We use two simple methods to derive four important explicit graphical solutions of the curve shortening flow in the plane. They are well-known as the circle, hairclip, paperclip, and grim reaper solutions of the curve...We use two simple methods to derive four important explicit graphical solutions of the curve shortening flow in the plane. They are well-known as the circle, hairclip, paperclip, and grim reaper solutions of the curve shortening flow. By the methods, one can also see that the hairclip and the paperclip solutions both converge to the grim reaper solutions as t → -∞.展开更多
In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight...In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.展开更多
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 manifold(M,g) with some...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 manifold(M,g) with some natural assumptions,we prove the planar phenomenon when the curve shortening flow blows up.展开更多
It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a poi...It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.展开更多
基金supported by MoST of Taiwan under grant number 105-2115-M-007-013supported by NSF of Jiangsu Province(BK20161412)the Postdoctoral Science Foundation of China(2016T90399,2014M561542)
文摘We use two simple methods to derive four important explicit graphical solutions of the curve shortening flow in the plane. They are well-known as the circle, hairclip, paperclip, and grim reaper solutions of the curve shortening flow. By the methods, one can also see that the hairclip and the paperclip solutions both converge to the grim reaper solutions as t → -∞.
文摘In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.
基金Supported by NSFC (Grant No. 11721101)National Key Research and Development Project SQ2020YFA070080。
文摘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 manifold(M,g) with some natural assumptions,we prove the planar phenomenon when the curve shortening flow blows up.
基金Supported by NatiOnal Natural Science Foundation of China （Grant No. 10671022） and Doctoral Programme Foundation of Institute of Higher Education of China （Grant No. 20060027023）
文摘It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.
