Translating solutions to Lagrangian mean curvature flow (Q2849025)

The following theorem is proved:NEWLINENEWLINENEWLINETheorem. There exist a finite set \(\{\overline\theta_1,\dots, \overline\theta_N\}\) and Lagrangian planes \(P_1,\dots, P_N\) such that, after passing to a subsequence, we have for every compactly supported smooth function \(\phi\), every \(f\) in \(C^2\), and every \(s\leq 0\) NEWLINENEWLINE\[NEWLINE\lim_{i\to\infty}\, \int_{\Sigma^i_s} f(\theta_{i,s})\,\phi\,d\mu= \sum^N_{j=1} m_j f(\overline\theta_j) \int_{P_j}\phi\,d\mu,NEWLINE\]NEWLINE where \(m_j\) denotes the multiplicity of \(P_j\). The set \((\overline\theta_1,\dots, \overline\theta_N)\) does not depend on the sequence of rescales chosen.NEWLINENEWLINENEWLINEThe purpose of the paper is to give conditions that exclude the existence of non-trivial translating solutions to Lagrangian mean curvature flow.NEWLINENEWLINENEWLINETwo open questions are proposed. In particular, one of them addresses the issue of uniqueness of translating solutions. Suppose that \((L_t)_{t\in\mathbb R}\) and \((L^*_t)_{t\in\mathbb R}\) are two translating solutions which are almost calibrated and exact. If after blow-down they produce the same self-expander, do they have to differ only by a rigid motion?