Counting ends on shrinkers

In this paper we apply a geometric covering method to study the number of ends on shrinkers. On one hand, we prove that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree. On the other hand, we prove that any complete non-compact shrinker with certain volume comparison condition has finitely many ends. Some special cases of shrinkers are also discussed.
Comment: 25 pages, 2 figures added