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Parabolicity of invariant Surfaces

Andrea Del Prete, Vicent Gimeno i Garcia

TL;DR

This work addresses the problem of determining when a complete surface that is invariant under a Killing flow is parabolic. It develops a practical, unified criterion that relates parabolicity to a generating curve $\gamma$ and the norm $\mu=\|\xi\|$ of the Killing field along $\gamma$, via a Killing-submersion reduction and conformal changes. The authors prove that $S$ must have at most two zeros of $\xi$, leading to plane or sphere topologies with rotational symmetry, and they provide a Laplacian-based framework to translate parabolicity questions to the one-dimensional base. The results are then applied to invariant surfaces in homogeneous 3-manifolds, notably Sol$_3$ and $\mathbb{E}^3(\kappa,\tau)$ spaces, yielding concrete parabolicity conclusions for minimal and constant mean curvature surfaces and illustrating the approach with explicit examples such as vertical cylinders and Heisenberg-type geometries.

Abstract

We present a clear and practical way to characterize the parabolicity of a complete immersed surface that is invariant with respect to a Killing vector field of the ambient space.

Parabolicity of invariant Surfaces

TL;DR

This work addresses the problem of determining when a complete surface that is invariant under a Killing flow is parabolic. It develops a practical, unified criterion that relates parabolicity to a generating curve and the norm of the Killing field along , via a Killing-submersion reduction and conformal changes. The authors prove that must have at most two zeros of , leading to plane or sphere topologies with rotational symmetry, and they provide a Laplacian-based framework to translate parabolicity questions to the one-dimensional base. The results are then applied to invariant surfaces in homogeneous 3-manifolds, notably Sol and spaces, yielding concrete parabolicity conclusions for minimal and constant mean curvature surfaces and illustrating the approach with explicit examples such as vertical cylinders and Heisenberg-type geometries.

Abstract

We present a clear and practical way to characterize the parabolicity of a complete immersed surface that is invariant with respect to a Killing vector field of the ambient space.
Paper Structure (5 sections, 8 theorems, 40 equations, 2 figures)

This paper contains 5 sections, 8 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem: Parabolicity of surfaces admiting a Killing submersion
  3. Application of the Main Theorem: Parabolicity of invariant surfaces in homogeneous 3-manifolds
  4. Parabolicity of invariant surfaces with constant mean curvature in $Sol_3$
  5. Parabolicity of vertical cylinders in Killing submersion

Key Result

Proposition 2.1

Let $\mathbb{E}$ be a $n$-dimensional Riemannian manifold which admits a complete Killing vector field $\xi\in\mathfrak{X}(\mathbb{E})$. Assume that an immersed complete regular surface $S\subset\mathbb{E}$ is invariant with respect to the one-parameter group of isometries of $\mathbb{E}$ associated

Figures (2)

  • Figure 1: If $\xi(p)=0$, $\xi(q)\neq0$ and $\gamma$ is the minimal geodesic segment joining $p$ to $q$, then there is no $p'\in\gamma$ with $\xi(p')=0$.
  • Figure 2: If $S$ has two points $p$ and $p'$ where $\xi$ vanish, then the points $p$ and $p'$ are antipodals points of a rotationally symmetric sphere.

Theorems & Definitions (16)

  • Remark 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 6 more