Proves solutions to Fu-Yau equation on Kähler manifolds.
problem Existence of solutions to the Fu-Yau equation.
method Proof of existence on compact Kähler manifolds.
result Existence of non-trivial solutions to modified Strominger system.
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Derives Li & Yau estimates for heat equations on manifolds.
problem Analyzing positive solutions of semilinear heat equations on manifolds.
method Adapts Li & Yau estimates to derive new inequalities.
result Derives Harnack inequality and discusses monotonicity, convexity, decay estimates.
The Calabi Conjecturemath.DG
Explores the geometric aspects of the Calabi Conjecture using PDE theory.
problem Existence of specific metrics in geometry.
method Nonlinear Elliptic PDE theory, focusing on Yau's solution.
result Proves the existence of a class of metrics with deep physical implications.
Study Fu-Yau equation on astheno-Kähler manifolds, proving existence and uniqueness.
problem Existence and uniqueness of solutions to Fu-Yau equation on astheno-Kähler manifolds.
method Analysis of Fu-Yau equation on compact Hermitian manifolds, proving existence and uniqueness for astheno-Kähler manifolds.
result Proved existence and uniqueness of solutions for Fu-Yau equation on astheno-Kähler manifolds.
Paper improves Li-Yau gradient estimates for heat equation solutions.
problem Improving Li-Yau gradient estimates for heat equation solutions.
method Different argument from Li-Xu and Qian; obtains time-dependent estimates.
result Improves Li-Yau type gradient estimates for positive solutions of heat equation.
Paper improves Li-Yau gradient estimates for heat equation solutions.
problem Improving Li-Yau gradient estimates for heat equation solutions.
method Obtained new Li-Yau type gradient estimates with time-dependent parameters.
result Improved Li-Yau gradient estimates for heat equation solutions.
Proves estimates for Kähler-Ricci flow solutions.
problem Positive solutions to Kähler-Ricci flow.
method Matrix Li-Yau-Hamilton estimates coupled with flow.
result Monotonicity formula derived.
Fu-Yau Hessian Equationsmath.DG
Solves Fu-Yau equation in arbitrary dimensions and slope.
problem Solving the Fu-Yau equation in arbitrary dimensions and slope.
method Introduced a more stringent ellipticity condition to solve the equation.
result Improved solutions for known and open cases of the Fu-Yau equation.
New obstruction found for Hull-Strominger system solutions.
problem Existence of solutions to the Hull-Strominger system.
method Construction of Hermitian-Einstein metric on string algebroid.
result Definition of Futaki invariants obstructing solutions.
We prove that a given Calabi-Yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the Strominger system provided that the second Chern class of the vector bundle is equal to the second Chern class of the tangent bundle. If the Calabi-Yau threefold has strict SU(3) holonomy then the eq…
Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.
problem Solving the form type Calabi-Yau equation on complex manifolds.
method Defined astheno-Ricci curvature and proved existence of solution under non-positive curvature condition.
result Existence of solution for Calabi-Yau equation with non-positive astheno-Ricci curvature.
In this paper we introduce a new equation on the compact Kahler manifolds. Solution of this equation corresponds to the Calabi-Yau metric. New equation differs from the Monge--Ampere equation considered by Calabi and Yau.
Derives gradient bounds for f-heat equations on manifolds with Bakry-Emery Ricci curvature.
problem Gradient estimates for positive solutions of f-heat equations on manifolds with specific curvature conditions.
method Applies Li-Yau gradient estimates to positive solutions of the f-heat equation on closed manifolds with Bakry-Emery Ricci curvature bounded below.
result Derives Li-Yau gradient bounds for positive solutions of the f-heat equation.
Sharp gradient estimates found for heat equation on hyperbolic spaces.
problem Finding sharp gradient estimates for heat equations on hyperbolic spaces.
method Introduced a general form of Li-Yau type gradient estimate and used explicit heat kernel expressions.
result Sharp Li-Yau type gradient estimates obtained for heat equation on hyperbolic spaces.
