We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …
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48 results for “Ricci-flat Kähler ALE metric”
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
problem Stability and instability of ALE Ricci-flat metrics.
method Use of a modified Perelman's λ-functional and Lojasiewicz inequality.
result Demonstrates dynamical instability of ALE Ricci-flat metrics.
New functional proves mass positivity for ALE metrics.
problem Proving mass positivity for ALE metrics with Ricci-flat deformations.
method Introduced a new functional and proved its monotonicity and Lojasiewicz-Simon inequality.
result Established that small perturbations of Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass.
In this paper we prove that for a complete, connected and oriented Käler affine manifold of dimension if it is Kähler affine Ricci flat or the Khler affine scalar curvature (), then the universal covering manifold of is isometric to the Euclidean n-space $…
Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classificatio…
ALE Ricci-flat manifolds stable under Ricci flow if integrable.
problem Stability of ALE Ricci-flat manifolds under Ricci flow.
method Adapting Tian's approach for closed manifolds, proving integrability for ALE Calabi-Yau manifolds.
result ALE Ricci-flat manifolds are dynamically stable if integrable.
The paper classifies 4D Ricci-flat ALE manifolds and their properties.
problem Characterizing 4D Ricci-flat ALE manifolds and their properties.
method Analyzing the correspondence between ALE gravitational instantons and Kähler orbifolds, and proving properties of these structures.
result There is a one-to-one correspondence between Hermitian non-Kähler ALE gravitational instantons and Bach-flat Kähler orbifolds in 2D.
The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.
problem Understanding the convergence behavior of Ricci-flat conifolds.
method Analyzing the Lichnerowicz Laplacian and tensor fields on cones, computing indicial roots and metric convergence orders.
result Lower bounds for metric convergence orders on Ricci-flat conifolds.
Proves unique ALE instanton with toric Hermitian structure.
problem Classify Ricci flat ALE instantons with toric Hermitian non-Kähler structure.
method Direct global analysis of Tod form in Weyl-Papapetrou coordinates, avoiding toric Kähler geometry.
result Eguchi-Hanson instanton is the only smooth, Ricci flat, ALE instanton with toric Hermitian non-Kähler structure.
We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are metrics obtained by Tian-Yau techniques. The proof relies on a construction of good compactifications of Q-Gorenstein deformations of quotient surface singularities…
Two conjectures on Ricci-flat metrics verified under specific conditions.
problem Properties of Ricci-flat metrics on complex manifolds.
method Analyzing conjectures and verifying them under specific conditions.
result Conjectures verified for certain types of metrics on complex surfaces.
We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup…
Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical Kähler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat Kähler cone admits a Ricci-flat Kä…
The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
Constructs Einstein metrics on manifolds with specific orbits.
problem Finding Einstein metrics on manifolds with given orbits.
method Continuous families of metrics constructed using vector bundles and .
result Recovery of metrics and .
New Ricci flows found with Einstein orbifolds at infinity.
problem Ancient and immortal Ricci flows with Einstein orbifolds at infinity.
method Continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows constructed.
result Found continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows on specific manifolds.
New Einstein RCD spaces found with cone singularities.
problem Existence of Einstein RCD spaces with cone singularities.
method Characterization of RCD spaces and cone singularity analysis.
result Existence of smooth non-compact 4-manifolds with ALE Ricci-flat RCD(0,4) metrics.
Stability and rigidity of Ricci-flat ALE manifolds proven.
problem Stability and rigidity of Ricci-flat ALE manifolds.
method Proved stability and rigidity of ALE manifolds with a parallel spinor under Ricci flow, given initial metrics close in .
result Strong decay rates prove positive scalar curvature rigidity in for each .
Study properties of para-Kähler manifolds with conformal Einstein soliton metrics.
problem Properties of para-Kähler manifolds with conformal Einstein soliton metrics.
method Investigated curvature properties of para-Kähler manifolds admitting conformal Einstein soliton.
result Certain curvature properties of para-Kähler manifolds were studied.
