On the J-Flow in Higher Dimensions and the Lower Boundedness of the Mabuchi Energy

By B. Weinkove. 2003

104 citations recorded by the source corpus.

The J-flow is a parabolic flow on Kahler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under a certain condition on the initial data, the J-flow converges to a critical metric. This is a generalization to higher dimensions of the author's previous work on Kahler surfaces. A corollary of this is the lower boundedness of the Mabuchi energy on Kahler classes satisfying a certain inequality when the first Chern class of the manifold is negative.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ca49a0c63a1115d12e9a6126af2f4bc5b7145fb0

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Taming symplectic forms and the Calabi–Yau equation (2007, methodology): “We make use of a Moser iteration argument as in [18], but applied to the exponential of φ in a similar way to [16], to obtain a C estimate of φ depending on Iα(φ).”
Nakai-Moishezon criterions for complex Hessian equations (2020, background): “Finally, we extend the result of [23, 19, 14] for constant scalar curvature Kähler metrics on smooth minimal models.”
On a class of fully nonlinear flows in Kähler geometry (2009, methodology): “Following [19], we prove by contradiction.”
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.”
Fully non-linear elliptic equations on compact Hermitian manifolds (2015, methodology): “At this point we can either use the C-estimate Proposition 10 to obtain an estimate for trαg, or this C -estimate can be directly derived from (149) as in Weinkove [42] or Tosatti-Weinkove [38].”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “Necessary and sufficient conditions for existence of solutions for k = n − 1 were given by Song–Weinkove [51] using a parabolic method (see [11,68,69]).”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, result): “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): “This involves solving a certain J-equation on the productM×M and obtaining currents that concentrate along the diagonal ∆ ⊂M ×M .”
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.”