Mean Curvature Flow with Bounded Gauss Image

Abstract

We study the mean curvature flow of a complete space-like submanifold in pseudo-Euclidean space with bounded Gauss image and bounded curvature. We establish a relevant maximum principle for our setting. Then, we can obtain the "confinable property" of the Gauss images and curvature estimates under the mean curvature flow. Thus we prove a corresponding long time existence result.Mathematics Subject Classification (2010). Primary 53C44.

Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

“…Using the similar calculation of section 5 in [34], we prove the completeness of the induced metric g if B has an upper bound. To the simplicity, here we shall use theorem 2.1 of Prof. Xin in [29] to obtain the completeness. Proof: Without loss of generality, we assume that the origin 0 ∈ M. From Proposition 3.1 of [20], we know that the pseudo-distance function z = X, X on M is a non-negative proper function.…”

“…Then ∇ ⊥ H also has a bound. By Theorem 2.1 in [29], we have for some k > 0, the set {z ≤ k} is compact, then there is a constant c depending only on the dimension m and the bounds of mean curvature and its covariant derivatives such that for all x ∈ M with z(x) ≤ k 2 , |∇z| ≤ c(z + 1). which forces M to be complete with respect to the induced metric.…”

“…The MCF in higher codimension has been studied extensively in the last few years (cf. [6,24,26,28,29]). Translating soliton and self-shrinker play an important role in analysis of singularities in MCF.…”

Some results on space-like self-shrinkers

“…which is obtained by replacing e i by e α in w. We also have w iαjβ by replacing e j by e β in w i α . We obtain In [19] we already calculate ∇ 2 B for general space-like n−submanifolds in R m+n m .…”

“…There are many interesting results on mean curvature flow on space-like hypersurfaces in certain Lorentzian manifolds [10,11,12,13]. For higher codimension we refer to the previous work of the second author [19].…”