We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang–Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated [Formula: see text]-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins–Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a [Formula: see text]-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.

中文翻译：

---

拉格朗日相算子的 Tan-concavity 特性及其在切线拉格朗日相流中的应用

我们探索拉格朗日相位算子的 tan 凹度，以研究变形的 Hermitian Yang-Mills (dHYM) 度量。只要度量几乎被校准，这个新属性就弥补了拉格朗日相位算子缺乏凹度。作为一个应用程序，我们在几乎校准的 [公式：见文本] 形式的空间上引入了切线拉格朗日相流 (TLPF)，该形式适合 Collins-Yau 最近发现的 dHYM 度量的 GIT 框架。TLPF 有一些线束平均曲率流看不到的特殊性质（即图的拉格朗日平均曲率流的镜像）。我们表明，从任何初始数据开始的 TLPF 在所有正时间都存在。此外，我们表明，假设存在 [公式：见文本]-子解，TLPF 平滑地收敛到 dHYM 度量，