scholarly journals Regularity estimates of solutions to complex Monge–Ampère equations on Hermitian manifolds

2011 ◽  
Vol 260 (7) ◽  
pp. 2004-2026 ◽  
Author(s):  
Xi Zhang ◽  
Xiangwen Zhang
Keyword(s):  
Estimates Of Solutions ◽  
Hermitian Manifolds ◽  
Regularity Estimates ◽  
Monge Ampere

Weak solutions of the Monge–Ampère equation on compact Hermitian manifolds

2017 ◽  
Vol 28 (09) ◽  
pp. 1740002
Author(s):  
Sławomir Kołodziej
Keyword(s):  
Weak Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Stability Of Solutions ◽  
Hermitian Manifolds ◽  
Priori Estimates ◽  
Existence And Stability ◽  
Monge Ampere

In this paper, we describe how pluripotential methods can be applied to study weak solutions of the complex Monge–Ampère equation on compact Hermitian manifolds. We indicate the differences between Kähler and non-Kähler setting. The results include a priori estimates, existence and stability of solutions.

Gradient estimates for Monge–Ampère type equations on compact almost Hermitian manifolds with boundary

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Masaya Kawamura
Keyword(s):  
Nonlinear Equations ◽  
Smooth Solution ◽  
A Priori ◽  
Gradient Estimates ◽  
Manifolds With Boundary ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Monge Ampere

Abstract We investigate Monge–Ampère type fully nonlinear equations on compact almost Hermitian manifolds with boundary and show a priori gradient estimates for a smooth solution of these equations.

scholarly journals A Parabolic Monge–Ampère Type Equation of Gauduchon Metrics

2017 ◽  
Vol 2019 (17) ◽  
pp. 5497-5538 ◽  
Author(s):  
Tao Zheng
Keyword(s):  
Smooth Function ◽  
Existence And Uniqueness ◽  
Type Equation ◽  
Uniqueness Of Solution ◽  
Long Time Existence ◽  
Hermitian Manifolds ◽  
Long Time ◽  
Monge Ampere ◽  
Time Existence

Abstract We prove the long time existence and uniqueness of solution to a parabolic Monge–Ampère type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth topology as $t$ approaches infinity which, up to scaling, is the solution to a Monge–Ampère type equation. This gives a parabolic proof of the Gauduchon conjecture based on the solution of Székelyhidi, Tosatti, and Weinkove to this conjecture.

scholarly journals Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Sławomir Kołodziej ◽  
Ngoc Cuong Nguyen
Keyword(s):  
Hermitian Manifold ◽  
General Measure ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Hermitian Manifolds ◽  
Right Hand ◽  
The Right ◽  
Monge Ampere ◽  
Compact Hermitian Manifold

AbstractWe prove the existence of a continuous quasi-plurisubharmonic solution to the Monge–Ampère equation on a compact Hermitian manifold for a very general measure on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh–Nguyen–Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh–Nguyen.

scholarly journals Hermitian metrics, (n-1,n-1) forms and Monge–Ampère equations

2019 ◽  
Vol 2019 (755) ◽  
pp. 67-101 ◽  
Author(s):  
Valentino Tosatti ◽  
Ben Weinkove
Keyword(s):  
Second Order ◽  
Order Estimate ◽  
Kahler Manifolds ◽  
Complex Manifolds ◽  
Smooth Solutions ◽  
Hermitian Metrics ◽  
Plurisubharmonic Functions ◽  
Hermitian Manifolds ◽  
Monge Ampere ◽  
Compact Complex Manifolds

AbstractWe show existence of unique smooth solutions to the Monge–Ampère equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi–Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kähler manifolds: those satisfying the Jost–Yau condition known as Astheno–Kähler. Gauduchon conjectured in 1984 that a Calabi–Yau theorem for Gauduchon metrics holds on all compact complex manifolds. We discuss another Monge–Ampère equation, recently introduced by Popovici, and show that the full Gauduchon conjecture can be reduced to a second-order estimate of Hou–Ma–Wu type.

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