Bilinear forms and extremal K�hler vector fields associated with K�hler classes

Futaki, Akito; Mabuchi, Toshiki

doi:10.1007/bf01446626

Cited by 113 publications

(139 citation statements)

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“…After changing the Kähler metric by a function v, the functions φ 1 , φ 2 are changed by E 1 (v), E 2 (v) [Futaki and Mabuchi 1995]. Since we always consider the case in which the metrics are invariant under a maximal compact group of the holomorphic automorphism group and J E 1 , J E 2 are Killing vector fields, the potentials of E 1 and E 2 are always real.…”

Section: General Results On Short Time Existencementioning

confidence: 99%

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Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles

Guan¹

2007

Pacific J. Math.

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For a certain class of completions of ‫ރ‬ * -bundles, we show that the existence of Calabi extremal metrics is equivalent to geodesic stability of the Kähler class, and we prove the exponential C ∞ convergence of the modified Calabi flow whenever the extremal metric exists, assuming that the manifold has hypersurface ends. In particular, we solve the problem of convergence of the modified Calabi flow on the almost homogeneous manifolds with two hypersurface ends which we dealt with in a 1995 Transactions paper. As a byproduct, we found a family of Kähler metrics, called extremal soliton metrics, interpolating the extremal metrics and the generalized quasiEinstein metrics. We also proved the existence of these metrics on compact almost homogeneous manifolds of two ends. For the completions of the ‫ރ‬ * -bundles we consider in this paper, we define what we call the generalized Mabuchi functional; the existence of extremal soliton metrics on these manifolds is again equivalent to the geodesic stability of the Kähler class with respect to this functional.

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Section: General Results On Short Time Existencementioning

confidence: 99%

“…where φ E is the function corresponding to the extremal vector field E in [Futaki and Mabuchi 1995]. In our case, φ E = a + bU − HR for the values a, b in (3) with c = 0.…”

Section: Geodesic Stabilitymentioning

confidence: 93%

Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles

Guan¹

2007

Pacific J. Math.

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“…It was originally introduced in this form by Futaki-Mabuchi [9]. To define the relative Futaki invariant, suppose that we have a torus action T on (V , L) commuting with α.…”

Section: Relative K-polystabilitymentioning

confidence: 99%

“…The strongest notion is K-polystability relative to the extremal C * -action. This is a C * -action χ defined by Futaki-Mabuchi [9] as follows. Fix a maximal torus of automorphisms T , and write t for its Lie algebra.…”

Section: Relative K-polystabilitymentioning

confidence: 99%

Relative K-stability of extremal metrics

Stoppa

Székelyhidi

2011

J. Eur. Math. Soc.

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“…. , W q ) κ ∈ C be the symmetric C-multilinear form as defined in [FM;p.209], where q is an arbitrary positive integer. Then by the same argument as above, we can easily show that u = {Z ∈ g; (Z, W 1 , .…”

Section: For Instance If M Is a Fano Manifold Then The Pair (M C 1mentioning

confidence: 99%

Vector Field Energies and Critical Metrics on Kähler Manifolds

Mabuchi¹

2001

Nagoya Mathematical Journal

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Abstract. Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionalsμ,ν whose critical points are extremal Kähler metrics or "Kähler-Einstein metrics" in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity ofν, the uniqueness of "Kähler-Einstein metrics" on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.

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Bilinear forms and extremal K�hler vector fields associated with K�hler classes

Cited by 113 publications

References 9 publications

Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles

Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles

Relative K-stability of extremal metrics

Vector Field Energies and Critical Metrics on Kähler Manifolds

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Bilinear forms and extremal K�hler vector fields associated with K�hler classes

Cited by 113 publications

References 9 publications

Extremal solitons and exponential C∞convergence of the modified Calabi flow on certain ℂP1Bundles

Extremal solitons and exponential C∞convergence of the modified Calabi flow on certain ℂP1Bundles

Relative K-stability of extremal metrics

Vector Field Energies and Critical Metrics on Kähler Manifolds

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Product

Resources

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Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles

Extremal solitons and exponential C^∞convergence of the modified Calabi flow on certain ℂP¹Bundles