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


Let uu be a smooth convex function in Rn\mathbb{R}^{n} and the graph Mβˆ‡uM_{\nabla u} of βˆ‡u\nabla u be a space-like translating soliton in pseudo-Euclidean space Rn2n\mathbb{R}^{2n}_{n} with a translating vector 1n(a1,a2,⋯ ,an;b1,b2,⋯ ,bn)\frac{1}{n}(a_{1}, a_{2}, \cdots, a_{n}; b_{1}, b_{2}, \cdots, b_{n}), then the function uu satisfies det⁑D2u=exp⁑{βˆ‘i=1nβˆ’aiβˆ‚uβˆ‚xi+βˆ‘i=1nbixi+c}onRn \det D^{2}u=\exp \left\{ \sum_{i=1}^n- a_i\frac{\partial u}{\partial x_{i}} +\sum_{i=1}^n b_ix_i+c\right\} \qquad \hbox{on}\qquad\mathbb R^n where aia_i, bib_i and cc are constants. The Bernstein type results are obtained in the course of the arguments.Comment: 9 page

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