On the convergence and singularities of the J‐Flow with applications to the Mabuchi energy

By Jian Song, B. Weinkove. 2004

209 citations recorded by the source corpus.

The J‐flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kähler manifolds with two Kähler metrics. It is the gradient flow of the J‐functional that appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the J‐flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all Kähler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant‐scalar‐curvature Kähler metrics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a90b7346e16be2315a12aa644d81dfb8433cae07

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “Thus we shall require the following version of the Evans-Krylov theorem for uniformly parabolic and concave equations, with the precise dependence of constants spelled out, and which can be proved using the arguments of Trudinger [60], and more particularly Tosatti-Weinkove [53] and Gill [17].”
Geodesic rays and stability in the cscK problem (2020, background): “It is a result by Song-Weinkove [64] that if the two conditions are satisfied, the corresponding J-equation is solvable, and by [28] the Archimedean JKX+ L-energy is proper, which implies the last statement.”
Convergence of the $J$-flow on toric manifolds (2014, methodology): “It was shown by Song-Weinkove [19] that this flow converges whenever there exists a metric ω ∈ Ω satisfying Λωα = c, where the constant c only depends on the classes [α],Ω, and is determined by”
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.”
Space of Kähler metrics III – On the lower bound of the Calabi energy and geodesic distance (2006, background): “For higher dimension K¨ahler manifolds, readers are referred to Song-WeinKove[ 3 ] . 3The diameter bound can be replaced by a bound on Sobleve constant.”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, unknown): “As our topological condition, namely that −µ(X,L, 0)L−KX is ample, is a stronger condition on L than needed by Song-Weinkove and Li-Shi-Yao, we do not provide the details.”
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.”
A second order estimate for general complex Hessian equations (2015, background): “…al. 2015] D. H. Phong, S. Picard, and X. Zhang, “On estimates for the Fu–Yau generalization of a Strominger system”, preprint, 2015. arXiv 1507.08193 [Song and Weinkove 2008] J. Song and B. Weinkove, “On the convergence and singularities of the J -flow with applications to the Mabuchi energy”,…”
Degenerate complex Hessian equations on compact K\"ahler manifolds (2014, background): “Some similar non-linear elliptic equations of have appeared in the study of geometric deformation flows such as the J-flow (see [38]).”
Fully non-linear elliptic equations on compact Hermitian manifolds (2015, unknown): “Fixing this value of c, if k = n− 1 then Song-Weinkove [29] showed that a solution exists if there is a metric χ′ = χ+ √ −1∂∂u satisfying (9) ncχ′n−1 − (n− 1)χ′n−2 ∧ α > 0 in the sense of positivity of (n − 1, n − 1)-forms.”
The Dirichlet problem for degenerate complex Monge-Ampere equations (2009, background): “,[12, 13, 19, 37]) and for singular canonical metrics and degenerate equations on Kähler manifolds [14, 17, 21, 30, 31, 35, 36].”