Some estimates for the generalized Abreu equation

Li, An-Min; Lian, Zhao; Sheng, Li

doi:10.1016/j.difgeo.2016.06.002

Cited by 9 publications

(21 citation statements)

References 11 publications

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“…We note that a very similar equation for a toric Kähler metric on

V

appears in the construction of Kähler manifolds without multiplicities, see . We refer the Reader to for a comprehensive study of equation on a toric variety, for arbitrary weight functions

v (p) > 0

and

w (p)

, which is referred to as the generalized Abreu equation .…”

Section: Examplesmentioning

confidence: 99%

Kähler metrics with constant weighted scalar curvature and weighted K‐stability

Lahdili

2019

Proc. London Math. Soc.

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We introduce a notion of a Kähler metric with constant weighted scalar curvature on a compact Kähler manifold X, depending on a fixed real torus double-struckT in the reduced group of automorphisms of X, and two smooth (weight) functions v>0 and normalw, defined on the momentum image (with respect to a given Kähler class α on X) of X in the dual Lie algebra of double-struckT. We show that a number of known results obstructing the existence of constant scalar curvature Kähler (cscK) metrics can be extended to the weighted setting. In particular, we introduce a functional Mnormalv,normalw on the space of double-struckT‐invariant Kähler metrics in α, extending the Mabuchi energy in the cscK case, and show that if α is Hodge, then constant weighted scalar curvature metrics in α are minima of Mnormalv,normalw. We define a (v,w)‐weighted Futaki invariant of a double-struckT‐compatible smooth Kähler test configuration associated to (X,α,T), and show that the boundedness from below of the (v,w)‐weighted Mabuchi functional Mnormalv,normalw implies a suitable notion of a (v,w)‐weighted K‐semistability.

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“…We note that a very similar equation for a toric Kähler metric on

V

appears in the construction of Kähler manifolds without multiplicities, see . We refer the Reader to for a comprehensive study of equation on a toric variety, for arbitrary weight functions

v (p) > 0

and

w (p)

, which is referred to as the generalized Abreu equation .…”

Section: Examplesmentioning

confidence: 99%

Kähler metrics with constant weighted scalar curvature and weighted K‐stability

Lahdili

2019

Proc. London Math. Soc.

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“…This was proved by Li in an unpublished paper [11]. For the readers convenience we give the proof in Appendix A.…”

Section: Differential Inequalities Of δφmentioning

confidence: 96%

A regularity theorem of the Abreu equation

Chen¹,

Li²,

Sheng³

et al. 2015

Journal of Mathematical Analysis and Applications

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“…22 h ≥ C 3 . By Theorem 2.7, Theorem 2.8 and by using the same method as in Subsections §5.3- §5.4 [8] (see [22] for more explanation), we can prove the following result.…”

Section: Estimates Of Riemannian Distances On ∂∆mentioning

confidence: 81%

“…For general dimension, Li, Lian, and Sheng [21] proved that the uniform stability is the necessary condition for the solvability of the generalized Abreu equation and implies the interior regularity. Some estimates and differential inequalities were established in [22] and [24].…”

Section: Introductionmentioning

confidence: 99%