$K$-energy maps integrating Futaki invariants

Mabuchi, Toshiki

doi:10.2748/tmj/1178228410

Cited by 227 publications

(246 citation statements)

References 6 publications

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“…We now see how the argument in [7] adapts to obtain the following Lemma 1. The definition of M above is independent of the curve t → ϕ(t).…”

Section: (T)(s(t) − π(T)s(t))dµ(t) mentioning

confidence: 97%

“…Inspired by the work of Donaldson on Yang-Mills connections on stable bundles, Mabuchi [7] introduced the K-energy functional on Ω + for polarized manifolds with positive first Chern class. The critical points of this functional are Kähler Einstein metrics, and it has played a significant role in the study of these metrics for this type of manifolds [1,4].…”

Section: Let (Mmentioning

confidence: 99%

“…We now adapt ideas of T. Mabuchi [7] to produce another characterization of extremal Kähler metrics. Mabuchi was interested in the study of Kähler Einstein metrics on manifolds with positive first Chern class.…”

Section: Santiago R Simancamentioning

confidence: 99%

“…Calabi computed the Euler-Lagrange equation, and showed that the critical metrics of this functional are those for which the gradient of the scalar curvature is a realholomorphic vector field. He called these metrics extremal.Inspired by the work of Donaldson on Yang-Mills connections on stable bundles, Mabuchi [7] introduced the K-energy functional on Ω + for polarized manifolds with positive first Chern class. The critical points of this functional are Kähler Einstein metrics, and it has played a significant role in the study of these metrics for this type of manifolds [1,4].…”

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confidence: 99%

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A {𝐾}-energy characterization of extremal Kähler metrics

Simanca

1999

Proc. Amer. Math. Soc.

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Abstract. We show that for any polarized compact Kähler manifold, the extremal Kähler metrics that represent the given cohomology class can be characterized as critical points of a suitably defined K-energy functional. Let (M,JHere s ω and dµ ω are the scalar curvature and volume form of ω, respectively. Calabi computed the Euler-Lagrange equation, and showed that the critical metrics of this functional are those for which the gradient of the scalar curvature is a realholomorphic vector field. He called these metrics extremal.Inspired by the work of Donaldson on Yang-Mills connections on stable bundles, Mabuchi [7] introduced the K-energy functional on Ω + for polarized manifolds with positive first Chern class. The critical points of this functional are Kähler Einstein metrics, and it has played a significant role in the study of these metrics for this type of manifolds [1,4]. The purpose of this note is to show that extremal Kähler metrics can also be defined as critical points of a functional analogous to the K-energy.Let G be a maximal compact subgroup of the biholomorphism group of (M, J), and let g be a Kähler metric on M with Kähler class Ω. Without loss of generality, we assume that g is G-invariant. We denote by L 2 k,G the real Hilbert space of Ginvariant real-valued functions of class L 2 k , and consider G-invariant deformations of this metric preserving the Kähler class:In this expression, the condition k > n ensures that L 2 k,G is a Banach algebra, making the scalar curvature ofω a well-defined function in the space.

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“…We now see how the argument in [7] adapts to obtain the following Lemma 1. The definition of M above is independent of the curve t → ϕ(t).…”

Section: (T)(s(t) − π(T)s(t))dµ(t) mentioning

confidence: 97%

Section: Let (Mmentioning

confidence: 99%

Section: Santiago R Simancamentioning

confidence: 99%

mentioning

confidence: 99%

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A {𝐾}-energy characterization of extremal Kähler metrics

Simanca

1999

Proc. Amer. Math. Soc.

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“…The main ingredient in the construction of the path integral is to approximate the infinite dimensional space of Kähler metrics M at fixed Kähler class by finite dimensional subspaces M n of so-called Bergman metrics. These subspaces are characterized by an integer n such that lim n→∞ M n = M in a very precise sense [6,9]. The path integral over M is then regularized by a finite dimensional integral over M n .…”

Section: A Note On Some Of Our Original Motivationsmentioning

confidence: 99%

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

Bilal

Ferrari

2013

Nuclear Physics B

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We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.

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

Song

Weinkove

2007

Comm Pure Appl Math

144

186

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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.We also study the singularities of the J -flow and, under certain conditions (which always hold for dimension 2) derive some estimates away from a subvariety. We discuss the conjectural remark of Donaldson that if the J -flow does not converge on a Kähler surface, then it should blow up over some curves of negative self-intersection.

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$K$-energy maps integrating Futaki invariants

Cited by 227 publications

References 6 publications

A {𝐾}-energy characterization of extremal Kähler metrics

A {𝐾}-energy characterization of extremal Kähler metrics

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

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

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