UNIQUENESS OF EXTREMAL KÄHLER METRICS FOR AN INTEGRAL KÄHLER CLASS

2004 ◽  
Vol 15 (06) ◽  
pp. 531-546 ◽  
Author(s):  
TOSHIKI MABUCHI
Keyword(s):  
Scalar Curvature ◽  
Recent Result ◽  
Complex Manifold ◽  
Constant Scalar Curvature ◽  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kahler Metric

For an integral Kähler class on a compact connected complex manifold, an extremal Kähler metric, if any, in the class is unique up to the action of Aut 0(M). This generalizes a recent result of Donaldson (see [4] for cases of metrics of constant scalar curvature) and that of Chen [3] for c1(M)≤0.

scholarly journals Remarks on extremal Kähler metrics on ruled manifolds

1992 ◽  
Vol 126 ◽  
pp. 89-101 ◽  
Author(s):  
Akira Fujiki
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Einstein Metric ◽  
Kähler Metrics ◽  
Algebraic Subgroup ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kahler Metric ◽  
Compact Kähler Manifold

Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

PRECOMPACTNESS OF THE CALABI ENERGY

1996 ◽  
Vol 07 (02) ◽  
pp. 245-254 ◽  
Author(s):  
SANTIAGO R. SIMANCA
Keyword(s):  
Continuous Function ◽  
Scalar Curvature ◽  
Complex Manifold ◽  
Vector Fields ◽  
Convergent Sequence ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Holomorphic Vector Fields ◽  
Extremal Kähler Metrics ◽  
Kahler Metric

For any complex manifold of Kähler type, the L2-norm of the scalar curvature of an extremal Kähler metric is a continuous function of the Kähler class. In particular, if a convergent sequence of Kähler classes are represented by extremal Kähler metrics, the corresponding sequence of L2-norms of the scalar curvatures is convergent. Similarly, the sequence of holomorphic vector fields associated with a sequence of extremal Kähler metrics with converging Kähler classes is convergent.

Partially regular and cscK metrics

2020 ◽  
Vol 31 (10) ◽  
pp. 2050079
Author(s):  
Andrea Loi ◽  
Fabio Zuddas
Keyword(s):  
Scalar Curvature ◽  
Complex Manifold ◽  
Positive Constant ◽  
Bergman Kernel ◽  
Constant Scalar Curvature ◽  
Kähler Metric ◽  
Kähler Form ◽  
Kahler Metric

A Kähler metric [Formula: see text] with integral Kähler form is said to be partially regular if the partial Bergman kernel associated to [Formula: see text] is a positive constant for all integer [Formula: see text] sufficiently large. The aim of this paper is to prove that for all [Formula: see text] there exists an [Formula: see text]-dimensional complex manifold equipped with strictly partially regular and cscK metric [Formula: see text]. Further, for [Formula: see text], the (constant) scalar curvature of [Formula: see text] can be chosen to be zero, positive or negative.

scholarly journals Curvature and the c-projective mobility of Kähler metrics with hamiltonian 2-forms

2016 ◽  
Vol 152 (8) ◽  
pp. 1555-1575 ◽  
Author(s):  
David M. J. Calderbank ◽  
Vladimir S. Matveev ◽  
Stefan Rosemann
Keyword(s):  
Necessary Conditions ◽  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Local Classification ◽  
Complex Cone ◽  
Kahler Metric ◽  
Cone Metric

The mobility of a Kähler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kähler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kähler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kähler metrics with hamiltonian 2-forms, we describe explicitly the Kähler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric.

scholarly journals KÄHLER GEOMETRY OF TORIC VARIETIES AND EXTREMAL METRICS

1998 ◽  
Vol 09 (06) ◽  
pp. 641-651 ◽  
Author(s):  
MIGUEL ABREU
Keyword(s):  
Scalar Curvature ◽  
Convex Polytopes ◽  
Constant Scalar Curvature ◽  
Affine Function ◽  
Combinatorial Identity ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Total Integral ◽  
First Chern Class ◽  
Moment Polytope

A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn. Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kähler metrics on X, using only data on Δ. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝn. A construction, due to Calabi, of a 1-parameter family of extremal Kähler metrics of non-constant scalar curvature on [Formula: see text] is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known relation between the total integral of the scalar curvature of a Kähler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kähler class.

Sign in / Sign up

Export Citation Format

Share Document