no. 1
Potential theory on almost complex manifolds
[Théorie du potentiel sur les variétés presque-complexes]
Harvey, F. Reese1 ; Lawson, H. Blaine2
1 Department of Mathematics RICE University Houston, TX 77005-1982 (USA)
2 Department of Mathematics Stony Brook University Stony Brook, NY 11794-3651 (USA)
Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 171-210.

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.

DOI : 10.5802/aif.2928
Classification : 32Q60, 32U05, 35J57
Keywords: Almost complex manifold, pseudo-holomorphic curve, plurisubharmonic function, viscosity solution, Dirichlet problem, complex Monge-Ampère equation, pluripotential theory
Mot clés : Variété presque complexe, courbe pseudo-holomorphe, fonction pluri-sousharmonique, solution de viscosité, problème de Dirichlet, equation de Monge-Ampère complexe, théorie pluripotentielle

Harvey, F. Reese 1 ; Lawson, H. Blaine 2

1 Department of Mathematics RICE University Houston, TX 77005-1982 (USA)
2 Department of Mathematics Stony Brook University Stony Brook, NY 11794-3651 (USA) 
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Harvey, F. Reese; Lawson, H. Blaine. Potential theory on almost complex manifolds. Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 171-210. doi : 10.5802/aif.2928. https://aif.centre-mersenne.org/articles/10.5802/aif.2928/

[1] Bedford, Eric; Taylor, B. A. The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Volume 37 (1976) no. 1, pp. 1-44 | DOI | EuDML | MR | Zbl

[2] Caffarelli, L.; Kohn, J. J.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations, Comm. Pure Appl. Math., Volume 38 (1985) no. 2, pp. 209-252 | DOI | MR | Zbl

[3] Harvey, F. Reese; Lawson, H. Blaine Jr. Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math., Volume 62 (2009) no. 3, pp. 396-443 | DOI | MR | Zbl

[4] Harvey, F. Reese; Lawson, H. Blaine Jr. Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds, J. Differential Geom., Volume 88 (2011) no. 3, pp. 395-482 http://projecteuclid.org/euclid.jdg/1321366356 | MR | Zbl

[5] Harvey, F. Reese; Lawson, H. Blaine Jr. The equivalence of viscosity and distributional subsolutions for convex subequations—a strong Bellman principle, Bull. Braz. Math. Soc. (N.S.), Volume 44 (2013) no. 4, pp. 621-652 | DOI | MR | Zbl

[6] Harvey, F. Reese; Lawson, H. Blaine Jr. Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry, Surveys in differential geometry. Geometry and topology (Surv. Differ. Geom.), Volume 18, Int. Press, Somerville, MA, 2013, pp. 103-156 | DOI | MR | Zbl

[7] Harvey, F. Reese; Lawson, H. Blaine Jr. The restriction theorem for fully nonlinear subequations., Ann. Inst. Fourier, Volume 64 (2014) no. 1, pp. 217-265 | DOI | EuDML | MR | Zbl

[8] Nijenhuis, Albert; Woolf, William B. Some integration problems in almost-complex and complex manifolds., Ann. of Math. (2), Volume 77 (1963), pp. 424-489 | DOI | MR | Zbl

[9] Pali, Nefton Fonctions plurisousharmoniques et courants positifs de type (1,1) sur une variété presque complexe, Manuscripta Math., Volume 118 (2005) no. 3, pp. 311-337 | DOI | MR | Zbl

[10] Pliś, Szymon The Monge-Ampère equation on almost complex manifolds, Math. Z., Volume 276 (2014) no. 3-4, pp. 969-983 | DOI | MR | Zbl

[11] Bonfiglioli, Andrea; Lanconelli, Ermanno Subharmonic functions in sub-Riemannian settings, J. Eur. Math. Soc. (JEMS), Volume 15 (2013) no. 2, pp. 387-441 | DOI | MR | Zbl

[12] Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), Volume 27 (1992) no. 1, pp. 1-67 | DOI | MR | Zbl

[13] Garabedian, P. R. Partial differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964, pp. xii+672 | MR | Zbl

[14] Harvey, F. Reese; Lawson, H. Blaine Jr. The equivalence of viscosity and distributional subsolutions for convex subequations—a strong Bellman principle, Bull. Braz. Math. Soc. (N.S.), Volume 44 (2013) no. 4, pp. 621-652 | DOI | MR | Zbl

[15] Hervé, Rose-Marie; Hervé, Michel Les fonctions surharmoniques dans l’axiomatique de M. Brelot associées à un opérateur elliptique dégénéré, Ann. Inst. Fourier (Grenoble), Volume 22 (1972) no. 2, pp. 131-145 | DOI | Numdam | MR | Zbl

[16] Hörmander, Lars An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966, pp. x+208 | MR | Zbl

[17] Hörmander, Lars The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274, Springer-Verlag, Berlin, 1985, pp. viii+525 (Pseudodifferential operators) | MR | Zbl

[18] Ishii, Hitoshi On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., Volume 38 (1995) no. 1, pp. 101-120 http://www.math.kobe-u.ac.jp/~fe/xml/mr1341739.xml | MR | Zbl

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