Perelman proves compact and complete noncompact shrinking breather properties.
problem Proving properties of shrinking breathers in both compact and noncompact settings.
method Using Perelman's entropy formula and L-geometry, constructing ancient solutions.
result Complete noncompact case proven without technical assumptions.
Paper classifies 3D breathers and generalizes Ricci soliton results.
problem Classifying and understanding Ricci solitons and breathers.
method Developed a condition for the existence of asymptotic shrinking gradient Ricci solitons.
result Every complete shrinking Ricci soliton with bounded Ricci curvature is gradient.
Two new proofs show Ricci flow breathers are special solutions.
problem Characterize solutions to Ricci flow on closed manifolds.
method Use singularity models and ancient solutions to show they are gradient Ricci solitons.
result Ricci flow breathers are gradient Ricci solitons.
Suppose M is a compact n-dimensional manifold, n≥2, with a metric gij(x,t) that evolves by the Ricci flow ∂tgij=−2Rij in M×(0,T). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality.
New nontrivial breathers found for Ricci flow on noncompact manifolds.
problem Perelman's theorem that nontrivial breathers don't exist for closed manifolds.
method Constructing nontrivial expanding breathers in all dimensions for noncompact manifolds.
result Nontrivial breathers found for Ricci flow on noncompact manifolds.
No breathers found for mean curvature flow in noncompact space.
problem Existence of breathers in mean curvature flow.
method Proved no breather theorems for mean curvature flow in noncompact Euclidean space.
result Breathers must be solitonic solutions under certain conditions.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
Let $\M$ be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This…
In this article we give a brief survey of breather and soliton solutions to the Ricci flow and prove a no breather and soliton theorem for homogeneous solutions.
In this note, we construct families of functionals of the type of F-functional and W-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using t…
Kahler-Ricci flow long-time behavior and initial data
problem Relationship between Kahler-Ricci flow and initial data
method Investigate long-time behavior
result Asymptotic profiles and non-trivial breathers
Under suitable conditions near infinity and assuming boundedness of curvature tensor, we prove a no breathers theorem in the spirit of Ivey-Perelman for some noncompact Ricci flows. These include Ricci flows on asymptotically flat (AF) manifolds with positive scalar curvature. Since the method for the compact case face…
In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of Fk-func…
We explain how to apply techniques from integrable systems to construct 2k-soliton homoclinic wave maps from the periodic Minkowski space S1×R1 to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the -1 flow equation associated to a compact …
In this paper we define the torsion flow, a CR analogue of the Ricci flow. For homogeneous CR manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior. We also derive monotonicity formulas for CR entropy functionals. As an application, we classify torsion breathers.
We study the Ricci flow on Riemannian groupoids. We assume that these groupoids are closed and that the space of orbits is compact and connected. We prove the short time existence and uniqueness of the Ricci flow on these groupoids. We also define a F-functional and derive the corresponding results for steady breathers…
In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eig…
A novel class of integrable surfaces is recorded. This class of O surfaces is shown to include and generalize classical surfaces such as isothermic, constant mean curvature, minimal, `linear' Weingarten, Guichard and Petot surfaces and surfaces of constant Gaussian curvature. It is demonstrated that the construction of…
New energy functional bounds Ricci flows on ancient spaces.
problem Bounding Ricci flows on ancient spaces.
method Introducing a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows.
result Provides an upper bound for the ordinary λ-functional.
We present a 2x2 Lax representation for discrete circular nets of constant negative Gauß curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Bäcklund transformations and an associated family. All the membe…
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.
Study proves shrinking doughnuts via modified curvature flow.
problem Existence of self-shrinking tori under mean curvature flow.
method Variational methods and modified curvature flow.
result Established an upper bound for the weighted energy of shrinking doughnuts.
In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton R4 or the round cyl…
The study shows no non-constant positive f-harmonic functions on complete gradient shrinking Ricci solitons.
problem Existence of non-constant positive f-harmonic functions on gradient shrinking Ricci solitons. method Proved the non-existence of non-constant positive f-harmonic functions using Liouville-type theorems. result No non-constant positive f-harmonic functions on complete gradient shrinking Ricci solitons. Study on shrinking solitons of generalized Ricci flow.
problem Characterizing shrinking solitons in generalized Ricci flow.
method Analyzing gradient shrinking solitons and pluriclosed solitons on compact manifolds.
result First non-trivial shrinking generalized soliton constructed.
Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.
problem Classifying shrinking Ricci solitons without curvature or volume restrictions.
method Proving the sharp Li-Yau equality for conjugate heat kernel on shrinking Ricci solitons.
result Several estimates and classification of four-dimensional, non-compact shrinking Ricci solitons.
5D shrinking solitons with bounded curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons.
method Proving rigidity for solitons with bounded curvature.
result 5D shrinking gradient Ricci solitons with bounded curvature are rigid.
