Related papers: Which shapes can appear in a Curve Shortening Flow…
A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real…
Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are…
This paper proves that, at the first singular time for a smoothly immersed surface moving by mean curvature flow in a n-manifold, each tangent flow is given by a smooth, branched shrinker, possibly with multiplicity. If n=3 and if the…
Certain exact solutions of the Einstein field equations over nonsimply-connected manifolds are reviewed. These solutions are spherically symmetric and have no curvature singularity. They provide a regularization of the standard…
In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the…
We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is $C^3$-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become…
In this paper we prove that a certain class of embedded unknotted curves in $\mathbb{R}^3$ evolving under curve shortening flow do not form singularities Type II before collapsing to a point. Our proof uses tools of the minimal surface…
Problems in exponential asymptotics are typically characterized by divergence of the associated asymptotic expansion in the form of a factorial divided by a power. In this paper, we demonstrate that in certain classes of problems that…
We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak…
We prove that the only closed, embedded ancient solutions to the curve shortening flow on $\mathbb{S}^2$ are equators or shrinking circles, starting at an equator at time $t=-\infty$ and collapsing to the north pole at time $t=0$. To obtain…
We construct a formal asymptotic series expansion for a general solution of the Brans-Dicke equations with a fluid source near a sudden singularity. This solution contains eleven independent arbitrary functions of the spatial coordinates as…
In this paper we prove that the generic singularities of mean curvature flow of closed embedded surfaces in $\mathbb R^3$ modeled by closed self-shrinkers with multiplicity has multiplicity one. Together with the previous result by…
Streamlines, vortex lines and magnetic flux tubes in turbulent fluids and plasmas display a great amount of coiling, twisting and linking, raising the question as to whether their topological complexity (continually created and destroyed by…
Several non-local curvature flows for plane curves with a general rotation number are discussed in this work. The types of flows include the area-preserving flow and the length-preserving flow. We have a relatively good understanding of…
We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of…
The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on…
We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency…
We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic fixed point in a flow on $\real^3$ originally described by Shil'nikov. By modifying how the invariant manifolds wrap…
This paper studies singularities of mean curvature flows with integral mean curvature bounds $H \in L^\infty L^p_{loc}$ for some $p \in ( n, \infty]$. For such flows, any tangent flow is given by the flow of a stationary cone $\mathbf{C}$.…
The line bundle mean curvature flow is a complex analogue of the mean curvature flow for Lagrangian graphs, with fixed points solving the deformed Hermitian-Yang-Mills equation. In this paper we construct two distinct examples of…