Uniformizes complete noncompact Kähler manifolds with maximal volume growth.
problem Uniformizing complete noncompact Kähler manifolds with maximal volume growth.
method Proved biholomorphic equivalence to Cn for specific manifolds. result Confirms Yau's uniformization conjecture for maximal volume growth.
Ricci flow shows PIC1 manifolds with maximal volume growth are like Euclidean space.
problem Characterizing complete PIC1 manifolds with maximal volume growth.
method Ricci flow with local curvature estimates.
result PIC1 manifolds with maximal volume growth are diffeomorphic to \(\mathbb{R}^n\).
Harmonic functions on Calabi-Yau manifolds with maximal volume growth are studied.
problem Characterizing harmonic functions on Calabi-Yau manifolds with maximal volume growth.
method Proved a Liouville type theorem for harmonic 1-forms, using a new local L2 estimate of the exterior derivative. result Subquadratic harmonic functions on Calabi-Yau manifolds with maximal volume growth are the real parts of holomorphic functions.
In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman's reduced volume of Ricci flow. We show that if (Mn,g) is an noncompact complete Ricci flat manifold with maximal volume growth satisfying ∣Rm∣(x)→0 as d(x)=dg(x,p)→∞, then Mn has the quadratic curvature dec…
Paper improves Stein property for certain Kahler manifolds.
problem Complete open Kahler manifolds with positive bisectional curvature being Stein.
method Restricts volume growth condition to a weaker one.
result Improves previous observation on Stein property.
Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cu…
New examples of Calabi-Yau 3-folds with unique properties.
problem Finding new Calabi-Yau 3-folds with specific properties.
method Constructing complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones with orbifold singularities.
result Examples of Calabi-Yau 3-folds with maximal volume growth and orbifold singularities.
The study bounds harmonic functions on manifolds with nonnegative Ricci curvature.
problem Bounding harmonic functions on manifolds with specific curvature properties.
method Analyzing the asymptotic volume ratio and eigenvalue counting function.
result Sharp upper bounds for harmonic functions with polynomial growth.
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.
Let M be a complete Kähler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and M admits a nonconstant holomorphic function with polynomial growth, we prove M must be of maximal volume growth. This confirms a conjecture of Ni. There are two essential ingredients in the p…
Existence criteria for Chern-Ricci flows on noncompact manifolds established.
problem Existence and properties of Chern-Ricci flows on noncompact complex manifolds.
method Generalization of results for Kahler-Ricci flows to Chern-Ricci flows, existence criteria established.
result Existence of complete Kahler metrics with nonnegative and bounded bisectional curvature on noncompact complex manifolds.
New volume comparison theorem for gradient Ricci almost solitons.
problem Volume comparison and rigidity of gradient Ricci almost solitons.
method Established a new volume comparison theorem with Bakry-Emery Ricci curvature.
result New volume rigidity result for gradient Ricci almost solitons.
In this paper we obtain three results concerning the geometry of complete noncompact positively curved Kähler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact Kähler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex…
Paper proves almost rigidity theorem and applies it to study RCD(0,N) spaces.
problem Understanding the structure of noncompact RCD(0,N) spaces with linear volume growth.
method Developed an almost rigidity theorem and applied it to study RCD(0, N) spaces.
result Obtained sublinear growth of diameter of geodesic spheres and non-existence of harmonic functions with polynomial growth.
Optimizes dimension estimate for holomorphic functions on Kähler manifolds.
problem Determining the optimal dimension for holomorphic functions with polynomial growth.
method Analyzes Kähler manifolds with non-negative holomorphic bisectional curvature.
result Identifies the specific gap and optimal dimension for maximal volume growth.
We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifold…
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
New Calabi-Yau metrics found on C^n for n>=3.
problem Finding Calabi-Yau metrics on complex manifolds.
method Constructing metrics with maximal volume growth and singular tangent cones.
result Infinitely many complete Calabi-Yau metrics on C^n.
