Prescribed $T$-curvature flow on the four-dimensional unit ball

TL;DR

This paper investigates the prescribed T-curvature problem on the four-dimensional unit ball using a curvature flow approach, establishing existence and convergence results by combining geometric inequalities and Morse theory.

Contribution

It introduces a novel combination of Ache-Chang's inequality with Morse theory to prove existence and exponential convergence of the T-curvature flow on the 4D unit ball.

Findings

Existence of extremal metrics under strong Morse inequalities

Exponential convergence of the T-curvature flow from a flat metric

Explicit expression for the extremal metric derived by Ndiaye-Sun

Abstract

In this paper, we study the prescribed $T$-curvature problem on the unit ball $\mathbb{B}^4$ of $\mathbb{R} ^4$ via the $T$-curvature flow approach. By combining Ache-Chang's inequality with the Morse-theoretic approach of Malchiodi-Struwe, we establish existence results under strong Morse-type inequalities at infinity. As a byproduct of our argument, we also prove the exponential convergence of the $T$-curvature flow on $\mathbb{B}^4$, starting from a $Q$-flat and minimal metric conformal to the standard Euclidean metric, to an extremal metric of Ache-Chang's inequality whose explicit expression was derived by Ndiaye-Sun.