On the lower bound of the Mabuchi energy and its application

By Xiuxiong Chen. 2000

236 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bde8be23f61d8c9436baf2a3d64191a070a2b773

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Citations in the corpus

A variational approach to the Yau–Tian–Donaldson conjecture (2015, background): “Indeed, denoting by Ent( μ ) the entropy of a measure μ relative to V − 1 ω n 0 , the Chen–Tian formula [Che00b, Tia00] expresses M( u ) − Ent(MA( u )) as linear combination of terms of the form As a result, there exist A, B > 0 with | M( u ) − Ent(MA( u )) | ≤ A J( u ) + B (see e.g. Lemma A.2).”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, background): “According to [7], E can be extended to a continuous function on all C1,1 potentials in H.”
Greatest lower bounds on the Ricci curvature of Fano manifolds (2009, methodology): “By the same computation as in Chen [4], we have M(ω) + (1 − t)Jα(ω) = D + ∫”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “5 The J flow and quotient Hessian flows The J-flow on Kähler manifolds was introduced independently by Donaldson [9] and Chen [4].”
Geometry of Kähler metrics and foliations by holomorphic discs (2005, methodology): “[9], [8]).”
Stability, energy functionals, and Kahler-Einstein metrics (2002, unknown): “on the canonical bundle K= ΛnT∗ X, and then the Ricci curvature is just the curvature of this metric. There is another more direct way of deﬁning the Mabuchi energy, pointed out by Tian [T1] and Chen [C1], which does not require an integral along paths: νω(φ) = 1 vol(X) Z X log( ωn φ ωn )ωn φ + hω(”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, methodology): “The non-Archimedean Mabuchi functional is now defined so as to satisfy the analogue of the Chen–Tian formula (see [20] and also [5, Proposition 3.”
Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities (2012, unknown): “+ǫ |S|2 h ·ωn+n R1 0 dt R X ϕ˙(t)(ω0 −ωt) ∧ωn−1 t ≥ R X log|S|2 h·(ωnϕ−ωn 0 ) +n R 1 0 dt R X ϕ˙(t)(ω0 −ωt) ∧ωn− t −C3 = Jχ(ϕ) −C3. It is well-known that the K-energy has an explicit expression (c.f. [7]): E(ϕ) = Z X log ωn ϕ ωn 0 ωn 0 +I(ϕ) +Q(ϕ), where I(ϕ) = n Z 1 0 dt Z X ϕ˙(t)ωn t, and Q(ϕ) = −”
Long time existence of Minimizing Movement solutions of Calabi flow (2012, background): “Specifically, it follows from a calculation in ([5] pg.”
Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016, methodology): “e have H(˚) 0, with equality i ˚ ˚ ref is constant. 8 SEBASTIEN BOUCKSOM, TOMOYUKI HISAMOTO, AND MATTIAS JONSSON (v) The Mabuchi functional (or K-energy) can now be dened via the Chen-Tian formula [Che00] (see also [BB14, Proposition 3.1]) as M(˚) = H(˚) + R(˚) + SE (˚): (1.11) These functionals van”
Convergence of the J-flow on Kähler surfaces (2003, methodology): “χjχilg ij. By the maximum principle for parabolic equations, (2.2) implies that inf M (Λχ0ω) ≤ Λχω≤ sup M (Λχ0ω), (2.3) which gives a lower bound for χ, χij ≥ 1 supM(Λχ0ω) gij. (2.4) The J-functional [C1] is deﬁned by Jω,χ0(φ 0,φ 1) = Z 1 0 Z M ∂φt ∂t ω∧χn−1 φt (n− 1)! dt, where {φt} is a path in H ”
Space of Kähler metrics III – On the lower bound of the Calabi energy and geodesic distance (2006, methodology): “, the detailed expression of the K energy functional [10]), we obtain ∫”
Regularity of weak minimizers of the K-energy and applications to properness and K-stability (2016, methodology): “The χ -twisted K-energy is deﬁned as When restricted to H ω , the above formula for the K-energy was originally introduced by Chen–Tian [Ch1], with a similar formula already appearing in [Ti1].”
On the convergence and singularities of the J‐Flow with applications to the Mabuchi energy (2004, methodology): “du weinkove@math.harvard.edu 1. Introduction The J-ﬂow is a parabolic ﬂow on Ka¨hler manifolds with two Ka¨hler classes. It was discovered by Donaldson [Do1] in the setting of moment maps and by Chen [Ch2] as the gradient ﬂow of the J-functional appearing in his formula for the Mabuchi energy [Ma1].”
Convergence of the $J$-flow on toric manifolds (2014, methodology): “The J-flow, introduced by Donaldson [9] and Chen [6, 7] is the parabolic equation ∂ ∂t ωt = − √ −1∂∂Λωtα, with initial condition ω0 ∈ Ω.”
On the constant scalar curvature Kähler metrics (II)—Existence results (2020, methodology): “Following [12], we can write down an explicit formula for K(φ):”