Boundary behavior of Kähler-Einstein metric of negative ricci curvature over quasi-projective manifolds with boundary of general type

2021 ◽  
Vol 44 (1) ◽  
Author(s):  
Shin Kikuta
Keyword(s):  
Ricci Curvature ◽  
General Type ◽  
Boundary Behavior ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Projective Manifolds

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

2021 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Boundary Condition ◽  
Smooth Boundary ◽  
Type Theorem ◽  
Type Inequality ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Einstein Manifolds ◽  
Conformal Class ◽  
Totally Geodesic ◽  
Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

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).

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 A note on Lang's conjecture for quotients of bounded domains

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Sébastien Boucksom ◽  
Simone Diverio
Keyword(s):  
Plurisubharmonic Function ◽  
General Type ◽  
Universal Cover ◽  
Projective Manifold ◽  
Bounded Domains ◽  
Final Version ◽  
Lang's Conjecture ◽  
Projective Manifolds ◽  
Lang’S Conjecture ◽  
Complex Projective

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains. Comment: 10 pages, no figures, comments are welcome. v3: following suggestions made by the referee, the exposition has been improved all along the paper, we added a variant of Theorem A which includes manifolds whose universal cover admits a bounded psh function which is strictly psh just at one point, and we added a section of examples. Final version, to appear on \'Epijournal G\'eom. Alg\'ebrique

scholarly journals Comparison Geometry of Manifolds with Boundary under a Lower Weighted Ricci Curvature Bound

2018 ◽  
Vol 72 (1) ◽  
pp. 243-280
Author(s):  
Yohei Sakurai
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Suitable Condition ◽  
Manifolds With Boundary ◽  
Weighted Mean ◽  
Curvature Condition ◽  
Curvature Bound ◽  
Comparison Geometry ◽  
Weighted Mean Curvature ◽  
Weighted Ricci Curvature

AbstractWe study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.

scholarly journals Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire

2014 ◽  
Vol 151 (2) ◽  
pp. 351-376 ◽  
Author(s):  
Fréderic Campana ◽  
Benoît Claudon ◽  
Philippe Eyssidieux
Keyword(s):  
General Type ◽  
Linear Representation ◽  
Kähler Manifolds ◽  
Fundamental Groups ◽  
Kahler Manifolds ◽  
Shafarevich Conjecture ◽  
Classical Work ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Holomorphic Convexity

AbstractWe extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kähler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture.

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