Researchers define and analyze the renormalized volume of 4D Ricci-flat ALE spaces.
problem Understanding the properties of 4D Ricci-flat ALE spaces.
method Introducing a new definition of renormalized volume and proving its properties.
result The renormalized volume is always less than or equal to zero, with equality for specific spaces.
The author has proved that a crepant resolution Y of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,\R). These manifolds are generalizations of the Ricci-flat ALE Kähler spaces known by the work of P. Kronheimer, D. Joyce and others. …
The study shows that positive scalar curvature 4-manifolds can be desingularized.
problem Proving the non-existence of singular positive scalar curvature metrics.
method Explicit constructions from scalar-flat Kähler ALE surfaces and desingularization process.
result Desingularization of positive scalar curvature 4-manifolds is possible.
Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.
problem Behavior of harmonic 2-forms on K3 surfaces with Ricci-flat metrics.
method Analysis of convergence of harmonic forms to flat 4D orbifold.
result Decomposition of harmonic 2-forms into converging subspaces.
Let G be a nontrivial finite subgroup of U(m) acting freely on C^m - 0. Then C^m/G has an isolated quotient singularity at 0. Let X be a resolution of C^m/G, and g a Kahler metric on X. We say that g is Asymptotically Locally Euclidean (ALE) if it is asymptotic in a certain way to the Euclidean metric on C^m/G. In this…
We show that any asymptotically locally Euclidean (ALE) metric which is obstruction-flat or extended obstruction-flat must be ALE of a certain optimal order. Moreover, our proof applies to very general elliptic systems and in any dimension . The proof is based on the technique of Cheeger-Tian for Ricci-flat m…
This is the sequel to "Asymptotically Locally Euclidean metrics with holonomy SU(m)", math.AG/9905041. Let G be a subgroup of U(m), and X a resolution of C^m/G. We define a special class of Kahler metrics g on X called Quasi Asymptotically Locally Euclidean (QALE) metrics. These satisfy a complicated asymptotic conditi…
The paper shows how to create scalar flat metrics with very large ADM mass.
problem Understanding the ADM mass of scalar flat Kähler ALE spaces.
method Blowing up points in ALE spaces to increase ADM mass.
result It is possible to produce scalar flat metrics with arbitrarily large ADM mass.
The paper studies Ricci curvature on Kähler-Ricci flow.
problem Analyzing Ricci curvature on Kähler-Ricci flow.
method Examining n-dimensional compact Kähler manifolds with semi-ample canonical line bundles under Kähler Ricci Flow.
result Ricci curvature converges to negative of generalized Kähler Einstein metric locally away from singular set.
Simplified proof of stability for Ricci flow near ALE metrics.
problem Stability of Ricci flow near ALE metrics with integrable deformations.
method Equivalence between integrability and almost-orthogonality property of Ricci-DeTurck tensor, analysis in weighted Holder spaces.
result Dynamical stability of Ricci flow near linearly stable Ricci-flat ALE metrics.
The paper proves conditions for Kähler manifolds with negative curvature.
problem Conditions for existence of Kähler-Einstein metrics and holomorphic curves.
method Analyzes compact Kähler manifolds homotopic to negatively curved Riemannian manifolds.
result Compact Kähler manifolds with negative curvature admit Kähler-Einstein metrics of general type.
The paper studies the geometry of flat manifolds with controlled holonomy.
problem Investigating the geometry of asymptotically flat manifolds with specific properties.
method Analyzes torus fibrations and Hitchin-Thorpe inequalities for Ricci-flat 4-manifolds.
result Proves that certain flat metrics on 4-manifolds are isometric to Euclidean or Taub-NUT.
Two new proofs provide Eguchi-Hanson metrics as ALE bubbles for Kummer constructions of K3 metrics.
problem Constructing Ricci-flat Kähler metrics on the K3 surface with special holonomy.
method Singular perturbation and weighted function space analysis.
result Large families of compact hyper-Kähler orbifolds as volume non-collapsed limits of Kummer constructions.
