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.
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48 results for “Ancient flow”
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 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.
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.
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.
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 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 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.
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.
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 .
Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in .
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.
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.
The study disproves rotating ancient flows in 4D space.
problem The existence of rotating ancient flows in .
method Analysis of ancient noncollapsed flows in .
result Nonexistence of rotating ancient flows among ancient noncollapsed flows in .
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.
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
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.
Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domainsmath.DG
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 pancake solutions found for curvature flows.
problem Finding unique ancient solutions to curvature flows.
method Constructing and analyzing -invariant ancient solutions.
result Unique -invariant ancient solutions found.
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 solutions to Ricci flow in higher dimensions are mostly cylinders or solitons.
problem Uniqueness of ancient -solutions in higher dimensions.
method Analysis of ancient -solutions with specific properties.
result The only noncompact ancient -solutions are cylinders, quotients, or the Bryant soliton.
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.
We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinkin…
Ancient Ricci flows on non-collapsed manifolds have finite fundamental groups.
problem Understanding the fundamental groups of ancient Ricci flows.
method Analyzing the structure of ancient Ricci flows and their tangent flows.
result The fundamental group of non-collapsed ancient Ricci flows is finite and a quotient of the regular part's fundamental group.
Proves rigidity of ancient solutions in mean curvature flow.
problem Rigidity of ancient solutions in mean curvature flow.
method Point-wise estimate for second fundamental form.
result Rigidity theorem of ancient solutions in codimension one.
Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.
problem Rigidity of ancient solutions to mean curvature flow with convex Gauss image.
method Refined curvature estimates.
result Better rigidity theorems for ancient solutions in higher codimension.
Compact, non-convex curve flows are created.
problem Creating compact, non-convex ancient solutions for curve shortening flow.
method Constructed an ancient solution asymptotic to Yin-Yang curve.
result Compact, non-convex ancient solutions for curve shortening flow are demonstrated.
We study properly immersed ancient solutions of the codimension one mean curvature flow in -dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or a…
Proves planarity and convexity for ancient solutions of mean curvature flow.
problem Ancient solutions of mean curvature flow in higher codimension.
method Parabolically scale-invariant variation of planarity estimate, convexity proof for pinched solutions.
result Characterizes certain pinched complete ancient solutions and shrinkers in higher codimension.
Study proves existence and uniqueness of ancient flows from cones.
problem Existence and uniqueness of ancient rescaled mean curvature flows.
method Proved existence and uniqueness using strong uniqueness theorem.
result Proved existence and uniqueness of ancient flows from cones.
We consider an embedded convex ancient solution to the curve shortening flow in . We prove that there are only two possibilities: the family is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II …
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.
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.
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.
Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, we prove codimension bounds for ancient mean curvature flows by their tangent flow at , generalizing a theorem for cylinders in [CM19b]. In the case of the -covered circle, we apply this bound to prove a stron…
Ancient flows converge fast with finite curvature and convexity.
problem Understanding ancient mean curvature flows with finite curvature.
method Established exponentially fast convergence and finite curvature properties.
result Ancient flows have finite total curvature and finite mass drop.
The paper classifies flows of ancient curves in 2D space.
problem Classifying closed convex flows by curvature powers.
method Sub-affine-critical powers of curvature for flow classification.
result Ancient flows converge exponentially to smooth shrinkers.
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.
Ecker's and Huisken's quantities agree for ancient mean curvature flows.
problem Understanding the finiteness of integral quantities for ancient mean curvature flows.
method Comparison of Ecker's and Huisken's integral quantities.
result Finiteness of Ecker's integral quantity implies finiteness of entropy at infinity.
Ancient solutions and translators identified for Lagrangian flow.
problem Characterizing ancient solutions and translators of Lagrangian mean curvature flow.
method Analyzing almost calibrated, exact, ancient solutions with specific geometric properties.
result All ancient solutions with entropy less than 3 are special Lagrangian, planes, or translators in \(\mathbb{C}^2\).
New proof classifies ancient flows in 3D space.
problem Classifying ancient noncollapsed flows in .
method Combining neck theorem and Harnack inequality rigidity.
result Directly establishes self-similarity of flows.
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.
Study higher-dimensional Ricci flow solutions, proving uniqueness.
problem Classifying ancient solutions to the Ricci flow on .
method Extending [13] to higher dimensions, proving uniqueness.
result Ancient solutions are either shrinking spheres or Type II solutions.
We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinea…
Study on Ricci flow of invariant metrics on spheres, finding new ancient solutions.
problem Analyzing the Ricci flow of Sp(n+1)-invariant metrics on spheres.
method Determine forward and ancient solutions, classify them, and classify their behavior under flow.
result Exhibit a new one-parameter family of ancient solutions on spheres with larger isometry groups.
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.
Classifies ancient solutions to curvature flows, finding two main types.
problem Classifying ancient solutions to fully nonlinear curvature flows.
method Natural conditions on speed, convexity, noncollapsing, uniform two-convexity.
result Exactly two possibilities: self-similarly shrinking cylinder or rotationally symmetric translating soliton.
New Ricci flows found with Einstein orbifolds at infinity.
problem Ancient and immortal Ricci flows with Einstein orbifolds at infinity.
method Continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows constructed.
result Found continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows on specific manifolds.
Unique ancient convex flow in a ball with free boundary found.
problem Classifying convex ancient free boundary mean curvature flows in the ball.
method Proof of existence and uniqueness in every dimension.
result A unique (modulo rotations and translations) convex ancient mean curvature flow found.