[Paper Review] Prescribed Scalar Curvature with Minimal Boundary Mean Curvature on $S^4_+$

Hichem Chtioui, Khalil El Mehdi|ArXiv.org|Jun 7, 2003

Geometric Analysis and Curvature Flows20 references9 citations

TL;DR

This paper establishes existence results for the prescribed scalar curvature problem with minimal boundary mean curvature on the 4-dimensional half-sphere $S^4_+$, using critical points at infinity theory and topological methods. It proves that under suitable non-degeneracy and sign conditions on the curvature function $K$, a positive solution exists, particularly when the global maximum of $K$ lies outside a critical set $\mathcal{F}^+$, or when the topology of critical points at infinity prevents contractibility.

ABSTRACT

This paper is devoted to the prescribed scalar curvature under minimal boundary mean curvature on the standard four dimensional half sphere. Using topological methods from the theory of critical points at infinity, we prove some existence results. These methods were first introduced by A. Bahri.

Motivation & Objective

• To establish existence conditions for positive solutions to the prescribed scalar curvature problem on the 4-dimensional half-sphere $S^4_+$ with zero boundary mean curvature.
• To analyze the lack of compactness in the associated variational problem due to the critical Sobolev exponent.
• To use topological invariants from the theory of critical points at infinity to detect solutions when standard compactness fails.
• To extend prior results from lower dimensions to the four-dimensional case, focusing on balance phenomena between self-interaction and interaction of blow-up profiles.
• To provide sufficient topological conditions on the curvature function $K$ ensuring existence of solutions, even when the global maximum of $K$ lies in a critical set $\mathcal{F}^+$.

Proposed method

• Employing the critical points at infinity theory developed by Bahri and others to analyze the asymptotic behavior of Palais-Smale sequences for the Euler-Lagrange functional associated with the problem.
• Defining the set $\mathcal{F}^+ = \{ y \in S^4_+ \mid \nabla K(y) = 0 \text{ and } \frac{-\Delta K(y)}{3K(y)} + 4H(y,y) > 0 \}$, which captures critical points contributing to the topology of the functional at infinity.
• Using the Green's function $G(x,y)$ and its regular part $H(x,y)$ to model the singular behavior of solutions near blow-up points.
• Applying deformation retracts and homology arguments to show that the existence of a solution is obstructed only if the critical set at infinity induces nontrivial topology.
• Using the fact that $\Sigma^+$ (the sublevel set of the functional) is contractible to deduce that the critical points at infinity must also be contractible if no solution exists, leading to contradiction under topological assumptions.
• Applying degree theory and asymptotic expansions of the functional $J(u)$ to relate the value of $J$ to the curvature function $K$, particularly near the threshold $c_1 = \frac{3}{2}S_4^{1/2}$.

Experimental results

Research questions

• RQ1 Under what conditions on the curvature function $K$ does the prescribed scalar curvature problem with minimal boundary mean curvature on $S^4_+$ admit a positive solution?
• RQ2 How does the topology of the critical points at infinity influence the solvability of the problem when the Palais-Smale condition fails?
• RQ3 What role does the balance between self-interaction and interaction of blow-up masses play in the four-dimensional case?
• RQ4 Can the existence of a solution be guaranteed when the global maximum of $K$ lies in the set $\mathcal{F}^+$, where the standard topological obstruction vanishes?
• RQ5 What is the Morse index of the solution under the topological conditions of Theorem 1.5 and Theorem 1.6?

Key findings

• If the global maximum $y_0$ of $K$ does not lie in $\mathcal{F}^+$, then problem (1) has a positive solution, as the topological obstruction from critical points at infinity is avoided.
• When $y_0 \in \mathcal{F}^+$, a solution exists if the set $\mathcal{F}^+$ is finite and the matrix $M(\tau_s)$ associated with $s$-tuples of points in $\mathcal{F}^+$ is non-degenerate, ensuring nontrivial topology.
• Under the assumption that $\|K - 1\|_{L^\infty(S^4_+)}$ is sufficiently small, the functional $J$ has no critical points at infinity above level $c_1 = \frac{3}{2}S_4^{1/2}$, and the existence of a solution follows from the non-contractibility of the critical set at infinity.
• The solution obtained under the smallness assumption on $K$ has Morse index at least $m$, where $m$ is the number of critical points in $\mathcal{F}^+$, as shown via degree-theoretic arguments.
• If the functional $J$ has no critical points at infinity with two or more masses, and $\mathcal{F}^+$ is non-empty, then the existence of a solution follows from the non-contractibility of the critical set at infinity, contradicting the assumption of no solution.
• The critical points at infinity are parametrized by $X \times [A, \infty)$, where $X = \bigcup_{y \in \mathcal{F}^+} \overline{W_s(y)}$, and their topological structure determines the solvability of the problem.