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| THE EXISTENCE OF CLOSE GEODESICS ON A COMPLETE RIEM ANNIAN MANIFOLD |
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Citation： |
Li Jiangfan.THE EXISTENCE OF CLOSE GEODESICS ON A COMPLETE RIEM ANNIAN MANIFOLD[J].Chinese Annals of Mathematics B,1989,10(1):85~93 |
| Page view： 732
Net amount： 757 |
Authors： |
Li Jiangfan; |
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| Abstract： |
This paper studies the existence of closed geodesics in the homotopy class of a give closed curve. Let M be a complete Riemannian manifold without boundary, $\[f \in {C^1}({S^1},M)\]$ Look at $\[{S^1}\]$ as $\[[0,2\pi ]/\{ 0,2\pi \} \]$. The following results are proved:
A. The initial value problem of heat equation $\[{\partial _i}{f_i} = \tau ({f_i}),{f_0} = f\]$ always admits a global solution.
B. (Existence of closed geodesics). If there exists a compact set $\[K \subset M\]$ such that
$\[f({S^1}) \cap K \ne \phi \]$ and
$$\[E(f) \le \frac{1}{\pi }i{(\partial K)^2}\]$$
then there exists a closed geodesic homotopie to f. If
$$\[E(f) \le \frac{1}{\pi }i{(M\backslash K)^2}\]$$,
then the closed geodesic is minimal.
C. Some estimates abont injective radius are obtained.
Some example is found showing that the inequalities in B are sharp. |
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