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.
Study of twistor spaces and minitwistor spaces for ALE gravitational instantons.
problem Characterizing the geometry of ALE gravitational instantons of type Amodd. method Analyzing the base locus of linear systems and using distinguished twistor lines.
result Explicit determination of images of certain twistor lines and description of real minitwistor lines.
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.
This paper is the sequel of our previous article "From ALE to ALF gravitational instantons", where we constructed ALF hyperkahler metrics on minimal resolutions of dihedral Kleinian singularities. In the present article we generalize the construction to smooth deformations of these Kleinian singularities, with help of …
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 explicitly calculate the analogue of the 't Hooft SU(2) Yang--Mills instantons on Gibbons--Hawking multi-centered gravitational instantons which come in two parallel families: the multi-Eguchi--Hanson, or A_k ALE gravitational instantons and the multi-Taub--NUT, or A_k ALF gravitational instantons. We …
Classification of toric self-dual Einstein gravitational instantons with negative cosmological constant
problem Classification of toric self-dual Einstein gravitational instantons
method Proving that if the conformal Kähler structure associated to one of the torus Killing fields is global and extends to an ALE manifold with no additional fixed points, then the corresponding self-dual Einstein instanton is precisely given by the infinite class of multipole solutions
result The classification of toric self-dual Einstein gravitational instantons is precisely given by the infinite class of multipole solutions
Sharp convergence theorem for Yang-Mills flow on ALE manifolds proved.
problem Proving convergence of Yang-Mills flow on ALE gravitational instantons.
method Noncompact version of the 'parabolic gap theorem'.
result Sharp convergence theorem for Yang-Mills flow on ALE 4-manifolds.
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 study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is R4 or $\mathrm{R}^3 …
In this paper, we study gravitational instantons (i.e., complete hyperkäler 4-manifolds with faster than quadratic curvature decay). We prove three main theorems: 1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH. 2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact comple…
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.
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.
We give an original analytic construction of hyperkahler ALF metrics on some ALE spaces of dihedral type, namely the spaces corresponding to minimal resolutions of Kleinian quotients relative to some binary dihedral group.
Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-v…
Classifies gravitational instantons with quadratic volume growth.
problem Classifying gravitational instantons with specific growth properties.
method Proves a classification theorem for ALG∗ gravitational instantons, determines topology, and proves existence of uniform coordinates. result Proves a relationship between ALG gravitational instantons of different orders.
Survey of recent progress in gravitational instantons
problem Classification of hyperkähler and Hermitian gravitational instantons
method Survey of recent progress in gravitational instantons
result Survey of recent progress in gravitational instantons
The paper proves unique characterization of gravitational instantons with specific volume growth.
problem Characterizing gravitational instantons with quadratic volume growth.
method Defining a period mapping and proving its surjectivity and openness.
result The periods uniquely characterize ALG∗ and ALG gravitational instantons up to diffeomorphism. Period maps surjective for certain gravitational instantons.
problem Understanding the uniformization of gravitational instantons.
method Uniformization theorems by Chen--Chen, Chen--Viaclovsky, Collins--Jacob--Lin, and Hein--Sun--Viaclovsky--Zhang.
result Period maps surjective for ALH*, ALG, and ALG* gravitational instantons.
Gravitational instantons are non-Kähler, providing a counterexample to Euclidean Black Hole Uniqueness.
problem Classical Euclidean Black Hole Uniqueness conjecture
method Analyzing the Chen-Teo gravitational instanton
result The Chen-Teo instanton is Hermitian and non-Kähler
Authors prove Torelli theorem for a specific type of gravitational instantons.
problem Identifying diffeomorphism types of gravitational instantons.
method Using mirror special Lagrangian fibrations and recent work on rational elliptic surfaces.
result Identify 10 diffeomorphism types of ALHb∗ gravitational instantons. This is our third paper in a series on the gravitational instantons. In this paper, we classify ALG and ALH gravitational instantons. In ALG case, we extend Hein's construction slightly and show that it's the only ALG gravitational instanton. In ALH case, we prove a Torelli-type theorem.
The Higgs field growth is studied on special geometric spaces, confirming a conjecture.
problem Growth of the Higgs field in special geometric spaces.
method Analyzing θ-Kapustin-Witten equations on ALX spaces. result Finite energy solutions on ALE and ALF instantons have vanishing commutator and flat connection.
Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cu…
This is our second paper in a series to study gravitational instantons, i.e. complete hyperkäler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instant…
Proves infinitesimal rigidity of Hermitian gravitational instantons.
problem Understanding the moduli space of Hermitian gravitational instantons.
method Proof of infinitesimal rigidity and integrability using boundary conditions and conformal Kähler properties.
result Completes the understanding of Hermitian gravitational instantons, both compact and non-compact.
Proves uniqueness of certain S1-symmetric gravitational instantons.
problem Proving uniqueness of S1-symmetric gravitational instantons. method Using a divergence identity and results from the G-signature theorem. result Proof of the S1-symmetric Euclidean Black Hole Uniqueness conjecture. Here we survey questions and results on the Hodge theory of hyperkaehler quotients, motivated by certain S-duality considerations in string theory. The problems include L^2 harmonic forms, Betti numbers and mixed Hodge structures on the moduli spaces of Yang-Mills instantons on ALE gravitational instantons, magnetic mo…
Study on mode stability of gravitational instantons of type D.
problem Proving mode stability of gravitational instantons of type D.
method Analogous to Lorentzian case, analyze Weyl curvature scalars satisfying a separable Teukolsky equation.
result Prove mode stability, showing no solutions compatible with regularity and asymptotic flatness.
