We consider the question of whether solutions of variants of Teichmüller harmonic map flow from surfaces $M$ to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least 2, as well as the flow from cylinders, we prove that such a finite‐time degeneration must occur in situations where the image of thin collars is ‘stretching out’ at a rate of at least $inj{(M,g)}^{-(\frac{1}{4}+\delta )}$, and we construct targets in which the flow from cylinders must indeed degenerate in finite time. For the rescaled Teichmüller harmonic map flow, the condition that the image stretches out is not only sufficient but also necessary and we prove the following sharp result: Solutions of the rescaled flow cannot degenerate in finite time if the image stretches out at a rate of no more than ${|\log (inj(M,g))|}^{\frac{1}{2}}$, but must degenerate in finite time if it stretches out at a rate of at least ${|\log (inj(M,g))|}^{\frac{1}{2}+\delta }$ for some $\delta >0$.

中文翻译：

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Teichmüller谐波图流变体的有限时间退化

我们考虑以下问题：Teichmüller谐波映射的变体的解是否从表面流动 $\mathrm{中号}$一般目标可以在有限时间内退化。对于至少2类的闭合表面的原始流动以及圆柱体的流动，我们证明了这样的有限时间退化必须发生在细颈圈的图像以≤最小$注射{（\mathrm{中号}，G）}^{-（\frac{\mathrm{1个}}{4}+\delta ）}$，我们构造了目标，在这些目标中，圆柱体的流量实际上必须在有限的时间内退化。对于重新缩放的Teichmüller调和图流，图像伸展的条件不仅足够，而且也是必要的，我们证明了以下清晰的结果：如果图像以零的速率伸展，则重新缩放的流的解不会在有限的时间内退化。多于${|\mathrm{日志}（注射（\mathrm{中号}，G））|}^{\frac{\mathrm{1个}}{2}}$，但如果延伸速度至少为，则必须在有限的时间内退化 ${|\mathrm{日志}（注射（\mathrm{中号}，G））|}^{\frac{\mathrm{1个}}{2}+\delta }$ 对于一些 $\delta >0$。