GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES
GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES
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DOI：
10.1017/fms.2014.19
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发表时间：
2014-04-01
影响因子：
1.7
通讯作者：
Mumford, David
中科院分区：
文献类型：
--
作者：
Bruveris, Martins;Michor, Peter W.;Mumford, David
研究了浸入平面曲线空间上Sobolev型度量的性质。我们证明了Sobolev型度量的测地线方程具有2阶和更高的常系数，对于光滑的初始数据以及某些Sobolev空间中的初始数据是全局适定的。因此，具有这样一个度量的闭平面曲线空间是测地线完备的。我们发现的曲率和其衍生物的测地线距离的下限。
We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.