Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one

Guan, Daniel; Chen, Xiuxiong

doi:10.4310/ajm.2000.v4.n4.a6

Cited by 20 publications

(44 citation statements)

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“…We prove the converse in [Guan ≥ 2011a]. In [Guan 2002;2006;Guan and Chen 2000] we dealt with some examples, and in [Guan 2009] we dealt with the two most conceptually difficult series of manifolds.…”

Section: Introductionmentioning

confidence: 99%

Type II almost-homogeneous manifolds of cohomogeneity one

Guan¹

2011

Pacific J. Math.

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This paper continues our investigation on the existence of extremal metrics of the general affine and type II almost-homogeneous manifolds of cohomogeneity one. It deals with the general type II cases with hypersurface ends: more precisely, with manifolds having certain ‫ރ‬ P n ‫ރ(×‬ P n ) * -or ‫ރ‬ P 2 -bundle structures. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics.

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Section: Introductionmentioning

confidence: 99%

Type II almost-homogeneous manifolds of cohomogeneity one

Guan¹

2011

Pacific J. Math.

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“…Let the open orbit be G/U, K be the maximal connected compact subgroup of G, L be the generic isotropic subgroup of K , i.e., K /L is a generic K orbit. We have that (see [20,Theorem 1]): Proposition 2.1.…”

Section: Preliminariesmentioning

confidence: 99%

“…By our construction, Q 2 is a Q 1 bundle over Q. If there is another connected complex Lie group G 1 acting on M and containning G, then by [1] and [20] (see also [19]), G 1 is semisimple. G 1 also acts transitively on Q 2 .…”

Section: Using Mathematica Withmentioning

confidence: 99%

“…We say that a manifold is of cohomogeneity one if the maximal compact subgroup has a (real) hypersurface orbit. In [20] and [14], we reduced compact complex almost homogeneous manifolds of cohomogeneity one into three types of manifolds. We denote the manifold by M and let G be a complex subgroup of its automorphism group which has an open orbit on M. Let us assume first that M is simply connected.…”

Section: Preliminariesmentioning

confidence: 99%

“…

Abstract: This paper is one in a series generalizing our results in [12,14,15,20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends.

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Affine compact almost-homogeneous manifolds of cohomogeneity one

Guan

2009

Open Mathematics

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This paper is one in a series generalizing our results in [12,14,15,20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

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Constant Scalar Curvature Equation and Regularity of Its Weak Solution

Zeng

2018

Comm Pure Appl Math

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In this paper we study the constant scalar curvature equation (CSCK), a nonlinear fourth‐order elliptic equation, and its weak solutions on Kähler manifolds. We first define the notion of a weak solution of CSCK for an L∞ Kähler metric. The main result is to show that such a weak solution (with uniform L∞ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of K‐energy minimizers. The new technical ingredient is a W2, 2 regularity result for the Laplacian equation Δgu=f on Kähler manifolds, where the metric has only L∞ coefficients. It is well‐known that such a W2, 2 regularity (W2, p regularity for any p > 1) fails in general (except for dimension 2) for uniform elliptic equations of the form aij∂ij2u=f for aij ∊ L∞ without certain smallness assumptions on the local oscillation of aij. We observe that the Kähler condition plays an essential role in obtaining a W2, 2 regularity for elliptic equations with only L∞ elliptic coefficients on compact manifolds. © 2017 Wiley Periodicals, Inc.

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Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one

Abstract: Dedicated to Professor L. K. Hua, who was the teacher of Professor J. Q. Zhong. Zhong was the teacher of Guan in China and led him into this topic. Guan still have copies of papers of Calabi and Futaki which Zhong showed him in 1986.

Cited by 20 publications

References 14 publications

Type II almost-homogeneous manifolds of cohomogeneity one

Type II almost-homogeneous manifolds of cohomogeneity one

Affine compact almost-homogeneous manifolds of cohomogeneity one

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

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