Curve Shortening Flow in a Riemannian Manifold
نویسندگان
چکیده
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 global behavior of the flow. In particular, we show the following results. 1). Let M be a compact locally symmetric space. If the curve shortening flow exists for infinite time, and lim t→∞ L(γt) > 0, then for every n > 0, lim t→∞ sup(| DT ∂s |) = 0. In particular, the limiting curve exists and is a closed geodesic in M. 2). For γ0 is a ramp, we have a global flow and the flow converges to a geodesic in C∞ norm.
منابع مشابه
Evolution of the first eigenvalue of buckling problem on Riemannian manifold under Ricci flow
Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evol...
متن کاملShortening Curves on Surfaces
METHODS of shortening a curve in a manifold have been used to establish the existence of closed geodesics, and in particular of simple closed geodesics on 2-spheres. For this purpose, a curve evolution process should (a) not increase the number of self-intersections of a curve, (b) exist for all time or until a curve collapses to a point, (c) shorten curves sufficiently fast so that curves whic...
متن کاملB-sub-manifolds and Their Stability
In this paper, we introduce a concept of B-minimal sub-manifolds and discuss the stability of such a sub-manifold in a Riemannian manifold (M, g). Assume B(x) is a smooth function on M . By definition, we call a sub-manifold Σ B-minimal in (M, g) if the product sub-manifold Σ × S1 is a minimal sub-manifold in a warped product Riemannian manifold (M×S1, g+ e2B(x)dt2), so its stability is closely...
متن کاملON THE LIFTS OF SEMI-RIEMANNIAN METRICS
In this paper, we extend Sasaki metric for tangent bundle of a Riemannian manifold and Sasaki-Mok metric for the frame bundle of a Riemannian manifold [I] to the case of a semi-Riemannian vector bundle over a semi- Riemannian manifold. In fact, if E is a semi-Riemannian vector bundle over a semi-Riemannian manifold M, then by using an arbitrary (linear) connection on E, we can make E, as a...
متن کاملACTION OF SEMISIMPLE ISOMERY GROUPS ON SOME RIEMANNIAN MANIFOLDS OF NONPOSITIVE CURVATURE
A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...
متن کامل