Uniform estimates for Calabi-Yau degenerations proved.
problem Calabi-Yau degenerations of polarised algebraic manifolds.
method Uniform Skoda and L∞-estimates for Kähler potentials. result Uniform Skoda type estimate and L∞-estimate for Calabi-Yau Kähler potentials proved. The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.
problem Uniform Yau-Tian-Donaldson conjecture for polarized toric manifolds.
method Combinatorial sufficient condition for relative K-polystability.
result Uniform relative K-polystability condition established.
Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.
problem Bounding the diameter of Calabi-Yau fibrations near singular fibers.
method Uniform diameter bound proof for Calabi-Yau fibrations with canonical singular fibers.
result Uniform diameter bounds for all fibres in suitable rescaling.
Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.
problem Understanding uniformization of Klt pairs under equality in Miyaoka-Yau inequality.
method Analyzing Kähler klt pairs with specific conditions and using orbifold Miyaoka-Yau inequality.
result Orbifold universal cover is either the unit ball or affine space.
Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.
problem Existence of Kähler-Einstein metrics with arbitrary polarizations.
method Quantization techniques and pluripotential theory.
result Uniform Yau-Tian-Donaldson theorem for Kähler-Einstein metrics.
Yau's uniformization conjecture states: a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to $\ce^n$. The Kähler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this ar…
Let Mn be a complete noncompact Kähler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that M is biholomorphic to Cn. This confirms Yau's uniformization conjecture when M has maximal volume growth.
Proves estimates for Calabi-Yau metrics as Kahler classes shrink.
problem Estimates for Calabi-Yau metrics under shrinking Kahler classes.
method Proves asymptotic expansion in terms of powers of fiber diameter with uniform C^k-estimates.
result Uniform estimates for all orders of derivatives of Calabi-Yau metrics.
Uniformizes klt pairs using bounded symmetric domains.
problem Characterizing klt pairs uniformizable by bounded symmetric domains.
method Determines conditions for uniformization using Miyaoka-Yau-type inequalities.
result Characterizations of orbifold quotients of polydisc and classical bounded symmetric domains.
New approach for uniform estimates in complex equations.
problem Uniform estimates for solutions to complex Monge-Ampere equations.
method Efficient new approach to uniform estimates.
result Efficient method for uniform estimates in geometric PDEs.
We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general …
Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.
problem K-stability of Calabi-Yau fibrations over curves.
method Adiabatic uniform K-stability and log-twisted K-stability of base curves.
result Uniform K-stability of Calabi-Yau fibrations if and only if base curves are K-stable.
Simplified proof and new C0 estimate for Kähler-Einstein metrics.
problem Existence of Kähler-Einstein metrics on Calabi-Yau manifolds.
method Alternative C0 a priori estimate for the Monge-Ampère equation. result Established a new uniform bound for the solution of the Monge-Ampère equation.
Continuity of delta invariant leads to uniform Kähler-Einstein metrics.
problem Continuity of delta invariant in Kähler and twisted Kähler-Einstein metrics.
method Analytic delta invariant and uniform Yau-Tian-Donaldson theorem.
result Uniform Yau-Tian-Donaldson theorem for twisted Kähler-Einstein metrics.
The paper establishes a correspondence for projective bundles over curves using test configurations and extremal metrics.
problem Establishing a correspondence for projective bundles over curves using test configurations and extremal metrics.
method Constructing compatible test configurations and using the generalized Calabi ansatz.
result The relative uniform stability of \( (\mathbb{P}(E),[ω]) \) implies the existence of an extremal metric.
We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments a…
The paper confirms a specific type of Sasakian manifold's structure.
problem Characterizing Sasakian manifolds with nonnegative transverse bisectional curvature.
method Analyzing the Sasakian analogue of Yau's uniformization conjecture.
result 5-dimensional Sasakian manifolds with positive transverse bisectional curvature are CR-biholomorphic to the standard Heisenberg group.
Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
problem Proving Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
method Analyzing Monge-Ampère equations corresponding to generalized and twisted Kähler-Ricci g-solitons, proving stability conditions.
result Existence of solutions is equivalent to equivariantly uniform Θ-twisted g-Ding-stability.
Proves constant scalar curvature Kähler metrics are very general.
problem Existence of constant scalar curvature Kähler metrics on smooth polarized varieties.
method Combining uniform arc K-stability and algebraic properties in families.
result The constant scalar curvature Kähler locus is very general.
We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on ℓ-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamenta…
In this article we study the limiting behavior of the Kähler Ricci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient Kähler-Ricci soliton is biholomorphic to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for complete non-…
Uniformizes varieties with log-canonical singularities using ball quotients.
problem Uniformizing complex projective varieties with log-canonical singularities.
method Criteria based on Miyaoka-Yau inequality and log-resolutions of singularities.
result Criteria for isomorphism to Baily-Borel-Mok compactifications.
Researchers introduce new energies to study constant scalar curvature metrics.
problem Understanding constant scalar curvature metrics on compact Kähler manifolds.
method Introduced a family of Kβ energies using Berman's quantization and intersection theory. Combined with non-Archimedean techniques, provided a uniform Yau-Tian-Donaldson correspondence. result Uniform Yau-Tian-Donaldson correspondence characterizes the existence of a unique constant scalar curvature Kähler metric.
