On the entire self-shrinking solutions to Lagrangian mean curvature flow II

Abstract

We prove the rigidity of entire graphic Lagrangian self-shrinkers in \(({\mathbb {R}}^{2n}, g_\tau )\), where \(g_\tau =\sin \tau \,\delta _0+\cos \tau \,g_0\) is a linear combination of the Euclidean metric \(\delta _0\) and the pseudo metric \(g_0=2\sum _i dx_idy_i\) with \(\tau \in (0,\frac{\pi }{2})\), complementing the previous results for \(\tau =0\) and \(\tau =\frac{\pi }{2}\); actually we obtain Bernstein theorems for three corresponding nonlinear elliptic equations between the Monge–Ampère equation (\(\tau =0\)) and the special Lagrangian equation (\(\tau =\frac{\pi }{2}\)). Moreover, we find Bernstein theorem fails when \(\tau \in (-\frac{\pi }{4},0)\) and entire graphic spacelike self-shrinker in Minkowski spaces share this non-rigidity property.