Variants on Alexandrov reflection principle and other applications of maximum principle

Sa Earp, Ricardo; Toubiana, Eric

Sa Earp, Ricardo ; Toubiana, Eric

Séminaire de théorie spectrale et géométrie, Tome 19 (2000-2001), pp. 93-121.

MR Zbl

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     author = {Sa Earp, Ricardo and Toubiana, Eric},
     title = {Variants on {Alexandrov} reflection principle and other applications of maximum principle},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {93--121},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {19},
     year = {2000-2001},
     mrnumber = {1909079},
     zbl = {1011.53005},
     language = {en},
     url = {http://www.numdam.org/item/TSG_2000-2001__19__93_0/}
}

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Sa Earp, Ricardo; Toubiana, Eric. Variants on Alexandrov reflection principle and other applications of maximum principle. Séminaire de théorie spectrale et géométrie, Tome 19 (2000-2001), pp. 93-121. http://www.numdam.org/item/TSG_2000-2001__19__93_0/

Bibliographie
Cité par

[1] U. Abresch. Constant mean curvature tori in terms of elliptic fonctions. J. Reine Ang. Math. 374,169-192 ( 1987) | MR | Zbl

[2] R. Aiyama and K. Akutagawa. Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space, J. Math. Soc. Japan, 52, no.4, 877-898 ( 2000). | MR | Zbl

[3] R. Aiyama and K. Akutagawa. Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3 sphere, Tohoku Math. J. 52, 95-105 ( 2000). | MR | Zbl

[4] A.D. Alexandrov. Uniqueness theorems for surfaces in the large. I, (Russian) Vestnik Leningrad Univ. Math. 11,5-17 ( 1956). | MR | Zbl

[5] T. Aubin Some nonlinear problems in Riemannian geometry. Springer ( 1998). | MR | Zbl

[6] J.L. Barbosa. Constant mean curvature surfaces boundedby a plane curve. Mat. Contemp. 1,3-15 ( 1991). | MR | Zbl

[7] J.L. Barbosa and R. Bérard. Eigenvalue and "twisted" eigenvalues problems, applications to CMC- surfaces. To appear in J. Math. Pures and Appl. | Zbl

[8] J.L. Barbosa, R. Sa Earp. New results on prescribed mean curvature hypersurfacesin Space Forms. An Acad. Bras. Cl. 67, No 1,1-5 ( 1995). | Zbl

[9] J.L. Barbosa and R. Sa Earp. Prescribed mean curvature hypersurfaces in H n+1 (-1 ) with convexplanar boundary, I. Geom. Dedicata, 71, 61-74 ( 1998). | MR | Zbl

[10] J.L. Barbosa and R. Sa Earp. Prescribed mean curvature hypersurfaces in Hn+1 with convex planar boundary, II. Séminaire de théorie spectrale et géométrie de Grenoble 16, 43-79 ( 1998). | EuDML | Numdam | Zbl

[11] P. Bérard and L. Hauswirth. General curvature estimatesfor stable H-surfaces immersed into space form. J. Math. Pures Appl. 78, 667-700 ( 1999). | MR | Zbl

[12] R. Bérard, L. Lima and W. Rossman. Index growth of hypersurfaces with constant mean curvature. To appear in Math. Z. | MR | Zbl

[13] H. Brezis and J.-M. Coron. Multiple solutions of H-systems and Rellich's conjecture. Commun. Pures Appl. Math. XXXVII, 149-187, 1984. | MR | Zbl

[14] A.I. Bobenko. All constant mean curvature tori in R3, S3, H3 in terms of elliptic functions. Math. Ann. 290,209-245 ( 1991). | EuDML | MR | Zbl

[15] F. Braga Brito and R. Sa Earp. Geometrie configurations of constant mean curvature surfaces with planar boundary. An. Acad. Bras. Ci. 63, No 1, 5-19 ( 1991). | MR | Zbl

[16] F. Braga Brito, W. Meeks Iii, H. Rosenberg and R. Sa Earp. Structure theorems for constant mean curvature surfaces bounded by a planar curve. Indiana Univ. Math. J. 40, No 1, 333 343 ( 1991). | MR | Zbl

[17] F. Braga Brito and R. Sa Earp. On the Structure of certain Weingarten surfaces with boundary a circle. An. Fac. Sci. Toulouse VI, No 2, 243-255 ( 1997). | EuDML | Numdam | MR | Zbl

[18] R. Bryant. Surfaces of mean curvature one in hyperbolic space, Asterisque 154-155 , Soc. Math de France, 321-347 ( 1987). | Numdam | MR | Zbl

[19] R. Bryant. Complex analysis and a class of Weingarten surfaces. Preprint.

