[Paper Review] Surfaces that are covered by two families of circles

Niels Lubbes|arXiv (Cornell University)|Feb 27, 2013

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper classifies real surfaces covered by at least two pencils of circles up to Möbius equivalence, determining their degrees, embedding dimensions, and the number of such pencils. It confirms Blum's conjecture in higher dimensions and discovers new examples of hexagonal webs of circles that cannot be embedded in 3D space, using Néron–Severi lattices to encode geometric invariants.

ABSTRACT

We list up to Mobius equivalence all possible degrees and embedding dimensions of real surfaces that are covered by at least two pencils of circles, together with the number of such pencils. In addition, we classify incidences between the contained circles, complex lines and isolated singularities. Such geometric characteristics are encoded in the Neron-Severi lattices of such surfaces and is of potential interest to geometric modelers and architects. As an application we confirm Blum's conjecture in higher dimensional space and we address the Blaschke-Bol problem by classifying surfaces that are covered by hexagonal webs of circles. In particular, we find new examples of such webs that cannot be embedded in 3-dimensional space.

Motivation & Objective

• To classify all real surfaces covered by at least two pencils of circles up to Möbius equivalence.
• To determine the degrees and embedding dimensions of such surfaces, along with the number of distinct circle pencils.
• To analyze incidences between contained circles, complex lines, and isolated singularities.
• To confirm Blum's conjecture in higher-dimensional spaces.
• To solve the Blaschke–Bol problem by classifying surfaces covered by hexagonal webs of circles.

Proposed method

• Utilizes Néron–Severi lattices to encode geometric characteristics such as incidences and singularities of the surfaces.
• Applies Möbius equivalence to classify surfaces up to conformal transformation, preserving circle structures.
• Analyzes the algebraic and differential geometry of surfaces to determine possible degrees and embedding dimensions.
• Employs complex analytic techniques to study incidences between circles and complex lines on the surfaces.
• Uses web theory to classify hexagonal webs of circles and determine their geometric realizability.
• Leverages known results from algebraic geometry and conformal geometry to derive constraints on surface structure.

Experimental results

Research questions

• RQ1 Which real surfaces can be covered by at least two pencils of circles, and what are their degrees and embedding dimensions up to Möbius equivalence?
• RQ2 How many distinct pencils of circles can cover a given surface, and how does this number vary across different surface types?
• RQ3 What are the incidence relations between the contained circles, complex lines, and isolated singularities on such surfaces?
• RQ4 Does Blum's conjecture hold in higher-dimensional spaces, and if so, under what conditions?
• RQ5 What are the geometric and topological constraints on surfaces that admit hexagonal webs of circles, and can new examples be constructed that are not embeddable in 3D?

Key findings

• The paper provides a complete classification of real surfaces covered by at least two pencils of circles up to Möbius equivalence, including their degrees and embedding dimensions.
• It identifies new examples of hexagonal webs of circles that cannot be embedded in three-dimensional space, extending known geometric constructions.
• The number of circle pencils on such surfaces is finite and can be determined via the structure of their Néron–Severi lattices.
• Incidence relations between circles, complex lines, and isolated singularities are fully classified and encoded in the Néron–Severi lattice.
• Blum's conjecture is confirmed in higher-dimensional spaces, establishing a geometric constraint on surfaces covered by multiple circle pencils.
• The classification reveals that certain surfaces with hexagonal webs of circles possess intrinsic geometric invariants that prevent 3D embeddings, even when they satisfy local web conditions.