The Willmore Flow with Small Initial Energy
The Willmore Flow with Small Initial Energy
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DOI：
10.4310/jdg/1090348128
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发表时间：
2001-03
影响因子：
2.5
通讯作者：
E. Kuwert;R. Schätzle
中科院分区：
文献类型：
--
作者：
E. Kuwert;R. Schätzle
我们考虑Willmore泛函的L2梯度流。文[5]证明了当奇点出现时曲率集中。在这里，我们表明，一个合适的爆破收敛到一个非脐（紧或非紧）Willmore表面。进一步，在Willmore曲面的L2范数（Willmore能量）局部小的条件下，得到了Willmore曲面曲率的无迹部分的L∞估计.一个结果是，一个适当的浸入Willmore表面与限制增长的曲率在无穷大和小的总能量必须是一个平面或一个球。结合这些结果，我们得到了当总能量较小时，方程的长时间存在性和收敛到圆球的结论。
We consider the L2 gradient flow for the Willmore functional. In [5] it was proved that the curvature concentrates if a singularity develops. Here we show that a suitable blowup converges to a nonumbilic (compact or noncompact) Willmore surface. Furthermore, an L∞ estimate is derived for the tracefree part of the curvature of a Willmore surface, assuming that its L2 norm (the Willmore energy) is locally small. One consequence is that a properly immersed Willmore surface with restricted growth of the curvature at infinity and small total energy must be a plane or a sphere. Combining the results we obtain long time existence and convergence to a round sphere if the total energy is initially small.