Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow

TL;DR

This paper introduces the tan-concavity property of the Lagrangian phase operator and applies it to define a new flow, the tangent Lagrangian phase flow, which converges to deformed Hermitian Yang-Mills metrics under certain conditions.

Contribution

The paper establishes the tan-concavity of the Lagrangian phase operator and introduces the tangent Lagrangian phase flow, providing a new approach to prove existence of dHYM metrics.

Findings

TLPF exists for all positive time from any initial data.

TLPF converges smoothly to a dHYM metric under a C-subsolution.

New proof for the existence of dHYM metrics in the highest branch.

Abstract

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.