[20] L. Caffarelli, B. Gidas and J. Spruck. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. XLII, 271-297 ( 1989). | MR | Zbl

[21] L. Caffarelli, L. Nirenberg, J. Spruck. The Dirichlet problemfor nonlinear second-order elliptic equations III. Functions of the eigenvalues of the Hessian. Acta Math. 155, 261-301 ( 1985). | MR | Zbl

[22] L. Caffarelli, L. Nirenberg, J. Spruck. Nonlinear second-order elliptic equations V. The Dirichlet problem for Weingarten surfaces. Comm. Pure Appl. Math. XLI, 47-70 ( 1988). | MR | Zbl

[23] P. Castillon. Sur les sous-variétés à courbure moyenne constante dans l'espace hyerbolique. Doctoral Thesis, Université Joseph Fourier (Grenoble I) ( 1995).

[24] S.S. Chern. On special W-surface. Trans. Amer. Math. Soc. 783-786 ( 1955). | Zbl

[25] R. Collin, Topologie et courbure des surfaces minima les proprement plongées de R3. Ann. Math. 2nd Series 145, 1-31 ( 1997). | MR | Zbl

[26] R. Collin, L. Hauswirth and H. Rosenberg. The geometry of finite topology Bryant surfaces.To appear in Annals of mathematics. | Zbl

[27] S.Y. Cheng and S.-T. Yau. Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. XXVIII, 333-354 ( 1975). | MR | Zbl

[28] M. Dajczer and M. Do Carmo. Helicoidal surfaces with constant mean curvature. Tôhoku Math J. 34 , 425-435 ( 1982). | MR | Zbl

[29] M.P. Do Carmo, H.B. Jr. Lawson On Alexandrov - Bernstein theorems in hyperbolic space. Duke Math. J. 50, No. 4 ( 1983). | MR | Zbl

[30] M. Do Carmo, J. Gomes, G. Thorbergsson. The influence of the boundary behaviour on hypersurfaces with constant mean curvature in Hn+1. Comm. Math. Helvitici 61, 429-491 ( 1986). | EuDML | MR | Zbl

[31] F. Duzaar and K. Steffen. The Plateau problem for parametric surfaces with prescribed mean curvature In: Geometric Analysis and Calculus of Variations (edited by J. Jürgen), pp.13-70, International Press ( 1996). | MR | Zbl

[32] R. Finn. Remarks relevant to minimal surfaces and to surfaces of prescribed mean curvature. J.Analyse Math.14, 139-160 ( 1965). | MR | Zbl

[33] B. Gidas, W-M Ni and L. Nirenberg. Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209-243 ( 1979). | MR | Zbl

[34] D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Springer ( l983). | Zbl

[35] J.M. Gomes. Sobre hipersuperficies comcurvatura média constante no espaço hiperbólico. Doctoral thesis, IMPA ( 1985).

[36] R.D. Gulliver. The Plateau problem for surfaces of prescribed mean curvature in a Riemannian manifold. J. Diff. Geom. 8, 317-330 ( 1972). | MR | Zbl

[37] R.D. Gulliver. Regularity of minimizing surfaces of prescribed mean curvature. Ann. of Math. 97, No. 2, 275-305 ( 1973). | MR | Zbl

[38] P. Hartman, W. Wintner. Umbilical points and W-surfaces. Amer. J. Math. 76, 502-508 ( 1954). | MR | Zbl

[39] E. Hebey. Introduction à l'analyse non linéaire sur les variétés. Diderot Editeur ( 1997). | Zbl

[40] S. Hildebrandt. On the Plateau problem for surfaces of constant mean curvature.Commun. Pure Appl. Math. XXIII, 97-114 ( 1970). | MR | Zbl

[41] D. Hoffman and H. Karcher. Complete embedded surfaces of finite total curvature. Geometry V (R. Osserman, ed.), Springer, 5-93 ( 1997). | MR | Zbl

[42] J. Hounie and M.L. Leite. The maximum principle for hypersurfaces with vanishing curvature functions. J. Diff. Geom. 41, 247-258 ( 1995). | MR | Zbl

[43] J. Hounie and M.L. Leite. Two-ended hypersurfaces with zero scalar curvature. Indiana Univ. Math. J. 48, No. 3, 867-882 ( 1999). | MR | Zbl

[44] D. Hoffman and W. Meeks. The strong halfspace theorem for minimal surfaces. Invent iones Math. 101, 373-377 ( 1990). | EuDML | MR | Zbl

