拉格朗日相算子的坦凹性及其在切线拉格朗日相流中的应用

Ryosuke Takahashi
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引用次数: 15

摘要

我们探讨了拉格朗日相位算符的坦凹性,用于研究变形厄米杨-米尔斯(dHYM)度量。这个新的性质弥补了拉格朗日相位算符的凹性不足,只要度规几乎是校准的。作为一个应用,我们在几乎校准的$(1,1)$-形式空间上引入了切线拉格朗日相流(TLPF),它适合于Collins-Yau最近发现的用于dHYM指标的GIT框架。TLPF具有线束平均曲率流(即图的拉格朗日平均曲率流的镜像)所没有的一些特殊性质。我们证明了从任何初始数据出发的TLPF在所有正时间内都存在。此外,我们还证明了在假设$C$-子解存在的情况下,TLPF平滑地收敛于一个dHYM度量,从而给出了dHYM度量在最高分支上存在的一个新的证明。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow
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 $(1,1)$-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 $C$-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.
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