Nous passons en revue certains résultats récents sur l’existence et l’unicité des sphères à courbure moyenne constante dans les variétés riemanniennes homogènes simplement connexes de dimension et leurs liens avec le problème isopérimétrique dans ces variétés.
This is a survey on some recent results about the existence and the uniqueness of constant mean curvature spheres in simply connected homogeneous Riemannian 3-manifolds and their relation to the isoperimetric problem in these manifolds.
Mot clés : Courbure moyenne, variété riemannienne homogène, problème isopérimétrique, théorème de Hopf, théorème d’Alexandrov
Mots clés : Mean curvature, homogeneous Riemannian manifold, isoperimetric problem, Hopf theorem, Alexandrov theorem
@article{TSG_2009-2010__28__13_0, author = {Daniel, Beno{\^\i}t}, title = {Sph\`eres \`a courbure moyenne constante et probl\`eme isop\'erim\'etrique dans les vari\'et\'es homog\`enes}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {13--27}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {28}, year = {2009-2010}, doi = {10.5802/tsg.276}, language = {fr}, url = {http://archive.numdam.org/articles/10.5802/tsg.276/} }
TY - JOUR AU - Daniel, Benoît TI - Sphères à courbure moyenne constante et problème isopérimétrique dans les variétés homogènes JO - Séminaire de théorie spectrale et géométrie PY - 2009-2010 SP - 13 EP - 27 VL - 28 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/tsg.276/ DO - 10.5802/tsg.276 LA - fr ID - TSG_2009-2010__28__13_0 ER -
%0 Journal Article %A Daniel, Benoît %T Sphères à courbure moyenne constante et problème isopérimétrique dans les variétés homogènes %J Séminaire de théorie spectrale et géométrie %D 2009-2010 %P 13-27 %V 28 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/tsg.276/ %R 10.5802/tsg.276 %G fr %F TSG_2009-2010__28__13_0
Daniel, Benoît. Sphères à courbure moyenne constante et problème isopérimétrique dans les variétés homogènes. Séminaire de théorie spectrale et géométrie, Tome 28 (2009-2010), pp. 13-27. doi : 10.5802/tsg.276. http://archive.numdam.org/articles/10.5802/tsg.276/
[1] A Hopf differential for constant mean curvature surfaces in and , Acta Math., Volume 193 (2004) no. 2, pp. 141-174 | MR | Zbl
[2] Generalized Hopf differentials, Mat. Contemp., Volume 28 (2005), pp. 1-28 | MR | Zbl
[3] A theorem of Hopf and the Cauchy-Riemann inequality, Comm. Anal. Geom., Volume 15 (2007) no. 2, pp. 283-298 | MR | Zbl
[4] A characteristic property of spheres, Ann. Mat. Pura Appl. (4), Volume 58 (1962), pp. 303-315 | MR | Zbl
[5] Stability of hypersurfaces with constant mean curvature, Math. Z., Volume 185 (1984) no. 3, pp. 339-353 | MR | Zbl
[6] Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z., Volume 197 (1988) no. 1, pp. 123-138 | MR | Zbl
[7] Geometric structures on 3-manifolds, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 93-164 | MR | Zbl
[8] A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier (Grenoble), Volume 51 (2001) no. 6, pp. 1763-1827 | MR | Zbl
[9] Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv., Volume 82 (2007) no. 1, pp. 87-131 | MR | Zbl
[10] Constant mean curvature surfaces in homogeneous manifolds (2009) (Livre en préparation. Version préliminaire publiée par le Korea Institute for Advanced Study)
[11] Existence and uniqueness of constant mean curvature spheres in (2008) (Prépublication, arXiv :0812.3059)
[12] On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J., Volume 32 (1983) no. 1, pp. 25-37 | MR | Zbl
[13] Differential geometry in the large, Lecture Notes in Mathematics, 1000, Springer-Verlag, Berlin, 1989 (Notes taken by Peter Lax and John W. Gray, With a preface by S. S. Chern, With a preface by K. Voss) | MR | Zbl
[14] On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I, Invent. Math., Volume 98 (1989) no. 1, pp. 39-58 | MR | Zbl
[15] Growth of connected locally compact groups, J. Functional Analysis, Volume 12 (1973), pp. 113-127 | MR | Zbl
[16] Constant mean curvature surfaces constructed by fusing Wente tori, Invent. Math., Volume 119 (1995) no. 3, pp. 443-518 | MR | Zbl
[17] Constant mean curvature spheres in (2010) (Prépublication)
[18] Curvatures of left invariant metrics on Lie groups, Advances in Math., Volume 21 (1976) no. 3, pp. 293-329 | MR | Zbl
[19] Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc., Volume 355 (2003) no. 12, p. 5041-5052 (electronic) | MR | Zbl
[20] Geometric measure theory, Elsevier/Academic Press, Amsterdam, 2009 (A beginner’s guide) | MR | Zbl
[21] The isoperimetric problem in spherical cylinders, Ann. Global Anal. Geom., Volume 26 (2004) no. 4, pp. 333-354 | MR | Zbl
[22] Uma introdução à simetrização em análise e geometria, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2005 (25o Colóquio Brasileiro de Matemática. [25th Brazilian Mathematics Colloquium]) | MR
[23] Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J., Volume 48 (1999) no. 4, pp. 1357-1394 | MR | Zbl
[24] The isoperimetric profile of homogeneous Riemannian manifolds, J. Differential Geom., Volume 54 (2000) no. 2, pp. 255-302 | MR | Zbl
[25] Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., Volume 26 (1977) no. 3, pp. 459-472 | MR | Zbl
[26] The isoperimetric problem, Global theory of minimal surfaces (Clay Math. Proc.), Volume 2, Amer. Math. Soc., Providence, RI, 2005, pp. 175-209 | MR | Zbl
[27] Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionenzahl, Math. Z., Volume 49 (1943), pp. 1-109 | MR | Zbl
[28] The geometries of -manifolds, Bull. London Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | MR | Zbl
[29] On stable constant mean curvature surfaces in and , Trans. Amer. Math. Soc., Volume 362 (2010) no. 6, pp. 2845-2857 | MR | Zbl
[30] Compact stable constant mean curvature surfaces in the Berger spheres (2009) (Prépublication, arXiv :0906.1439)
[31] Counterexample to a conjecture of H. Hopf, Pacific J. Math., Volume 121 (1986) no. 1, pp. 193-243 | MR | Zbl
Cité par Sources :