Sur les variétés presque complexes, historiquement on a étudié les courbes pseudo-holomorphes beaucoup plus intensément que leurs objets « duals », les fonctions pluri-sousharmoniques. Ces fonctions sont définies en général par la condition que les restrictions aux courbes pseudo-holomorphes soient sous-harmoniques. Dans cet article les fonctions pluri-sousharmoniques sont définies en utilisant la théorie de viscosité avec une version du hessien complexe qui existe sur toutes les variétés presque complexes. Trois théorèmes sont démontrés. Le premier est un théorème de restriction qui établit l’équivalence de notre définition avec la définition « standard ». Dans le deuxième théorème, en utilisant nos définitions de viscosité, le problème de Dirichlet pour l’équation Monge-Ampère complexe est résolu dans les deux cas, homogène et inhomogène. Finalement, on démontre que les fonctions pluri-sousharmoniques considerées ici coïncident de façon précise avec les distributions pluri-sousharmoniques. En particulier, cela démontre une conjecture de Nefton Pali.
On almost complex manifolds the pseudo-holomorphic curves have been much more intensely studied than their “dual” objects, the plurisubharmonic functions. These functions are standardly defined by requiring that the restriction to each pseudo-holomorphic curve be subharmonic. In this paper plurisubharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proven. The first is a restriction theorem which establishes the equivalence of our definition with the “standard” definition. In the second theorem, using our “viscosity” definitions, the Dirichlet problem is solved for the complex Monge-Ampère equation in both the homogeneous and inhomogeneous forms. Finally, it is shown that the plurisubharmonic functions considered here agree in a precise way with the plurisubharmonic distributions. In particular, this proves a conjecture of Nefton Pali.
@article{AIF_2015__65_1_171_0,
author = {Harvey, F. Reese and Lawson, H. Blaine},
title = {Potential theory on almost complex manifolds},
journal = {Annales de l'Institut Fourier},
volume = {65},
year = {2015},
pages = {171-210},
doi = {10.5802/aif.2928},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2015__65_1_171_0}
}
Harvey, F. Reese; Lawson, H. Blaine. Potential theory on almost complex manifolds. Annales de l'Institut Fourier, Tome 65 (2015) pp. 171-210. doi : 10.5802/aif.2928. http://gdmltest.u-ga.fr/item/AIF_2015__65_1_171_0/
[1] The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Tome 37 (1976) no. 1, pp. 1-44 | Article | MR 445006 | Zbl 0315.31007
[2] The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations, Comm. Pure Appl. Math., Tome 38 (1985) no. 2, pp. 209-252 | Article | MR 780073 | Zbl 0598.35048
[3] Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math., Tome 62 (2009) no. 3, pp. 396-443 | Article | MR 2487853 | Zbl 1173.35062
[4] Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds, J. Differential Geom., Tome 88 (2011) no. 3, pp. 395-482 http://projecteuclid.org/euclid.jdg/1321366356 | MR 2844439 | Zbl 1235.53042
[5] The equivalence of viscosity and distributional subsolutions for convex subequations—a strong Bellman principle, Bull. Braz. Math. Soc. (N.S.), Tome 44 (2013) no. 4, pp. 621-652 | Article | MR 3167126 | Zbl 1296.35032
[6] Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry, Surveys in differential geometry. Geometry and topology, Int. Press, Somerville, MA (Surv. Differ. Geom.) Tome 18 (2013), pp. 103-156 | Article | MR 3087918 | Zbl 1323.35036
[7] The restriction theorem for fully nonlinear subequations., Ann. Inst. Fourier, Tome 64 (2014) no. 1, pp. 217-265 | Article | MR 3330548 | Zbl 1320.32037
[8] Some integration problems in almost-complex and complex manifolds., Ann. of Math. (2), Tome 77 (1963), pp. 424-489 | Article | MR 149505 | Zbl 0115.16103
[9] Fonctions plurisousharmoniques et courants positifs de type sur une variété presque complexe, Manuscripta Math., Tome 118 (2005) no. 3, pp. 311-337 | Article | MR 2183042 | Zbl 1089.32033
[10] The Monge-Ampère equation on almost complex manifolds, Math. Z., Tome 276 (2014) no. 3-4, pp. 969-983 | Article | MR 3175167 | Zbl 1295.32053
[11] Subharmonic functions in sub-Riemannian settings, J. Eur. Math. Soc. (JEMS), Tome 15 (2013) no. 2, pp. 387-441 | Article | MR 3017042 | Zbl 1270.31002
[12] User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), Tome 27 (1992) no. 1, pp. 1-67 | Article | MR 1118699 | Zbl 0755.35015
[13] Partial differential equations, John Wiley & Sons, Inc., New York-London-Sydney (1964), pp. xii+672 | MR 162045 | Zbl 0124.30501
[14] The equivalence of viscosity and distributional subsolutions for convex subequations—a strong Bellman principle, Bull. Braz. Math. Soc. (N.S.), Tome 44 (2013) no. 4, pp. 621-652 | Article | MR 3167126 | Zbl 1296.35032
[15] Les fonctions surharmoniques dans l’axiomatique de M. Brelot associées à un opérateur elliptique dégénéré, Ann. Inst. Fourier (Grenoble), Tome 22 (1972) no. 2, pp. 131-145 | Article | Numdam | MR 377092 | Zbl 0224.31014
[16] An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London (1966), pp. x+208 | MR 203075 | Zbl 0271.32001
[17] The analysis of linear partial differential operators. III, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 274 (1985), pp. viii+525 (Pseudodifferential operators) | MR 781536 | Zbl 0601.35001
[18] On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., Tome 38 (1995) no. 1, pp. 101-120 http://www.math.kobe-u.ac.jp/~fe/xml/mr1341739.xml | MR 1341739 | Zbl 0833.35053