Study Kapustin--Witten equations on ALE and ALF spaces, proving asymptotic estimates and vanishing theorems.
problem Analyzing solutions to Kapustin--Witten equations on specific gravitational instantons.
method Proving asymptotic estimates and using them to prove vanishing theorems.
result Proves vanishing theorems for certain cases of Kapustin--Witten equations.
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.
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.
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.
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.
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 …
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…
Study of D2 ALF manifolds via hyperkahler quotients of affine spaces.
problem Resolving flat orbifold quotients of R4. method Infinite-dimensional generalization of Kronheimer's construction, hyperkahler quotients of affine spaces, singular equivariant instantons, stability of Nahm data.
result Construction of the family of D2 ALF manifolds as a deformation of the flat orbifold (R3×S1)/Z2. 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.
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…
Classifies instantons on ALF multi-Taub-NUT spaces and ties them to bow solutions.
problem Classifying instantons on specific spaces.
method Uses Nahm's equations and bow solutions to construct instantons.
result Constructs holomorphic instanton bundles on multi-Taub-NUT spaces.
Instantons on ALF spaces constructed from bow data.
problem Constructing instantons on Asymptotically Locally Flat spaces.
method Using bow data to represent ALF spaces and their moduli spaces, constructing anti-self-dual connections.
result Anti-self-dual connections on ALF spaces are instantons with finite action.
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…
We compute the equivariant elliptic genera of several classes of ALE and ALF manifolds using localization in gauged linear sigma models. In the sigma model computation the equivariant action corresponds to chemical potentials for U(1) currents and the elliptic genera exhibit interesting pole structure as a function of …
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…
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.
Ricci flow preserves ALF structure on high-dimensional manifolds.
problem Preserving ALF structure under Ricci flow on high-dimensional manifolds.
method Developed a weighted Fredholm framework and a renormalized functional λ_ALF.
result Ricci flow preserves ALF structure on ALF n-manifolds with n≥4.
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 …
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.
Classifies 4D toric Hermitian ALF metrics with conical singularities.
problem Classifying specific types of 4D Riemannian metrics.
method Explicit formulas provided for classification.
result Examples of metrics with conical singularities have infinitely many distinct topologies.
The study examines deformations of Ricci-flat ALF spaces, showing they must be Hermitian.
problem Classifying and understanding deformations of Ricci-flat ALF spaces.
method Assuming suitable fall-off conditions, the authors show that deformations must be Hermitian and carry a non-trivial Killing vector field.
result The new Ricci-flat metric must belong to the family of previously classified metrics under mild additional hypotheses.
The paper constructs ALF Calabi-Yau metrics on specific manifolds.
problem Creating higher-dimensional ALF Calabi-Yau metrics.
method Using Taub-NUT deformation of hyperkähler cones and crepant resolutions.
result Existence of ALF Calabi-Yau metrics on certain manifolds.
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. In this paper we prove that over an asymptotically locally flat (ALF) Riemannian four-manifold the energy of an "admissible" SU(2) Yang--Mills is always integer. This result sharpens the previously known energy identity for such Yang--Mills instantons over ALF geometries. Furthermore we demonstrate that this statement …
In this paper, we introduce the notions of ALF Weyl connection and of associated mass, and we prouve the positive mass theorem for the ALF Weyl structures.
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.
Study describes ALF instantons with conical singularities.
problem Understanding ALF instantons with conical singularities.
method Applied techniques from previous work.
result Only 4D subfamilies can be smoothly compactified with conical singularities.
Study relationships between submanifolds and ambient Kahler 4-manifolds' fundamental groups.
problem Relationships between submanifolds and fundamental groups of Kahler 4-manifolds.
method Analyzes fundamental groups of embedded Levi-flat or pseudoconvex submanifolds in Kahler 4-manifolds.
result Fundamental group of M4 determined by the fundamental group of compact embedded Levi-flat or pseudoconvex submanifolds. Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its mo…
Proves positive mass theorems for specific ALF and ALG manifolds.
problem Proving positive mass theorems for ALF and ALG manifolds.
method Reduces the problem to non-existence of positive scalar curvature metrics on closed manifolds.
result Shows that incompressible conditions for S1 and T2 are sufficient for nonnegativity of mass. We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 …
Study on asymptotic behavior of Taub-NUT type solitons and construction of new ALF Calabi-Yau metrics.
problem Asymptotic behavior of steady gradient Kähler-Ricci solitons of Taub-NUT type.
method Determination of asymptotic cone, special case analysis, and construction of new metrics using Tian-Yau-Hein method.
result Construction of new ALF Calabi-Yau metrics on quotients of the Taub-NUT type soliton.
We prove positive mass theorems on ALF manifolds, i.e. complete noncompact manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length.
ALF reduces network parameters and operations by 70% and 61%, respectively, on embedded hardware.
problem Efficient deployment of deep learning models on resource-constrained hardware.
method Autoencoder-based low-rank filter-sharing technique.
result ALF achieves significant compression with minimal accuracy loss.
The study classifies Einstein metrics on a specific total space, revealing various behaviors and transitions.
problem Classifying conformally Kähler, U(2)-invariant, Einstein metrics. method Drawing on Derdziński's results from the 80s, the study classifies metrics on O(−m) for all m∈N. result The study discovers infinitely many 1-parameter families of Einstein metrics with different behaviors, including asymptotically hyperbolic, ALF, and cone angle limits.
We construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which …
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.
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.
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
We study inverse scattering for Δg+V on (X,g) a conformally compact manifold with metric g, with variable sectional curvature $-\alf^2(y)$ at the boundary and V∈C∞(X) not vanishing at the boundary. We prove that the scattering matrix at a fixed energies (λ1, λ2) in a suitable subset of $\mc$, de…
This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.
problem Characterize all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces.
method Unified construction using axi-symmetric harmonic functions and methods from scalar-flat Kähler metrics.
result All such metrics are ALF and belong to known families.
We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat metrics asymptotic to the product of a real line with a flat 3-manifold.
The paper proves mass nonnegativity for certain asymptotically locally flat manifolds.
problem Proving mass nonnegativity for asymptotically locally flat manifolds.
method Using positive mass theorems and scalar curvature assumptions.
result Mass is nonnegative for specified asymptotically locally flat manifolds.
This paper studies the space of L2 harmonic forms and L2 harmonic spinors on Taub-bolt, a Ricci-flat Riemannian 4-manifold of ALF type. We prove that the space of harmonic square-integrable 2-forms on Taub-bolt is 2-dimensional and construct a basis. We explicitly find a 2-parameter family of L2 zero mod…
New proof of instability for certain Einstein metrics.
problem Einstein metrics on specific 4-manifolds.
method Proving instability of conformally Kähler, Einstein metrics.
result Proven instability of certain Einstein metrics.
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…
Study shows instantons and monopoles decompose into U(1) components.
problem Understanding the asymptotic structure of instantons and monopoles.
method Analysis of multi-centered Taub-NUT manifolds, calorons, and monopoles on R^3.
result Instantons and monopoles asymptotically decompose into U(1) components.
Gravitational instantons are constructed as superpositions of Atiyah-Hitchin and Taub-NUT geometries.
problem Constructing gravitational instantons from Atiyah-Hitchin and Taub-NUT geometries.
method A gluing construction that captures the superposition of moduli spaces of centred SU(2) monopoles and Taub-NUT manifolds.
result Gravitational instantons are explicitly shown to be superpositions of Atiyah-Hitchin and Taub-NUT geometries.