Estimates for complex Monge-Ampère equations lead to insights on moduli spaces and singular metrics.
problem Uniform estimates for complex Monge-Ampère equations on Kähler manifolds.
method Refined techniques to control degenerate equations and analyze families of singular Kähler-Einstein metrics.
result Uniform integrability properties and insights into moduli spaces of stable varieties.
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.
Derives formulas for Monge-Ampère measures and reduces complex conjectures to simpler existence problems.
problem Complex Monge-Ampère measures and their applications in algebraic geometry.
method Derives formulas and reduces conjectures to simpler existence problems.
result Reduces uniform Yau-Tian-Donaldson conjecture to existence of approximate decompositions.
The paper studies invariant weighted Bergman metrics on domains.
problem Investigating invariant weighted Bergman metrics under biholomorphisms.
method Introducing invariant weight assignments, using Bergman's minimum integral method and domain version of Tian-Yau-Zelditch expansion.
result Uniform convergence of weighted Bergman kernels and metrics on uniform squeezing domains.
Study on Kähler-Ricci flow's Hölder regularity on compact manifolds.
problem Hölder regularity of Kähler-Ricci flow on compact Kähler manifolds.
method Adapting Hein-Tosatti's method for collapsing Calabi-Yau metrics, uniform spatial Hölder estimate obtained for all time.
result Uniform spatial Hölder estimate of Kähler-Ricci flow for all time.
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.
Paper confirms conjecture for projective manifolds in supercritical phase.
problem Stability condition for deformed Hermitian-Yang-Mills equation.
method Establishes stability result not involving uniform constants.
result Confirms conjecture for projective manifolds in supercritical phase.
Proves finitely generated associated graded rings for valuations on log Fano pairs.
problem Stability thresholds of log Fano pairs.
method Proves finite generation of associated graded rings for valuations.
result Log Fano pairs are uniformly K-stable if their stability threshold is less than a certain value.
This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kähler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. A…
In this note, we prove an existence result on exhaustion functions adapting the method by L.-F. Tam. Then we apply it to prove short-time existence of Ricci flow and study Yau's uniformization conjecture using similar method as Fei He and Lee-Tam.
Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.
problem Existence of Kähler-Einstein metrics on manifolds with big anticanonical class.
method Developed a theory of Deligne functionals and slope formulas for singular metrics, proving a slope formula for the Ding functional in the big setting.
result Existence of a unique Kähler-Einstein metric implies uniform Ding stability.
Proves uniform K-stability is open in Kähler cone.
problem Stability of Kähler metrics in complex geometry.
method Introduced new norm on test configurations and estimates for non-archimedean energy functionals.
result Uniform K-stability is an open condition in the Kähler cone.
We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Grom…
Uniform RC-positivity results for direct image bundles.
problem Understanding the relation between rational connectedness and RC-positivity.
method Analyzing vector bundles and their direct images, using weak RC-positivity as a starting point.
result Uniform RC-positivity of direct image bundles under weak RC-positivity conditions.
We study the uniformization conjecture of Yau by using the Gromov-Haudorff convergence. As a consequence, we confirm Yau's finite generation conjecture. More precisely, on a complete noncompact Kähler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely genera…
Let X be any Q-Fano variety and Aut(X)0 be the identity component of the automorphism group of X. Let G be a connected reductive subgroup of Aut(X)0 that contains a maximal torus of Aut(X)0. We prove that X admits a Kähler-Einstein metric if and only if $X…
Let $({\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$, and a Poincaré inequality without assuming the Ricci curvature is bound…
Proves existence of Kähler-Einstein metrics in big cohomology classes.
problem Existence of Kähler-Einstein metrics in big cohomology classes.
method Using a divisorial stability condition and Fujita-Odaka type delta invariants, building up from scratch the theory of pluripotential theory.
result Uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics in the big cohomology class setting.
We prove the following result: if a Q-Fano variety is uniformly K-stable, then it admits a Kähler-Einstein metric. We achieve this by modifying Berman-Boucksom-Jonsson's strategy with appropriate perturbative arguments and non-Archimedean estimates. The idea of using the perturbation is motivated by our prev…
Constructs a new type of metric for elliptic surfaces.
problem Finding a canonical metric on elliptic surfaces.
method Explicit construction of semi-flat constant scalar curvature Kähler current.
result Uniqueness of the semi-flat cscK current under certain conditions.
The paper proves a special case of Yau's conjecture for Kähler surfaces.
problem Uniformization of complete noncompact Kähler surfaces with positive sectional curvature.
method Proves a complex Monge-Ampère equation to construct a plurisubharmonic weight function.
result A complete noncompact Kähler surface with positive and bounded sectional curvature is biholomorphic to \(\mathbb{C}^2\).
The paper introduces a new type of Ricci flow on graphs to study their curvature.
problem Understanding the curvature of graphs and their convergence properties.
method Proposes a weighted Forman and Lin-Lu-Yau Ricci flow on graphs and proves the existence and uniqueness of solutions.
result The normalized curvature flow on trees converges to a constant curvature metric.
We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archime…
We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. …
New stability criteria for Fano varieties using generalized b-divisors.
problem Characterizing uniform K-stability in Fano varieties. method Introducing a new function ildeδ and formalism for K-stability, proving stability conditions for Kähler-Einstein metrics. result Existence of a unique Kähler-Einstein metric implies uniform D-log K-stability when ildeδ(D)>1.