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.
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.
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.
We study the convergence behavior of the general inverse σk-flow on Kähler manifolds with initial metrics satisfying the Calabi Ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic singularities along negatively self-intersected sub-varieties are formed as a result of …
The study creates Kähler-Einstein metrics with conical singularities.
problem Producing Kähler-Einstein metrics with specific singularities.
method Using the Calabi ansatz for metrics with conical singularities.
result Created regular Calabi-Yau cones and Kähler-Einstein metrics with negative Ricci.
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.
New metrics found on Hirzebruch surfaces solve complex geometry questions.
problem Finding new conformally Kähler, Einstein-Maxwell metrics on Hirzebruch surfaces.
method Used special Killing potentials found by Futaki and Ono.
result Proved existence of new conformally Kähler, Einstein-Maxwell metrics.
Study Kähler-Einstein edge metrics on Hirzebruch surfaces, verifying a conjecture and finding a rigid singularity.
problem Verifying a conjecture about Kähler-Einstein edge metrics on Hirzebruch surfaces.
method Using the Calabi ansatz, constructing a family of metrics and studying their angle deformation.
result Verification of a conjecture and finding a rigid singularity.
Method finds approximate Ricci-flat metrics on Calabi-Yau manifolds.
problem Finding analytic Kähler potentials for Calabi-Yau manifolds.
method Numerically calculating Ricci-flat Kähler potentials via machine learning and fitting to Donaldson's Ansatz.
result Simple analytic expressions for approximately Ricci-flat Kähler potentials are found, including explicit dependence on complex structure parameter.
New complete Calabi-Yau metrics found in complex space.
problem Finding metrics on complex spaces with specific conditions.
method Generalized Calabi ansatz, non-archimedean Monge-Ampère equation.
result Complete Calabi-Yau metrics constructed in Fano manifolds.
Study on special Kähler metrics with specific asymptotic properties.
problem Finding complete constant scalar curvature Kähler metrics with Poincaré-Mok-Yau asymptotic property.
method Use of Calabi's ansatz and moment construction.
result Provide examples of such metrics on non-compact Kähler manifolds.
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.
We study and construct non-abelian hermitian Yang-Mills (HYM) instantons on Calabi-Yau cones. By means of a particular isometry preserving ansatz, the HYM equations are reduced to a novel Higgs-Yang-Mills flow on the Einstein-Kahler base. For any 2d-dimensional Calabi-Yau cone, we find explicit solutions of the flow eq…
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.
Formula found for Calabi-Yau metrics on flag variety bundles.
problem Finding unique asymptotically conical Calabi-Yau metrics on flag variety canonical bundles.
method Generalizing the Calabi Ansatz to Kähler classes, deriving a simple formula.
result Explicit formula for metrics on canonical bundles of flag varieties.
The paper constructs Einstein metrics on holomorphic bundles.
problem Finding complete conformally Kähler Einstein metrics on holomorphic bundles.
method Explicit momentum construction via ODE methods and Calabi ansatz.
result Non-trivial complete conformally Kähler Einstein metrics on certain holomorphic bundles are found.
The paper describes Calabi-Yau metrics on complex flag manifolds using Lie theory.
problem Finding complete Calabi-Yau metrics on canonical bundles of complex flag manifolds.
method Using Lie theory and the Calabi ansatz technique to provide explicit examples of noncompact complete Calabi-Yau manifolds.
result Explicit examples of noncompact complete Calabi-Yau manifolds, including canonical bundles of non-toric flag manifolds.
Study explores Laplacian coflow versions on Calabi-Yau 7-manifolds.
problem Analyzing G2-structures on Calabi-Yau 7-manifolds. method Reduced Ansätze for the Laplacian coflow on various Calabi-Yau 7-manifolds.
result Obtained a modified Kähler-Ricci flow.
Study constructs complete Kahler-Einstein metrics near isolated log-canonical singularities.
problem Constructing complete Kahler-Einstein metrics near isolated log-canonical singularities.
method Two approaches: uniformization by a complex ball and Calabi Ansatz.
result Two metrics are shown to be the same, providing a local model.
An ansatz of Calabi allows construction of Kahler metrics in an Hermitian disk bundle over a Kahler manifold. We attempt to give a definitive treatment of this ansatz, with the following results: We give curvature conditions on the disk bundle that guarantee existence of families of complete Kahler metrics of constant …
Joyce's criterion for sLag smoothings extended to non-compact, non-transverse intersections.
problem Existence of special Lagrangian smoothings for non-compact, non-transverse intersections.
method Leung-Yau-Zaslow transform, deformed Hermitian Yang-Mills connections, Calabi ansatz, mean curvature flow, Bridgeland stability conditions.
result Existence of sLag smoothings on stable loci with slope inequality.
Study of singularity formation in dHYM flow on a blown-up CP^3.
problem Analyzing singularity formation in the dHYM flow on a blown-up CP^3.
method Using Calabi ansatz, the paper studies the singularity formation of the dHYM cotangent flow on the one-point blow up of CP^3.
result An explicit example of singularity formation along the exceptional divisor, with a limit satisfying the corresponding singular dHYM equation.