Researchers construct explicit bundles for ALF metrics, revealing rational patching matrices for gravitational instantons.
problem Constructing explicit toric Ricci-flat metrics and their associated bundles.
method Explicit construction of patching matrices for ALF metrics and gravitational instantons.
result Rational form of patching matrices for gravitational instantons in the Chen--Teo family.
Review of gravitational instantons in physics.
problem Understanding solutions to Einstein equations in four dimensions.
method Lecture review of examples.
result Examples of gravitational instantons.
Gravitational instantons collapse to a punctured plane with a special Kahler metric.
problem The collapse of gravitational instantons from a complex structure limit.
method Analysis of a sequence of ALH*-gravitational instantons and their collapse to a punctured plane.
result The moduli space of pointed ALH*-gravitational instantons collapses to a punctured plane with a special Kahler metric.
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
Charge measurements for instantons and gravitational perturbations.
problem Evaluating charges in Hermitian non-Kähler Einstein 4-manifolds and their perturbations.
method Evaluation of charges via Killing spinors and perturbation analysis of gravitational instantons.
result Generic gravitational perturbations admit a closed 2-form measuring the charge change.
We construct ALE Calabi-Yau metrics with cone singularities along the exceptional set of resolutions of Cn/Γ with non-positive discrepancies. In particular, this includes the case of the minimal resolution of two dimensional quotient singularities for any finite subgroup Γ⊂U(2) acting freely on t…
The paper studies gravitational instantons with flat limits and finds elliptic regularity estimates.
problem Analyzing gravitational instantons with flat limits using elliptic analysis.
method Establishing elliptic regularity estimates and showing uniform constants for a family of metrics.
result The Laplacian is Fredholm and an isomorphism between specific weighted spaces.
The paper constructs gravitational instantons with unique collapse patterns.
problem Creating gravitational instantons with specific geometric properties.
method Using gluing construction, the metric collapses to quotients of R^3 with specific patterns.
result The constructed gravitational instantons have unique collapse patterns and exceptional points.
Researchers solve field equations for special gravitational instantons.
problem Solving field equations for conformally Kähler Riemannian four-manifolds.
method Developed a framework to solve the field equations for generalised gravitational instantons using conformal self-duality and cosmological Einstein-Maxwell.
result Found conformally self-dual and Einstein-Maxwell generalisations of specific geometries.
New gravitational instantons and non-holomorphic minimal spheres discovered.
problem Existence of non-holomorphic minimal spheres in ALF spaces.
method Gluing construction of ALF gravitational instantons using specific metrics.
result Existence of non-holomorphic minimal spheres in a general class of ALF spaces.
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…
Proves stability of gravitational instantons, proving operator positivity.
problem Stability of gravitational instantons.
method Riemannian analog of black hole mode stability for Hermitian, non-self-dual gravitational instantons.
result Teukolsky equation is a positive definite operator on Hermitian, non-self-dual gravitational instantons.
The paper connects hyperpolygon spaces to Higgs bundle moduli spaces via degenerations.
problem Modeling hyperkähler 4-manifolds and their degenerations.
method Using parabolic SL(2,C)-Higgs bundles and Nakajima quiver varieties.
result ALG-D4 spaces degenerate to ALE-D4 spaces under a limit. New polystability theory connects Calabi-Yau varieties to gravitational instantons.
problem Understanding the structure of Calabi-Yau manifolds and their metrics.
method Introducing a new concept of poly-stability and relating it to gravitational instantons.
result Polystability is equivalent to the existence of certain gravitational instantons.
Proves uniqueness and existence of toric gravitational instantons.
problem Proves uniqueness and existence of four-dimensional asymptotically flat, Ricci-flat, toric gravitational instantons.
method Adapting black hole uniqueness theorems to a harmonic map formulation of Ricci-flat metrics with torus symmetry.
result Proves that instantons are uniquely characterised by their rod structure and that for every admissible rod structure, there exists a smooth instanton.
Classifies instantons on a specific gravitational instanton and computes partition functions.
problem Classifying finite energy harmonic 2-forms and anti-self-dual Yang-Mills instantons.
method Analyzes U(1)-bundles and computes instantons explicitly. result Unique anti-self-dual Yang-Mills instantons exist and are described explicitly.
This paper proves Hitchin moduli spaces are ALG gravitational instantons.
problem Proving Hitchin moduli spaces are ALG gravitational instantons.
method Computing Torelli parameters for each Hitchin moduli space corresponding to different parabolic data.
result All Hitchin moduli spaces studied are ALG-D4 gravitational instantons. We construct a family of instanton metric obtained from new exact singular solutions for minimal surfaces by noticing the correspondence between minimal surfaces in the three dimesional Euclidean space and gravitational instantons possessing two killing vectors. By Calabi's correspondence, we derive a family of explici…
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).