Unstable CMC spheres and outlying CMC spheres in AF 3-manifolds

TL;DR

This paper develops a nonlinear ODE method to construct constant mean curvature (CMC) surfaces in symmetric Riemannian manifolds, demonstrating the existence of unstable and outlying CMC spheres in asymptotically Schwarzschild spaces, highlighting limitations in stability conditions.

Contribution

Introduces a new nonlinear ODE approach for constructing CMC surfaces, specifically unstable and outlying spheres in asymptotically Schwarzschild manifolds.

Findings

Constructed unstable CMC spheres in asymptotically Schwarzschild manifolds.

Showed that stability conditions in prior work cannot be universally removed.

Demonstrated the existence of outlying CMC spheres in the same setting.

Abstract

In this paper, we introduce a non linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like $g_{ij}=(1+\frac{1}{l})^{2}\delta_{ij}+O(l^{-2})$. The existence of unstable CMC spheres tells us that the stability condition in Qing-Tian's work [Qing-Tian-CMC] can not be removed generally.