On the rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space III

Abstract

Let f be a smooth strictly convex solution ofdefined on R n , where a i , b i and c are constants, then the graph M ∇ f of ∇ f is a spacelike translating soliton for mean curvature flow in pseudo-Euclidean space R 2n n with the indefinite metric dx i dy i . In this paper, we classify the entire solutions of the PDE above for dimension n = 1 and show every entire classical strictly convex solution (n ≥ 2) must be a quadratic polynomial under a decay condition on the hessian (D 2 f ).

Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

“…This result provides another proof of a nonexistence theorem for complete spacelike translating solitons in [8], and a simple proof of rigidity theorem in [33]. Secondly, we generalize the rigidity theorem of entire spacelike Lagrangian translating solitons in [34] to spacelike translating solitons with general codimensions. As a directly application of theorem, we obtain two interesting corollaries in terms of Gauss image.…”

“…Notice that the Gauss image of spacelike graphic submanifold M m in R m+n n is bounded if and only if the induced metric g = det(g ij ) is bounded (see [31]). Therefore, it is easy to see that example (1.6) above and example (1.7) in [34] have boundless Gauss images. As a directly application of theorem 5, we have By relaxing the bound of the Gauss image to controlled growth, we also get a more general corollary from theorem 5 as Prof. Dong generalized a rigidity theorem for spacelike graph with parallel mean curvature in [15].…”

“…Theorem 4 [34]: Let f (x) be an entire smooth strictly convex function on R n (n ≥ 2) and its graph M ∇f = {(x, ∇f (x))} be a translating soliton in R 2n n . If there exists a number ǫ > 0 such that the induced metric (D 2 f ) satisfies…”

“…Here we shall use the idea in [34] to generalize Theorem 4 to spacelike graphic translating solitons in pseudo-Euclidean space R m+n n .…”

“…In [17], Huang and Xu investigated the rigidity problem of entire spacelike translating soliton graph (x, ∇f ) in R 2n n with some symmetry conditions. Motivating by [5,13,16,17], Xu-Zhu [34] proved a rigidity theorem of entire convex translating solutions for mean curvature flow in R 2n n under a decay condition on the induced metric (D 2 f ) and provided a class of nontrivial entire spaclike Lagrangian translating solitons.…”