New ansatz for generalized Kähler surfaces derived from hyperKähler ansatz.
problem Deriving a new ansatz for generalized Kähler surfaces.
method Generalized Gibbons-Hawking ansatz for nondegenerate Poisson structure with biholomorphic S1 action. result Classification of all complete solutions with smallest symmetry group.
We describe a quaternionic-based Ansatz generalizing the Gibbons-Hawking Ansatz to a class of hyperkähler metrics with hidden symmetries. We then apply it to obtain explicit expressions for gravitational instanton metrics of type Dk.
We show that a complete simply-connected hyperkaehler 4-manifold with an isometric triholomorphic circle action is obtained from the Gibbons-Hawking ansatz with some suitable harmonic function.
Barrier methods classify minimal submanifolds in hyperkaehler spaces.
problem Classifying compact minimal submanifolds in hyperkaehler spaces.
method Barrier argument and strong stability condition analysis.
result Results towards a classification of compact minimal submanifolds.
We discuss the Ricci-flat `model metrics' on C2 with cone singularities along the conic {zw=1} constructed by Donaldson using the Gibbons-Hawking ansatz over wedges in R3. In particular we describe their asymptotic behavior at infinity and compute their energies.
Analogous to 4D, new Ansatz for quaternionic Kähler spaces with a free action.
problem Describing the geometry of quaternionic Kähler spaces with a free action.
method Gibbons-Hawking-like Ansatz based on quaternionic Kähler moment map.
result Explicit equivariant completion of twistor space construction.
A homogeneous Gibbons-Hawking ansatz is described, leading to 4-dimensional hyperkahler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkahler metrics that are, in a suitable sense, comp…
The abstract discusses special Lagrangians and their flow, proving conjectures and observing related phenomena.
problem Existence and long-time existence of special Lagrangian representatives and Lagrangian mean curvature flow.
method Gibbons-Hawking ansatz, circle-invariant hyperkaehler 4-manifolds, Calabi-Yau 2-folds, Thomas conjecture, Thomas-Yau conjecture.
result Proves versions of the Thomas conjecture and Thomas-Yau conjecture.
Study complex structures of hyperkähler manifolds with infinite type.
problem Understanding complex structures of hyperkähler four-manifolds with infinite topological type.
method Analyzing the Gibbons-Hawking ansatz and proving biholomorphic relationships to hypersurfaces or minimal resolutions of singular surfaces.
result For almost all complex structures, the manifold is biholomorphic to a hypersurface in \(\mathbb{C}^3\). For the remaining, it is biholomorphic to the minimal resolution of a singular surface under certain conditions.
Complete Calabi-Yau metrics on C^{N+1} are constructed.
problem Constructing complete Calabi-Yau metrics on C^{N+1} for N >= 3.
method Generalized Gibbons-Hawking ansatz with gluing procedure to overcome volume-form defects.
result Calabi-Yau metrics with tangent cone R^N are constructed.
Study Spin(7)-manifolds with a 4-torus action using a symmetric matrix ansatz.
problem Characterize Spin(7)-manifolds with a 4-torus action. method Provide a Gibbons-Hawking type ansatz using a symmetric 4imes4-matrix of functions. result First known Spin(7)-manifolds with a rank 4 symmetry group and full holonomy. Existence of Ricci flat metric on Kummer K3 surface proven.
problem Proving existence of Ricci flat metric on Kummer K3 surface.
method General strategy of Donaldson's gluing construction, compact elliptic theory on usual Hölder and Sobolev spaces, explicit isometry to Gibbons-Hawking ansatz.
result Existence of a Ricci flat metric on the Kummer K3 surface.
We give a new construction of Ricci-flat self-dual metrics which is a natural extension of the Gibbons--Hawking ansatz. We also give characterisations of both these constructions, and explain how they come from harmonic morphisms.
We use a generalization of the Gibbons-Hawking ansatz to study the behavior of certain non-compact Calabi-Yau manifolds in the large complex structure limit. This analysis provides an intermediate step toward proving the metric collapse conjecture for toric hypersurfaces and complete intersections.
