On a class of fully nonlinear flows in Kähler geometry

By H. Fang, Mijia Lai, Xinan Ma. 2009

116 citations recorded by the source corpus.

Abstract In this paper, we study a class of fully nonlinear metric flows on Kähler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the result of Song–Weinkove. As a consequence, under the given condition, we solve the corresponding Euler equation, which is fully nonlinear of Monge–Ampère type. As an application, we also discuss a complex Monge–Ampère type equation including terms of mixed degrees, which was first posed by Chen.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e938b5fc2540898f087caac3d88f6eebf9474d15

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Fully non-linear elliptic equations on compact Hermitian manifolds (2015, methodology): “This result was later generalized by Fang-Lai-Ma [12] to general k, and existence results for general k and non-constant h on Hermitian manifolds were obtained by Guan-Sun [20], Sun [32].”
Convergence of the $J$-flow on toric manifolds (2014, methodology): “In [13] it was shown that a solution to these special cases of equation (7) exist if and only if there is a metric χ ∈ Ω satisfying a positivity condition analogous to (2).”
Nakai-Moishezon criterions for complex Hessian equations (2020, background): “Both the J-equation and the J-flow are extensively studied in [3, 4, 22, 23, 19, 13, 20, 16, 7, 21, 12, 6].”
Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds (2014, unknown): “[9] who extended the result of [38] to f = (σn/σk) 1 n−k for 1 ≤ k ≤ n − 1; see also [16–18].”
The J-flow and stability (2013, background): “In analogy with the result of Song-Weinkove [15], it is shown in [9] that a necessary and sufficient condition for Equation (106) to have a solution is the following: There is a metric ω ∈ [ω] such that (107) cω − ( n− 1 k ) ω ∧ α > 0, in the sense of positivity of (n− 1, n− 1)-forms.”
A numerical criterion for generalised Monge-Ampère equations on projective manifolds (2020, methodology): “This involves solving a certain J-equation on the productM×M and obtaining currents that concentrate along the diagonal ∆ ⊂M ×M .”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “This would be useful in providing a unified approach to parabolic Hölder estimates for parabolic flows in complex geometry, including theKähler–Ricci flow, the Chern–Ricci flow, the J-flow and their generalizations (see [8,9,11,14,23,29,46,59,68] for example).”