (1986), 257–267 | DOI | Zbl
[20] El Soufi A., Ilias S., “Riemannian manifolds admitting isometric immersions by their first eigenfunctions”, Pac. J. Math., 195:1 (2000), 91–99 | DOI | Zbl
[21] El Soufi A., Ilias S., “Laplacian eigenvalue functionals and metric deformations on compact manifolds”, J. Geom. Phys., 58:1 (2008), 89–104 ; arXiv: math/0701777 [math.MG] | DOI | Zbl
[22] Gulliver R.D., \textup {II}, Osserman R., Royden H.L., “A theory of branched immersions of surfaces”, Amer. J. Math., 95:4 (1973), 750–812 | DOI | Zbl
[23] Hassannezhad A., “Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem”, J. Funct. Anal., 261:12 (2011), 3419–3436 | DOI | Zbl
[24] Hersch J., “Quatre propriétés isopérimétriques de membranes sphériques homogènes”, C. r. Acad. sci. Paris. Sér. A, 270 (1970), 1645–1648 | Zbl
[25] Jakobson D., Levitin M., Nadirashvili N., Nigam N., Polterovich I., “How large can the first eigenvalue be on a surface of genus two?”, Int. Math. Res. Not., 2005:63 (2005), 3967–3985 ; arXiv: math/0509398 [math.SP] | DOI | Zbl
[26] Jakobson D., Nadirashvili N., Polterovich I., “Extremal metric for the first eigenvalue on a Klein bottle”, Can. J. Math., 58:2 (2006), 381–400 ; arXiv: math/0311484 [math.SP] | DOI | Zbl
[27] M. A. Karpukhin, “Nonmaximality of known extremal metrics on torus and Klein bottle”, Sb. Math., 204:12 (2013), 1728–1744 | DOI | DOI | Zbl
[28] Karpukhin M.A., “Spectral properties of bipolar surfaces to Otsuki tori”, J. Spectr. Theory, 4:1 (2014), 87–111 ; arXiv: 1205.6316 [math.DG] | DOI | Zbl
[29] Karpukhin M., “Spectral properties of a family of minimal tori of revolution in five-dimensional sphere”, Can. Math. Bull., 58:2 (2015), 285–296 ; arXiv: 1301.2483 [math.DG] | DOI | Zbl
[30] Karpukhin M., “Upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces”, Int. Math. Res. Not., 2016:20 (2016), 6200–6209 ; arXiv: 1503.08493 [math.DG] | DOI | Zbl
[31] Karpukhin M., Kokarev G., Polterovich I., “Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces”, Ann. Inst. Fourier, 64:6 (2014), 2481–2502 | DOI | Zbl
[32] Karpukhin M., Nadirashvili N., Penskoi A.V., Polterovich I., An isoperimetric inequality for Laplace eigenvalues on the sphere, E-print, 2017, arXiv: 1706.05713 [math.DG] | Zbl
[33] Kokarev G., “Variational aspects of Laplace eigenvalues on Riemannian surfaces”, Adv. Math., 258 (2014), 191–239 ; arXiv: 1103.2448 [math.SP] | DOI | Zbl
[34] Kokarev G., “On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces”, Anal. PDE, 7:6 (2014), 1397–1420 | DOI | Zbl
[35] Korevaar N., “Upper bounds for eigenvalues of conformal metrics”, J. Diff. Geom., 37:1 (1993), 73–93 | DOI | Zbl
[36] Lapointe H., “Spectral properties of bipolar minimal surfaces in $\mathbb S^4$”, Diff. Geom. Appl., 26:1 (2008), 9–22 ; arXiv: math/0511443 [math.DG] | DOI | Zbl
[37] Li P., Yau S.-T., “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 | DOI | Zbl
[38] Montiel S., Ros A., “Minimal immersions of surfaces by the first eigenfunctions and conformal area”, Invent. math., 83:1 (1986), 153–166 | DOI | Zbl
[39] N. S. Nadirashvili, “Multiple eigenvalues of the Laplace operator”, Math. USSR, Sb., 61:1 (1988), 225–238 | DOI | Zbl
[40] Nadirashvili N., “Berger's isoperimetric problem and minimal immersions of surfaces”, Geom. Funct. Anal., 6:5 (1996), 877–897 | DOI | Zbl
[41] Nadirashvili N., “Isoperimetric inequality for the second eigenvalue of a sphere”, J. Diff. Geom., 61:2 (2002), 335–340 | DOI | Zbl
[42] Nadirashvili N.S., Penskoi A.V., “An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane”, Geom. Funct. Anal., 28:5 (2018), 1368–1393 ; arXiv: 1608.07334 [math.DG] | DOI | Zbl
[43] Nadirashvili N., Sire Y., “Maximization of higher order eigenvalues and applications”, Moscow Math. J., 15:4 (2015), 767–775 ; arXiv: 1504.07465 [math.DG] | DOI | Zbl
[44] Nadirashvili N., Sire Y., “Isoperimetric inequality for the third eigenvalue of the Laplace–Beltrami operator on $\mathbb S^2$”, J. Diff. Geom., 107:3 (2017), 561–571 ; arXiv: 1506.07017 [math.DG] | DOI | Zbl
[45] Nayatani S., Shoda T., Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian, E-print, 2017, arXiv: 1704.06384 [math.DG] | Zbl
[46] Penskoi A.V., “Extremal spectral properties of Lawson tau-surfaces and the Lamé equation”, Moscow Math. J., 12:1 (2012), 173–192 ; arXiv: 1009.0285 [math.SP] | DOI | Zbl
[47] Penskoi A.V., “Extremal spectral properties of Otsuki tori”, Math. Nachr., 286:4 (2013), 379–391 ; arXiv: 1108.5160 [math.SP] | DOI | Zbl
[48] A. V. Penskoi, “Extremal metrics for eigenvalues of the Laplace–Beltrami operator on surfaces”, Russ. Math. Surv., 68:6 (2013), 1073–1130 | DOI | DOI | Zbl
[49] Penskoi A.V., “Generalized Lawson tori and Klein bottles”, J. Geom. Anal., 25:4 (2015), 2645–2666 ; arXiv: 1308.1628 [math.DG] | DOI | Zbl
[50] Petrides R., “Maximization of the second conformal eigenvalue of spheres”, Proc. Amer. Math. Soc., 142:7 (2014), 2385–2394 ; arXiv: 1206.0229 [math.SP] | DOI | Zbl
[51] Petrides R., “On the existence of metrics which maximize Laplace eigenvalues on surfaces”, Int. Math. Res. Not., 2018:14 (2018), 4261–4355 | DOI | Zbl
[52] Sher D.A., “Conic degeneration and the determinant of the Laplacian”, J. anal. math., 126 (2015), 175–226 ; arXiv: 1208.1809 [math.AP] | DOI | Zbl
[53] Takahashi T., “Minimal immersions of Riemannian manifolds”, J. Math. Soc. Jpn., 18:4 (1966), 380–385 | DOI | Zbl
[54] Yang P.C., Yau S.-T., “Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds”, Ann. Sc. Norm. Super. Pisa. Cl. Sci., 7:1 (1980), 55–63