On an \(a\) \(priori\) \(L^\infty\) estimate for a class of Monge-Ampère type equations on compact almost Hermitian manifolds

Masaya Kawamura

doi:10.56754/0719-0646.2402.0239

DOI:

https://doi.org/10.56754/0719-0646.2402.0239

Abstract

We investigate Monge-Ampère type equations on almost Hermitian manifolds and show an \(a\) \(priori\) \(L^\infty\) estimate for a smooth solution of these equations.

Keywords

Monge-Ampère type equation , almost Hermitian manifold , Chern connection

Masaya Kawamura Department of General Education, National Institute of Technology, Kagawa College 355, Chokushi-cho, Takamatsu, Kagawa, Japan, 761-8058.

Pages: 239–261
Date Published: 2022-08-22
Vol. 24 No. 2 (2022)

L. Chen, “Hessian equations of Krylov type on Kähler manifolds”, preprint, arXiv:2107.12035v3, 2021.

J. Chu, V. Tosatti and B. Weinkove, “The Monge-Ampère equation for non-integrable almost complex structures”, J. Eur. Math. Soc., vol. 21, no. 7, pp. 1949–1984, 2019.

P. Gauduchon, “Le théorème de l‘excentricité nulle”, C. R. Acad. Sci. Paris S Ìer. A-B, vol. 285, no. 5, pp. A387–A390, 1977.

A. Newlander and L. Nirenberg, “Complex analytic coordinates in almost complex manifolds”, Ann. of Math. (2), vol. 65, pp. 391–404, 1957.

W. Sun, “On a class of fully nonlinear elliptic equations on closed Hermitian manifolds”, J. Geom. Anal., vol. 26, no. 3, pp. 2459–2473, 2016.

W. Sun, “On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: Lâˆž estimate”, Comm. Pure Appl. Math., vol. 70, no. 1, pp. 172–199, 2017.

V. Tosatti and B. Weinkove, “Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds”, Asian J. Math., vol. 14, pp. 19–40, 2010.

V. Tosatti and B. Weinkove, “The complex Monge-Ampère equation on compact Hermitian manifolds”, J. Amer. Math. Soc., vol. 23, pp. 1187–1195, 2010.

Q. Tu and N. Xiang, “The Dirichlet problem for mixed Hessian equations on Hermitian manifolds”, preprint, arXiv:2201.05030v1, 2022.

L. Vezzoni, “On Hermitian curvature flow on almost complex manifolds”, Differential Geom. Appl., vol. 29, no. 5, pp. 709–722, 2011.

C.-J. Yu, “Nonpositively curved almost Hermitian metrics on products of compact almost complex manifolds”, Acta Math. Sin., vol. 31, no. 1, pp. 61–70, 2015.

J. Zhang, “Monge-Ampère type equations on almost Hermitian manifolds”, preprint, arXiv:2101.00380, 2022.

T. Zheng, “An almost complex Chern-Ricci flow”, J. Geom. Anal., vol. 28, no. 3, pp. 2129– 2165, 2018.