We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative bounded holomorphic bisectional curvature and maximal volume growth is biholomorphic to complex Euclide…
Compactifies Kähler manifolds with nonnegative Ricci curvature.
problem Compactify Kähler manifolds with nonnegative Ricci curvature.
method Proves compactification theorems for complete Kähler manifolds with nonnegative Ricci curvature.
result Proves that a specific type of Kähler Ricci flat manifold is a crepant resolution of a normal affine algebraic variety.
The paper proves properties of geometric flows on noncompact manifolds.
problem Existence criteria for geometric flows on noncompact affine Riemannian manifolds.
method Obtained existence criteria through a geometric flow on noncompact affine Riemannian manifolds.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature and bounded geometry are diffeomorphic to \(\mathbb{R}^n\) if their tangent bundle has maximal volume growth.
Estimates the dimension of harmonic functions with polynomial growth on manifolds.
problem Estimating the dimension of harmonic functions with polynomial growth on manifolds.
method Analyzes the asymptotic behavior of hd(M) for large d on manifolds with maximal volume growth and unique tangent cone at infinity. result Obtains estimates of hd(M) in terms of d, n, and α. Study on positive scalar curvature and its impact on Ricci limit spaces.
problem The influence of uniformly positive scalar curvature on Ricci limit spaces.
method Investigates uniformly positive scalar curvature on non-collapsed Ricci limit spaces.
result Proves a limit space splits at most n-2 lines or R-factors.
New methods compute geometry of hyperKähler metrics at infinity.
problem Understanding the geometry of hyperKähler metrics at infinity.
method Quasi-asymptotically conical metrics, Taub-NUT deformations, compactification by manifolds with corners.
result Identifies unique tangent cones and cohomology groups.
In this paper we prove that a nonflat Kähler-Ricci soliton of the Ricci flow on a complex two-dimensional Kähler manifold with nonnegative holomorphic bisectional curvature can not be of maximal volume growth.
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 growth rates of harmonic functions on curved surfaces.
problem Understanding the growth rates of harmonic functions on curved surfaces.
method Gradient estimate and frequency analysis on complete surfaces and manifolds with non-negative curvature.
result Existence and properties of nonconstant polynomial growth harmonic functions on manifolds with maximal volume growth.
New metrics on C^3 defy uniqueness, differing even at infinity.
problem Non-uniqueness of Calabi-Yau metrics with maximal volume growth.
method Constructed a family of inequivalent metrics on C^3.
result First example of non-uniqueness in Calabi-Yau metrics asymptotic to a fixed cone.
Study on Funk geometry volume growth and polytope flags, verifying conjectures.
problem Volume minimization in Funk geometry and polytopes.
method Analyzing Holmes--Thompson volume, studying asymptotics, computing coefficients.
result Second highest volume coefficient minimized by unique center point, maximized by regular polygons.
Quiver varieties' geometry at infinity studied using Nakajima metric.
problem Understanding the geometry at infinity of quiver varieties.
method Using Melrose's approach to study the geometry at infinity of the Nakajima metric on reduced Hilbert schemes.
result Quiver varieties are quasi-asymptotically conical under generic conditions.
Paper proves existence of nonconstant CR-holomorphic functions in Sasakian manifolds.
problem Proving existence of nonconstant CR-holomorphic functions in Sasakian manifolds.
method Analyzing Sasakian manifolds with specific curvature properties.
result First step towards CR analogue of Yau uniformization conjecture.
The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.
problem Proving a sharp mean value inequality for non-negative superharmonic functions.
method Develops a new sharp mean value inequality and an explicit formula for weighted scalar curvature.
result The new inequality removes the radius restriction of Schoen-Yau's result and provides an explicit formula for integral of weighted scalar curvature.
Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.
problem Estimating volume growth for gradient Ricci solitons.
method Survey and prove new volume growth estimates.
result New volume growth estimates for expanding gradient Ricci solitons.
