scholarly journals Maxwell-Einstein metrics on Certain CP1 Bundles

2021 ◽  
Author(s):  
Daniel Guan
Keyword(s):  
Fiber Bundle ◽  
Einstein Metrics ◽  
Einstein Metric

Abstract In this paper, we prove that for certain fiber bundle there is a Maxwell-Einstein metric conformally related to any given Kähler class.

scholarly journals Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junyan Cao ◽  
Henri Guenancia ◽  
Mihai Păun
Keyword(s):  
General Type ◽  
Einstein Metrics ◽  
Kodaira Dimension ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
Fiber Space ◽  
Generic Fiber ◽  
Lelong Numbers ◽  
Kähler Einstein Metrics

Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1

1993 ◽  
Vol 114 (3) ◽  
pp. 443-451
Author(s):  
Al Vitter
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Vector Bundles ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Holomorphic Vector Bundles ◽  
Kähler Form ◽  
Points Of View ◽  
The Stability ◽  
Complex Projective

Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

Homogeneous Einstein Finsler Metrics on -dimensional Spheres

2018 ◽  
Vol 62 (3) ◽  
pp. 509-523
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Constant Sectional Curvature ◽  
Einstein Metric ◽  
Finsler Metrics ◽  
Dimensional Sphere ◽  
Order Ordinary Differential Equation ◽  
Canonical Metric ◽  
Homogeneous Einstein Metrics

AbstractIn this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.

scholarly journals Ricci-flat and Einstein pseudoriemannian nilmanifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 170-193 ◽  
Author(s):  
Diego Conti ◽  
Federico A. Rossi
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Ricci Flat ◽  
Open Questions ◽  
Expository Paper ◽  
New Criterion

AbstractThis is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.

scholarly journals Smooth structures and Einstein metrics on

2009 ◽  
Vol 147 (2) ◽  
pp. 409-417 ◽  
Author(s):  
RAREŞ RǍSDEACONU ◽  
IOANA ŞUVAINA
Keyword(s):  
Scalar Curvature ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Smooth Structure ◽  
Smooth Structures ◽  
Higher Dimensional

AbstractWe show that each of the topological 4-manifolds $\bcp^2\# k\overline{\bcp^2}$, for k = 5, 6, 7, 8 admits a smooth structure which has an Einstein metric of scalar curvature s > 0, a smooth structure which carries an Einstein metric with s < 0 and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We also exhibit new examples of higher dimensional manifolds carrying Einstein metrics of both positive and negative scalar curvature.

THE SASAKI–RICCI FLOW

2010 ◽  
Vol 21 (07) ◽  
pp. 951-969 ◽  
Author(s):  
KNUT SMOCZYK ◽  
GUOFANG WANG ◽  
YONGBING ZHANG
Keyword(s):  
Ricci Flow ◽  
Einstein Metrics ◽  
Positive Case ◽  
Einstein Metric ◽  
Well Posedness ◽  
Long Time Existence ◽  
Null Case ◽  
Long Time ◽  
Time Existence

In this paper, we introduce the Sasaki–Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki–Einstein metric. Hence it is an odd-dimensional counterpart of the Kähler–Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Cao's results on the Kähler–Ricci flow.

scholarly journals Dynamical construction of Kähler-Einstein metrics

2010 ◽  
Vol 199 ◽  
pp. 107-122
Author(s):  
Hajime Tsuji
Keyword(s):  
Einstein Metrics ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
General Fiber ◽  
New Construction ◽  
Kähler Einstein Metrics ◽  
Singular Hermitian Metric ◽  
Projective Morphism

AbstractIn this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphismf:X→Swith connected fibers such that a general fiber has an ample canonical bundle, and for a positive integerm, we construct a canonical singular Hermitian metrichE,monwith semipositive curvature in the sense of Nakano.

scholarly journals KÄHLER–EINSTEIN METRICS WITH CONE SINGULARITIES ON KLT PAIRS

2013 ◽  
Vol 24 (05) ◽  
pp. 1350035 ◽  
Author(s):  
HENRI GUENANCIA
Keyword(s):  
Einstein Metrics ◽  
Einstein Metric ◽  
Technical Assumption ◽  
Kähler Einstein Metrics ◽  
Smooth Locus

We prove that any Kähler–Einstein metric attached to a klt pair (X, D) has cone singularities along D on the log-smooth locus of the pair, under some technical assumption on the cone angles.

Ricci curvature in Kähler geometry

2017 ◽  
Vol 28 (09) ◽  
pp. 1740009
Author(s):  
Song Sun
Keyword(s):  
Ricci Curvature ◽  
Einstein Metrics ◽  
Complex Analysis ◽  
Einstein Metric ◽  
Tangent Cones ◽  
Kahler Geometry ◽  
Fano Manifolds ◽  
Winter School ◽  
Kähler Einstein Metrics ◽  
Hausdorff Limits

These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.

scholarly journals DEGENERATION OF KÄHLER–EINSTEIN MANIFOLDS I: THE NORMAL CROSSING CASE

2004 ◽  
Vol 06 (02) ◽  
pp. 301-313
Author(s):  
WEI-DONG RUAN
Keyword(s):  
Einstein Metrics ◽  
Total Space ◽  
Kähler Manifolds ◽  
Einstein Metric ◽  
Kahler Manifolds ◽  
Einstein Manifolds ◽  
Normal Crossing ◽  
Smooth Part ◽  
Kähler Einstein Metrics

In this paper we prove that the Kähler–Einstein metrics for a degeneration family of Kähler manifolds with ample canonical bundles converge in the sense of Cheeger–Gromov to the complete Kähler–Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil–Peterson metric in this case.

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