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.
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 λALE 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.
We prove that if an ALE Ricci-flat manifold (M,g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we s…
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…
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.
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 introduce a natural definition of the renormalized volume of a 4-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of 4-dimensional Ricci-fla…
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.
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.
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.
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…
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…
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 Lp∩L∞. result Strong decay rates prove positive scalar curvature rigidity in Lp for each p∈[1,n−2n). 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…
We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an n-dimensional complex manifold such that the an+1 coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assu…
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…
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.
Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
problem Understanding the rigidity of Ricci-flat manifolds with specific curvature decay.
method Analyzing the gradient of the Green function and using curvature decay conditions.
result Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
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. …
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.
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.
In this paper, we study the gradient Ricci soliton equation on a complete Riemannian manifold. We show that under a natural decay condition on the Ricci curvature, the Ricci soliton is Ricci-flat and ALE.
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 R4m+4. result Recovery of Spin(7) metrics A8 and B8. The paper defines curvature at infinity for flat manifolds.
problem Defining curvature at the boundary of flat manifolds.
method Constructing coordinates at infinity for asymptotically flat ends.
result A Weyl tensor and renormalized volume defined at infinity.
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.
Study toric gravitational instantons using rod structures and inequalities.
problem Classify toric ALE/ALF instantons.
method Express signature in terms of rod structure, apply Hitchin-Thorpe inequalities, analyze rod structures with three turning points.
result Necessary conditions for rod structures of toric ALE/ALF instantons.
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…
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.
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 n≥3. The proof is based on the technique of Cheeger-Tian for Ricci-flat m…
In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat 4-manifolds with curvature decay and controlled holonomy. As a…
Study connects G2-structures to flat connections on compact 3-manifolds.
problem Understanding moduli spaces of G2-structures and flat connections. method Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. result Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. 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…
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.
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 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.
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.
Sharp mass bounds for ALE and ALF toric 4-manifolds.
problem Establishing lower bounds for the mass of ALE and ALF toric 4-manifolds.
method Using gravitational instantons and conical angle defects, the mass is bounded below by a sum of the mass of the corresponding instanton and an expression determined by conical angle defects.
result The mass of an ALE or ALF toric 4-manifold is not less than the mass of the corresponding gravitational instanton.
In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples…
This paper classifies Calabi-Yau manifolds near cones.
problem Classifying Calabi-Yau manifolds near cones.
method Complete classification of smooth complete Calabi-Yau manifolds asymptotic to a given cone.
result Classification of all smooth complete Calabi-Yau manifolds near a given cone.
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.
Formula derived for mass of almost Kähler manifolds, extending previous results.
problem Calculating mass in almost Kähler geometry.
method SpinC adaptation of Witten's proof, extending previous complex-geometric methods. result Explicit formula for ADM mass in terms of Hermitian scalar curvature and topological data.
New rigidity results for tensors on non-compact manifolds with curvature conditions.
problem Rigidity phenomena for tensors on non-compact Riemannian manifolds.
method Extending Bochner technique to non-compact settings, using Lichnerowicz Laplacian.
result Vanishing and rigidity of curvature tensors on Ricci-flat and Einstein manifolds.
Study 4D steady gradient Ricci solitons reducing to 3D manifolds.
problem Understanding 4D steady gradient Ricci solitons that reduce to 3D.
method Analyzing asymptotic geometry and curvature properties.
result 4D solitons either reduce to spherical space forms or the 3D Bryant soliton.
There are three main components to this article: (i) A formula for the eta invariant of the signature complex for any finite subgroup of SO(4) acting freely on S3 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…
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
New graphs with maximum degree 4 found to be Ricci-flat.
problem Characterizing Ricci-flat graphs with maximum degree 4.
method Defined Ricci curvature on graphs and used previous results to find all such graphs.
result All Ricci-flat graphs with maximum degree at most 4 were determined.
Constructing Einstein analogues with a non-zero cosmological constant
problem Constructing an Einstein analogue with a non-zero cosmological constant
method Proving the solution is either the Plebański-Demiański metric or has an anti-self-dual Weyl tensor
result For λ < 0, there is a conformal infinity separating two asymptotically hyperbolic metrics; one is globally conformal to an ALE scalar-flat Kähler metric; gravitational instantons with different topologies are constructed; the geometry is a 4-pole solution in the Calderbank-Pedersen classification