Degenerate J-flow on compact K\"ahler manifolds

TL;DR

This paper investigates a degenerate twisted J-flow on compact K"ahler manifolds, proving its existence, uniqueness, convergence to a weak solution, and implications for the properness of the Mabuchi K-energy in specific K"ahler classes.

Contribution

It establishes the long-time existence, uniqueness, and convergence of the degenerate twisted J-flow, confirming a conjecture by Song-Weinkove and linking it to Mabuchi K-energy properness.

Findings

The degenerate twisted J-flow exists for all time on compact K"ahler manifolds.

The flow converges to a weak solution of the degenerate twisted J-equation.

Properness of the Mabuchi K-energy twisted by a semi-positive form is established in certain K"ahler classes.

Abstract

In this note, we study a degenerate twisted J-flow on compact K\"ahler manifolds. We show that it exists for all time, it is unique and converges to a weak solution of a degenerate twisted J-equation. In particular, this confirms an expectation formulated by Song-Weinkove for the J-flow. As a consequence, we establish the properness of the Mabuchi K-energy twisted by a certain semi-positive closed (1,1)-form for K\"ahler classes in a certain subcone.