We study the limiting behavior of the Kahler-Ricci flow on P(OPn⊕OPn(−1)⊕(m+1)), assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to Pn or contracts a subvariety of c…
The paper finds explicit instantons on a specific 6-manifold.
problem Finding explicit solutions to the Hermitian Yang-Mills equations.
method Dimensional reduction and reformulation of the equations on quotient spaces.
result Explicit abelian instantons and deformed Hermitian Yang-Mills connections are found.
Researchers solve Calabi-Yau equation on Kodaira-Thurston manifold.
problem Solving the Calabi-Yau equation on a specific manifold.
method Using an invariant almost-Kaehler structure and an ansatz to reduce the problem to a Monge-Ampère equation.
result The problem is reduced to a Monge-Ampère equation under certain conditions.
We consider cones over manifolds admitting real Killing spinors and instanton equations on connections on vector bundles over these manifolds. Such cones are manifolds with special (reduced) holonomy. We generalize the scalar ansatz for a connection proposed by Harland and Nolle in such a way that instantons are parame…
The study examines metrics with non-generic holonomy on manifolds.
problem Characterizing metrics with non-generic holonomy on compact manifolds.
method Analyzes conformal classes of metrics with non-generic holonomy.
result Identifies conditions under which metrics have non-generic holonomy.
We extend Calabi ansatz over Kähler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat Kähler metric on Kähler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat Kähler metric on the toric Kähler cone manifold…
Constructive method for contact structures on homogeneous manifolds.
problem Describing contact structures on compact homogeneous contact manifolds.
method Using representation theory of Lie algebras to compute contact forms and Sasaki-Einstein structures.
result Explicit examples of crepant resolutions of Calabi-Yau cones.
We construct gradient Kähler-Ricci solitons on Ricci-flat Kähler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the resul…
Solves a conjecture about hyperKähler manifolds using quaternionic Monge-Ampère equation.
problem Proving the solvability of a conjecture about hyperKähler manifolds with torsion.
method Solving quaternionic Monge-Ampère equation on hyperKähler manifolds without assuming flatness.
result Proves the conjecture for hyperKähler manifolds with trivial canonical bundle.
Constructs complete metrics and solitons on complex vector bundles.
problem Finding complete metrics and solitons on complex vector bundles.
method Employing the theory of hamiltonian 2-forms and constructing metrics on total spaces of vector bundles.
result Obtains new examples of asymptotically conical Kähler shrinkers, Calabi-Yau metrics, and steady solitons.
Compact Ricci-flat manifolds with conical singularities are found.
problem Finding compact Ricci-flat manifolds with conical singularities.
method Analyzing flat limits of smooth polarized Calabi-Yau manifolds and using the Calabi ansatz.
result The existence of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.
Introduces new equations linking Kähler-Einstein and Hermitian-Yang-Mills theories.
problem Existence of solutions to coupled Kähler-Einstein and Hermitian-Yang-Mills equations.
method Moment map interpretation, Futaki invariant, Matsushima-Lichnerowicz theorem, deformation results.
result Nontrivial solutions produced under certain conditions.
Study solutions and singularities of G2-structures flows on specific manifolds.
problem Investigate singularities and solutions of G2-structures flows.
method Explicit solutions and singularities of Ricci-harmonic flow, Ricci-like flows, and negative gradient flow of G2-structures on specific manifolds.
result First examples of Type I singularities of Ricci-harmonic flow and Type IIb and Type III singularities of Ricci-like flows.
Establishes metrics with positive curvature on projective line bundles.
problem Existence of complete Kähler metrics with semi-positive holomorphic sectional curvature.
method Calabi's Ansatz and product approach.
result Existence of complete Kähler metrics with many zeroes.
The paper constructs Ricci-flat Kähler manifolds with specific decay properties.
problem Finding Ricci-flat Kähler manifolds with controlled decay rates.
method Geometric existence proof and construction of ansatz.
result Existence of 39 distinct Ricci-flat Kähler 3-folds with specific asymptotic angles.
We use the twistorial construction of D-instantons in Calabi-Yau compactifications of type II string theory to compute an explicit expression for the metric on the hypermultiplet moduli space affected by these non-perturbative corrections. In this way we obtain an exact quaternion-Kahler metric which is a non-trivial d…
Study geometric operators on Tian-Yau spaces, finding L2 harmonic forms and asymptotic regularity.
problem Analyzing geometric elliptic operators on Tian-Yau spaces.
method Use a-pseudodifferential calculus to determine L2 harmonic forms and asymptotic regularity. result Determine the space of L2 harmonic forms and refined asymptotic regularity of ALH* structures. Generalizes gravitational instanton metrics to a new class.
problem Finding explicit expressions for gravitational instantons.
method Quaternionic-based Ansatz generalizing Gibbons-Hawking Ansatz.
result Explicit expressions for Dk type gravitational instantons. 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.
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.
We classify superpotentials for cohomogeneity one Ricci solitons, finding new ones and proving non-existence in higher dimensions.
problem Classifying superpotentials for cohomogeneity one Ricci solitons.
method Hamiltonian system analysis and specific ansatz constructions.
result New superpotential found and non-existence proven in higher dimensions.
We consider an extension of the results of S. Bando, R. Kobyashi, G. Tian, and S. T. Yau on the existence of Ricci-flat Kähler metrics on quasi-projective varieties Y=X\D with α[D]=c_1(X), α>1. The requirement that D admit a Kähler-Einstein metric is generalized to the condition that the link S in the normal bundle of …
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.
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.
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.
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…