Minimal Graphs in ${ℍ}^n\times ℝ$ and ${ℝ}^{n+1}$

Sà Earp, Ricardo; Toubiana, Eric

Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2373-2402.

Nous construisons des barrières géométriques dans ${ℍ}^n\times ℝ.$

Nous prouvons l’existence et l’unicité d’une solution de l’équation du graphe vertical minimal sur l’intérieur d’un polyhèdre convexe de ${ℍ}^n$ qui se prolonge sur l’intérieur de chaque face, prenant la valeur infinie sur une face et la valeur zéro sur les autres faces.

Dans ${ℍ}^n\times ℝ$, nous résolvons le problème de Dirichlet pour l’équation du graphe vertical minimal sur un domaine $C^0$ convexe $\Omega \subset {ℍ}^n$ prenant des données continues arbitraires sur le bord fini et le bord asymptotique de $\Omega$.

Nous prouvons l’existence d’une autre hypersurface de type Scherk, donnée par la solution de l’équation du graphe vertical minimal sur l’intérieur d’un certain polyhèdre admissible prenant alternativement les valeurs $+\infty$ et $-\infty$ sur les faces adjacentes.

Nous établissons des resultats analogues pour des graphes minimaux dans ${ℝ}^{n+1}$.

We construct geometric barriers for minimal graphs in ${ℍ}^n\times ℝ.$

We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in ${ℍ}^n$ extending continuously to the interior of each face, taking infinite boundary data on one face and zero boundary value data on the other faces.

In ${ℍ}^n\times ℝ$, we solve the Dirichlet problem for the vertical minimal equation in a $C^0$ convex domain $\Omega \subset {ℍ}^n$ taking arbitrarily continuous finite boundary and asymptotic boundary data.

We prove the existence of another Scherk type hypersurface, given by the solution of the vertical minimal equation in the interior of certain admissible polyhedron taking alternatively infinite values $+\infty$ and $-\infty$ on adjacent faces of this polyhedron.

We establish analogous results for minimal graphs when the ambient is the Euclidean space ${ℝ}^{n+1}$.

Classification : 53C42, 35J25
Keywords: Dirichlet problem, minimal equation, vertical graph, Perron process, barrier, convex domain, asymptotic boundary, translation hypersurface, Scherk hypersurface
Mot clés : problème de Dirichlet, équation de surface minimale, graphe vertical, processus de Perron, barrière, domaine convexe, bord asymptotique, hypersurface de translation, hypersurface de Scherk

[1] Anderson, Michael T. Complete minimal varieties in hyperbolic space, Invent. Math., Volume 69 (1982) no. 3, pp. 477-494 | DOI | MR | Zbl

[2] Anderson, Michael T. Complete minimal hypersurfaces in hyperbolic $n$-manifolds, Comment. Math. Helv., Volume 58 (1983) no. 2, pp. 264-290 | DOI | MR | Zbl

[3] Bérard, Pierre; Earp, R. Sa Minimal hypersurfaces in ${ℍ}^n\times ℝ$, total curvature and index (ArXiv: 0808.3838v1)

[4] Courant, R.; Hilbert, D. Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons Inc., New York, 1989 (Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience Publication) | MR | Zbl

[5] Coutant, A. Hypersurfaces de type Scherk (Univ. Paris 12)

[6] Earp, R. Sa Parabolic and Hyperbolic Screw motion in ${ℍ}^2\times ℝ$, Journ. Austra. Math. Soc., Volume 85 (2008), pp. 113-143 (DOI 10.1017/S1446788708000013) | DOI | MR | Zbl

[7] Earp, R. Sa; Toubiana, E. Existence and uniqueness of minimal graphs in hyperbolic space, Asian J. Math., Volume 4 (2000), pp. 669-694 | MR | Zbl

[8] Earp, R. Sa; Toubiana, E. An asymptotic theorem for minimal surfaces and existence results for minimal graphs in ${ℍ}^2\times ℝ$, Math. Annalen, Volume 342 (2008) no. 2, pp. 309-331 (DOI 10.1007/s00208-008-0237-0) | DOI | MR | Zbl

[9] Earp, R. Sa; Toubiana, E. Introduction à la géométrie hyperbolique et aux surfaces de Riemann, Cassini, 2009 | Zbl

[10] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983 | MR | Zbl

[11] Hauswirth, Laurent; Rosenberg, Harold; Spruck, Joel Infinite boundary value problems for constant mean curvature graphs in ${ℍ}^2\times ℝ$ and ${𝕊}^2\times ℝ$, Amer. J. Math., Volume 131 (2009) no. 1, pp. 195-226 | MR | Zbl

[12] Jenkins, Howard; Serrin, James The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., Volume 229 (1968), pp. 170-187 | DOI | MR | Zbl

[13] Mazet, L.; Rodriguez, M. M.; Rosenberg, H. The Dirichlet problem for the minimal surface equation – with possible infinite boundary data – over domains in a Riemannian manifold (1–60, arXiv: 0806.0498v1, 2008)

[14] Nelli, Barbara; Rosenberg, Harold Minimal surfaces in ${ℍ}^2\times ℝ$, Bull. Braz. Math. Soc. (N.S.), Volume 33 (2002) no. 2, pp. 263-292 Errata: Bull. Braz. Math. Soc. (N.S.) 38 (2007), no. 4, 661–664 | MR | Zbl

[15] Ou, Ye-Lin $p$-harmonic functions and the minimal graph equation in a Riemannian manifold, Illinois J. Math., Volume 49 (2005) no. 3, p. 911-927 (electronic) http://projecteuclid.org/getRecord?id=euclid.ijm/1258138228 | MR | Zbl

[16] Spruck, J. Interior gradient estimates and existence theorems for constant mean curvature graphs in $M^n\times ℝ$, Pure Appl. Math. Q., Volume 3 (2007) no. 3, Special Issue: In honor of Leon Simon. Part 2, pp. 785-800 | MR | Zbl