  In the study of the curve shortening flow on general closed curves, Abresch
and Langer posed a conjecture that the homothetic curves can be regarded as
saddle points between multi-folded circles and some singular curves. In other
words, these homothetic curves are the watershed between curves with a
nonsingular future and those with singular future along the flow. In this
article, we provide an affirmitive proof to this conjecture.
