On a Class of Fully Nonlinear Elliptic Equations on Closed Hermitian Manifolds II: L^∞ Estimate

Abstract

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of the cone condition, we derive the L∞ estimate directly. As an application, we solve the complex quotient equations on closed Kähler manifolds. © 2016 Wiley Periodicals, Inc.

Parabolic complex Monge–Ampère type equations on closed Hermitian manifolds

“…Applying the technique in [20,21], we obtain the following lemma on closed Hermitian manifolds. We recall the following two lemmas proven in [21].…”

“…Their results were extended to the complex Monge-Ampère type equations by Fang, Lai and Ma [8] also using parabolic flow method. More general cases were treated by Guan and the author [10], the author [18,20] on Hermitian manifolds using continuity method. Later, the author [19] reproduced the results in [18] by parabolic flow method.…”

The Parabolic Flows for Complex Quotient Equations

“…Without loss of generality, we may assume that sup M u = −2λ in this paper. To obtain the second order estimate, we improve a key lemma in [25] to fit in with the parabolic case.…”

“…There are many results regarding different flows on closed complex manifolds, and we refer readers to [8,12,13,19,24,30,35,36]. In this paper, we preliminarily explore the parabolic flow equation (1.1) for complex quotient equations, and reprove the solvability of the corresponding complex quotient equations (see [26,25]).…”

On uniform estimate of complex elliptic equations on closed Hermitian manifolds

“…In the work of the author [12], we provide a direct uniform estimate for the two types of equations on closed Kähler manifolds. In this paper, we further develop the technique in [12] and apply a key lemma by Zhang [18] to extend the estimate to general Hermitian cases. The lemma in [18] helps us to improve our result in an early version of this paper.…”