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

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.