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

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.

💡 Research Summary

The paper addresses the prescribed T‑curvature problem on the four‑dimensional unit ball (B^{4}\subset\mathbb{R}^{4}). Given a smooth positive function (f) on the boundary sphere (S^{3}), the goal is to find a metric (g) conformal to the Euclidean metric (g_{B^{4}}) such that the Q‑curvature vanishes in the interior, the T‑curvature equals (f) on the boundary, and the boundary is minimal (mean curvature zero). This problem is a higher‑order analogue of the classical Nirenberg problem and involves the Paneitz operator (P_{4}^{g}) together with its boundary counterpart (P_{3,b}^{g}) introduced by Chang and Qing.

The authors first rewrite the geometric prescription as a fourth‑order semilinear elliptic boundary value problem for a conformal factor (u). By introducing the adapted metric (g^{*}=e^{2\rho}g_{B^{4}}) (Ache–Chang) and setting (w=u-\rho), the system becomes \