Let M be a complete n-dimensional Riemannian manifold, if the sobolev inqualities hold on M, then the geodesic ball has maximal volume growth; if the Ricci curvature of M is nonnegative, and one of the general Sobolev inequalities holds on M, then M is diffeomorphic to Rn.
The study provides volume growth estimates for specific types of manifolds.
problem Estimating volume growth for Ricci solitons and quasi-Einstein manifolds.
method Similar to classical results, the study proves volume growth estimates for gradient Ricci solitons and quasi-Einstein manifolds.
result Sharp volume growth estimates for gradient shrinking Ricci solitons and upper bound volume growth estimates for quasi-Einstein manifolds.
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…
Study submanifolds in gradient Ricci solitons with bounded curvature, proving volume growth properties.
problem Volume growth of submanifolds in gradient Ricci solitons with bounded weighted mean curvature.
method Analyzing submanifolds in shrinking gradient Ricci solitons with bounded weighted mean curvature vector.
result Proves polynomial and at least linear volume growth for submanifolds under certain conditions.
The study proves that certain noncompact Hessian manifolds are diffeomorphic to R^n.
problem Characterizing complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature.
method Using a geometric flow on noncompact affine Riemannian manifolds, constructing Hessian metrics, and proving diffeomorphism.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature are diffeomorphic to R^n if their tangent bundle has maximal volume growth.
Paper proves volume growth estimate for steady gradient Ricci solitons.
problem Estimating the volume growth of steady gradient Ricci solitons.
method Proved a volume growth estimate using Nash entropy.
result Volume growth rate is no smaller than $r^{ rac{n+1}{2}}$.
Study on volume growth, Liouville theorems, and f-harmonic functions in Ricci shrinkers.
problem Volume growth and Liouville theorems in Ricci shrinkers.
method Analysis of f-harmonic functions and volume comparison properties. result Integral properties and local gradient estimates of f-harmonic functions on Ricci shrinkers. Two rigidity theorems for manifolds with nonnegative Ricci curvature and specific volume growth.
problem Characterizing manifolds with nonnegative Ricci curvature and specific volume growth properties.
method Rigidity theorems based on volume growth and existence of harmonic functions.
result Conditions for the Riemannian universal cover to have Euclidean volume growth and existence of nonconstant linear growth harmonic functions.
Study noncompact RCD(0,N) spaces with linear volume growth, proving diameter bounds and a splitting theorem.
problem Understanding non-compact RCD(0, N) spaces with linear volume growth.
method Analyzing properties of level sets and applying geometric inequalities.
result Diameter of level sets of a Busemann function grows at most linearly.
Study shows how volume growth affects homological torsion in 3-manifolds.
problem Understanding homological torsion growth in 3-manifolds.
method Examined abelian covers of alternating links and computed homological torsion growth explicitly.
result Explicit computation of homological torsion growth in terms of volume growth.
Study volume growth and asymptotic cones of nonnegative Ricci curvature manifolds.
problem Whether the volume growth order of manifolds is greater than or equal to the dimension of their asymptotic cones.
method Analyzing asymptotic cones and volume growth conditions, extending Sormani's results.
result Existence of asymptotic cones with upper box dimension at most equal to the volume growth order.
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…
Study on volume growth of horospheres in specific Heintze groups.
problem Volume growth of horospheres in diagonalizable Heintze groups.
method Analysis of volume growth in Heintze groups with diagonalizable structure.
result Two distinct isometry and quasi-isometry classes of horospheres with explicit volume growth calculations.
Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.
problem Understanding if manifolds with specific curvature bounds and volume growth must be of finite topological type.
method Constructs a family of (2+n)−dimensional open manifolds with positive Ricci curvature and sectional curvature bounds. result Volume growth can be arbitrarily close to quadratic, and Betti numbers are infinite.