The axisymmetric $\sigma_k$-Nirenberg problem

TL;DR

This paper investigates the existence, compactness, and blow-up behavior of solutions to the axisymmetric $\sigma_k$-Nirenberg problem on spheres, relating solution properties to the curvature function's behavior near poles.

Contribution

It provides new criteria for solution existence and compactness based on the flatness order of the prescribed curvature function at poles, extending understanding of the $\sigma_k$-Nirenberg problem.

Findings

Solution set is non-empty when flatness order exceeds a threshold.

Explicit constant $C(K)$ determines solution existence at critical flatness.

Blow-up sequences exist for certain curvature behaviors, indicating non-compactness.

Abstract

We study the problem of prescribing $\sigma_k$-curvature for a conformal metric on the standard sphere $\mathbb{S}^n$ with $2 \leq k < n/2$ and $n \geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $\beta \in [2,n)$. We prove among other things the following statements. (1) When $\beta>n-2k$, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When $ \beta = n-2k$, there is an explicit positive constant $C(K)$ associated with $K$. If $C(K)>1$, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If $C(K)<1$, the solution set is compact but the total degree counting is $0$, and the solution set is sometimes empty and sometimes non-empty. (3) When $\frac{2}{n-2k}\le \beta < n-2k$, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When $\beta < \frac{n-2k}{2}$, there exists $K$ for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of $\beta$, there exists also some $K$ for which the solution set is empty.