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

Abstract

Let f be a smooth strictly convex solution of $$\begin{aligned} \det \left( \frac{\partial ^{2}f}{\partial x_{i}\partial x_{j}}\right) =\exp \left\{ \sum _{i=1}^n- a_i\frac{\partial f}{\partial x_{i}} +\sum _{i=1}^n b_ix_i+c\right\} \end{aligned}$$ 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 classify the entire solutions of the PDE above for dimension \(n=1\) and show every entire classical strictly convex solution \((n\ge 2)\) must be a quadratic polynomial under a decay condition on the hessian \((D^2f)\).