[Paper Review] Linear Weingarten surfaces in Euclidean and hyperbolic space

Rafael López|ArXiv.org|Jun 17, 2009

Mathematics and Applications9 references3 citations

TL;DR

This paper classifies linear Weingarten surfaces in Euclidean and hyperbolic 3-space, focusing on cyclic, rotational, and parabolic surfaces. It proves that non-rotational cyclic linear Weingarten surfaces in ℝ³ are either Riemann examples (H=0) or generalized cones (K=0), and constructs a family of complete, non-embedded, periodic hyperbolic rotational surfaces in ℝ³ and parabolic complete surfaces in ℍ³ satisfying aH + bK = c.

ABSTRACT

In this paper we review some author's results about Weingarten surfaces in Euclidean space $ ^3$ and hyperbolic space $\h^3$. We stress here in the search of examples of linear Weingarten surfaces that satisfy a certain geometric property. First, we consider Weingarten surfaces in $ ^3$ that are foliated by circles, proving that the surface is rotational, a Riemann example or a generalized cone. Next we classify rotational surfaces in $ ^3$ of hyperbolic type showing that there exist surfaces that are complete. Finally, we study linear Weingarten surfaces in $\h^3$ that are invariant by a group of parabolic isometries, obtaining its classification.

Motivation & Objective

• To classify linear Weingarten surfaces in ℝ³ and ℍ³ under geometric symmetry constraints such as rotational invariance or cyclic foliation by circles.
• To determine whether non-rotational cyclic surfaces can satisfy linear Weingarten relations, extending known results for constant mean curvature surfaces.
• To construct and characterize complete linear Weingarten surfaces in ℝ³ and ℍ³, particularly in the hyperbolic and parabolic types.
• To establish existence and geometric properties of complete surfaces satisfying aH + bK = c in both Euclidean and hyperbolic 3-space.

Proposed method

• Analyzes surfaces foliated by circles (cyclic surfaces) in ℝ³ using differential geometry of principal curvatures and curvature relations.
• Reduces the problem to ordinary differential equations for the profile curve in rotational and parabolic surface cases.
• Applies maximum principles and asymptotic analysis to study behavior at infinity and completeness.
• Uses symmetry and ODE techniques to classify solutions of the Weingarten equation aH + bK = c in ℝ³ and ℍ³.
• Employs curvature-based classification via the discriminant Δ = a² + 4bc to distinguish elliptic, parabolic, and hyperbolic types.
• Applies L’Hôpital’s rule and limit analysis to resolve singularities in solutions and prove existence of complete surfaces.

Experimental results

Research questions

• RQ1 Are there non-rotational cyclic surfaces in ℝ³ that satisfy a linear Weingarten relation aκ₁ + bκ₂ = c or aH + bK = c?
• RQ2 Can complete, non-embedded, periodic rotational surfaces exist in ℝ³ for hyperbolic linear Weingarten relations (Δ < 0)?
• RQ3 Do parabolic complete surfaces exist in hyperbolic 3-space ℍ³ satisfying aH + bK = c for constant a, b, c?
• RQ4 What geometric properties characterize the profile curves of rotational linear Weingarten surfaces in ℝ³ and ℍ³?
• RQ5 How does the sign of the discriminant Δ = a² + 4bc affect the completeness and topology of linear Weingarten surfaces?

Key findings

• Non-rotational cyclic linear Weingarten surfaces in ℝ³ are either Riemann examples (H = 0) or generalized cones (K = 0), with no other examples existing.
• A family of complete, non-embedded, periodic hyperbolic rotational linear Weingarten surfaces exists in ℝ³ when Δ < 0, contradicting Hilbert’s theorem for constant negative curvature.
• Parabolic complete surfaces satisfying aH + bK = c exist in hyperbolic 3-space ℍ³, extending known results on rotational surfaces in symmetric spaces.
• For a+2b<0, solutions to the profile curve ODE reach a finite endpoint s= s̄ with z(s̄)=0 and θ(s̄) satisfying a cos θ₁ - b sin²θ₁ -1 = 0.
• When a+2b>0, the profile curve α(s) is symmetric about s₀ where θ(s₀)=π, and the curve is invariant under horizontal translations.
• In all cases, the profile curve is a vertical graph with concave z(s) and lim_{s→s̄} θ′(s) = ±∞, indicating singular behavior at the boundary.