Curve Shortening Flow preserves circularity for convex projections.
problem Understanding the behavior of curves under Curve Shortening Flow.
method Contradiction argument and analysis of tangent flows.
result Smooth curves with convex projections become asymptotically circular under Curve Shortening Flow.
In this paper we classify convex compact ancient solutions to the affine curve shortening flow: namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking ellipse. The method combines a rescaling argument inspired by \cite{Wang}, affine invariance of the equation and monotonicit…
Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.
problem Analyzing the behavior of space curves under curve shortening flow in R3. method Analysis of properties of space curves evolved by the curve shortening flow, including convexity preservation and avoidance principle.
result Orthogonal projections of space curves remain convex, and the Avoidance principle is shown for spherical curves.
We prove that the only closed, embedded ancient solutions to the curve shortening flow on S2 are equators or shrinking circles, starting at an equator at time t=−∞ and collapsing to the north pole at time t=0. To obtain the result, we first prove a Harnack inequality for the curve shortening flow o…
New method finds closed timelike geodesics on Lorentzian manifolds.
problem Existence of closed timelike geodesics in Lorentzian geometry.
method Introducing timelike geodesic homotopy and combining with a local length argument.
result Provides new results on the existence of closed timelike geodesics.
This article gives solutions to the exercises in Bestvina and Feighn's paper on Sela's work on limit groups. We prove that all constructible limit groups are limit groups and give an account of the shortening argument of Rips and Sela.
We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve shortening flow: If the isoperimetric profile of the region enclosed by the initial curve is greater than that of some `model' convex region with exactly four vertices and with reflection symmetry in bo…
Generators found for automorphisms of special groups.
problem Characterizing automorphisms of special groups.
method Using a novel hierarchical shortening argument, the paper provides generators for automorphisms of special groups.
result Generators for automorphisms of special groups are found, including Dehn twists and pseudo-twists.
The paper proves the existence of surfaces of section for geodesic flows on closed surfaces.
problem Existence of surfaces of section for geodesic flows on closed surfaces.
method Study of configurations of simple closed geodesics and use of the curve shortening flow.
result Construction of surfaces of section that intersect or have hyperbolic components in their boundary.
Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal dis…
We consider applications of the theory of balanced weight filtrations and iterated logarithms, initiated in arXiv:1706.01073, to PDEs. The main result is a complete description of the asymptotics of the Yang--Mills flow on the space of metrics on a holomorphic bundle over a Riemann surface. A key ingredient in the argu…
Classifies self-similar curve shortening flows in hyperbolic 2-space.
problem Classifying self-similar curve shortening flows in hyperbolic 2-space.
method Analyzes and classifies solutions in hyperbolic 2-space.
result Completes the classification of self-similar curve shortening flows in constant curvature model spaces in 2-dimensions.
Curve shortening flow is not unique on certain metrics.
problem Non-uniqueness of curve shortening flow on specific metrics.
method Formulated a uniqueness conjecture and constructed a non-static solution.
result Curve shortening flow is not unique on a non-flat metric on the plane.
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.
Unique ancient solutions found for anisotropic curve shortening flow.
problem Finding unique solutions for anisotropic curve shortening flow.
method Constructing translating and ancient solutions under given conditions.
result Unique ancient and translating solutions found for anisotropic curve shortening flow.
The paper studies curve shortening flows on non-convex surfaces.
problem Behavior of curve shortening flows on non-convex surfaces.
method Defined a graph property and proved its preservation under curve shortening flow.
result The curve becomes a graph after a finite time under the curve shortening flow.
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.
X.-J. Wang proved a series of remarkable results on the structure of convex ancient solutions to mean curvature flow. Some of his results do not appear to be widely known, however, possibly due to the technical nature of his arguments and his exploitation of methods which are not widely used in mean curvature flow. In …
New method shortens and straightens curves, proving convergence and well-posedness.
problem Shortening and straightening of curves.
method Conceptual shift in curve shortening to tangent aligning, variational study of geometric flows.
result Proves convergence to a straight line and global well-posedness for various geometric flows.
Curve shortening flow increases annulus modulus.
problem Behavior of annulus modulus under curve shortening flow.
method Nested curves evolving under curve shortening flow.
result Modulus of enclosed annulus is monotonically increasing.
Curve shortening flow converges to a point with entropy bound.
problem Analyzing the behavior of curves under shortening flow near singularities.
method Analyzes blow-up limits and uses entropy bounds to prove convergence.
result Initial curves with entropy bound converge to a round point in finite time.
Curve shortening flow shrinks curves to points.
problem The behavior of curves under curve shortening flow.
method Nonlinear partial differential equations, maximum principle, monotonicity formulas, Harnack inequalities, blowup analysis.
result The curve shortening flow shrinks any closed embedded curve in the plane to a round point.
