Study ancient caloric functions on graphs, extending a theorem from manifolds.
problem Bounding the dimension of ancient caloric functions on graphs.
method Extending Colding and Minicozzi's theorem to graphs.
result Dimension of ancient caloric functions is bounded by growth degree and graph dimension.
For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions…
Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.
problem Bounding ancient solutions to biharmonic heat equation.
method Using polynomial volume growth and dimensions of biharmonic functions.
result Ancient solutions are bounded by polynomial dimensions.
Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.
problem Analyzing ancient caloric functions on manifolds beyond volume doubling.
method Time polynomial structure result on ancient caloric functions with polynomial growth.
result Finiteness result for ancient caloric functions is essentially sharp, except for multi-end cases.
The study examines polynomial growth functions on gradient shrinking Ricci solitons.
problem Characterizing harmonic and caloric functions with polynomial growth on gradient shrinking Ricci solitons.
method Analysis of polynomial growth functions under different curvature conditions.
result Finite dimensional estimates for harmonic and caloric functions with polynomial growth.
Study ancient solutions on graphs with unbounded Laplacians, generalizing previous results.
problem Understanding ancient solutions on graphs with unbounded Laplacians.
method Generalizing Colding and Minicozzi's theorem and Hua's result to graphs with unbounded Laplacians.
result The dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth.
Study on heat equation and eigenfunctions on RCD spaces, proving unique continuation.
problem Unique continuation for caloric functions and eigenfunctions on RCD spaces.
method Establish weak unique continuation theorem for caloric functions and eigenfunctions on compact RCD(K,2) spaces.
result Existence of non-trivial eigenfunctions and caloric solutions vanishing up to infinite order at one point.
Ancient flows of elliptic functionals classified in various dimensions.
problem Classifying ancient solutions to gradient flows of elliptic functionals.
method Analyzing closed ancient solutions in Riemannian manifolds.
result Ancient solutions classified in multiple dimensions and cases.
Ancient Ricci flows are identified without curvature sign condition.
problem Identifying type II ancient Ricci flows and their backward limits.
method Using a size condition of the sharp log Sobolev functional near infinity.
result Rigidity result for ancient Ricci flows without sign condition on curvatures.
Global Schrödinger map flows to Kähler manifolds proved for high dimensions with small data.
problem Global existence of Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces.
method Decay estimates of moving frame dependent quantities in caloric gauge setting, combined with a bootstrap-iteration scheme.
result Global existence of Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces for high dimensions.
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.
Classifies ancient and expanding Ricci flows with specific groups.
problem Classifying ancient and expanding Ricci flows with certain groups.
method Uses a renormalized λALE-functional to control the large-scale behavior of Perelman's μ-functional.
result Identifies hyperkähler ALE metrics as the only spin ancient Ricci flows with specific groups.
Ancient convex solutions to flow equations are limited to simple shapes.
problem Characterizing ancient convex solutions to flow equations.
method Analyzing mean curvature flow and curvature functions of convex hypersurfaces.
result Ancient convex solutions to flow equations are limited to spherical, cylindrical, or planar shapes.
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.
Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.
problem Bounding Nash entropy in ancient Ricci flows.
method Uniformly bounded Nash entropy implies uniform bounds on the ν-functional, leading to uniform logarithmic and Sobolev inequalities.
result Uniform logarithmic and Sobolev inequalities on ancient Ricci flows with bounded Nash entropy.
New convex ancient solutions found for flows by high powers of curvature.
problem Existence of closed convex ancient solutions to curvature flows.
method Proves existence of closed convex ancient solutions with specific curvature flow speeds.
result Existence of non-homothetic convex ancient solutions for flows by high powers of curvature.
Ancient Ricci flows with asymptotic solitons have uniform bounds and inequalities.
problem Bounding and understanding ancient Ricci flows with asymptotic solitons.
method Analyzing asymptotic solitons, proving uniform bounds on Perelman's ν-functional, and showing Nash entropy bounds.
result Uniform bounds on Perelman's ν-functional and logarithmic/Sobolev inequalities for ancient solutions.
Study polynomial growth harmonic functions on infinite penny graphs.
problem Finite-dimensional property of polynomial growth harmonic functions on infinite penny graphs.
method Asymptotically sharp dimensional estimate for ancient solutions of the heat equation.
result Proved the asymptotically sharp dimensional estimate.
Ancient solutions of heat equation with exponential growth are analytic in time.
problem Analyticity of solutions to the heat equation in time.
method Proving analyticity for ancient solutions with exponential growth.
result Ancient solutions with exponential growth are analytic in time.
We consider the evolution of hypersurfaces on the unit sphere Sn+1 by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such a…
New translations defined; curve shortening flow solved in hyperbolic plane.
problem Solving curve shortening flow in hyperbolic geometry.
method Introduced new translations, solved equations, analyzed ancient solutions.
result Explicit solutions and area estimates for ancient solutions.
