An Obata singular theorem for stratified spaces

Mondello, Ilaria

doi:10.1090/tran/7105

An Obata singular theorem for stratified spaces
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Trans. Amer. Math. Soc. 370 (2018), 4147-4175 Request permission

Abstract:

Consider a stratified space with a positive Ricci lower bound on the regular set and no cone angle larger than $2\pi$. For such stratified space we know that the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension. We prove here an Obata rigidity result when the equality is attained: the lower bound of the spectrum is attained if and only if the stratified space is isometric to a spherical suspension. Moreover, we show that the diameter is at most equal to $\pi$, and it is equivalent for the diameter to be equal to $\pi$ and for the first non-zero eigenvalue of the Laplacian to be equal to the dimension. We finally give a consequence of these results related to the Yamabe problem. Consider an Einstein stratified space without cone angles larger than $2\pi$: if there is a metric conformal to the Einstein metric and with constant scalar curvature, then it is an Einstein metric as well. Furthermore, if its conformal factor is not a constant, then the space is isometric to a spherical suspension.

References

Kazuo Akutagawa, Gilles Carron, and Rafe Mazzeo, The Yamabe problem on stratified spaces, Geom. Funct. Anal. 24 (2014), no. 4, 1039–1079. MR 3248479, DOI 10.1007/s00039-014-0298-z
Kazuo Akutagawa, Gilles Carron, and Rafe Mazzeo, Hölder regularity of solutions for Schrödinger operators on stratified spaces, J. Funct. Anal. 269 (2015), no. 3, 815–840. MR 3350731, DOI 10.1016/j.jfa.2015.02.003
Pierre Albin, Éric Leichtnam, Rafe Mazzeo, and Paolo Piazza, The signature package on Witt spaces, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 2, 241–310 (English, with English and French summaries). MR 2977620, DOI 10.24033/asens.2165
D. Bakry and M. Ledoux, Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator, Duke Math. J. 85 (1996), no. 1, 253–270. MR 1412446, DOI 10.1215/S0012-7094-96-08511-7
Dominique Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes, Lectures on probability theory (Saint-Flour, 1992) Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, pp. 1–114 (French). MR 1307413, DOI 10.1007/BFb0073872
Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
Vincent Bour and Gilles Carron, Optimal integral pinching results, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 41–70 (English, with English and French summaries). MR 3335838, DOI 10.24033/asens.2238
Jean-Pierre Bourguignon and Jean-Pierre Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc. 301 (1987), no. 2, 723–736. MR 882712, DOI 10.1090/S0002-9947-1987-0882712-7
Nicola Gigli, An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature, Anal. Geom. Metr. Spaces 2 (2014), no. 1, 169–213. MR 3210895, DOI 10.2478/agms-2014-0006
Saïd Ilias, Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 151–165 (French). MR 699492
Christian Ketterer, Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl. (9) 103 (2015), no. 5, 1228–1275 (English, with English and French summaries). MR 3333056, DOI 10.1016/j.matpur.2014.10.011
Christian Ketterer, Obata’s rigidity theorem for metric measure spaces, Anal. Geom. Metr. Spaces 3 (2015), no. 1, 278–295. MR 3403434, DOI 10.1515/agms-2015-0016
John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880, DOI 10.1090/S0273-0979-1987-15514-5
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
Ilaria Mondello, The local Yamabe constant of Einstein stratified spaces, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 1, 249–275. MR 3592686, DOI 10.1016/j.anihpc.2015.12.001
Ilaria Mondello, The Yamabe problem on stratified spaces, HAL Id : tel-01204671 (2015).
Sebastián Montiel, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ. Math. J. 48 (1999), no. 2, 711–748. MR 1722814, DOI 10.1512/iumj.1999.48.1562
Morio Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340. MR 142086, DOI 10.2969/jmsj/01430333
Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258. MR 303464
Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR 994021, DOI 10.1007/BFb0089180
Yoshihiro Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275. MR 174022, DOI 10.1090/S0002-9947-1965-0174022-6
Jeff A. Viaclovsky, Monopole metrics and the orbifold Yamabe problem, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 7, 2503–2543 (2011) (English, with English and French summaries). MR 2866998