Complete self-shrinkers of the mean curvature flow

Abstract

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for L-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on polynomial volume growth, which is assumed in Cao and Li [5]. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on polynomial volume growth. For complete proper self-shrinkers of dimension 2 and 3, we give a classification of them under assumption of constant squared norm of the second fundamental form.

Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

“…Similarly, we also get the following result, which compares with Theorem 1.2 in [13]. We set u = −H and we use (8.37) to obtain…”

“…Having chosen a local unit normal N to the immersion, we have where we recall here that T = X ⊤ is the tangential component of X. Observe that (4.1) implies also that In particular, ifM is Einstein we deduce As a first application of the latter equation we have the following result, which compares with Theorem 1.1 in [13].…”

“…Corollary 8.11 covers the codimension 1 case in R n+1 extending [8] and it will be given also in a general codimension case but with more stringent assumptions in Corollary 8. 13. Section 9 deals with applications of a Simons' type formula for mean curvature flow solitons in a space of constant sectional curvature.…”

Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

“…There is a plenty of works on the classification and uniqueness problem for translating soliton and self-shrinker in Euclidean space (cf. [2,4,9,11,14,18,21,23,25,27]). On the other hand, there are many works on the rigidity problem for complete spacelike submanifolds in pseudo-Euclidean space.…”

“…If p * ∈ B R 0 (0), then for any x ∈ B a (0),F (x) ≤ a 4 max B R 0 (0) B If p * ∈ B a (0)\B R 0 (0), by assumption g ii ≤ m |x| ǫ ,we get[(a 2 − |x| 2 ) 2 B 2 ](p * ) ≤ 1 m C 0 a 4 + C 2 a 3 + mnC 1 ǫ a 4 . (5 9). …”