[45] H. Hopf. Differential geometry in the large. Lect. Notes in Math., Springer, 1000 ( 1983). | MR | Zbl

[46] W-Y Hsiang. On generalization of theorems of A. D. Alexandrov and C. Delaunay on hypersurfaces of constant mean curvature. Duke Math. J. 49, No.3 ( 1982). | MR | Zbl

[47] N. Kapouleas. Complete constant mean curvature surfaces in Euclidean three space. Ann. Math. 131, 239-330 ( 1990). | MR | Zbl

[48] N. Kapouleas. Compact constant mean curvature surfaces in Euclidean three-space. J. Diff. Geom. 33, 683-715 ( 1991). | MR | Zbl

[49] K. Kenmotsu. Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245, 89-99 ( 1979). | EuDML | MR | Zbl

[50] N. Korevaar, R. Kusner and B. Solomon. The structure of complete embedded surfaces with constant mean curvature. J. Diff. Geom. 30, 465-503 ( 1989). | MR | Zbl

[51] N. Korevaar, R. Kusner, W.H. Meeks Iii and B. Solomon. Constant mean curvature surfacesin hyperbolic Space. Amer. J. Math. 114, 1-143 ( 1992). | MR | Zbl

[52] M. Koiso, Symmetry of hypersurfaces of constant mean curvature with symmetric boundary. Math. Z. 191,567-574 ( 1986). | MR | Zbl

[53] M. Kokubu. Weierstrass representation for minimal surfaces in hyperbolic space, Tohoku Math. J. (2) 49, no. 3, 367-377 ( 1997). | MR | Zbl

[54] N. Korevaar. Sphere theorems via Alexandrov for constant Weingarten curvature hyper surfaces-appendix to a note of A. Ros. J. Diff. Geom. 27, 221-223 ( 1988). | Zbl

[55] N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen. Refined asymptotics for constant scalar curvature metries with isolated singularities. Invent. Math. 135, 233-272 ( 1999). | MR | Zbl

[56] R. Kusner. Global Geometry of Extremal Surfaces in Three-Space, Doctoral Thesis, University of California, Berkeley ( 1985).

[57] B. Lawson. Complete minimal surfaces in S3. Ann. Math. 92, 335-374 ( 1970). | MR | Zbl

[58] B. Lawson. Lectures on minimal submanifolds. Secon edition, Publish or Perish ( 1980). | Zbl

[59] R. Langevin and H. Rosenberg. A maximum principle at infinity for minimal surfaces and applications. Duke Math. J. 57, 819-828 ( 1988). | MR | Zbl

[60] C. Li Mononicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains. Commun. Part. Diff. Eq. 16, No 2&3, 491-526 ( 1991). | MR | Zbl

[61] C. Li Mononicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains. Commun. Part. Diff. Eq. 16, No 4&5, 585-615 ( 1991). | MR | Zbl

[62] G. Levitt, H. Rosenberg. Symmetry of constant mean curvature hypersurfaces in hyperbolic space. Duke Math. J. 52, No. 1 ( 1985). | MR | Zbl

[63] R. López and S. Montiel, Constant mean curvalure dises with bounded area. Proc. Amer Math. Soc. 123, 1555-1558 ( 1995). | MR | Zbl

[64] F.J. Lopez and A. Ros. On embedded complete minimal surfaces of genus zero. J. Diff. Geom. 33, No 1, 293 300 ( 1991). | MR | Zbl

[65] J. Mccuan. Symmetry via spherical reflection. To appear in J. Diff. Anal. | MR | Zbl

[66] W. Meeks Iii. The topology and geometry of embedded surfaces of constant mean curvature. J. Diff. Geom. 27,539-552 ( 1988). | MR | Zbl

[67] W. Meeks Iii and H. Rosenberg. The maximum principle al infinity for minimal surfaces in flat 3-manifolds. Comm. Math. Helv. 65, 255-270 ( 1990). | MR | Zbl

[68] W. Meeks Iii and H. Rosenberg. The geometry and conformal structure of properly embedded minimal surfaces of finite topology in F3. Invent. Math. 114, 625-639 ( 1993). | MR | Zbl

[69] W. Meeks Iii and S.-T. Yau. The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z.179,151-168 ( 1982). | MR | Zbl

[70] R. Molzon. Symmetry and overdetermined boundary value problems. Forum Math. 3,143-156 ( 1991). | MR | Zbl

[71] C. Jr. Morrey The problem of Plateau on a Riemannian manifold. Ann. of Math. 49, No. 4, 807-851, 1948. | MR | Zbl