Study proves existence and properties of shrinkers in area-preserving curve-shortening flow.
problem Existence and properties of shrinkers in area-preserving curve-shortening flow.
method Using known results on λ-curves, we prove existence of non-circular shrinkers and deduce a saddle-point property.
result Existence and properties of shrinkers in area-preserving curve-shortening flow, including a saddle-point property.
Curve shortening problem solved via Schwarz function.
problem Solving the curve shortening problem in the z-plane. method Using the Schwarz function to solve the differential equation StSz=Szz. result Explicit solutions for known curve shortening flow shapes can be recovered.
Study shows how a curve shortens to a half-circle under specific flow.
problem Stability of a semi-circle under curve shortening flow.
method Sharp rate of convergence for a free-boundary curve shortening flow in a convex domain.
result Established a sharp rate of convergence to a round half-point.
Modified curve shortening flow constructs λ-Angenent curve.
problem Constructing λ-Angenent curve. method Modified curve shortening flow
result Constructs λ-Angenent curve. 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.
Characterizes rotational solitons for curve shortening flow on revolution surfaces.
problem Understanding the behavior of curves under curve shortening flow on revolution surfaces.
method Characterization and asymptotic behavior analysis.
result Asymptotic behavior of rotational solitons to parallel geodesics.
Curve shortening flow's regularity depends on initial conditions after a certain time.
problem Understanding the regularity of evolving curves under curve shortening flow.
method Proposing and proving principles of controllable regularity based on initial conditions.
result No regularity estimate holds before a specific time, A/π. 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.
Ancient solutions to curve shortening flow are constructed and analyzed.
problem Constructing ancient solutions to curve shortening flow.
method Analyzing the rotating Yin-Yang soliton and Grim Reaper translating soliton to approximate the solution.
result An ancient solution to planar curve shortening is constructed and analyzed.
New distance comparison principle for curve shortening flow in higher dimensions.
problem Understanding curve shortening flow in higher dimensions.
method Established a variant of Huisken's distance comparison principle.
result Symmetric curve shortening flow with one-to-one convex projection develops Type I singularities and becomes asymptotically circular.
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.
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.
Curve shortening in metric-affine plane shrinks convex curves to points.
problem Shortening curves in non-Euclidean spaces.
method Curve shortening flow in metric-affine plane with geometric conditions.
result Closed convex curves in metric-affine plane shrink to points in finite time.
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
Motivated by Legendrian curve shortening flows in R3, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a point at the first singular time.
Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
We consider an embedded convex ancient solution Γt to the curve shortening flow in R2. We prove that there are only two possibilities: the family Γt 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 …
Study on curve shortening flow with boundary conditions, proving convergence or contraction.
problem Analyzing curve shortening flow with free boundaries.
method Introduced a reflected chord-arc profile and obtained chord-arc estimates.
result Proved that flows either converge to a critical chord or contract to a round half-point.
New estimate for Curve Shortening Flow improves graphical solutions.
problem Improving regularity estimates for Curve Shortening Flow.
method Generalizing delayed parabolic regularity for Curve Shortening Flow.
result Interior graphical estimate for Curve Shortening Flow.
Sharp chord-arc estimates for curve shortening flow on spheres.
problem Understanding the behavior of curves on spheres under curve shortening flow.
method Proving sharp chord-arc estimates and curvature control.
result Simple spherical curves either contract to points or converge to great circles.
We give a classification of all self-similar solutions to the curve shortening flow in the plane.
Study inverse curve shortening flow on hyperbolic plane, classifying solitons.
problem Understanding the behavior of curves in hyperbolic geometry under a specific flow.
method Classifying solitons with respect to vector fields and studying their properties.
result Parabolic solitons are graphs on the y-axis, conformal solitons on the x-axis.
The study finds solitons for curve shortening flow on hyperbolic plane.
problem Characterizing solitons for curve shortening flow on hyperbolic plane.
method Characterization using geodesic curvature and inner product with fixed vector in Minkowski space.
result Existence of 2-parameter family of soliton solutions on 2D hyperbolic plane.
Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.
Study curve shortening flows on specific surfaces, proving properties and existence.
problem Analyzing curve shortening flows on rotational surfaces with negative Gauss curvatures.
method Assume negative Gauss curvatures and conditions on Gauss curvature and curve curvature. Prove curve remains a graph and establish flow properties.
result Prove the curve remains a graph over parallels and establish long-time existence of the flow.
In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call "grid peeling") is the convex-layer decomposition of subsets G⊂Z2 of the integer grid, previous…