Ancient curve flows classified into specific types.
problem Classifying ancient finite-entropy curve shortening flows.
method Proving flow types through mathematical analysis.
result Ancient flows are one of several specific types.
In this paper, we prove that any non-flat ancient solution to Kähler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional curvature must be compact. Both results generalize the corresponding earlier resul…
Ancient solutions to mean curvature flow have unique shapes.
problem Understanding unique shapes of ancient solutions.
method Proved a Bernstein theorem for ancient solutions.
result Ancient solutions to mean curvature flow have unique shapes.
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.
Ancient pancakes solve mean curvature flow problem.
problem Mean curvature flow problem
method Constructing an embedded ancient solution as a stack of pancakes
result Embedded ancient solution to mean curvature flow
Ancient solutions to Ricci flow in higher dimensions are mostly cylinders or solitons.
problem Uniqueness of ancient κ-solutions in higher dimensions. method Analysis of ancient κ-solutions with specific properties. result The only noncompact ancient κ-solutions are cylinders, quotients, or the Bryant soliton. Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.
problem Characterizing ancient solutions on an infinite strip with polynomial and exponential growth.
method Analyzing parabolic equations on an infinite strip, proving properties of ancient solutions.
result Ancient solutions on the strip are constant if they grow polynomially, and have a finite-dimensional space for slower exponential growth.
Let n≥3 and m=n+2n−2. We construct 5-parameters, 4-parameters, 3-parameters ancient solutions of the equation vt=(vm)xx+v−vm, v>0, in R×(−∞,T) for some T∈R. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient so…
Ancient Ricci flow found from Taub-Bolt metric.
problem Existence of ancient Ricci flow solutions.
method Analysis of Ricci flow from Taub-Bolt metric.
result Non-trivial ancient solution to Ricci flow found.
Ancient curves span halfplanes via flow.
problem Ancient solutions to Curve Shortening Flow.
method Constructing infinite family of solutions.
result Spanning halfplane with ancient curves.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
Ancient solutions to heat equations on graphs are shown to be analytic in time.
problem Analyzing the analyticity of ancient solutions to heat equations on graphs.
method Proving time analyticity under a sharp growth condition.
result Ancient solutions to heat equations on graphs are time analytic under certain conditions.
Classifies ancient convex curves in convex domains.
problem Ancient convex curve shortening flows on convex domains.
method Classification of convex ancient solutions.
result Ancient convex curves in convex domains classified.
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 ancient solutions to 3D Ricci flow.
problem Classifying ancient solutions to 3D Ricci flow.
method Proves uniqueness of solutions based on classification criteria.
result Ancient solutions are either shrinking spheres or Perelman's Type II solutions.
Ancient symplectic solutions to mean curvature flow are flat.
problem Understanding ancient solutions to mean curvature flow in symplectic geometry.
method Using a complex phase map to prove a Bernstein theorem.
result Ancient solutions to the symplectic mean curvature flow are flat.
The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.
problem Understanding ancient solutions to curvature flows in bounded and unbounded regions.
method Constructing pancake-like and sausage-like ancient compact solutions.
result Ancient solutions to curvature flows in bounded and unbounded regions.
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
New ancient solutions found for curvature flow in 2D.
problem Ancient solutions for curvature flow in 2D.
method Constructing and classifying convex ancient solutions.
result All convex ancient solutions classified for α∈(32,1). Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in R2. method Determined Morse indices and kernels of the linearized operator of shrinkers. Constructed flows using unstable eigenfunctions.
result Existence of ancient flows with finite entropy.
Study ancient Ricci flow solutions, proving unique asymptotic behavior.
problem Understanding unique asymptotics of compact ancient solutions to 3D Ricci flow.
method Analyzing noncollapsed compact ancient solutions, proving asymptotic behavior.
result Proves unique asymptotic behavior for compact ancient solutions.
New ancient solutions to mean curvature flow in high dimensions identified.
problem Finding ancient solutions to mean curvature flow in high codimension.
method Constructing compact ancient solutions to mean curvature flow in Euclidean space with high codimension.
result Characterization of the asymptotic behavior of constructed ancient solutions.
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.
New ancient curve shortening flows created from grim reapers.
problem Ancient curve shortening flows in 3D space.
method Built from translating grim reapers in perpendicular planes.
result Constructed new nonplanar ancient solutions.
Ancient mean curvature flows get codimension bounds from their tangent flow.
problem Understanding the limiting behavior of ancient mean curvature flows.
method Proving codimension bounds using the tangent flow at −∞. result Ancient mean curvature flows are rigid to their tangent flow at −∞. Ancient pancake solutions found for curvature flows.
problem Finding unique ancient solutions to curvature flows.
method Constructing and analyzing O(1)imesO(n)-invariant ancient solutions. result Unique O(n)-invariant ancient solutions found. Ancient solutions in 3D are mostly cylinders or symmetric.
problem Understanding the structure of ancient solutions to the Ricci flow in 3D.
method Detailed proof of compact ancient κ-solutions being rotationally symmetric.
result Compact ancient κ-solutions are rotationally symmetric.