Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities

TL;DR

This paper investigates the asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities, establishing radial symmetry and asymptotic similarity to radial solutions, extending classical results to more general curvature equations.

Contribution

It proves that solutions with isolated singularities are asymptotic to radial solutions, generalizing classical Yamabe results and providing new asymptotic approximation techniques.

Findings

Solutions are asymptotic to radial solutions near singularities

Extension of classical Yamabe results to $\sigma_k$-curvature equations

Provides alternative proof via linearized operator analysis

Abstract

$\sigma_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $\sigma_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $\sigma_k$-Yamabe equation is asymptotic to a radial solution to the same equation on $\mathbb R^n \setminus \{0\}$. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and $\sigma_k$ curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.