We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kahler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric flow in terms of the ambient canonical bundle, (ii) a modified volume functional which takes into account the totally real condition. We discuss short-time existence for our flow and show it couples well with the Streets-Tian symplectic curvature flow for almost Kahler manifolds. We also discuss possible applications to Lagrangian submanifolds and calibrated geometry.

中文翻译：

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从拉格朗日几何到完全真实的几何：耦合流和校准

我们表明，拉格朗日平均曲率流的特性是更普遍现象的特例，涉及环境空间的几何流和完全真实的子流形之间的耦合。两种流动都由环境 Ricci 曲率驱动，或者在非 Kahler 情况下，由其类似物驱动。为此，我们探索完全真实子流形的几何形状，定义（i）根据环境规范丛定义的新几何流，（ii）考虑完全真实条件的修改体积泛函。我们讨论了我们的流的短时存在性，并表明它与几乎 Kahler 流形的 Streets-Tian 辛曲率流很好地耦合。我们还讨论了拉格朗日子流形和校准几何的可能应用。