相关论文: Nonconvex ancient solutions to Curve Shortening Fl…
Suppose that $\mathcal{M}$ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in $\mathbb{C}^n$. We show that if $\mathcal{M}$ has a blow-down given by the static union of two Lagrangian subspaces with…
In this paper, we study a new area-preserving curvature flow for closed convex planar curves. This flow will decrease the length of the evolving curve and make the curve more and more circular during the evolution process. And finally, the…
By carrying out refined curvature estimates, we prove better rigidity theorems of complete noncompact ancient solutions to the mean curvature flow in higher codimension under various Gauss image restriction.
We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a…
In this paper we introduce a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow. The length estimates we obtain are used to prove results about the level set flow in…
In this paper, we investigate the evolution of spacelike curves in Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which…
We provide some counterexamples concerning the uniqueness and regularity of weak solutions to the initial-boundary value problem for gradient flows of certain strongly polyconvex functionals by showing that such a problem can possess a…
Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this…
In 1995, Hamilton introduced a Harnack inequality for convex solutions of the mean curvature flow. In this paper we prove an alternative Harnack inequality for curve shortening flow, i.e. one-dimensional mean curvature flow, that does not…
We study the inverse curve shortening flow in the hyperbolic plane $\h^2$. We classify all solitons with respect to parabolic and conformal vector fields of $\h^2$. In the upper half-plane model of $\h^2$, we prove that parabolic solitons…
We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions…
We construct a class of compact ancient solutions to the mean curvature flow in Euclidean space with high codimension. In particular, we construct higher codimensional ancient curve shortening flows. Moreover, we characterize the asymptotic…
A recent article by Li and Lv considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a…
We show that every $n$-dimensional, $\kappa$-noncollapsed, noncompact, complete ancient solution to the Ricci flow with uniformly PIC for $n=4$ or $n\ge 12$ has weakly PIC$_2$ and bounded curvature. Combining this with earlier results, we…
We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family…
We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. Firstly, we prove an analogue to Huisken's distance…
We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses.
In this paper we introduce two $1/\kappa^{n}$-type ($n\ge1$) curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. And finally, the evolving curves…
In [4], we proved that every noncompact ancient $\kappa$-solution to the Ricci flow in dimension $3$ is either locally isometric to a family of shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we announced that…
We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity…