Proves local noncollapsing estimate for mean curvature flow.
problem Ensuring noncollapsing in mean curvature flow.
method Combining local estimate with earlier work on ancient solutions.
result Ancient convex solutions that sweep out entire space are noncollapsed.
Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.
problem Volume noncollapsing for Kähler metrics induced by complex Monge-Ampère equations.
method Proves local volume noncollapsing estimate with Ricci curvature lower bound.
result Establishes diameter and gradient estimates for Kähler metrics.
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.
Study confirms conjectures on Ricci limit spaces and their topological properties.
problem Understanding the topological structure of noncollapsed Ricci limit spaces.
method Analysis of tangent cones and application of manifold recognition theorems.
result Cross-sections of tangent cones at points in 4D spaces are homeomorphic to a fixed spherical space form.
Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.
problem Understanding the asymptotic geometry of ancient Ricci flows with nonnegative Ricci curvature.
method Analyze tangent flows at infinity and use estimates for noncollapsed F-limit metric solitons.
result Two dichotomy theorems for ancient Ricci flows: either the asymptotic volume ratio is zero or every tangent flow is a Ricci flat cone.
New proof classifies ancient flows in 3D space.
problem Classifying ancient noncollapsed flows in R3. method Combining neck theorem and Harnack inequality rigidity.
result Directly establishes self-similarity of flows.
In our previous work we showed that for an ancient solution to the Ricci flow with nonnegative curvature operator, assuming bounded geometry on one time slice, bounded entropy implies noncollapsing on all scales. In this paper we prove the implication in the other direction, that for an ancient solution with bounded no…
In this paper, we analyze the asymptotic behavior of κ-noncollapsed and positively curved steady Ricci solitons and prove that any n-dimensional κ-noncollapsed steady Kähler-Ricci soliton with non-negative sectional curvature must be flat.
Ancient solutions to Kähler Ricci flow classified completely.
problem Ancient solutions to Kähler Ricci flow with nonnegative bisectional curvature.
method Complete classification of κ-noncollapsed, complete ancient solutions.
result Classification of all ancient solutions to Kähler Ricci flow with nonnegative bisectional curvature.
Paper classifies singularity models for 3D hypersurfaces in R^4.
problem Classifying singularity models for 3D hypersurfaces in R^4.
method Proving classification through mathematical proof.
result All noncollapsed translating hypersurfaces in R^4 are classified.
Paper disproves potential singularity models for 3D hypersurfaces in R^4.
problem Noncollapsed wing-like flows as singularity models for mean curvature flow in R^4.
method Fine bubble-sheet analysis generalizing fine neck analysis.
result Ancient noncollapsed flows in R^4 are always simple geometric shapes, not wedges.
Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative c…
Study on ancient Ricci flows with positive curvature, proving noncollapsedness.
problem Characterizing ancient Ricci flows with positive sectional curvature.
method Analyzing complete and noncompact Type I ancient Ricci flows with positive sectional curvature.
result Ancient solutions are noncollapsed on all scales in complete and noncompact cases, and in even-dimensional closed cases.
In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.
We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality.
The study characterizes and rules out collapsing in convex ancient mean curvature flow.
problem Characterizing and ruling out collapsing in convex ancient mean curvature flow.
method Characterization and counterexamples.
result Collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane.
Enhanced estimates for ancient ovals and translators in 3D and 4D.
problem Sharp estimates for ancient ovals and translators.
method Derivation of gradient and Hessian estimates.
result Sharp gradient and Hessian estimates for ancient ovals and translators.
Establishes 4D regularity for certain metric spaces.
problem Noncollapsed sequences of metrics with Ricci tensor bounds.
method A priori L2 curvature estimates.
result Diffeomorphism finiteness and rigidity theorems.
We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound are topologically spheres. As an application we show that for any finite dimensional Alexandrov space Xn with n≥5 there exists an Alexandrov space Y homeomorphic to X which can not be o…
The paper studies how certain surfaces evolve in space without collapsing.
problem Evolution of surfaces with inhomogeneous speeds without collapsing.
method Analyzes curvature flows with a specific speed function and structural conditions.
result Establishes exterior noncollapsing estimates for the flow.
Quantitative estimate for curvature in mean curvature flow.
problem Estimating curvature in mean curvature flow.
method Proving a curvature estimate for smooth convex ancient flows.
result Curvature grows at most quadratically in terms of rescaled extrinsic distance.
A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension 3 which have positive sectional curvature and are κ-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in R3, and prove that the rotationally symm…
Sharp bound on singular set dimension for specific geometric problems.
problem Hausdorff dimension of singular set in free boundary problems.
method Analysis of noncollapsed limits of manifolds with Ricci curvature bounds.
result Dimension bound of singular set is n−5. The paper classifies noncollapsed translators in 4D space.
problem Classifying entire convex translators in 4D space.
method Developed Fredholm theory and used Lyapunov-Schmidt reduction.
result The one-parameter family of translators is uniquely determined.
