Study on Calabi-Yau metrics and their degenerations.
problem Understanding degenerations of Ricci-flat Kahler metrics on Calabi-Yau manifolds.
method Survey of recent results on questions about degenerations.
result Survey of recent results on degenerations of metrics.
Holomorphic connections on Calabi-Yau manifolds are flat.
problem Existence of holomorphic connections on Calabi-Yau manifolds.
method Proving the existence of flat holomorphic connections for holomorphic vector bundles.
result Holomorphic vector bundles over compact Kähler Calabi-Yau manifolds admit flat holomorphic connections.
Schoen-Yau's zero mass theorem stability remains an open question.
problem Geometric stability of Schoen-Yau's zero mass theorem.
method Review of geometric stability, examples, and convergence notions.
result Open question on geometric stability of Schoen-Yau's zero mass theorem.
Constructs hyperbolic affine spheres and Calabi-Yau metrics.
problem Constructing non-isometric Calabi-Yau metrics.
method Non-Abelian Hodge theory for parabolic Higgs bundles.
result Answers a question about Calabi-Yau metrics.
Paper proves edge-connectivity equals minimum degree for graphs with non-negative curvature.
problem Edge-connectivity vs. minimum degree in graphs with non-negative curvature.
method Analyzes finite connected graphs with non-negative Lin-Lu-Yau curvature.
result Edge-connectivity equals minimum degree for graphs with non-negative curvature.
The Wu-Yau theorem is verified for negative curvature, and new examples of Kähler-Einstein metrics are found.
problem The Wu-Yau theorem and its positive analog.
method Examples and conjectures to verify the Wu-Yau theorem and its positive analog.
result New examples of Kähler-Einstein metrics without negative holomorphic sectional curvature.
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.
Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
problem Characterizing domains with Kähler-Einstein Bergman metrics.
method Asymptotics of derivatives of the Bergman kernel along critically tangent paths.
result Two-dimensional pseudoconvex domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
The paper proves bounded domains are biholomorphic to the unit ball.
problem Covering manifolds with finite volume.
method Analyzes C2 and C1,ε boundary conditions for bounded domains. result Bounded domains are biholomorphic to the unit ball under certain conditions.
Constructs noncommutative spaces for D-branes on complex algebraic spaces.
problem Mathematical model for D-branes on noncommutative spaces.
method Toric geometry, Azumaya schemes, invertible sheaves.
result Embeds algebraic Calabi-Yau spaces into soft noncommutative schemes.
Given a polarized family of varieties over the unit disc, smooth except over the origin and with smooth fibers Calabi-Yau, we show that the origin lies at finite Weil-Petersson distance if and only if after a finite base change the family is birational to one with central fiber a Calabi-Yau variety with at worst canoni…
Survey on metric SYZ conjecture and non-archimedean geometry.
problem Existence of special Lagrangian fibrations on Calabi-Yau manifolds.
method Pluripotential theory and non-archimedean geometry.
result Subtleties and open questions in the conjectural picture.
Flow deforms curves to match an embedded target.
problem Evolve curves to match an embedded target curve.
method Target flow with curve shortening and forcing term.
result Smooth convergence to the target curve.
Study on Kähler-Einstein metrics on log Calabi-Yau families, proving local triviality and discrediting a metric conjecture.
problem Understanding Kähler-Einstein metrics on log Calabi-Yau families.
method Investigation of relative Ricci-flat Kähler metrics, focusing on Kodaira dimension zero.
result Disproof of a folklore conjecture about the semipositivity of relative flat metrics.
Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.
problem Proves Yau-Tian-Donaldson conjecture for a specific class of manifolds.
method Uses holomorphic actions of compact Lie groups and combinatorial conditions.
result Equivalence of K-uniform stability and K-stability for spherical varieties.
Solves embedding problem for 5D manifolds into Calabi-Yau 3-folds.
problem Embedding a 5D manifold into a Calabi-Yau 3-fold with a specific 3-form.
method Defines 'strongly pseudoconvex' 3-forms and shows solvability of embedding problem for these forms under certain conditions.
result Perturbative embedding problem can be solved for closed strongly pseudoconvex 3-forms if a vector space of obstructions vanishes.
The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
problem Existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
method Explicit K-stability condition, degeneration, and asymptotic cone analysis.
result Uniqueness of K-invariant Calabi-Yau metrics on affine spherical manifolds. This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special La…
Auxiliary equations improve bounds in symplectic geometry.
problem Improving bounds for the Calabi-Yau equation.
method Adapted Monge-Ampère equations for symplectic geometry.
result Reduced the bound from exponential to L1. Improved heat equation estimates without gradient curvature assumption.
problem Improving Hamilton's matrix Harnack estimate for heat equation without gradient curvature assumption.
method New ingredients include a sharp Li-Yau estimate, a suitable vector field construction, and integral arguments.
result Removed the gradient curvature assumption in Hamilton's estimate for heat equation.
New minimal surface doublings of Clifford Torus improve bounds on minimal surfaces in S^3.
problem Proving bounds on minimal surfaces in S^3 with fixed genus.
method Applying a general theorem to produce new minimal doublings of the Clifford Torus, using min-max methods, and verifying Yau's conjecture.
result Improved quadratic lower bound for the number of embedded minimal surfaces in S^3 with prescribed genus.
In this paper, we derive a partial result related to a question of Yau: "Does a simply-connected complete Kähler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?" Main Theorem. Let M2n be a simply-connected complete Kähler manifold M with negative sectional curvature …
Study stability of pseudo-Kähler and neutral Calabi-Yau manifolds, finding stability in 2D but failing in higher dimensions.
problem Stability of compact pseudo-Kähler and neutral Calabi-Yau manifolds.
method Analysis of stability through deformation theory and construction of counterexamples.
result Stability of compact pseudo-Kähler surfaces but failure in higher dimensions.
