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.
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.
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.
Modified curve shortening flow constructs λ-Angenent curve.
problem Constructing λ-Angenent curve. method Modified curve shortening flow
result Constructs λ-Angenent curve. 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 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.
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.
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.
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 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 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.
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.
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.
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 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.
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. 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 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 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.
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.
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.
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.
New Harnack inequality for curve shortening flow without convexity.
problem Proving a Harnack inequality for curve shortening flow without convexity.
method Developed a new Harnack inequality for one-dimensional mean curvature flow (curve shortening flow) that doesn't require convexity.
result Explicit time by which an initial curve becomes graphical under curve shortening flow.
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.
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 …
New method approximates anisotropic curve shortening flow.
problem Approximating anisotropic curve shortening flow.
method Weak formulation and finite element approximation.
result Optimal H1-error bound for approximation. 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.
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.
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.
The article analyzes the stability of a curve shortening flow for planar networks.
problem Stability analysis of anisotropic curve shortening flow for planar networks.
method Used Lojasiewicz-Simon gradient inequality to derive stability results.
result For initial data close to an energy minimizer, the flow exists globally and converges to a different energy minimum.
Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.
problem Spectral stability of Dirichlet eigenvalues on an evolving annulus.
method Variational formulas, Rellich-type identities, and harmonic capacity methods.
result Established quantitative bounds comparing the spectrum of the evolving annulus with a flat cylinder.
In this paper, we consider affine self-similar solutions for the affine curve shortening flow in the Euclidean plane. We obtain the equations of all affine self-similar solutions up to affine transformations and solve the equations or give descriptions of the solutions for the degenerate case. Some new special solution…
Paper proves unique tangent flow at infinity for entropy-limited curve shortening.
problem Proving uniqueness of tangent flows for finite-entropy curve shortening.
method Rescaled backward convergence to a line, entropy analysis, and geometric properties.
result Ancient smooth curve shortening flow has a unique tangent flow at infinity.
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.
We study curve shortening flows in two types of warped product manifolds. These manifolds are S1×N with two types of warped metrics where S1 is the unit circle in R2 and N is a closed Riemannian manifold. If the initial curve is a graph over S1, then its curve shortening flow exists for all times an…
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 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.
Existence of translating solutions shown for curve diffusion flow.
problem Existence of translating solutions for curve diffusion flow.
method Higher order curve shortening flow approach.
result Properly immersed translating solutions exist.
Study curve shortening flow in high dimensions with boundary constraints.
problem Understanding the behavior of curves in high-dimensional spaces with boundary conditions.
method Used curvature and higher-derivative estimates, Stahl-type maximum principle, and blow-up analysis.
result Flow converges to a shrinking semicircle model or has only semicircle boundary singularities in low entropy regimes.
New results on geodesic flows using curve shortening flow.
problem Stability and existence of geodesics in Riemannian surfaces.
method Curve shortening flow approach.
result Existence of infinitely many geodesics intersecting a given one in every primitive class.