[72] B. Nelli, H. Rosenberg. Some remarks on embedded hypersurfaces in Hyperbolic Space of constant mean curvature and spherical boundary. Ann. Glob. An. and Geom. 13, 23-30 ( 1995) | MR | Zbl

[73] B. Nelli and R. Sa Earp. Some Properties of Hypersurfaces of Prescribed Mean Curvature in Hn+1. Bull. Sc. Math. 120. No 6, 537-553 ( 1996). | MR | Zbl

[74] B. Nelli and J. Spruck. On the existence and uniqueness of constant mean curvature hypersurfaces in hyperbolic space.Geometric Analysis and Calculus of Variations, International Press, J. Jost (Ed.), 253-266 ( 1996). | MR | Zbl

[75] J. Ordóňes. Superficies helicoidais com curvatura constante no espaço de formas tridimensionais. Doctoral Thesis, PUC-Rio ( 1995).

[76] U. Pinkall and I. Stirling. On the classification of constant mean curvature tori. Ann. Math. 130, 407-451 ( 1989). | MR | Zbl

[77] M.H. Protter and H. F. Weinberger. Maximum principles in differential equations. Elglewood Cliffs., New Jersey Prentice-Hall ( 1967). | MR | Zbl

[78] J. Ratcliffe. Foundations of hyperbolic manifolds. Springer ( 1999). | MR | Zbl

[79] H. Rosenberg. Hypersurfaces of constant curvature in space forms. Bull. Sc. Math. 2e série 117, 211 -239 ( 1993). | MR | Zbl

[80] H. Rosenberg. Some recent developments in the theory of properly embedded minimal surfaces in R3. Séminaire Bourbaki, 44ème année, No 759 ( 1991-92). | Numdam | Zbl

[81] L. Rodriguez and H. Rosenberg. Half-space theorems for mean curvature one surfaces in hyperbolic space. Proc. Amer. Math. Soc. 126, No 9, 2755-2762 ( 1998). | MR | Zbl

[82] A. Ros and H. Rosenberg. Constant mean curvature surfaces in a half-space with boundary in the boundary of the half-space. J. Diff. Geom. 44, 807-817 ( 1996) | MR | Zbl

[83] H. Rosenberg and R. Sa Earp. Some remarks on surfaces of prescribed mean curvature. Differential Geometry (Symposium in honor of M. do Carmo). Pitman monographs and surveys in Pure and Applied Mathematics, 123-148 ( 1991). | MR | Zbl

[84] H. Rosenberg and R. Sa Earp. Some Structure Theorems for Complete Constant Mean Curvature Surfaces with Boundary a Convex Curve. Proc. Amer. Math. Soc. 113, No 4, 1045-1053 ( 1991). | MR | Zbl

[85] H. Rosenberg and R. Sa Earp. The geometry of properly embedded special surfaces in R3; e.g.surfaces satisfying aH + bK = 1, where a and b are positive. Duke Math. J.73, No 2, 291-306 ( 1994). | MR | Zbl

[86] W. Rossman and K. Sato. Constant mean curvature surfaces in hyperbolic 3-space with two ends. J. Exp.Math., No 1,101-1197 ( 1998). | MR | Zbl

[87] W. Rossman, M. Umehara and K. Yamada. Irreducible constant mean curvature l surfaces in hyperbolic space with positive genus. Tôhoku Math. J., 449-484, 49 ( 1997). | MR | Zbl

[88] R. Sa Earp. Recent developments on the structure of compact surfaces with planar boundary In: The Problem of Plateau (edited by Th.M. Rassias), pp. 245-257, World Scientific ( 1992). | MR | Zbl

[89] R. Sa Earp. On two mean curvature equations in hyperbolic space. In: New Approaches in Nonlinear Analysis (editedby Th.Rassias), Hadronic Press, U.S.A., 171-182 ( 1999). | Zbl

[90] R. Sa Earp and E. Toubiana. Sur les surfaces de Weingarten spéciales de type minimal Boletim da Socie dade Brasileira deMatemática, 26, No. 2, 129-148 ( 1995). | MR | Zbl

[91] R. Sa Earp and E. Toubiana. Classification des Surfaces Speciales de Revolution de Type Delaunay. Amer. J. Math. 121, No 3, 671-700 ( 1999). | MR | Zbl

[92] R. Sa Earp and E. Toubiana. Symmetry of properly embedded special Weingarten surfacesin H3. Trans. Amer. Math. Soc. 352, No 12, 4693-4711 ( 1999). | MR | Zbl

