Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rotational Ricci surfaces

Iury Domingos, Roney Santos, Feliciano Vitório

Abstract

We classify rotational surfaces in the three-dimensional Euclidean space whose Gaussian curvature $K$ satisfies \begin{equation*} KΔK - \|\nabla K\|^2-4K^3 = 0. \end{equation*} These surfaces are referred to as rotational Ricci surfaces. As an application, we show that there is a one-parameter family of such surfaces meeting the boundary of the unit Euclidean three-ball orthogonally. In addition, we show that this family interpolates a vertical geodesic and the critical catenoid.

Rotational Ricci surfaces

Abstract

We classify rotational surfaces in the three-dimensional Euclidean space whose Gaussian curvature satisfies \begin{equation*} KΔK - \|\nabla K\|^2-4K^3 = 0. \end{equation*} These surfaces are referred to as rotational Ricci surfaces. As an application, we show that there is a one-parameter family of such surfaces meeting the boundary of the unit Euclidean three-ball orthogonally. In addition, we show that this family interpolates a vertical geodesic and the critical catenoid.
Paper Structure (10 sections, 12 theorems, 84 equations, 4 figures)

This paper contains 10 sections, 12 theorems, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Ricci condition for rotational surfaces
  3. Preliminaries and first examples
  4. Reduction to a first order ODE
  5. The $(a,b,c)$-dependence of $\Sigma_f$
  6. Classification of rotational Ricci surfaces
  7. Case a=0
  8. Case b=0
  9. Case a/=0 and b/=0
  10. Rotational Ricci surfaces meeting $\partial B^3$ orthogonally

Key Result

Proposition 2.4

Let $\Sigma_f$ be a rotational Ricci surface, where $f:I \to \mathbb R$ is a positive smooth function. Let $s_*\in \bar{I}$ be such that $f$ is well-defined and positive at $s_*.$

Figures (4)

  • Figure 3.1: Catenoidal-Ricci surface and profile curves for $(c,d) = (0,1).$
  • Figure 3.2: Catenoidal-Ricci surface and profile curves for $(b,c)=(\frac{3}{4},0).$
  • Figure 3.3: Funnel-Ricci surface and profile curves for $c=-1$.
  • Figure 3.4: Funnel-Ricci surface and profile curves for $a=1$.

Theorems & Definitions (31)

  • Example 2.1: Complete case
  • Example 2.2: Extended case
  • Example 2.3: Non-extendable case
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 21 more