Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spherical Ricci tori with rotational symmetry

Iury Domingos, Irene. I. Onnis

TL;DR

This work addresses the construction and embedding of $c$-spherical Ricci metrics on closed surfaces, focusing on rotationally symmetric tori. The intrinsic approach reduces the spherical Ricci condition to an autonomous ODE for a warping function $f$, yielding a Hamiltonian structure with periodic orbits that generate a two-parameter family of torus metrics; in the explicit $c>0$ case, an exact form for $f_{m,\ell}$ is obtained and a broad class of non-isometric examples is exhibited. The authors then realize these metrics as induced metrics on compact rotational surfaces in $\mathbb{S}^3_c$ via a period-function analysis, proving the existence of embedded compact spherical Ricci surfaces by selecting parameter pairs with rational periods. By connecting these spherical Ricci tori to Delaunay-type and minimal/CMC surface theory in $\mathbb{S}^3_c$ through an Otsuki–Wei correspondence, the paper broadens the catalog of explicit spherical Ricci examples with rotational symmetry and enhances the geometric understanding of their immersions.

Abstract

In this article, we study $c$-spherical Ricci metrics, that is, Riemannian metrics whose Gaussian curvature $K$ satisfies \begin{equation*} (K - c)ΔK - |\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.

Spherical Ricci tori with rotational symmetry

TL;DR

This work addresses the construction and embedding of -spherical Ricci metrics on closed surfaces, focusing on rotationally symmetric tori. The intrinsic approach reduces the spherical Ricci condition to an autonomous ODE for a warping function , yielding a Hamiltonian structure with periodic orbits that generate a two-parameter family of torus metrics; in the explicit case, an exact form for is obtained and a broad class of non-isometric examples is exhibited. The authors then realize these metrics as induced metrics on compact rotational surfaces in via a period-function analysis, proving the existence of embedded compact spherical Ricci surfaces by selecting parameter pairs with rational periods. By connecting these spherical Ricci tori to Delaunay-type and minimal/CMC surface theory in through an Otsuki–Wei correspondence, the paper broadens the catalog of explicit spherical Ricci examples with rotational symmetry and enhances the geometric understanding of their immersions.

Abstract

In this article, we study -spherical Ricci metrics, that is, Riemannian metrics whose Gaussian curvature satisfies \begin{equation*} (K - c)ΔK - |\nabla K|^2 - 4K(K - c)^2 = 0, \end{equation*} for some . 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 , establishing the existence of embedded compact spherical Ricci surfaces by controlling a period function associated with the isometric immersion.
Paper Structure (13 sections, 9 theorems, 75 equations, 5 figures)

This paper contains 13 sections, 9 theorems, 75 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Ricci metrics with rotational symmetry
  4. SO(2)-invariant metrics
  5. An associated autonomous system
  6. Constructing generalized Ricci tori
  7. Generalized Ricci tori of type (a,0,c) with ac>0
  8. The spherical Ricci tori
  9. An explicit family of solutions
  10. Spherical Ricci tori immersed in the 3-sphere
  11. Rotational surfaces
  12. Immersing spherical Ricci tori into S3c
  13. Some explicit examples

Key Result

Proposition 3.1

Let $a, c, m \in \mathbb R^*$ have the same sign. Then the level set $\{(x,y):E_m(x, y) = \ell\}$ is a compact smooth curve surrounding the equilibrium point $(x_*, 0)$ if and only if where $P_m(x)$ is the potential function of system.

Figures (5)

  • Figure 4.1: The set $\Lambda_c$ of parameters.
  • Figure 5.1: The admissible set $\Lambda'_c$.
  • Figure 5.2: The admissible set $\Lambda'_1$: (I) metrics induced on minimal rotational surfaces in $\mathbb{S}^3$; (II) the induced metric of the Clifford torus; (III) metrics induced on Delaunay surfaces in $\mathbb{R}^3$.
  • Figure 5.3: Generating curve and embedded $1$-spherical Ricci torus with $m = 0.51$ and $\ell = 0.73$.
  • Figure 5.4: Generating curve and immersed $1$-spherical Ricci torus with $m = 0.75$ and $\ell \approx 0.87$.

Theorems & Definitions (28)

  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • ...and 18 more