Ancient solutions of the mean curvature flow

Abstract

In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t → 0 collapse to a round point, but for t → −∞ become more and more oval: near the center they have asymptotic shrinkers modeled on round cylinders S j × R n−j and near the tips they have asymptotic translators modeled on Bowl j+1 × R n−j−1 . We also give a characterization of the round shrinking sphere among ancient α-Andrews flows. Our proofs are based on the recent estimates of Haslhofer-Kleiner [HK13].

Unique asymptotics of ancient convex mean curvature flow solutions

“…By symmetry we have ∇H ∞ = 0 at the origin. Thus, by the same proof of the equality case of Hamilton's Harnack inequality, which was in fact the observation made in [12],M t ∞ must be a translating soliton, and so it must be the Bowl soliton. Finally, due to the uniqueness of the Bowl with the mean curvature being one at the origin, the subsequential limit is actually a full limit.…”

“…Haslhofer and Hershkovits [12] provided recently more details on White's construction. If one represents R n+1 as R n+1 = R k × R l with k + l = n + 1, then the White-Haslhofer-Hershkovits construction proves the existence of an ancient solution M t with O(k)×O(l) symmetry.…”

Ancient Solutions of Geometric Flows with Curvature Pinching

“…Several other characterizations of the sphere have been obtained, e.g. in terms of a control on the diameter growth as t → −∞, or on the ratio between the outer and inner radius, see [34,37]. We remark that an equivalence similar to Theorem 1, involving pinching properties of the intrinsic curvature, has been obtained in [17] for ancient solutions of the Ricci flow.…”

“…However, it is possible to prove some rigidity results which provide a partial structural description of this class. The typical result of this kind is the following, proved in [37], see also [34], stating that an ancient solution with uniformly pinched principal curvatures is necessarily a shrinking sphere. Theorem 1.…”

Strong spherical rigidity of ancient solutions of expansive curvature flows

“…In the contractive case, various authors have found characterizations of the shrinking spheres as the unique ancient solutions which satisfy suitable geometric conditions, such as uniform convexity, uniform pinching of inner and outer radii, or a bound on the diameter growth, see [17,[20][21][22]. Our theorem can be regarded as a counterpart in the expanding case of these rigidity results for those speeds which admit ancient flows, since we obtain that the sphere is the only compact embedded ancient solution which comes out of a point.…”