Paper studies fundamental groups of certain Ricci solitons.
problem Understanding fundamental groups of specific Ricci solitons.
method Analyzes properties of complete steady gradient Ricci solitons with nonnegative sectional curvature.
result Fundamental groups of these solitons are either trivial or infinite.
Alexandrov immersed surfaces maintain their properties under mean curvature flow.
problem Preserving Alexandrov immersivity in mean curvature flow.
method Mean curvature flow techniques adapted for Alexandrov immersed, 2D surfaces.
result Mean curvature flow properties hold for Alexandrov immersed surfaces.
Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.
problem Developing an ABP approach to Sobolev and Michael-Simon inequalities under volume noncollapsing assumptions.
method Using a refinement of Brendle's contact-set argument to derive lower bounds for the volumes of geodesic balls.
result A Michael-Simon type inequality for immersed submanifolds with nonnegative sectional curvature and volume noncollapsing.
The paper shows that ancient noncollapsed mean curvature flows have a blowdown of at most n-2 dimensions.
problem Understanding the blowdown of ancient noncollapsed mean curvature flows.
method Fine cylindrical analysis and fine neck analysis generalization.
result The blowdown of ancient noncollapsed mean curvature flows is at most n-2 dimensional.
In this paper, we study κ-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the standard cylinder Sn−1×R. By making use of the properties of…
In this paper, we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally ε-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any κ-noncollapsed gradient steady Ricci soliton (Mn,g,f) with nonnegative curvature oper…
Classifies ancient noncollapsed flows in 4D space.
problem Classify all noncollapsed singularities of the mean curvature flow in R^4.
method Proves differential neck theorem, introduces new ideas like switch and differential Merle-Zaag dynamics.
result Classifies all ancient noncollapsed solutions in R^4.
Ancient solutions to Ricci flow with isotropic curvature conditions are classified.
problem Classifying ancient solutions to Ricci flow with isotropic curvature conditions.
method Analyzing properties of ancient solutions with isotropic curvature conditions.
result Ancient solutions to Ricci flow with isotropic curvature conditions are either shrinking cylinders or the Bryant soliton.
We discuss some geometric conditions under which a complete noncompact shrinking gradient Ricci soliton will split at infinity.
We prove that any noncompact κ-noncollapsed steady Ricci soliton with nonnegative curvature operator must be rotationally symmetric if it has a linear curvature decay.
The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.
problem Analyzing noncompact steady gradient Ricci solitons with nonnegative curvature operator.
method Examining the asymptotic behavior of noncompact κ-noncollapsed steady gradient Ricci solitons with nonnegative curvature operator away from a compact set.
result 4D noncompact κ-noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling.
In this paper we investigate the differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method that was introduced by the first two named authors in \cite{LaTi14}.
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 found a 6D manifold with specific geometric properties.
problem Finding a 6D manifold with Ricci curvature ≥ 0 and infinitely generated fundamental group.
method Built a smooth complete manifold with specific geometric properties.
result Found a 6D manifold with Ric≥0 and π1(M6)=Q/Z infinitely generated. The paper shows that certain Einstein orbifolds cannot be limits of smooth Einstein metrics.
problem Understanding the limits of smooth Einstein metrics on compact Einstein orbifolds.
method Analyzing sequences of compact Einstein manifolds and their limits, providing an explicit obstruction for certain orbifolds.
result Explicit obstruction for negative Einstein orbifolds appearing as limits of compact Einstein manifolds, which does not vanish for hyperbolic orbifolds.
This research shows that steady solitons in higher dimensions always reduce at infinity.
problem Characterizing steady solitons with nonnegative sectional curvature in higher dimensions.
method Dimension reduction analysis and tangent flow classification.
result Steady solitons in higher dimensions always reduce at infinity.
In this paper we define an orientation of a measured Gromov-Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncoll…
Proves curvature tensor convergence for smoothable spaces.
problem Curvature tensor behavior in smoothable Alexandrov spaces.
method Weak convergence of curvature tensors in noncollapsing sequences.
result Proves convergence of curvature tensors in smoothable Alexandrov spaces.
We show that if a CD(K,n) space (X,d,fHn) with n≥2 has curvature bounded from above by κ in the sense of Alexandrov then f=const.
In this note, we show that any n-dimensional κ-noncollapsed steady Kähler-Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in \cite{DZ2}.
The study disproves rotating ancient flows in 4D space.
problem The existence of rotating ancient flows in R4. method Analysis of ancient noncollapsed flows in R4. result Nonexistence of rotating ancient flows among ancient noncollapsed flows in R4. Ancient solutions of Ricci flow with Type I growth are classified.
problem Understanding ancient solutions of Ricci flow with specific curvature growth.
method Analyzing ancient solutions with Type I curvature growth in arbitrary dimensions.
result Ancient solutions with Type I growth are classified into specific types.
We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.