Pincement de la première valeur propre du laplacien pour les hypersurfaces et rigidité
Séminaire de théorie spectrale et géométrie, Tome 26 (2007-2008) , pp. 123-138.

Robert C. Reilly a obtenu des majorations de la première valeur propre du laplacien pour les hypersurfaces de l’espace euclidien. De plus, il a montré que le cas d’égalité dans ces majorations est atteint uniquement pour les sphères géodésiques. Dans cet exposé, nous nous intéressons au problème de pincement pour ces majorations. Nous montrons que si le cas d’égalité est presque atteint, alors l’hypersurface est proche d’une sphère, en un sens que nous préciserons. Nous déduisons ensuite des résultats pour les hypersurfaces presque ombiliques ainsi qu’une nouvelle caractérisation des sphères géodésiques.

Robert C. Reilly obtained upper bounds for the first eigenvalue of the Laplacian for hypersurfaces of Euclidean space. He also showed that the equality case of these upper bounds is attained only for geodesic spheres. In this talk, we are interested in the pinching problem for these inequalities. We show that if the equality is almost attained, then the hypersurface is close to a sphere. Then, we deduce results for almost umbilical hypersurfaces and a new characterization of geodesic spheres.

DOI : https://doi.org/10.5802/tsg.263
Classification : 53A07,  53C20,  53C21,  58C40
Mots clés : hypersurfaces, laplacien, valeurs propres, pincement, rigidité, courbure moyenne constante 
@article{TSG_2007-2008__26__123_0,
     author = {Julien Roth},
     title = {Pincement de la premi\`ere valeur propre du laplacien pour les hypersurfaces et rigidit\'e},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {123--138},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {26},
     year = {2007-2008},
     doi = {10.5802/tsg.263},
     zbl = {1187.58010},
     mrnumber = {2654600},
     language = {fr},
     url = {https://tsg.centre-mersenne.org/item/TSG_2007-2008__26__123_0/}
}
Julien Roth. Pincement de la première valeur propre du laplacien pour les hypersurfaces et rigidité. Séminaire de théorie spectrale et géométrie, Tome 26 (2007-2008) , pp. 123-138. doi : 10.5802/tsg.263. https://tsg.centre-mersenne.org/item/TSG_2007-2008__26__123_0/

[1] A.D. Alexandrov Uniqueness theorems for the surfaces in the large I, Vesnik Leningrad Univ., Volume 11 (1956), pp. 5-17 | MR 86338

[2] E. Aubry Diameter pinching in almost positive Ricci curvature (To appear) | Zbl 1167.53033

[3] B. Colbois; J.F. Grosjean A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space, Comment. Math. Helv., Volume 82 (2007), pp. 175-195 | MR 2296061 | Zbl 1112.53003

[4] A. Fialkow Hypersurfaces of a space of constant curvature, Ann. of Math., Volume 39 (1938), pp. 762-783 | MR 1503435 | Zbl 0020.06601

[5] J.-F. Grosjean Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces (math. DG/0709.0831)

[6] J.F. Grosjean Upper bounds for the first eigenvalue of the Laplacian on compact manifolds, Pac. J. Math., Volume 206 (2002) no. 1, pp. 93-111 | MR 1924820 | Zbl 1050.58024

[7] H. Hopf Differential Geometry in the large, Lecture Notes in Mathematics, Volume 1000, Springer-Verlag, Berlin, 1983 | MR 707850 | Zbl 0526.53002

[8] W.J. Hsiang; Z.U. Teng; W.C. Yu New examples of constant mean curvature immersions of (2k-1)-spheres into Euclidean 2k-space, Ann. Math. (2), Volume 117 (1983) no. 3, pp. 609-625 | MR 701257 | Zbl 0522.53052

[9] M. Jleli; F. Pacard An end-to-end construction for compact constant mean curvature surfaces, Pac. J. Math., Volume 221 (2005) no. 1, pp. 81-108 | MR 2194146 | Zbl 1110.53043

[10] S. Montiel; A. Ros Compact hypersurfaces : the Alexandrov theorem for higher order mean curvature, Pitman Monographs Surveys Pure Appl. Math., Volume 52 (1991), pp. 279-296 (in honor of M.P. Do Carmo ; edited by B. Lawson and K. Tenenblat) | MR 1173047 | Zbl 0723.53032

[11] R.C. Reilly On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv., Volume 52 (1977), pp. 525-533 | EuDML 139710 | MR 482597 | Zbl 0382.53038

[12] A. Ros Compact hypersurfaces with constant higher order mean curvatures, Revista Mat. Iberoamer., Volume 3 (1987), pp. 477-483 | MR 996826 | Zbl 0673.53003

[13] J. Roth Sphere Rigidity in the Euclidean space (à paraître dans Proceedings of the VIII International Colloquium in Differential Geometry, Santiago de Compostela, 2009, World Scientific)

[14] J. Roth Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of Euclidean space, Ann. Glob. Anal. Geom., Volume 33 (2008) no. 3, pp. 293-306 | MR 2390836 | Zbl 1151.53010

[15] T.Y. Thomas On closed space of constant mean curvature, Amer. J. Math., Volume 58 (1936), pp. 702-704 | MR 1507192

[16] H. Wente Counterexample to a conjecture of H. Hopf, Pac. J. Math., Volume 121 (1986) no. 1, pp. 193-243 | MR 815044 | Zbl 0586.53003