# ANNALES DE L'INSTITUT FOURIER

The Plateau problem for convex curvature functions
Annales de l'Institut Fourier to appear, , 66 p.

We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for LSC hypersurfaces of constant or prescribed curvature for general curvature functions inside general Hadamard manifolds modulo a single scalar condition. In particular, convex curvature functions of bounded type are fully treated.

Nous étudions le problème de Plateau paramétrique dans des variétés riemanniennes générales pour des hypersurfaces localement strictement convexes (LSC) et à courbure prescrite pour une classe générale de fonctions de courbure convexes. Nous établissons une condition scalaire pour l’existence de solutions dans le cas où il existe une barrière externe et la variété ambiante est une variété d’Hadamard

Accepted : 2018-02-05
Classification:  58E12,  35J25,  35J60,  53C21,  53C42
Keywords: Plateau problem, non-linear elliptic PDEs
@unpublished{AIF_0__0_0_A1_0,
author = {Smith, Graham},
title = {The Plateau problem for convex curvature functions},
note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Smith, Graham. The Plateau problem for convex curvature functions. Annales de l'Institut Fourier, to appear, 66 p.

[1] Caffarelli, Luis; Kohn, Joseph J.; Nirenberg, Louis; Spruck, Joel The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge–Ampère, and uniformly elliptic, equations, Commun. Pure Appl. Math., Tome 38 (1985) no. 2, pp. 209-252 | Article | MR 780073 | Zbl 0598.35048

[2] Caffarelli, Luis; Nirenberg, Louis; Spruck, Joel The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge–Ampère equation, Commun. Pure Appl. Math., Tome 37 (1984) no. 3, pp. 369-402 | Article | MR 3793362

[3] Caffarelli, Luis; Nirenberg, Louis; Spruck, Joel The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math., Tome 155 (1985) no. 3-4, pp. 261-301 | Article | MR 806416

[4] Caffarelli, Luis; Nirenberg, Louis; Spruck, Joel Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces, Commun. Pure Appl. Math., Tome 41 (1988) no. 1, pp. 47-70 | Article | MR 917124

[5] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Springer, Classics in Mathematics (2001), xiv+517 pages (Reprint of the 1998 edition) | Article | MR 1814364

[6] Guan, Bo; Spruck, Joel Boundary-value problems on ${S}^{n}$ for surfaces of constant Gauss curvature, Ann. Math., Tome 138 (1993) no. 3, pp. 601-624 | Article | MR 1247995

[7] Guan, Bo; Spruck, Joel The existence of hypersurfaces of constant Gauss curvature with prescribed boundary, J. Differ. Geom., Tome 62 (2002) no. 2, pp. 259-287 http://projecteuclid.org/euclid.jdg/1090950194 | Article | MR 1988505

[8] Guan, Bo; Spruck, Joel Locally convex hypersurfaces of constant curvature with boundary, Commun. Pure Appl. Math., Tome 57 (2004) no. 10, pp. 1311-1331 | Article | MR 2069725

[9] Guan, Bo; Spruck, Joel Convex hypersurfaces of constant curvature in hyperbolic space, Surveys in geometric analysis and relativity, International Press. (Advanced Lectures in Mathematics (ALM)) Tome 20 (2011), pp. 241-257 | Article | MR 2906928 | Zbl 0840.14027

[10] Ivočkina, Nina M. A priori estimate of ${|u|}_{{C}_{2}\left(\overline{\Omega }\right)}$ of convex solutions of the Dirichlet problem for the Monge–Ampère equation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., Tome 96 (1980), pp. 69-79 | MR 579472 | Zbl 0472.35040

[11] Labourie, François Un lemme de Morse pour les surfaces convexes, Invent. Math., Tome 141 (2000) no. 2, pp. 239-297 | Article | MR 1775215

[12] Rosenberg, Harold; Smith, Graham Degree Theory of Immersed Hypersurfaces (2010) (https://arxiv.org/abs/1010.1879) | Article

[13] Sheng, Weimin; Urbas, John; Wang, Xu-Jia Interior curvature bounds for a class of curvature equations, Duke Math. J., Tome 123 (2004) no. 2, pp. 235-264 | Article | MR 2066938

[14] Smith, Graham The non-linear Dirichlet problem in Hadamard manifolds (http://www.crm.cat/en/Publications/Publications/2009/Preprints/Pr870.pdf) | Article | Zbl 0765.14023

[15] Smith, Graham An Arzela–Ascoli theorem for immersed submanifolds, Ann. Fac. Sci. Toulouse, Math., Tome 16 (2007) no. 4, pp. 817-866 | Article | MR 2789720

[16] Smith, Graham Moduli of flat conformal structures of hyperbolic type, Geom. Dedicata, Tome 154 (2011), pp. 47-80 | Article | MR 2832711

[17] Smith, Graham Compactness results for immersions of prescribed Gaussian curvature I. Analytic aspects, Adv. Math., Tome 229 (2012) no. 2, pp. 731-769 | Article | MR 2855077

[18] Smith, Graham The non-linear Plateau problem in non-positively curved manifolds, Trans. Am. Math. Soc., Tome 365 (2013) no. 3, pp. 1109-1124 | Article | MR 3003259

[19] Smith, Graham Special Lagrangian curvature, Math. Ann., Tome 355 (2013) no. 1, pp. 57-95 | Article | MR 3004576

[20] Smith, Graham Compactness for immersions of prescribed Gaussian curvature II: geometric aspects, Geom. Dedicata, Tome 172 (2014), pp. 303-350 | Article | MR 3253784

[21] Smith, Graham Global singularity theory for the Gauss curvature equation, Sociedade Brasileira de Matemática, Ensaios Matemáticos, Tome 28 (2015), 114 pages | MR 3363142 | Zbl 0132.41301

[22] Spruck, Joel Fully nonlinear elliptic equations and applications to geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser (1995), pp. 1145-1152 | MR 1404014

[23] Trudinger, Neil S. On the Dirichlet problem for Hessian equations, Acta Math., Tome 175 (1995) no. 2, pp. 151-164 | Article | MR 1368245

[24] Trudinger, Neil S.; Wang, Xu-Jia On locally convex hypersurfaces with boundary, J. Reine Angew. Math., Tome 551 (2002), pp. 11-32 | Article | MR 1932171

[25] White, Brian The space of $m$-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J., Tome 36 (1987) no. 3, pp. 567-602 | Article | MR 905611

[26] White, Brian The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J., Tome 40 (1991) no. 1, pp. 161-200 | Article | MR 1101226