abstract:

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\Bbb{R}^2$ under the $\alpha$-curve shortening flow for exponents $\alpha>{1\over 2}$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $\alpha$-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case $\alpha=1$, and we prove for all exponents up to the critical case $\alpha>{1\over 2}$.

中文翻译：

---

曲线缩短流向平移孤子的收敛

摘要：

本文关注在指数$\alpha>{1\over 2}$的$\alpha$-curve缩短流下嵌入$\Bbb{R}^2$中的完全非紧凸曲线的渐近行为。我们表明，任何这样的曲线的两端都渐近于两条平行线，在 $\alpha$ 曲线缩短流下会聚到独特的平移孤子，其两端渐近于相同的平行线。即使在标准情况 $\alpha=1$ 中，这也是一个新结果，我们证明了所有指数直到临界情况 $\alpha>{1\over 2}$。