2015
DOI: 10.1007/s00526-015-0905-3
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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 ).

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Cited by 3 publications
(7 citation statements)
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“…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.…”
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confidence: 65%
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“…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.…”
mentioning
confidence: 65%
“…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].…”
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confidence: 93%
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