调和映射的共形热流的光滑性
Smoothness of conformal heat flow of harmonic maps
AI总结：
本文研究调和映射共形热流的光滑性，证明其无有限时间奇性，并在小能量或能量一致集中条件下给出长时间收敛结果。
摘要：
调和映射的共形热流是一个演化方程组，它将调和映射流与沿共形方向的度量演化结合在一起。已知该流存在全局弱解，并且除至多有限多个奇点外是光滑的。本文证明，与通常的调和映射流不同，该流不会发生有限时间奇性。若初始能量较小，则在某个时间序列 $t_n \to \infty$ 下，可得到映射到一点的一致收敛以及度量的共形因子的一致收敛。此外，在能量集中关于时间一致的假设下，证明存在时间序列 $t_n \to \infty$，使得 $f(\cdot,t_n)$ 在远离至多有限多个点的任意紧集上于 $W^{1,2}$ 中收敛到一个调和映射。
英文摘要：
The conformal heat flow of harmonic maps is a system of evolution equations combined with harmonic map flow with metric evolution in conformal direction. It is known that global weak solution of the flow exists and smooth except at mostly finitely many singular points. In this paper, we show that no finite time singularity occurs, unlike the usual harmonic map flow. And if the initial energy is small, we can obtain the uniform convergence of the map to a point and the conformal factor of the metric under some time sequence $t_n \to \infty$. Also, under the assumption that energy concentration is uniform in time, we show that there exists a sequence of time $t_n \to \infty$ such that $f(\cdot,t_n)$ converges to a harmonic map in $W^{1,2}$ on any compact set away from at most finitely many points.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
