Let f be a smooth strictly convex solution of

defined on \({\mathbb {R}}^{n}\), where \(a_i\), \(b_i\) and c are constants, then the graph \(M_{\nabla f}\) of \(\nabla f\) is a space-like translating soliton for mean curvature flow in pseudo-Euclidean space \({{\mathbb {R}}}^{2n}_{n}\) with the indefinite metric \(\sum dx_idy_i\). In this paper, we prove that there exists no a smooth strictly convex solution, defined on \({{\mathbb {R}}}^n\) unless \(\sum _{i=1}^{n} a_ib_i\ge 0\). Namely, there exists no entire spacelike Lagrangian translating soliton with a timelike translating vector in \({{\mathbb {R}}}^{2n}_{n}\)(with null coordinates).

令f为的光滑严格凸解

上定义\（{\ mathbb {R}} ^ {N} \） ，其中\（A_I \） ，\（b_i \）和Ç是常数，则该图形\（M _ {\ nabla F} \）的\ （\ nabla f \）是具有不确定度量\ {\的伪欧几里德空间\（{{\ mathbb {R}}} ^ {2n} _ {n} \）中平均曲率流的类空平移孤子和dx_idy_i \）。在本文中，我们证明除非存在\（\ sum _ {i = 1} ^ {n} a_ib_i \，否则不存在对\（{{\ mathbb {R}}} ^ n \）定义的光滑严格凸解。 ge 0 \）。即，不存在具有像时间一样的平移矢量的整个空间像拉格朗日平移孤子\（{{\ mathbb {R}}} ^ {2n} _ {n} \）（具有空坐标）。