Defocusing mKdV flow on centroaffine plane curves

On submaximal plane curves

We prove that a submaximal curve in P has sequence of multiplicities (μ, ν, . . . , ν), with μ < sν for every integer s with (s− 1)(s+ 2) ≥ 6.76( r − 1). This note is a sequel to [10], where a specialization method was developed in order to bound the degree of singular plane curves. The problem under consideration is, given a system of multiplicities (m) = (m1,m2, . . . ,mr) ∈ Z and points p1, ...

On Plane Polynomial Curves

We study some properties of generator sequences of planar semigroups and give a method of construction of plane curves with one place at infinity with given generator sequences. We also discuss similar questions for polynomial curves. ∗We express our deep gratitude and appreciation for to Professor Shreeram S. Abhyankar for being a constant source of our mathematical inspiration.

Integrable Flows for Starlike Curves in Centroaffine Space

We construct integrable hierarchies of flows for curves in centroaffine R through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and prove that the restricted hierarchy of flows on curves that project to conics in RP induces...

The polyharmonic heat flow of closed plane curves

Hyperbolic mean curvature flow: Evolution of plane curves

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves F : S × [0, T ) → R which satisfies the following evolution equation ∂F ∂t (u, t) = k(u, t) ~ N(u, t)− ▽ρ(u, t), ∀ (u, t) ∈ S1 × [0, T ) with the initial data F (u, 0) = F0(u) and ∂F ∂t (u, 0) = f(u) ~ N0, where k is the mean curvature an...