The paper studies Einstein orbifolds and their desingularization, identifying obstructions to perturbing approximate Einstein metrics.
problem Identifying obstructions to transforming approximate Einstein metrics into actual Einstein metrics on desingularized orbifolds.
method Analyzes the desingularization of Einstein orbifolds and identifies the first obstruction to perturbing an approximate Einstein metric to an actual Einstein metric.
result Identifies the first obstruction to transforming approximate Einstein metrics into actual Einstein metrics on desingularized orbifolds.
The study approximates Ricci solitons and quasi-Einstein metrics on toric surfaces.
problem Investigate Ricci solitons and quasi-Einstein metrics on toric Kähler manifolds.
method General numerical method applied to toric Kähler manifolds, focusing on two generalizations of Einstein metrics.
result Numerical approximations and solutions found for specific metrics on toric surfaces.
We develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on CP2♯2CP2, which is conformal to an Einstein metric (the Chen-LeBrun-Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various g…
Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
The paper approximates Einstein tensor using finite elements.
problem Approximating Einstein tensor for piecewise polynomial metrics.
method Finite element method applied to Riemannian metrics.
result Convergence rate of O(hr+1) in H−2(Ω)-norm. Study quantization of Kähler-Einstein metrics using balanced metrics.
problem Approximating Kähler-Einstein metrics by balanced metrics.
method Use canonical Bergman metrics and introduce algebro-geometric obstructions.
result Existence and weak convergence of balanced metrics for CKE manifolds.
The paper examines conditions for Kähler-Einstein metrics on deformations of Fano manifolds.
problem Conditions for Kähler-Einstein metrics on deformations of Fano manifolds.
method Analyzes necessary and sufficient conditions, approximates Weil-Petersson metric, describes plurisubharmonicity of energy functional.
result Provides new conditions for the existence of Kähler-Einstein metrics on deformations of Fano Kähler-Einstein manifolds.
Compact complex manifolds get special metrics that minimize a functional.
problem Finding special metrics on compact complex manifolds.
method Finite dimensional approximations and generalized balanced metrics.
result Special metrics minimize a modified Mabuchi functional.
New Einstein metrics constructed on complex line bundle over CP1.
problem Constructing SU(2)-invariant negative Einstein metrics on complex line bundles. method Rigorous numerics to approximate, then fixed-point methods to perturb to genuine Einstein metrics.
result Complete, asymptotically hyperbolic Einstein metrics constructed.
Researchers prove existence of a special Einstein metric on a 12-dimensional sphere.
problem Proving the existence of a non-round Einstein metric invariant under a specific group action.
method Numerical analysis techniques were used to produce an approximate Einstein metric, which was then perturbed into a true Einstein metric.
result A novel O(3)imesO(10)-invariant Einstein metric on S12 was successfully constructed. Constructs stable Hilbert bundles on curves using Diophantine approximation.
problem Constructing stable Hilbert bundles on complex projective curves.
method Investigating arithmetic properties of the upper half plane and applying Diophantine approximation to bound Hermitian-Einstein metrics.
result Constructs Hilbert bundles with Hermitian-Einstein metrics on curves of positive genus.
The purpose of this paper is to prove the uniqueness of conical Kähler-Einstein metrics, under the condition that the twisted Ding-functional is proper. This is a generalization of the author's previous work, and we shall first investigate the uniqueness of twisted Kähler-Einstein metrics, and then use these smooth p…
The study shows how certain singular Kähler metrics can define Kähler currents and RCD spaces.
problem Understanding singular Kähler-Einstein metrics and their properties.
method Analyzing the properties of singular Kähler-Einstein metrics and their approximations.
result Singular Kähler-Einstein metrics can define Kähler currents and RCD spaces under certain conditions.
Develops variational approach for Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
problem Existence and characterization of Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
method Variational approach, algebraic approximation of singularities, function α_ω, continuity method.
result Many K-stable manifolds admit all possible Kähler-Einstein metrics with prescribed singularities.
To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g on M×(−1,0). W…
We find a Sasaki-Einstein metric from a CFT state in AdS5.
problem Finding a Sasaki-Einstein metric from a CFT state.
method Using supergravity in AdS5 and a superconformal gauge theory in R3,1 in the t'Hooft limit. result Explicit finite N-approximations to the Sasaki-Einstein metric. For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern--Gauss--Bonnet formula. Relations between…
In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when M admits…
The Kobayashi-Hitchin correspondence is proven for twisted vector bundles on Kähler manifolds.
problem Proving the Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles.
method Proved the correspondence and approximate correspondence for twisted holomorphic vector bundles on compact Kähler manifolds.
result A twisted holomorphic vector bundle is g−polystable if and only if it is g−Hermite-Einstein, and g−semistable if and only if it is approximate g−Hermite-Einstein. We apply Tian's method in Kahler-Einstein problem to prove that a conic K\''ahler metric with lower Ricci curvature bound can be approximated by smooth K\''ahler metrics with the same lower Ricci curvature bound. Furthermore, conic singularities here can be along a simple normal crossing divisor.
This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.
New compact, negatively curved Einstein 4-manifolds found without being locally homogeneous.
problem Finding new compact, negatively curved Einstein manifolds of dimension 4.
method Construction via branched covers of hyperbolic 4-manifolds, interpolation, and perturbation.
result First examples of compact, negatively curved Einstein 4-manifolds that are not locally homogeneous.
New probabilistic method constructs Kähler-Einstein metrics and suggests zero-free properties of zeta functions.
problem Existence and explicit formulas for Kähler-Einstein metrics on Fano varieties.
method Probabilistic construction involving canonical random point processes.
result Zero-free properties of Archimedean zeta functions and their relation to Langlands program.
The existence of \emph{weak conical Kähler-Einstein} metrics along smooth hypersurfaces with angle between 0 and 2π is obtained by studying a smooth continuity method and a \emph{local Moser's iteration} technique. In the case of negative and zero Ricci curvature, the C0 estimate is unobstructed; while in the ca…
We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with s>23. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…
The global holomorphic α-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the α-invariant of toric Fano manifolds.
We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle F over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on F coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson.…
Bismut Einstein metrics on complex manifolds are Kähler Einstein or Bismut Ricci flat.
problem Characterizing Bismut Einstein metrics on compact complex manifolds.
method Observing the (2,0)-part of Bismut Ricci form and using it to prove properties of the metrics.
result Bismut Einstein metrics with non-zero Einstein constant are Kähler Einstein, and those with zero are Bismut Ricci flat.
Survey explains Kahler-Einstein metrics construction from interpolation problems.
problem Kahler-Einstein metrics on compact complex manifolds.
method Statistical mechanical construction from interpolation problems.
result Probabilistic construction of Kahler solutions to Einstein's equations.
Study investigates Einstein flow stability and convergence with matter sources.
problem Stability and convergence of Einstein flow with matter sources.
method Incorporates matter sources into the Einstein flow and examines stability and convergence.
result Similar conclusions can be drawn about the evolution of manifolds to approximate homogeneity and isotropy.
The study explores Einstein Kropina metrics on Lie groups and homogeneous spaces.
problem Investigating Einstein Kropina metrics on Lie groups and homogeneous spaces.
method Constructing Einstein Kropina metrics on Lie groups and homogeneous spaces using specific procedures.
result Classification and construction of Einstein Kropina metrics on various Lie groups and homogeneous spaces.
New findings on instability of Einstein metrics in warped products.
problem Investigating the stability of warped product Einstein metrics.
method Exploiting the relationship between warped product Einstein metrics, quasi-Einstein metrics, and Ricci solitons, introducing a new destabilising perturbation (the Ricci variation).
result Certain infinite families of warped product Einstein metrics are unstable in high dimensions.