Study solves Monge-Ampère equation for complete Calabi-Yau metrics.
problem Existence of complete Calabi-Yau metrics on log Calabi-Yau pairs.
method Free boundary Monge-Ampère equation, Legendre duality, existence proof in smooth and Hölder spaces.
result Existence and strict convexity of solutions in smooth and Hölder spaces.
Proves regularity for multiple membrane solutions.
problem Stationary solutions to the multiple membrane problem.
method Uses -regularity estimate.
result Proves regularity for stationary solutions.
Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.
problem Existence of special Lagrangian fibrations in Calabi-Yau hypersurfaces.
method Non-Archimedean and tropical Monge-Ampère equations on Berkovich and skeleton spaces, proving uniqueness and deriving solutions.
result Unique solutions to tropical and non-Archimedean Monge-Ampère equations, leading to existence of special Lagrangian fibrations.
Review of Yau's conjecture on zero sets of Laplace eigenfunctions.
problem Yau's conjecture on zero sets of Laplace eigenfunctions.
method Discussion of old and new results and methods related to the conjecture, including solutions and new results in smooth settings.
result Discussion of Donnelly and Fefferman's solution of the conjecture in the real-analytic Riemannian manifold case and new results in the smooth setting.
New complete Calabi-Yau metric on C^3 with maximal volume growth.
problem Modeling collapsing Calabi-Yau threefolds near nodal points.
method Existence result perturbed into an actual solution with correction of slowly decaying error terms.
result Complete Calabi-Yau metric on C^3 with maximal volume growth and non-standard geometry near singularity.
Simplified argument for second order estimate in quaternionic Calabi-Yau problem.
problem Second order estimates for quaternionic Calabi-Yau problem on hyperkähler manifolds.
method Simplified argument to derive the estimate.
result Simplified derivation of second order estimate.
Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.
problem Analyzing the behavior of G2-structures under Laplacian and Hitchin flows.
method Investigation of Laplacian and Hitchin flows on contact Calabi-Yau 7-manifolds.
result Ancient solutions of the Laplacian flow with finite time Type I singularity and immortal solutions of the Laplacian coflow with infinite time Type IIb singularity.
The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.
problem Constructing invariant Calabi-Yau structures on complexified symmetric spaces.
method Solutions of a Monge-Ampère type equation.
result Existence of solutions to the Monge-Ampère type equation.
Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.
problem Uniform K-stability and existence of cscK metrics.
method Special Fujita approximations and regularization of entropy functional.
result Uniformly K-stable polarized smooth projective varieties admit cscK metrics.
Solves Fu-Yau equation for negative slope parameters in arbitrary dimensions.
problem Solving the Fu-Yau equation for negative slope parameters in arbitrary dimensions.
method Solves the Fu-Yau equation for negative slope parameters in arbitrary dimensions.
result First non-trivial solutions of the Fu-Yau equation in any dimension strictly greater than 2.
We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the pro…
In this paper we derive Cheng-Yau, Li-Yau, Hamilton estimates for Riemannian manifolds with Bakry-Emery Ricci curvature bounded from below, and also global and local upper bounds, in terms of Bakry-Emery Ricci curvature, for the Hessian of positive and bounded solutions of the weighted heat equation on a closed Riemann…
Generalizes embeddedness result for extreme curves.
problem Embeddedness of solutions to the Plateau problem for extreme curves.
method Generalization of Meeks-Yau's result.
result Generalization of embeddedness result.
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.
Study of flat bundles from Calabi-Yau theory converging to Riemann-Hilbert solutions.
problem Understanding flat bundles and their limits in Calabi-Yau theory.
method Analysis of variations of BPS structures and convergence to Riemann-Hilbert problems.
result Expression for Gromov-Witten partition function in terms of confluent hypergeometric equations.
