The local Yamabe constant of Einstein stratified spaces

Mondello, Ilaria

doi:10.1016/j.anihpc.2015.12.001

Mondello, Ilaria

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 249-275.

Résumé

On a compact stratified space $(X, g)$ , a metric of constant scalar curvature exists in the conformal class of g if the scalar curvature $S_{g}$ satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the local Yamabe constant $Y_{ℓ} (X)$ . This latter is a conformal invariant introduced in the recent work of K. Akutagawa, G. Carron and R. Mazzeo. It depends on the local structure of X, in particular on its links, but its explicit value is unknown. We show that if the links satisfy a Ricci positive lower bound, then we can compute $Y_{ℓ} (X)$ . In order to achieve this, we prove a lower bound for the spectrum of the Laplacian, by extending a well-known theorem by A. Lichnerowicz, and a Sobolev inequality, inspired by a result due to D. Bakry. A particular stratified space, with one stratum of codimension 2 and cone angle bigger than 2π, must be handled separately – in this case we prove the existence of an Euclidean isoperimetric inequality.

MR Zbl

DOI : 10.1016/j.anihpc.2015.12.001

Mots clés : Geometric analysis, Stratified spaces, Yamabe constant, Sobolev inequality, Eigenvalues of the Laplacian, Isoperimetric profiles

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     title = {The local {Yamabe} constant of {Einstein} stratified spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     publisher = {Elsevier},
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     year = {2017},
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     zbl = {1355.53027},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2015.12.001/}
}

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Mondello, Ilaria. The local Yamabe constant of Einstein stratified spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 249-275. doi : 10.1016/j.anihpc.2015.12.001. http://www.numdam.org/articles/10.1016/j.anihpc.2015.12.001/

Bibliographie
Cité par

[1] Akutagawa, K.; Botvinnik, B. Yamabe metrics on cylindrical manifolds, Geom. Funct. Anal., Volume 13 (2003) no. 2, pp. 259–333 | DOI | MR | Zbl

[2] Akutagawa, K.; Carron, G.; Mazzeo, R. The Yamabe problem on stratified spaces, Geom. Funct. Anal., Volume 24 (2014) no. 4, pp. 1039–1079 | DOI | MR | Zbl

[3] Akutagawa, K.; Carron, G.; Mazzeo, R. Hölder regularity of solutions for Schrödinger operators on stratified spaces, J. Funct. Anal., Volume 269 (2015), pp. 815–840 | DOI | MR | Zbl

[4] Albin, P.; Leichtnam, É.; Mazzeo, R.; Piazza, P. The signature package on Witt spaces, Ann. Sci. Éc. Norm. Super. (4), Volume 45 (2012) no. 2, pp. 241–310 | Numdam | MR | Zbl

[5] Aubin, T. Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9), Volume 55 (1976) no. 3, pp. 269–296 | MR | Zbl

[6] Bacher, K.; Sturm, K.-T. Ricci bounds for Euclidean and spherical cones, Singular Phenomena and Scaling in Mathematical Models, Springer, Cham, 2014, pp. 3–23 | DOI | MR | Zbl

[7] Bakry, D. Lectures on Probability Theory, Lecture Notes in Math., Volume vol. 1581, Springer, Berlin (1994), pp. 1–114 (Saint-Flour, 1992) | DOI | MR | Zbl

[8] Besse, A.L. Einstein Manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008 (reprint of the 1987 edition) | MR | Zbl

[9] Bour, V. Flots géométriques d'ordre quatre et pincement intégral de la courbure, 2012 http://tel.archives-ouvertes.fr/tel-00771720

[10] Burago, D.; Burago, Y.; Ivanov, S. A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001 | DOI | MR | Zbl

[11] Cheeger, J. Spectral geometry of singular Riemannian spaces, J. Differ. Geom., Volume 18 (1983) no. 4, pp. 575–657 | DOI | MR | Zbl

[12] Croke, C.B. Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Éc. Norm. Super. (4), Volume 13 (1980) no. 4, pp. 419–435 | Numdam | MR | Zbl

[13] Gallot, S. Équations différentielles caractéristiques de la sphère, Ann. Sci. Éc. Norm. Super. (4), Volume 12 (1979) no. 2, pp. 235–267 | Numdam | MR | Zbl

[14] Goresky, M.; MacPherson, R. Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Results in Mathematics and Related Areas (3), vol. 14, Springer-Verlag, Berlin, 1988 | MR | Zbl

[15] Kleiner, B. An isoperimetric comparison theorem, Invent. Math., Volume 108 (1992) no. 1, pp. 37–47 | DOI | MR | Zbl

[16] Mather, J.N. Dynamical Systems, Proc. Sympos., Academic Press, New York (1973), pp. 195–232 (Univ. Bahia, Salvador, 1971) | MR | Zbl

[17] Obata, M. Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Jpn., Volume 14 (1962), pp. 333–340 | DOI | MR | Zbl

[18] Petean, J. Isoperimetric regions in spherical cones and Yamabe constants of $M \times S^{1}$ , Geom. Dedic., Volume 143 (2009), pp. 37–48 | DOI | MR | Zbl

[19] Petersen, P. Riemannian Geometry, Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006 | MR | Zbl

[20] Ritoré, M. Optimal isoperimetric inequalities for three-dimensional Cartan–Hadamard manifolds, Global Theory of Minimal Surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 395–404 | MR | Zbl

[21] Ros, A. The isoperimetric problem, Global Theory of Minimal Surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 175–209 | MR | Zbl

[22] Schoen, R. Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differ. Geom., Volume 20 (1984) no. 2, pp. 479–495 | DOI | MR | Zbl

[23] Schoen, R.; Yau, S.T. On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys., Volume 65 (1979) no. 1, pp. 45–76 | DOI | MR | Zbl

[24] Schoen, R.; Yau, S.T. Proof of the positive mass theorem. II, Commun. Math. Phys., Volume 79 (1981) no. 2, pp. 231–260 | DOI | MR | Zbl

[25] Schoen, R.; Yau, S.-T. Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math., Volume 92 (1988) no. 1, pp. 47–71 | DOI | MR | Zbl

[26] Talenti, G. Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), Volume 110 (1976), pp. 353–372 | DOI | MR | Zbl

[27] Thom, R. Ensembles et morphismes stratifiés, Bull. Am. Math. Soc., Volume 75 (1969), pp. 240–284 | DOI | MR | Zbl

[28] Trudinger, N.S. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa, Volume 3 (1968) no. 22, pp. 265–274 | Numdam | MR | Zbl

[29] Weil, A. Sur les surfaces à courbure négative, C. R. Acad. Sci. Paris, Volume 182 (1926) no. 4, pp. 1069–1071 | JFM

[30] Whitney, H. Complexes of manifolds, Proc. Natl. Acad. Sci. USA, Volume 33 (1947), pp. 10–11 | DOI | MR | Zbl

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