Spherical Ricci tori with rotational symmetry

TL;DR

This paper constructs a family of rotationally symmetric spherical Ricci metrics on tori, demonstrating their diverse realizations and embeddedness as surfaces in 3-spheres, advancing understanding of Ricci curvature conditions.

Contribution

It explicitly constructs a two-parameter family of rotationally symmetric spherical Ricci metrics on tori and shows their realization as embedded surfaces in the 3-sphere.

Findings

Infinite non-isometric examples on the same torus

Existence of embedded compact spherical Ricci surfaces

Control of a period function for isometric immersion

Abstract

In this article, we study $c$-spherical Ricci metrics, that is, Riemannian metrics whose Gaussian curvature $K$ satisfies \begin{equation*} (K - c)\Delta K - |\nabla K|^2 - 4K(K - c)^2 = 0, \end{equation*} for some $c>0$. We explicitly construct a two-parameter family of such metrics with rotational symmetry and show that infinitely many non-isometric examples can be realized on the same torus. Moreover, we investigate their realization as induced metrics on compact rotational surfaces in $\mathbb{S}^3$, establishing the existence of embedded compact spherical Ricci surfaces by controlling a period function associated with the isometric immersion.