The paper explores metric reduction in generalized geometry and constructs holomorphic bundles.
problem Metric reduction in generalized geometry and constructing holomorphic bundles.
method Investigation of Bismut connections and construction of metric generalized principal bundles.
result Construction of a family of generalized holomorphic line bundles over .
Existence proved for specific types of gravitational instantons.
problem Existence of toric ALE and ALF gravitational instantons.
method Established existence and uniqueness results for ALE and ALF gravitational instantons.
result Existence of a unique, Ricci-flat, toric ALE and ALF gravitational instanton for every admissible rod structure.
Holomorphic Euler number vanishes for certain Kähler manifolds.
problem Finding obstructions for Kähler manifolds with specific curvature properties.
method Vanishing theorem of Dolbeault-Morse-Novikov cohomology.
result Holomorphic Euler number of Kähler manifolds with almost nonnegative Ricci curvature vanishes.
Study classifies gravitational instantons based on their asymptotic geometry.
problem Classifying gravitational instantons based on their asymptotic properties.
method Investigation of asymptotic geometry of Hermitian non-Kähler Ricci-flat metrics.
result All Hermitian non-Kähler gravitational instantons can be compactified to log del Pezzo surfaces.
Positive scalar curvature implies small 2-systoles in Kähler manifolds
problem Finding topologically non-trivial 2-spheres with small area in Kähler manifolds
method Using index theoretic methods
result Quantitative upper bounds on the 2-systole
The paper characterizes Eguchi-Hanson space and its higher-dimensional analogs using Lichnerowicz Laplacian.
problem Characterizing complete Ricci-flat ALE orbifolds.
method Analytical proof using Lichnerowicz Laplacian and dimension constraints.
result Uniqueness of Eguchi-Hanson space and its higher-dimensional analogs among Ricci-flat Kähler ALE orbifolds.
Study connects -structures to flat connections on compact 3-manifolds.
problem Understanding moduli spaces of -structures and flat connections.
method Equivalence between moduli spaces of -structures and flat connections on compact 3-manifolds.
result Equivalence between moduli spaces of -structures and flat connections on compact 3-manifolds.
Study on Einstein deformations and desingularizations, finding some 4-orbifolds cannot be limits of smooth metrics.
problem Whether every Einstein 4-orbifold is a limit of smooth Einstein 4-manifolds.
method Analysis of integrability of deformations through variations of Schoen's Pohozaev identity and introduction of preserved integral quantities.
result Spherical and hyperbolic 4-orbifolds with the simplest singularities cannot be limits of smooth Einstein 4-manifolds.
Paper shows non-CSC HCMU metrics can't be isometrically immersed into 3D space forms.
problem Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
method Using moving frames to demonstrate the impossibility of isometric minimal immersion.
result Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifold…
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.
Given a compact constant scalar curvature Kaehler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kaehler Ricci-flat resolution, we find sufficient conditions on the position of the singular points to ensure the existence of a global constant scalar curvature Kaehler desingula…
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 examines complete Kähler manifolds with nonnegative Ricci curvature and discovers rigidity properties.
problem Characterizing and understanding properties of complete Kähler manifolds with nonnegative Ricci curvature.
method Analyzes volume growth, scalar curvature, and curvature decay to establish rigidity results.
result Complete Ricci flat Kähler manifolds with Euclidean volume growth are rigid, with unique tangent cones.
K{ä}hler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functionalmath.DG
Develops Kähler geometry on new varieties for canonical metrics.
problem No specific problem stated; focuses on new varieties.
method Introduces new varieties, develops Kähler geometry, associates convex functions with metrics.
result Provides expression for Mabuchi functional and combinatorial sufficient condition of properness.
There are three main components to this article: (i) A formula for the eta invariant of the signature complex for any finite subgroup of acting freely on is given. An application of this is a non-existence result for Ricci-flat ALE metrics on certain spaces. (ii) A formula for the orbifold correcti…