Convergence of the J-flow on Kähler surfaces

By B. Weinkove. 2003

82 citations recorded by the source corpus.

Donaldson defined a parabolic flow on Kahler manifolds which arises from considering the action of a group of symplectomorphisms on the space of smooth maps between manifolds. One can define a moment map for this action, and then consider the gradient flow of the square of its norm. Chen discovered the same flow from a different viewpoint and called it the J-flow, since it corresponds to the gradient flow of his J-functional, which is related to Mabuchi's K-energy. In this paper, we show that in the case of Kahler surfaces with two Kahler forms satisfying a certain inequality, the J-flow converges to a zero of the moment map.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3c9d77543d5660ba63ceb18e319654e19e7feca7

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The J-flow and stability (2013, background): “It was shown by Song-Weinkove [15] (see also Weinkove [22, 23]) that when a solution to Equation (2) exists, then the J-flow converges to this solution.”
On a class of fully nonlinear elliptic equations on Hermitian manifolds (2013, methodology): “It was studied by Chen [5], Weinkove [32,33], Song and Weinkove [28] using parabolic methods.”
A numerical criterion for generalised Monge-Ampère equations on projective manifolds (2020, methodology): “Introduction The Monge-Ampère equation and related PDE such as the J-equation [12, 5, 31, 41, 40, 7, 23, 13, 14, 42, 43, 11, 45, 18, 32] and the deformed Hermitian YangMills (dHYM) equation [20, 6, 8, 19, 44, 26, 15, 17, 24, 3, 4, 30, 16, 36, 37, 21] play an important role in Kähler geometry.”
Nakai-Moishezon criterions for complex Hessian equations (2020, methodology): “Both the J-equation and the J-flow are extensively studied in [3, 4, 22, 23, 19, 13, 20, 16, 7, 21, 12, 6].”
Greatest lower bounds on the Ricci curvature of Fano manifolds (2009, methodology): “Alternatively, for a more self-contained proof we can use Proposition 7.”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, methodology): “This method was employed in the Kähler case in [59], and a related argument was used in the almost complex setting in [55].”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, background): “Our results can be applied to prove the following generalization to quotient Hessian flows of the results of [63, 64, 12], as well as an alternative proof of a result of Székelyhidi [50, Proposition 22] on the Hessian quotient equations.”
On the convergence and singularities of the J‐Flow with applications to the Mabuchi energy (2004, background): “The second author showed in [ We1 , We2] that the J-flow converges to a critical metric under the condition”
$$C^{2,\alpha }$$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).”
On a class of fully nonlinear flows in Kähler geometry (2009, methodology): “Theorem 1.1 can be viewed as a finite interpolation between results of Yau [Y], Cao [C], Song-Weinkove [SW].”