Sharp gradient estimate on hyperbolic spaces derived from heat kernel.
problem Finding sharp Li-Yau type gradient estimates for positive solutions of heat equations.
method Introduced Li-Yau multiplier set and used recurrence relations of heat kernels on hyperbolic spaces.
result Optimal Li-Yau gradient estimate on hyperbolic spaces.
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…
The paper provides gradient estimates for a specific equation on Riemannian manifolds.
problem Gradient estimates for positive solutions to a specific equation on Riemannian manifolds.
method Obtained gradient bounds for positive solutions without depending on the solution's bounds or the Laplacian of the distance function.
result Gradient bound of a positive solution does not depend on the solution's bounds or the Laplacian of the distance function.
The Calabi-Yau equation on the Kodaira-Thurston manifold, viewed as an S^1-bundle over a 3-torusmath.DG
We prove that the Calabi-Yau equation on the Kodaira-Thurston manifold has a unique solution for every -invariant initial datum.
We propose a new construction of compact non-Kähler Calabi-Yau manifolds with balanced metrics and study the Strominger system on them. In particular, we obtain explicit solutions to the Strominger system with degeneracies on , where is an immersed minimal surface of genus in .
In the spirit of [10,2], we study the Calabi-Yau equation on -bundles over endowed with an invariant non-Lagrangian almost-Kähler structure showing that for -invariant initial data it reduces to a Monge-Ampère equation having a unique solution. In this way we prove that for every total space $M…
Solutions to Strominger system found for square of Kähler class.
problem Finding solutions to Strominger system with specific balanced classes.
method Deforming Calabi-Yau and Hermitian-Yang-Mills metrics.
result Classes that are squares of Kähler metrics admit solutions.
Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.
problem Solving the Calabi-Yau equation on symplectic manifolds.
method Global deformation of almost complex structures compatible with symplectic form, constructing measurable Lipschitz Kahler metric.
result Existence theorem for solutions to the one-form type Calabi-Yau equation on closed symplectic manifolds.
The paper derives Harnack inequalities for positive heat equation solutions on Finsler manifolds.
problem Deriving inequalities for heat equation solutions on Finsler manifolds.
method Generalizing Li-Yau type gradient estimates to Finsler geometry and applying to heat equation solutions.
result General gradient estimate for positive solutions of the heat equation on Finsler manifolds under curvature assumptions.
In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point in a spacetime , we consider a canonical family of surfaces approaching along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving …
The paper establishes estimates for heat equations and harmonic functions on metric spaces with curvature-dimension condition.
problem Analyzing geometric properties of metric measure spaces with curvature-dimension condition.
method Establishing local Li-Yau estimates and proving sharp Yau's gradient estimates for heat equations and harmonic functions.
result Sharp Li-Yau and gradient estimates for weak solutions of heat equations and harmonic functions on spaces.
Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifoldsmath.DG
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we …
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by stability proofs.
result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
Solves non-Archimedean Calabi-Yau equation on complex log pairs.
problem Non-Archimedean Monge-Ampère equation on Berkovich analytification.
method Solves complex Monge-Ampère equation, then adapts to non-Archimedean setting.
result Non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties.
Study existence of harmonic 1-forms on Calabi-Yau manifolds.
problem Tackles existence of harmonic 1-forms on Calabi-Yau manifolds.
method Uses neural networks to approximate metrics and harmonic 1-forms.
result Suggests existence of harmonic 1-forms on some Calabi-Yau manifolds.
Let be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kähler form and a holomorphic four-form . Under certain assumptions, we define Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifol…
We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type via monotonicity formulae.
The paper solves a generalized Kähler Calabi-Yau problem for nondegenerate structures.
problem Formulating a Calabi-Yau conjecture in generalized Kähler geometry for nondegenerate Poisson structures.
method Defining Hamiltonian deformation spaces, establishing uniqueness, and using GIT framework.
result All solutions are hyper-Kähler metrics, and the flow evolves within the given class.
In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized La…