A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of …
New minimal 2-spheres found in hyperkähler 4-manifolds, unstable and not holomorphic.
problem Characterizing stable minimal surfaces in hyperkähler 4-manifolds.
method Gluing construction using Scherk and Taub-NUT surfaces, harmonic map parametrization.
result Existence of unstable minimal 2-spheres with degree-1 Gauss lift, not holomorphic.
The paper studies critical points and flows of a G2-Hilbert functional on manifolds with circle actions.
problem Critical points and flows of the G2-Hilbert functional on manifolds with S1-actions. method Analysis of S1-invariant G2-structures, reduction to a 6-dimensional quotient, and derivation of a negative L2-gradient flow. result The unnormalized flow admits only trivial stationary configurations: flat connection, scalar-flat base metric, and constant fiber length.
We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic…
Study of harmonic maps and instantons in 4D.
problem Constructing non-spin, simply-connected Ricci-flat 4-manifolds.
method Non-perturbative approach ruling out conical singularities from axisymmetric harmonic maps.
result Systematic counterexamples to Riemannian black hole uniqueness conjecture.
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.
Local equivalence found between certain solitons and generalized Kähler-Ricci solitons.
problem Understanding and constructing generalized Kähler-Ricci solitons.
method Establishing local equivalence and extending to complete GKRS under natural conditions.
result Local classification and construction of new examples in all dimensions, especially in four dimensions.
We consider G2-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of T3-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz giving the geometr…
New Spin(7) metrics found from Kähler quotients.
problem Finding Spin(7) metrics from Kähler quotients. method Kähler reduction and PDEs on quotient manifolds.
result Infinitely many new explicit examples of Spin(7) metrics. We study the Hessian geometry of toric Gibbons-Hawking metrics and their phase change phenomena via the images of their moment maps.
It is well known that any 4-dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on R^3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Ki…
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 closed G2-structures with T3-symmetry, classifying them into types and deriving hypersymplectic structures.
problem Classify closed G2-structures with T3-symmetry and derive associated hypersymplectic structures.
method Decompose G2-structures into canonical forms, classify structures based on orbit isotropy, and derive hypersymplectic structures.
result Closed G2-structures with T3-symmetry are classified into two types, leading to specific hypersymplectic structures.
The paper constructs singularities for Lagrangian flow in Gibbons-Hawking spaces with vanishing mean curvature.
problem Infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons-Hawking spaces.
method One-parameter family of barrier curves and detailed asymptotic analysis.
result The mean curvature converges uniformly to zero, but the second fundamental form becomes unbounded.
This work presents a classification of all smooth 't Hooft-Jackiw-Nohl-Rebbi instantons over Gibbons-Hawking spaces. That is, we find all smooth SU(2) Yang-Mills instantons over these spaces which arise by conformal rescalings of the metric with suitable functions. Since the Gibbons-Hawking spaces are hyper-Kahler grav…
We consider a general 4n-dimensional quaternionic Kahler geometry with a free action of the torus T^(n+1). The toric action lifts onto the Swann bundle of the quaternionic Kahler space to a tri-holomorphic action that commutes with the standard H* action on the bundle. By matching Pedersen and Poon's generalized Gibbon…
Paper proves convergence of MDL to Einstein-Hilbert with boundary term.
problem Proving convergence of discrete MDL to continuous Einstein-Hilbert action.
method Proves \(Γ\)-convergence using diffeomorphism-natural discrete MDL-type functional.
result Identifies Carathéodory densities and obtains \(\liminf/\limsup\) bounds.
The paper studies quotient spaces of Spin(7) manifolds by S^1 actions and their G2 structures.
problem Understanding quotient spaces of Spin(7) manifolds by S^1 actions and their geometric properties.
method Derives equations relating intrinsic torsion of Spin(7)-structures to G2-structures, focusing on three torsion classes.
result Shows that the quotient space cannot have G2 holonomy unless the original manifold is a Calabi-Yau 4-fold.
