scholarly journals Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds

2021 ◽  
Vol 35 ◽  
pp. 129-161
Author(s):  
Bruno Premoselli
Keyword(s):  
Einstein Metrics ◽  
Hyperbolic Manifolds ◽  
Negatively Curved ◽  
Ramified Covers

scholarly journals ALGEBRAIC FIBER SPACES AND CURVATURE OF HIGHER DIRECT IMAGES

2020 ◽  
pp. 1-56
Author(s):  
Bo Berndtsson ◽  
Mihai Păun ◽  
Xu Wang
Keyword(s):  
Line Bundle ◽  
Open Subset ◽  
Fiber Space ◽  
Zariski Open Subset ◽  
Negatively Curved ◽  
Ramified Covers

Let $p:X\rightarrow Y$ be an algebraic fiber space, and let $L$ be a line bundle on $X$ . In this article, we obtain a curvature formula for the higher direct images of $\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$ restricted to a suitable Zariski open subset of $X$ . Our results are particularly meaningful if $L$ is semi-negatively curved on $X$ and strictly negative or trivial on smooth fibers of $p$ . Several applications are obtained, including a new proof of a result by Viehweg–Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi–Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers – and the complications that are induced by them.

scholarly journals Positivity in Kähler–Einstein theory

2015 ◽  
Vol 159 (2) ◽  
pp. 321-338 ◽  
Author(s):  
GABRIELE DI CERBO ◽  
LUCA F. DI CERBO
Keyword(s):  
Scalar Curvature ◽  
Einstein Metrics ◽  
Cone Angle ◽  
Positive Scalar Curvature ◽  
Einstein Theory ◽  
Uniform Bounds ◽  
Projective Varieties ◽  
Edge Type ◽  
Negatively Curved ◽  
Kähler Einstein Metrics

AbstractTian initiated the study of incomplete Kähler–Einstein metrics on quasi–projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1-α) for α∈ (0, 1). In this paper we study how the existence of such Kähler–Einstein metrics depends on α. We show that in the negative scalar curvature case, if such Kähler–Einstein metrics exist for all small cone-angles then they exist for every α∈((n+1)/(n+2), 1), wherenis the dimension. We also give a characterisation of the pairs that admit negatively curved cone-edge Kähler–Einstein metrics with cone angle close to 2π. Again if these metrics exist for all cone-angles close to 2π, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.

THE RENORMALIZED VOLUME AND UNIFORMIZATION OF CONFORMAL STRUCTURES

2016 ◽  
Vol 17 (4) ◽  
pp. 853-912 ◽  
Author(s):  
Colin Guillarmou ◽  
Sergiu Moroianu ◽  
Jean-Marc Schlenker
Keyword(s):  
Einstein Metrics ◽  
Conformal Class ◽  
Conformal Boundary ◽  
Negatively Curved ◽  
Non Linear Equation ◽  
Higher Dimensional ◽  
Dimensional Version ◽  
Cotangent Space ◽  
Conformal Structures ◽  
Asymptotically Hyperbolic

We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definition depends on the choice of metric $h_{0}$ on $\unicode[STIX]{x2202}M$ in the conformal class at infinity determined by $g$, we denote it by $\text{Vol}_{R}(M,g;h_{0})$. We show that $\text{Vol}_{R}(M,g;\cdot )$ is a functional admitting a ‘Polyakov type’ formula in the conformal class $[h_{0}]$ and we describe the critical points as solutions of some non-linear equation $v_{n}(h_{0})=\text{constant}$, satisfied in particular by Einstein metrics. When $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while if $n=4$ this amounts to solving the $\unicode[STIX]{x1D70E}_{2}$-Yamabe problem. Next, we consider the variation of $\text{Vol}_{R}(M,\cdot ;\cdot )$ along a curve of AHE metrics $g^{t}$ with boundary metric $h_{0}^{t}$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_{n}(h)=\text{constant}$ and $\text{Vol}(\unicode[STIX]{x2202}M,h)=1$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space ${\mathcal{T}}(\unicode[STIX]{x2202}M)$ of conformal structures on $\unicode[STIX]{x2202}M$. We obtain, as a consequence, a higher-dimensional version of McMullen’s quasi-Fuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.

Invariant Einstein metrics on $\mathrm{SU}(n)$ and complex Stiefel manifolds

2020 ◽  
Vol 72 (2) ◽  
pp. 161-210 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Yusuke Sakane ◽  
Marina Statha
Keyword(s):  
Einstein Metrics ◽  
Stiefel Manifolds

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

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