Potential theory on almost complex manifolds
Annales de l'Institut Fourier, Volume 65 (2015) no. 1, p. 171-210

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.

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.

DOI : https://doi.org/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
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {1},
     year = {2015},
     pages = {171-210},
     doi = {10.5802/aif.2928},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_1_171_0}
}
Harvey, F. Reese; Lawson, H. Blaine. Potential theory on almost complex manifolds. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 171-210. doi : 10.5802/aif.2928. http://www.numdam.org/item/AIF_2015__65_1_171_0/

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