We establish the general formalism for constructing metrics of Calabi-Yau (p+1)-folds in terms of that of a p-fold by adding a complex-line bundle. We present a few explicit low-lying examples. We further consider holomorphic linearization and obtain the six-dimensional analogue of the Gibbons-Hawking instanton. Whilst…
Expressive quantum circuits are harder to train due to flatter cost landscapes.
problem Designing quantum circuits that are both expressive and trainable.
method Deriving a relationship between expressibility and gradient magnitude, extending barren plateau phenomenon.
result Highly expressive ansätze exhibit flatter cost landscapes, making them harder to train.
A classification result for Ricci-flat anti-self-dual asymptotically locally Euclidean 4-manifolds is obtained: they are either hyperkähler (one of the gravitational instantons classified by Kronheimer), or they are a cyclic quotient of a Gibbons-Hawking space. The possible quotients are described in terms of the monop…
We study the Hessian geometry of toric multi-Taub-NUT metrics and their phase change phenomena via the images of their moment maps. This generalizes an earlier paper on toric Gibbons-Hawking metrics.
A new approach to quantum machine learning circuits reduces training difficulties.
problem Challenges in training deep quantum circuits due to flat training landscapes.
method Variable structure approach (VAns) to build ansatzes, applying rules for gate growth and removal.
result VAns successfully mitigates trainability and noise-related issues, improving performance in various applications.
Solutions to the n-dimensional Laplace equation which are constant on a central quadric are found. The associated twistor description of the case n=3 is used to characterise Gibbons-Hawking metrics with tri-holomorphic $SL(2, \C)$ symmetry.
Deep QMC ansatzes improve variational QMC accuracy.
problem Improving variational QMC accuracy with neural network ansatzes.
method Analysis of deep neural network ansatzes PauliNet and FermiNet convergence to fixed-node limit.
result Deep QMC ansatzes can reach fixed-node limit with large network sizes.
Study finds all hyper-Kähler 4-manifolds with specific symmetries and structures.
problem Characterizing hyper-Kähler 4-manifolds with conformal Kähler structures.
method Analyzing twistor elementary states and locally flat spaces, showing compatibility and incompatibility of complex structures.
result Only hyper-Kähler 4-metric with a non-constant Killing-Yano tensor is the half-flat Taub-NUT instanton.
It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutio…
MBQC linked to CQCA, yielding efficient Ansätze.
problem Quantum computation efficiency and Ansatz adaptation.
method Relating MBQC to CQCA and constructing Ansätze.
result MBQC Ansätze can lead to different performances on learning tasks.
Proves conjecture about special Lagrangians in G2-manifolds.
problem Existence of special Lagrangians in G2-manifolds.
method Solves real Monge-Ampère equation with singular right-hand side.
result Smoothness and asymptotic properties of special Lagrangians proved.
Study Ricci flow on CP1-bundles over Kähler-Einstein manifolds.
problem Preserving an initial metric on CP1-bundles.
method Ricci flow on CP1-bundles over a product of Kähler-Einstein manifolds.
result The ansatz is preserved along the Ricci flow.
The modified J-flow with Calabi ansatz shows convergence or blow-up behavior based on topological constants.
problem Analyzing the behavior of the modified J-flow with Calabi ansatz.
method Using the Calabi symmetry and studying the singularities of the flow.
result The modified J-flow with Calabi ansatz converges to a solution away from a variety, and blows up along the variety.
Develops methods to solve complex and real Hessian equations.
problem Solving complex and real Hessian equations on various domains.
method Introduces an ansatz to reduce PDEs to systems of ODEs, integrating via abelian integrals.
result Constructs entire solutions of arbitrary subcritical phase for dHYM/LYZ and special Lagrangian equations.
Constructs scalar-flat Kähler metrics with varying conical singularities.
problem Creating scalar-flat Kähler metrics with specific singularities.
method Using LeBrun's ansatz, constructs metrics with varying conical singularities.
result Constructs complete scalar-flat Kähler metrics with prescribed conical singularities.
Invariant Kähler metrics on line bundles are derived from the Calabi ansatz.
problem Finding invariant scalar-flat Kähler metrics on line bundles over generalized flag varieties.
method Proved using the Calabi ansatz and uniqueness in each Kähler class.
result Existence of a unique scalar-flat Kähler metric in each Kähler class.