Eternal solutions to almost calibrated Lagrangian and symplectic mean curvature flows (Q6542843)

The mean curvature flow has been studied extensively because of its intrinsic interest, and also as a tool for solving a variety of problems in differential geometry. In particular, mean curvature flow has been proposed as a way of proving the existence of special Lagrangian submanifolds of Calabi-Yau manifolds, because the Lagrangian condition is preserved by the flow, see [\textit{K. Smoczyk}, ``A canonical way to deform a Lagrangian submanifold'', Preprint, \url{arXiv:dg-ga/9605005}]. More generally, almost calibrated submanifolds remain so under the flow. The relevant definitions can be found in the paper.\N\N\textit{J. Chen} and \textit{J. Li} [Invent. Math. 156, No. 1, 25--51 (2004; Zbl 1059.53052)] and \textit{M.-T. Wang} [J. Differ. Geom. 57, No. 2, 301--338 (2001; Zbl 1035.53094)] showed that there is no finite time type I singularity for the almost calibrated mean curvature flow. Thus it is of interest to understand type II singularities, whose blow ups are always eternal solutions (they exist for all times \(-\infty<t<\infty\)), such as translating solutions.\N\NHere the authors show that any almost calibrated Lagrangian mean curvature flow of surfaces \(\Sigma_t\) in \(\mathbb{C}^2\) with Lagrangian angle \(\theta\) satisfying \(\cos\theta\ge \delta>0\) and \(|H|^2\ge \varepsilon|A|^2\) for all \(t\), with \(\varepsilon>1-\delta\), must be flat planes. Here \(H\) and \(A\) denote the mean curvature and second fundamental form respectively.\N\NAs a consequence of the Gauss equation \(2K=|H|^2-|A|^2\), they obtain the following. Any eternal mean curvature flow \(\Sigma_t\) in \(\mathbb{C}^2\) with Gauss curvature \(K\ge -\frac{1}{2}(1-\varepsilon)|A|^2\) for all \(t\), with \(\varepsilon>1-\delta\), cannot arise as a blow up flow of an almost calibrated mean curvature flow.\N\NThe authors also consider the symplectic mean curvature flow of surfaces in a Kähler-Einstein surface with Kähler form \(\omega\) and compatible complex structure \(J\). The property of being symplectic is preserved by the mean curvature flow, and by results of \textit{J. Chen} and \textit{J. Li} [Adv. Math. 163, No. 2, 287--309 (2001; Zbl 1002.53046)] and \textit{M.-T. Wang} [J. Differ. Geom. 57, No. 2, 301--338 (2001; Zbl 1035.53094)] there is no finite time type I singularity of the flow. The authors show, analogously to above, that any eternal symplectic mean curvature flow \(\Sigma_t\) in \(\mathbb{C}^2\) with Kähler angle \(\alpha\) satisfying \(\cos\alpha\ge \delta>0\) and \(\left| \overline{\nabla}J_{\Sigma_t}\right| \ge \varepsilon |A|^2\) for all \(t\), with \(\varepsilon>1-\delta\), must be flat planes.\N\NIt follows from this and the Ricci equation that any eternal mean curvature flow \(\Sigma_t\) in \(\mathbb{C}^2\) with normal curvature \(K_\Sigma^\perp\le \frac{1}{2}(1-\varepsilon)|A|^2\) for all \(t\), with \(\varepsilon>1-\delta\), cannot arise as the blow up of a symplectic mean curvature flow.\N\NThe results improve on previous work of \textit{X. Han} et al. [Chin. Ann. Math., Ser. B 32, No. 2, 223--240 (2011; Zbl 1221.53098)] and \textit{X. Han} and \textit{J. Sun} [Ann. Global Anal. Geom. 38, No. 2, 161--169 (2010; Zbl 1198.53073)].