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

Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun, Heming Wang

TL;DR

This work tackles the prescribed $T$-curvature problem on the 4-dimensional unit ball by evolving conformal metrics under a $T$-curvature flow and leveraging Ache–Chang's inequality together with Morse theory at infinity to obtain existence results under strong Morse-type obstructions. A normalized flow with adapted geometry is employed to handle boundary nonintrinsicity, and the energy functional $E_f$ is shown to decrease, guiding the analysis of convergence versus blow-up. The authors develop a detailed blow-up and spectral decomposition framework, culminating in a shadow flow that tracks bubble formation and concentrates at boundary critical points with negative Laplacian, while the flow’s limit encodes the prescribed boundary data via a Morse-theoretic argument. Under the stated hypotheses, the flow converges, and in a notable corollary, exponential convergence to an explicit extremal metric of the Ache-Chang inequality is established for suitable initial data, revealing both existence and dynamic stability in this higher-order conformal problem.

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.

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

TL;DR

This work tackles the prescribed -curvature problem on the 4-dimensional unit ball by evolving conformal metrics under a -curvature flow and leveraging Ache–Chang's inequality together with Morse theory at infinity to obtain existence results under strong Morse-type obstructions. A normalized flow with adapted geometry is employed to handle boundary nonintrinsicity, and the energy functional is shown to decrease, guiding the analysis of convergence versus blow-up. The authors develop a detailed blow-up and spectral decomposition framework, culminating in a shadow flow that tracks bubble formation and concentrates at boundary critical points with negative Laplacian, while the flow’s limit encodes the prescribed boundary data via a Morse-theoretic argument. Under the stated hypotheses, the flow converges, and in a notable corollary, exponential convergence to an explicit extremal metric of the Ache-Chang inequality is established for suitable initial data, revealing both existence and dynamic stability in this higher-order conformal problem.

Abstract

In this paper, we study the prescribed -curvature problem on the unit ball of via the -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 -curvature flow on , starting from a -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.
Paper Structure (13 sections, 28 theorems, 314 equations)

This paper contains 13 sections, 28 theorems, 314 equations.

Table of Contents

  1. Introduction
  2. The flow equation
  3. Asymptotic behavior of the normalized flow
  4. Normalized flow
  5. Curvature evolution and its $L^2$-convergence
  6. Blow-up analysis
  7. Spectral decomposition
  8. Upper bound on the changing rate of $F_2$
  9. Dominance of $B$ and the estimate of $F_2$
  10. The shadow flow
  11. Scaled stereographic projection
  12. Chracterization of the limit of the shadow flow
  13. Proof of the main results

Key Result

Theorem 1.1

Suppose that $f: \mathbb{S} ^3\to\mathbb{R}$ is a positive smooth function with only nondegenerate critical points such that $\Delta_{g_{\mathbb{S} ^3}}f(a)\neq 0$ at any critical point $a$. For $i=0,\ldots,3$, let where $\operatorname{morse}\mkern1mu(f,a)$ denotes the Morse index of $f$ at the critical point $a$. If the following system has no solution with coefficients $k_i\geq 0$, then proble

Theorems & Definitions (57)

  • Theorem 1.1
  • Remark 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • proof
  • Proposition 3.1
  • proof
  • ...and 47 more