scholarly journals Note on scalar curvature of extremal Kähler metrics on CP2#2CP2¯

2015 ◽  
Vol 38 ◽  
pp. 1-9
Author(s):  
Ti Yao Li
Keyword(s):  
Scalar Curvature ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics

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.

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.

ON THE CALABI ENERGY OF EXTREMAL KÄHLER METRICS

1995 ◽  
Vol 06 (06) ◽  
pp. 825-830 ◽  
Author(s):  
ANDREW D. HWANG
Keyword(s):  
Scalar Curvature ◽  
Global Minimum ◽  
A Priori ◽  
Critical Value ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Kähler Form ◽  
Extremal Kähler Metrics ◽  
Critical Metric ◽  
Kählerian Manifold

Fix a positive (1, 1)-class on a compact Kählerian manifold. Given a Kähler form representing this class, define its Calabi energy to be the L2-norm of its scalar curvature. This note proves that a critical metric for the Calabi energy, if any, is a global minimum among representatives of the chosen class, and that the critical value is determined a priori by the Kähler class. This answers affirmatively two questions of Calabi ([2, p. 99]).

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