[93] R. Sa Earp and E. Toubiana. Some applications of maximum principle to hypersurfaces in Euclidean and hyperbolic space. In: New Approaches in Nonlinear Analysis (edited by Th.Rassias), Hadronic Press, U.S.A., 183-202 ( 1999). | Zbl

[94] R. Sa Earp and E. Toubiana. Existence and uniqueness of minimal graphs in hyperbolic space. Asian Journal of Mathematics (new Journal), (Editors-in-Chief: S-T Yau (Harvard) e R. Chan (Hong Kong)),4, No. 3, 669-694, International Press ( 2000). | MR | Zbl

[95] R. Sa Earp and E. Toubiana. On the Geometry of Constant Mean Curvature One Surfaces in Hyperbolic Space. Illinois J. Math. 45, No 2 ( 2001). | MR | Zbl

[96] R. Sa Earp and E. Toubiana. Introduction à la géométrie hyperbolique et aux surfaces de Riemann. Diderot Editeur, Paris ( 1997). | Zbl

[97] R. Sa Earp and E. Toubiana. Meromorphic data for mean curvature one surfaces in hyperbolic space. Preprint.

[98] R. Sa Earp and E. Toubiana. A Weierstrass-Kenmotsu formula for prescribed mean curvature surfaces in hyperbolic space. Séminaire de Théorie Spectrale et Géométrie de l'Institut Fourier de Grenoble, 19, 9-23 ( 2001). | EuDML | Numdam | MR | Zbl

[99] R. Sa Earp and E. Toubiana. Meromorphic data for mean curvature one surfaces in hyperbolic space, II. Preprint. | Zbl

[100] B. Semmler. Surfaces de courbure moyenne constante dans les espaces euclidien et hyperbolic, Docto Thesis, Univ. Paris VII, ( 1997).

[101] J. Serrin. A symmetry problem in potential theory. Arch. Rat. Mech. Anal. 43,304-318 ( 1971). | MR | Zbl

[102] R. Schoen. Uniqueness, symmetry, and embeddedness of minimal surfaces. J. Diff. Geom. 18, 791-809 ( 1983). | MR | Zbl

[103] M. Soret Déformations de surfaces minimales. Thèse de Doctorat, Univ. de Paris VII ( 1993).

[104] A.J. Small. Surfaces of constant mean curvature 1 in H3 and algebraic curves on a quadric. Proc. AMS 122 No4, 1211-1220 ( 1994). | MR | Zbl

[105] B. Smyth. The generalization of Delaunay's theorem to constant mean curvature surfaces with conti nuous internai symmetry. Preprint.

[106] M. Spivak. A comprehensive introduction to differential geometry. Publish or Perish Volume IV, second edition ( 1979). | Zbl

[107] K. Steffen. Parametric surfaces of prescribed mean curvature In: Calculus of variations and geometrie evolution problems (edited by S. Hildebrant and M. Struwe), 211-265, Lecture notes in Mathematics 1713, Springer ( 1999). | MR | Zbl

[108] M. Struwe. Large H-surfaces via the Mountain-pass lemma. Math. Ann. 270, 441-459 ( 1985). | MR | Zbl

[109] M. Struwe. Plateau's problem and the calculus of variations. Math. Notes 35, Princeton Univ. Press ( 1989). | Zbl

[110] M. Umehara and K. Yamada. Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space. Annals of Math. 137, 611-638 ( 1993). | MR | Zbl

[111] M. Umehara and K. Yamada. A parametrization of the Weierstrass formulae and perturbation of some minimal surfaces in R3 into the hyperbolic 3-space. J. Reine Angew. Math. 432, 93-116 ( 1992). | MR | Zbl

[112] M. Umehara and K. Yamada. Surfaces of constant mean curvature c in H3(-c2) with prescribed Gauss map. Math. Ann. 304, 203-224 ( 1996). | MR | Zbl

[113] H. Wente. An existence theorem for surfaces of constant mean curvature. Math. Anal. Appl. 26, 318-344 ( 1969). | MR | Zbl

[114] H. Wente. A general existence theorem for surfaces of constant mean curvature. Math. Z. 120, 277-288 ( 1971). | MR | Zbl

[115] H. Wente. Large solutions to the volume constrained Plateau problem. Arch. Mech. Anal. 75, 59-77 ( 1980). | MR | Zbl

[116] H. Wente. A counter-example to the conjecture of H. Hopf. Pacific J. Math. 121,193-243 ( 1986). | Zbl

[117] S.-T Yau. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. XXVIII, 201-228 ( 1975). | MR | Zbl