Convergence of Curve Shortening Flow to Translating Soliton

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in ${\mathbb{R}}^2$ under the $\alpha$-curve shortening flow for exponents $\alpha >\frac{1}{2}$. 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 $\alpha >\frac{1}{2}$.