The paper proves bounded curvature for Kähler Ricci shrinker surfaces.
problem Understanding the curvature of Kähler Ricci shrinker surfaces.
method Proved bounded sectional curvature using earlier work.
result Complete classification of all Kähler Ricci shrinker surfaces.
Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.
problem Understanding the structure of ends in Ricci shrinkers, especially singular ones.
method Analyzes general and asymptotically conical ends, applies to weak convergence.
result No new conical end can form in the limit of sequences of Ricci shrinkers.
The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.
problem Understanding the structure and behavior of Ricci shrinkers.
method Proving rigidity and ε-regularity theorems for Ricci shrinkers using entropy and curvature.
result Non-compact Ricci shrinkers are asymptotic to cones under certain curvature conditions.
The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.
problem Understanding the relationship between Kähler-Ricci shrinkers and Fano fibrations.
method Using birational algebraic geometry, the paper proves properties of Kähler-Ricci shrinkers and formulates conjectures relating them to Fano fibrations.
result The existence of Kähler-Ricci shrinkers is conjectured to be related to K-stability of polarized Fano fibrations.
Paper improves heat kernel estimates on Ricci shrinkers.
problem Estimates on heat kernels for Ricci shrinkers.
method Improves estimates from previous work and extends recent progress.
result Theory of $\IF$-convergence holds on Ricci flows induced by Ricci shrinkers.
Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.
problem Proving uniqueness of Kähler Ricci shrinkers on toric orbifolds.
method Extending results from toric manifolds to toric orbifolds.
result Uniqueness of Kähler Ricci shrinkers on toric orbifolds established.
Proves K-polystability for Kähler-Ricci shrinkers with decaying curvature.
problem K-stability of Kähler-Ricci shrinkers with decaying curvature.
method Developed algebraic theory for Kähler-Ricci shrinkers and proved K-polystability.
result Existence of Kähler-Ricci shrinker metric implies K-polystability in decaying curvature case.
Round cylinders are rigid in Ricci shrinkers close to the standard product.
problem Proving rigidity of round cylinders in Ricci shrinkers.
method Proving isometry using pointed-Gromov-Hausdorff topology.
result Ricci shrinkers close to Sn−1imesR are isometric to Sn−1imesR. Local gaps in Ricci shrinkers depend only on dimension.
problem Understanding local properties of Ricci shrinkers.
method Proved local versions of Ricci curvature and entropy gap theorems.
result Local gaps depend only on dimension, not global entropy.
Close to complex projective spaces, Ricci shrinkers are rigid.
problem Rigidity of complex projective spaces in Ricci shrinkers.
method Proving isometry using Gromov-Hausdorff distance.
result Ricci shrinkers close to (CPN,gFS) are isometric to (CPN,gFS). Local gap theorem for Ricci shrinkers ensures flatness if certain functionals are close to zero.
problem Understanding the global geometry of Ricci shrinkers from local information.
method Proving a local gap theorem using the local μ-functional. result Ricci shrinkers are flat if the local μ-functional is close to zero. The paper studies topological properties of Ricci shrinkers using weighted L2 cohomology.
problem Proving topological results for smooth gradient Ricci shrinkers.
method Weighted L2 cohomology and extensions to mean curvature flow self-shrinkers. result Establishes upper bounds for Betti numbers, vanishing theorem for cohomology, and dichotomy for ends.
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.
problem Estimating the first eigenvalue of a Laplacian on self-shrinkers in Ricci shrinkers.
method Analyzes the drifted Laplacian on hypersurfaces in Ricci shrinkers, proving a lower bound for the first nonzero eigenvalue.
result Provides a lower bound for the first nonzero eigenvalue of the drifted Laplacian on embedded f-minimal hypersurfaces.
Uniform entropy bound for Ricci shrinkers with bounded curvature.
problem Bounding entropy for Ricci shrinkers with specific curvature constraints.
method Establishing uniform entropy bounds for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
result Uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
Study geometric structure of Ricci shrinker ends without global curvature assumptions.
problem Understand the geometric structure of Ricci shrinker ends without global curvature constraints.
method Analyze blow-up sequences of Ricci shrinkers at points with Type I scalar curvature bound, extending F-convergence theory.
result Limits of Ricci shrinkers at points with Type I scalar curvature bound split a line in four dimensions.
Paper proves rigidity of certain Ricci shrinkers.
problem Rigidity of Ricci shrinkers in specific spaces.
method Quantitative characterization, rigidity inequality, contraction and extension.
result Uniqueness of tangent flow for compact Ricci flows.
The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.
problem Linear stability of Gradient Ricci Shrinker metrics.
method Found commutator formulas and generalized a stability theorem.
result Generalized a necessary condition for linear stability.
Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.
problem Analyzing convergence of Ricci flow and harmonic map heat flow.
method Established long-time existence of harmonic map heat flow between Ricci flow and shrinker.
result Ricci flow converges exponentially to compact integrable shrinkers and at singularities modelled on the shrinker.
Compact shrinkers with curvature pinching conditions proven.
problem Ensuring shrinkers are compact under curvature pinching conditions.
method Various curvature pinching conditions applied to shrinkers with positive Ricci curvature and asymptotically nonnegative sectional curvature.
result Shrinkers with curvature pinching conditions are proven to be compact.
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.
Study classifies 4D shrinkers with nonnegative Ricci curvature.
problem Classifying 4D shrinkers with nonnegative Ricci curvature.
method Asymptotic analysis, eigenvalue evolution, Gauss-Bonnet-Chern formula, integration by parts.
result Classifies 4D shrinkers under specific curvature conditions.
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.
Eigenvalue bounds for Schrödinger operators on Ricci shrinkers and related manifolds.
problem Estimating eigenvalues of Schrödinger operators on Ricci shrinkers and related manifolds.
method Using Ricci shrinkers and Perelman's μ-functional, the paper derives lower bounds for the lowest eigenvalues of Schrödinger operators.
result Lower bounds for the lowest eigenvalues of Schrödinger operators on Ricci shrinkers and related manifolds, with equality conditions characterized.
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.
problem Compactness and energy concentration in gradient Ricci shrinkers.
method Bubble-tree convergence and local energy analysis.
result No energy concentrates in neck regions, leading to a local diffeomorphism finiteness theorem.
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…
The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.
problem Estimating dimensions and existence of holomorphic sections with polynomial growth on Kähler Ricci shrinkers.
method Proved upper bounds for dimensions and existence of sections using polynomial growth.
result Upper bounds for dimensions and existence of holomorphic sections with polynomial growth on Kähler Ricci shrinkers.
The paper proves continuity of Morse index for Ricci shrinkers.
problem Lower and upper semi-continuity of the Morse index for gradient Ricci shrinkers.
method Adapting and refining recent arguments on CMC hypersurfaces and polynomially weighted Sobolev spaces, with techniques for non-compact shrinkers.
result Identifies a condition ensuring the Morse index of asymptotically conical shrinkers is bounded below by the f-index of their asymptotic cone.
Proves Kähler-Ricci shrinkers are complex analytic varieties.
problem Characterizing singular Kähler-Ricci shrinkers.
method Analyzes limits of Kähler-Ricci flows and applies algebraic geometry.
result Singular Kähler-Ricci shrinkers are locally algebraic complex-analytic varieties.
Holomorphic functions grow polynomially on Kähler-Ricci shrinkers, proving ring finitely generated.
problem Understanding polynomial growth of holomorphic functions on Kähler-Ricci shrinkers.
method Analyzing scalar curvature conditions to prove finite generation of the ring of holomorphic functions.
result The ring of holomorphic functions with polynomial growth on Kähler-Ricci shrinkers is finitely generated.
Symmetries in shrinking Ricci solitons spread outward.
problem Understanding symmetries in shrinking Ricci solitons.
method Propagating approximate symmetries to larger scales.
result Symmetries in shrinking Ricci solitons spread outward.
Paper estimates Wasserstein distance for Ricci shrinkers.
problem Estimating Wasserstein distance for Ricci shrinkers.
method Analyzes Wasserstein distance between measures in tangent spaces.
result Provides upper estimate for Wasserstein distance.
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.
problem Characterizing κ-solutions with specific asymptotic behavior. method Analysis of Ricci flow in dimensions n≥4. result Uniformly Positive Isoperimetric Constant (PIC) for κ-solutions. 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…
Haslhofer and Müller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly be an orbifold Ricci shrinker. In this paper we prove a compactness theorem for non…
We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the loca…
We study integral and pointwise bounds on the curvature of gradient shrinking Ricci solitons. As applications we discuss gap and compactness results for gradient shrinkers.
In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact …
In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking Kähler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space. We also prove an analog of a result of H.-D. Cao and H. Li about the non-existence…
For complete shrinking (gradient) Ricci solitons, we observe a quantification of Wylie's result that the fundamental group is finite.
Study noncollapsed F-limit metric solitons, proving properties similar to smooth Ricci shrinkers.
problem Understanding noncollapsed F-limit metric solitons in Ricci flow.
method Systematic study and proving properties similar to smooth Ricci shrinkers.
result Proves quadratic lower bound for scalar curvature, local gap theorem, global Sobolev inequality, and optimal volume growth lower bound.
We prove surfaces are unknotted with specific properties.
problem Unknottedness of free boundary minimal surfaces and self-shrinkers.
method Introduced concepts of boundary graph and graph at infinity to prove unknottedness.
result Proved surfaces are unknotted with specific properties.