Study on Einstein deformations of negative Kähler Einstein metrics.
problem Understanding Einstein deformations of Kähler Einstein metrics.
method Relate second order Einstein deformation theory to complex geometry, gauge normalise, and use Taylor expansion.
result Taylor expansion to order two of an Einstein deformation is determined by h12 and the divergence of the Kodaira-Spencer bracket. The paper classifies and estimates invariant Einstein metrics on Ledger-Obata spaces.
problem Classifying and estimating invariant Einstein metrics on Ledger-Obata spaces.
method Investigates invariant Einstein metrics on Ledger-Obata spaces Fm/diag(F), focusing on F4/diag(F) and general cases. result Classifies invariant Einstein metrics on F4/diag(F) and estimates the number of such metrics on general Ledger-Obata spaces. We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einst…
The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that, if E is semi-stable, then Donaldson's functional is bounded from below. This implies that E admits an approximate Hermitian-Einstein structure, generalizing a classic result…
Paper discusses conditions for deforming coupled Kähler-Einstein metrics.
problem Conditions for deforming coupled Kähler-Einstein metrics.
method Analyzes deformation of coupled Kähler-Einstein metrics on Fano manifolds.
result Necessary and sufficient condition for deformation of coupled Kähler-Einstein metrics.
The study examines the stability of Einstein metrics on fiber bundles.
problem Linear stability of Einstein metrics on fiber bundles.
method Deriving instability conditions, studying Riemannian product structures, estimating coindices, and investigating circle bundle constructions.
result Obtained a rigidity result for linearly stable Einstein metrics of circle bundle constructions.
The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of Kähler Einstein metrics on Fano manifolds, based on Chen's weak C^{1,1} geodesic without using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the…
Homogeneous Einstein metrics on Euclidean spaces are shown to be Einstein solvmanifolds.
problem Characterizing homogeneous Einstein metrics on Euclidean spaces.
method Using periodic, integrally minimal foliations and geometric flow induced by the orbit-Einstein condition.
result Homogeneous Einstein metrics on Euclidean spaces are proven to be Einstein solvmanifolds.
Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
problem Characterize compact quasi-Einstein metrics with constant scalar curvature.
method Connection to Sasakian geometry and circle bundles over Einstein metrics.
result Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
The study finds positive Einstein metrics on complex manifolds and spheres.
problem Existence of positive Einstein metrics on complex manifolds and spheres.
method Investigation of cohomogeneity one metrics and use of known Einstein metrics.
result Existence of positive Einstein metrics on S4m+4 and S8. Formula found for Kähler-Einstein metric existence obstruction.
problem Obtaining Kähler-Einstein metrics on manifolds.
method Residue formula for Futaki-Zhang obstruction.
result Found coupled Kähler-Einstein metrics on a specific toric Fano manifold.
New Einstein metrics created by modifying hyperbolic infinity.
problem Creating new Einstein metrics.
method Perturbing conformal infinity of geometrically finite hyperbolic metrics and applying inverse function theorem.
result Construct new examples of Einstein metrics.
New examples found of complex manifolds with special metrics.
problem Existence of Kähler-Einstein metrics on certain complex manifolds.
method Using Hultgren's polytope formulation, constructing explicit examples of toric Fano manifolds.
result Found examples of projective bundles that admit coupled Kähler-Einstein metrics but no ordinary Kähler-Einstein metrics.
Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.
problem Characterizing log del Pezzo surfaces with specific geometric properties.
method Examining two classes of non-toric log del Pezzo surfaces and analyzing their geometric properties.
result Examples found that admit Kähler-Ricci solitons but not Sasaki-Einstein cone links.
Defines geodesic-Einstein metrics and their relation to nonlinear stabilities.
problem Nonlinear stabilities of line bundles over holomorphic fibrations.
method Introduces geodesic-Einstein metrics and a Donaldson type functional.
result Geodesic-Einstein metrics minimize the Donaldson type functional.
The paper finds Einstein-Randers metrics on specific homogeneous spaces.
problem Finding metrics on specific homogeneous spaces.
method Proved existence of Einstein metrics and then showed existence of Non-Riemannian Einstein-Randers metrics.
result Specific homogeneous spaces admit Non-Riemannian Einstein-Randers metrics.