arXiv research

The paper studies Ricci curvature on Kähler-Ricci flow.

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A cr…

The paper proves bounded curvature for Kähler Ricci shrinker surfaces.

Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…

The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.

The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.

Paper improves heat kernel estimates on Ricci shrinkers.

Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.

Proves K-polystability for Kähler-Ricci shrinkers with decaying curvature.

Round cylinders are rigid in Ricci shrinkers close to the standard product.

Local gaps in Ricci shrinkers depend only on dimension.

Close to complex projective spaces, Ricci shrinkers are rigid.

Local gap theorem for Ricci shrinkers ensures flatness if certain functionals are close to zero.

The paper studies topological properties of Ricci shrinkers using weighted $L^2$ cohomology.

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As applications, we obtain some curvature estimates of the Ricci shrinkers depending only on the non-collapsing constant.

Estimates the first eigenvalue of a Laplacian on self-shrinkers in Ricci shrinkers.

Uniform entropy bound for Ricci shrinkers with bounded curvature.

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

Study geometric structure of Ricci shrinker ends without global curvature assumptions.

Paper proves rigidity of certain Ricci shrinkers.

The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.

Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.

Compact shrinkers with curvature pinching conditions proven.

We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.

We show that recent work of Ni and Wilking yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kähler--Ricci shrinkers by Feldman, Ilmanen, and Knopf exhibit that this result is sharp.

Study classifies 4D shrinkers with nonnegative Ricci curvature.

Eigenvalue bounds for Schrödinger operators on Ricci shrinkers and related manifolds.

In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinker…

Proves convergence of gradient Ricci shrinkers with uniform bounds.

In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain sel…

In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estim…

In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is co…

In this paper we prove that for a complete, connected and oriented Käler affine manifold $(M,G)$ of dimension $n,$ if it is Kähler affine Ricci flat or the K$\ddot{a}$hler affine scalar curvature $S\equiv0,$ ($n\leq 5$), then the universal covering manifold $\widetilde{M}$ of $M$ is isometric to the Euclidean n-space $…

The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.

The paper proves continuity of Morse index for Ricci shrinkers.

For any complete noncompact K$\ddot{a}$hler manifold with nonnegative and bounded holomorphic bisectional curvature,we provide the necessary and sufficient condition for non-ancient solution to the Ricci flow in this paper.

Proves Kähler-Ricci shrinkers are complex analytic varieties.

Holomorphic functions grow polynomially on Kähler-Ricci shrinkers, proving ring finitely generated.

Symmetries in shrinking Ricci solitons spread outward.

Paper estimates Wasserstein distance for Ricci shrinkers.

By establishing two general quadratic inequalities, we obtain some inequalities related to Ricci curvatures for Lagrangian submanifolds of K$\ddot{\mathrm{a}}$hler QCH-manifolds, which generalize some results for Lagrangian submanifolds of complex space forms.

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If $g$ is not a local maxim…

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.

In this paper we consider a perturbation of the Ricci solitons equation proposed in \cite{jpb1} and studied in \cite{CaMa} and we classify noncompact gradient shrinkers with bounded nonnegative sectional curvature.

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finit…