New result on energy-genus bounds for 6D symplectic Calabi-Yau manifolds.
problem Energy-genus bounds for pseudo-holomorphic curves in almost complex manifolds.
method Compactness and regularity theorems for J-holomorphic currents.
result Established a new result in dimension 6 for symplectic Calabi-Yau 6-manifolds.
Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate singularities.
We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler-Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recas…
Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.
problem Collapsing geometry of Riemannian manifolds with Ricci curvature constraints.
method Locally bounded Ricci covering geometry and Ricci flow smoothing techniques.
result Volume collapsed Calabi-Yau manifolds admit Ricci-flat Kähler metrics and compatible Killing structures.
The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.
problem Properness of complete minimal hypersurfaces in higher dimensions.
method Chord-arc estimates and gluing techniques.
result Construction of a complete, improperly embedded minimal hypersurface in Rn+1 for every n≥3. The paper explores non-Kähler SYZ mirrors for solvmanifolds, proving cohomological properties and constructing new mirror pairs.
problem Understanding non-Kähler SYZ mirrors for solvmanifolds and their cohomological properties.
method Investigates geometric and cohomological properties, proving relationships and constructing mirror pairs.
result Proves the Fourier-Mukai transform exchanges type-A and type-B cycles, and provides criteria for non-Kähler SYZ mirror pairs.
D-branes on noncommutative spaces mimic string theory, offering new insights into mirror symmetry.
problem Exploring noncommutative mirror symmetry through D-branes on noncommutative Calabi-Yau spaces.
method Constructing noncommutative ringed spaces from local resolutions, realizing D-branes as morphisms, and defining kinetic energy.
result Dynamical D-branes on noncommutative spaces can be described by a Polyakov-like action, suggesting a bridge between string theory and noncommutative geometry.
Proves SYZ mirror symmetry for del Pezzo and rational elliptic surfaces.
problem Proving mirror symmetry for specific Calabi-Yau surfaces.
method Adapting Hein's work, constructing asymptotically semi-flat Calabi-Yau metrics, and defining a mirror map.
result Existence and uniqueness of Calabi-Yau metrics on Y∖D. We continue the study of the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, this states that if two Calabi-Yau manifolds X and Y are mirror partners, then X and Y have special Lagrangian torus fibrations which are dual to each other. Much work on this conjecture is necessarily of a speculative nature, a…
This paper is a survey of some recent progress on the study of Calabi's extremal Kähler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then tu…
The paper studies stability of extensions of tangent sheaves on Kähler-Einstein and Calabi-Yau pairs.
problem Stability of extensions of tangent sheaves on Kähler-Einstein and Calabi-Yau pairs.
method Analyzes the extension sheaf of the orbifold tangent sheaf by the structure sheaf, showing stability under certain conditions.
result The extension sheaf is slope semistable under specific conditions, generalizing Tian's result.
Paper answers a question about lifting positive Ricci curvature.
problem Lifting positive Ricci curvature from quotient of isometric actions.
method Cheeger deformation techniques and curvature estimates.
result Proves a result about positive Ricci curvature on quotient manifolds.
We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of R3 is discrete. This give…
This is the first part in a two-part series on complete Calabi-Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition O(r−n−ε) needed in earlier work to O(r−ε), relying on some new ideas about harm…
Study proves uniqueness of asymptotic limits for specific manifolds.
problem Proving uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth.
method Established using natural curvature and cross section assumptions.
result Uniqueness and exponential convergence rate for complete noncollapsed Ricci-flat manifolds with linear volume growth.
We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then give a new construction of an infinite family of symplectic Calabi-Yau surfaces with first Betti number b_1=2,3, along wi…
Let M be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to [0,∞)×X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M; in particular we show that there is no loss of generality in assumi…
Sharp gradient bound found for compact manifolds.
problem Finding a sharp gradient bound for compact manifolds.
method Using a specific function α=1 in the Li-Yau gradient bound.
result A sharp gradient bound is found for compact manifolds.
Survey on recent breakthrough linking curvature and Kobayashi hyperbolicity.
problem Connecting curvature properties to Kobayashi hyperbolicity in complex geometry.
method Detailed analysis of Wu-Yau theorem and its proof.
result A compact complex manifold with negative holomorphic sectional curvature admits a Kähler metric with negative Ricci curvature.
The famous pinching problem says that on a compact simply connected n-manifold if its sectional curvature satisfies Kmin>(1/4)Kmax>0, then the manifold is homeomorphic to the sphere. In [8, problem 12], S. T. Yau proposed the following problem: If we replace Kmax by the scalar curvature, can we deduc…
Minimal surfaces with negative curvature found in large spheres.
problem Existence of minimal surfaces with negative curvature in large dimensional spheres.
method Applied Song's strategy to closed Riemann surfaces with large automorphism groups, resulting in almost hyperbolic minimal surfaces.
result Existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension.
Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.
problem Understanding Bergman kernels on Kähler manifolds and their properties.
method Localization and expansion analysis of Bergman kernels.
result Answered Lu-Tian's question about Bergman kernels having no logarithmic singularity.
In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when M admits…
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…
Estimates symplectic Calabi-Yau equation using Cheng-Yau method.
problem Estimating solutions to the Calabi-Yau equation on symplectic 4-manifolds.
method Applies a Cheng-Yau type estimate in the symplectic setting.
result Proves an a priori estimate for the symplectic Calabi-Yau equation.