Abstract
We define the Monge–Ampère operator \({(i\partial {\bar{\partial }}u)^{2}}\) for continuous J-plurisubharmonic functions on four dimensional almost complex manifolds.
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Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge–Ampère equation. Invent. Math. 37, 1–44 (1976)
Bedford, E., Taylor, B.A.: Variational properties of the complex Monge–Ampère equation I. Dirichlet principle. Duke Math. J. 45, 375–403 (1978)
Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–41 (1982)
Bertrand, F.: Sharp estimates of the Kobayashi metric and Gromov hyperbolicity. J. Math. Anal. Appl. 345(2), 825–844 (2008)
Błocki, Z.: On the definition of the Monge–Ampère operator in \({\mathbb{C}}^2\). Math. Ann. 328, 415–423 (2004)
Błocki, Z.: Minicourse on Pluripotential Theory. University of Vienna (2012)
Cegrell, U.: The gradient lemma. Ann. Polon. Math. 91, 143–146 (2007)
Cieliebak, K., Eliashberg, Y.: Stein Structures: Existence and Flexibility. Contact and Symplectic Topology, Bolyai Society Mathematical Studies, pp. 357–388. Springer, Berlin (2014)
Gaussier, H., Sukhov, A.: Estimates of the Kobayashi–Royden metric in almost complex manifolds. Bull. Soc. Math. Fr. 133(2), 259–273 (2005)
Harvey, F.R., Lawson, H.B., Jr.: Potential theory on almost complex manifolds. Ann. Inst. Fourier (Grenoble) 65(1), 171–210 (2015)
Ivashkovich, S., Rosay, J.-P.: Schwarz-type lemmas for solutions of \(\overline{\partial }\)-inequalities and complete hyperbolicity of almost complex manifolds. Ann. Inst. Fourier (Grenoble) 54(7), 2387–2435 (2004)
Ivashkovich, S., Rosay, J.-P.: Boundary values and boundary uniqueness of J-holomorphic mappings. Int. Math. Res. Not. IMRN 17, 3839–3857 (2011)
Kuzman, U.: Poletsky theory of discs in almost complex manifolds. Complex Var. Elliptic Equ. 59(2), 262–270 (2014)
Pali, N.: Fonctions plurisousharmoniques et courants positifs de type \((1,1)\) sur une variété presque complexe. Manuscr. Math. 118(3), 311–337 (2005)
Pliś, S.: The Monge–Ampère equation on almost complex manifolds. Math. Z. 276(3–4), 969–983 (2014)
Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 175, 251–286 (1968)
Sukhov, A.: Regularized maximum of strictly plurisubharmonic functions on an almost complex manifold. Int. J. Math. 24(12), 1350097 (6 pp) (2013)
Acknowledgments
The author would like to express his gratitude to Z. Błocki for helpful discussions and advice during the work on this paper. The author was partially supported by the NCN Grant 2011/01/D/ST1/04192.
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Pliś, S. Monge–Ampère Operator on Four Dimensional Almost Complex Manifolds. J Geom Anal 26, 2503–2518 (2016). https://doi.org/10.1007/s12220-015-9635-1
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DOI: https://doi.org/10.1007/s12220-015-9635-1