Proves shrinking Ricci solitons near Kähler cones are Kähler.
problem Characterizing shrinking gradient Ricci solitons near Kähler cones.
method Analyzes asymptotic behavior and proves Kähler property.
result Proves shrinking gradient Ricci solitons are Kähler near Kähler cones.
Paper proves rigidity for Ricci solitons with specific conditions.
problem Understanding the properties of Ricci solitons under various conditions.
method Analyzes shrinking and expanding Ricci solitons with specific constraints.
result Compact shrinking Ricci solitons are Einstein if the potential function is controlled.
Compact shrinking solitons with positive curvature are proven to be compact.
problem Characterizing shrinking gradient Kähler-Ricci solitons with specific curvature properties.
method Analyzing properties of complete shrinking gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature.
result Compact shrinking solitons with positive curvature are proven to be compact.
New proof of shrinking gradient Ricci soliton rigidity.
problem Rigidity of shrinking gradient Ricci solitons.
method Maximum principle, maximum curvature condition.
result Shrinking gradient Ricci soliton with constant scalar curvature is isometric to a finite quotient of R^2 x S^2.
We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-nonc…
Compact shrinking Kähler-Ricci solitons with positive curvature are proven to be finite.
problem Characterizing shrinking Kähler-Ricci solitons with positive bisectional curvature.
method Alternative proof using Munteanu and Wang's argument.
result Compactness of shrinking gradient Kähler-Ricci solitons with positive bisectional curvature.
Study proves rigidity for solitons with uncertainty principle.
problem Rigidity of shrinking Ricci solitons with uncertainty principle.
method Proves rigidity theorems for shrinking gradient Ricci solitons.
result Proves rigidity theorems with sharp constant in R^n.
Complete shrinking soliton found on a specific complex surface.
problem Classifying complete shrinking gradient Kähler-Ricci solitons in two complex dimensions.
method Proved existence of a unique soliton with bounded scalar curvature on a specific blowup.
result Complete classification of such solitons in two complex dimensions.
Study Bernstein results for self-shrinking solutions in Lagrangian flow.
problem Understanding entire self-shrinking solutions in Lagrangian flow.
method Prior estimates and barriers construction.
result Showed Bernstein type results for self-shrinking solutions.
New findings on shrinking solitons with positive isotropic curvature.
problem Characterizing shrinking solitons with positive isotropic curvature.
method Analyzing properties of gradient shrinking solitons in dimensions 5 and above.
result Non-flat complete shrinking solitons with positive isotropic curvature are quotients of the round sphere or the cylinder.
Rigidity proven for specific gradient Ricci solitons.
problem Characterizing gradient shrinking Ricci solitons with certain tensor properties.
method Proving rigidity through fourth order divergence-free tensors and Weyl tensor conditions.
result Gradient shrinking Ricci solitons with specific tensor properties are rigid.
The study proves compactness and existence of entropy minimizers for self-shrinking surfaces.
problem Understanding entropy in higher-codimension mean curvature flow.
method Measure-theoretical techniques and rigidity results for self-shrinkers.
result Existence of entropy minimizers and improved rigidity results.
Isometries of cones embed in those of asymptotic shrinking Ricci solitons.
problem Characterizing isometries of asymptotically conical shrinking Ricci solitons.
method Analyzing the embedding of isometries from the cone's cross-section into the entire shrinker.
result Isometries of the cone's cross-section embed in the entire shrinker.
In this paper, we will give a local version of the Hamilton-Ivey type pinching estimate of the gradient shrinking soliton with vanishing Weyl tensor, and then give a complete classification on gradient shrinking solitons with vanishing Weyl tensor.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
The paper proves rigidity for self-shrinking spacelike graphs in pseudo-Euclidean space.
problem Understanding the rigidity of self-shrinking spacelike graphs in pseudo-Euclidean space.
method Volume growth estimate and Co-Area formula.
result Various rigidity results for spacelike entire self-shrinking graphs.
A new method shrinks a complex structure without much change.
problem Understanding the geometry of Bing's wild involution.
method Producing a counterintuitive construction to shrink the Bing decomposition without much change.
result A method to shrink a complex structure (Bing's decomposition) without much change.
Let (Y,d) be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2, Y is a smooth manifold satisfying a shrinking Ricci soliton equation.
Sharp upper diameter limit found for Ricci solitons.
problem Bounding the diameter of compact shrinking Ricci solitons.
method Used a sharp logarithmic Sobolev inequality and Vitali-type covering argument.
result Sharp upper diameter bound established in terms of scalar curvature and entropy.
The paper studies geometric properties of self-shrinkers in shrinking Ricci solitons.
problem Understanding geometric properties of self-shrinkers in specific geometric settings.
method Proved spectral properties of drifted Laplacian and used them to derive geometric properties.
result Described domains in the ambient space that cannot contain self-shrinkers.
Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.
problem Analyzing Schrödinger heat kernel on gradient shrinking Ricci solitons.
method Deriving sharp Gaussian upper bounds for the Schrödinger heat kernel.
result Sharp upper and lower bounds for eigenvalues of the Schrödinger operator.