Curve Shortening Flow In Riemannian Manifolds

Geometric ?ow belongs to the mathematic branch of using analysis methods to studyhow geometric objects evolve according to specific ways. Since 1980s, geometric ?ow hasalways been one of the most active research fields of geometric analysis. In recent years,a lot of important results have arisen in this branch. One of the most significant events isthat, in 2003, a young Russian mathematician, Perelman, published three sequent paperson the internet where experts believed that he maybe had successfully used the Ricci-Hamilton ?ow to solve the famous Poincare′ conjecture, which is no doubt the mostexciting news in this century, and has dramatically stimulated people's enthusiasm forgeometric ?ow. Our work is finished under such a background.In this paper, we deal with another important geometric ?ow, i.e. the mean curvature?ow in which case each point of the sub-manifold moves with an instant velocity equalto the mean curvature vector field at this point. For the sake of reducing the complexityof the problem, we only consider the motion of curve according to the mean curvature?ow in Riemannian manifolds. We usually call this special mean curvature ?ow as curveshortening ?ow, or in brief curve ?ow. Curve ?ow is of some interest in differentialgeometry, for instance in the problem of finding geodesics and minimal surfaces. Theinterest is not only theoretical since the motions by curvature appear in the modelling ofvarious phenomena as crystal growth, ?ame propagation and interfaces between phases.More recently, this ?ow has also appeared in the young field of image progressing whereit provides an efficient way to smooth curves representing the contours of the objects.The history of the research of curve ?ow is relatively short. But its developmentis quite fast. At the early time, Gage, Hamilton, Abresh, Langer, Grayson, etc, studiedthe curve ?ow on the Euclidean plane. Their work led to the settlement of the Jordanconjecture concerning the plane embedding curves. Late, Altschuler and Grayson coop-erated to study the curve ?ow in the 3-dimensional Euclidean space. It was these scholars'continuous efforts that made it fruitful in this area in a rather short time.Based on ancestors' excellent work, we further develop the theory of curve flowin general Riemannian manifolds. We begin with computing the first variation of thelength of curve, and then introduce the definition of curve ?ow. Then, after assumingthe curvature of the target satisfying some appropriate conditions, we prove the classicBernstein-type estimate with which we can prove the convergence of global ?ow in com-pact manifolds. To go further, we introduce the ramp curves and prove that the ramp curve?ow exists eternally. This result allow us to give an interesting proof of the well-knownLyusternik-Fet theorem by constructing ramp curve ?ows trickily. Finally, we analyze thecurve ?ow in S3(1) in detail.