## 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…