32 Citations
Curve shortening flow on singular Riemann surfaces
- Mathematics
- 2020
In this paper, we study curve shortening flow on Riemann surfaces with singular metrics. It turns out that this flow is governed by a degenerate quasilinear parabolic equation. Under natural…
The maximal curves and heat flow in fully affine geometry
- Mathematics
- 2022
In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture (cf. [10]) in 1977 that an affine maximal graph of a smooth, locally uniformly convex…
The Affine Shape of a Figure-Eight under the Curve Shortening Flow
- Mathematics
- 2021
We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when…
Some New Results in Geometric Analysis
- Mathematics
- 2021
This thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the point-wise curvature preserving…
Invariance of immersed Floer cohomology under Maslov flows
- MathematicsAlgebraic & Geometric Topology
- 2021
We show that immersed Lagrangian Floer cohomology in compact rational symplectic manifolds is invariant under Maslov flow; this includes coupled mean curvature/Kähler-Ricci flow in the sense of…
Legendrian curve shortening flow in R 3
- Mathematics
- 2018
This gives a partial answer to a conjecture of Grayson [12], which states that all figure-eight curves with zero signed area should shrink to a point under curve shortening flow. In particular, our…
Singularities of the Curve Shortening Flow in a Riemannian Manifold
- MathematicsActa Mathematica Sinica, English Series
- 2021
In this paper, the curve shortening flow in a general Riemannian manifold is studied, Altschuler’s results about the flow for space curves are generalized. For any n-dimensional (n ≥ 2) Riemannian…
Curve shortening flow in a 3-dimensional pseudohermitian manifold
- MathematicsCalculus of Variations and Partial Differential Equations
- 2021
In this paper, we introduce a curve shortening flow in a 3-dimensional pseudohermitian manifold with vanishing torsion. The flow preserves the Legendrian condition and decreases the length of curves.…
Invariant hypersurface flows in centro-affine geometry
- MathematicsScience China Mathematics
- 2021
In this paper, the invariant geometric flows for hypersurfaces in centro-affine geometry are explored. We first present evolution equations of the centro-affine invariants corresponding to the…
Existence of solution and asymptotic behavior for a class of parabolic equations
- MathematicsCommunications on Pure & Applied Analysis
- 2021
We prove existence and uniqueness of a positive solution for a class of quasilinear parabolic equations. We also show some maximum principles on the derivatives of the solution and study the…
References
SHOWING 1-7 OF 7 REFERENCES
A stable manifold theorem for the curve shortening equation
- Mathematics
- 1987
On presente une famille de solutions homothetiques de l'equation pour une courbe planaire ∂X/∂τ=KN et on demontre l'existence de varietes non lineaires stables et instables autour de telles solutions
The heat equation shrinking convex plane curves
- Mathematics
- 1986
Soient M et M' des varietes de Riemann et F:M→M' une application reguliere. Si M est une courbe convexe plongee dans le plan R 2 , l'equation de la chaleur contracte M a un point
The heat equation shrinks embedded plane curves to round points
- Mathematics
- 1987
Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N est son vecteur unite normal entrant. C(•,t) est…
The normalized curve shortening flow and homothetic solutions
- Mathematics
- 1986
The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient…