Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The classification of complete improper affine spheres with singularities of low total curvature and new examples

Jun Matsumoto

TL;DR

This work advances the global classification of complete improper affine fronts in $\mathbb{R}^3$ by total curvature, revealing that TC $\ge -6\pi$ forces genus zero, and that TC $= -8\pi$ can admit genus 1, with partial results and explicit Weierstrass data for these low-curvature cases. The authors develop an end-by-end analysis, linking embedded ends to equality in an Osserman-type inequality and identifying three asymptotic end-types (type-P, type-R, type-NR). They construct new embedded-end examples, including multi-end configurations, and provide a detailed taxonomy for TC $= -2\pi, -4\pi, -6\pi$ with explicit data, culminating in a genus-1 existence result for TC $= -8\pi$ and a new genus-1 example with TC $= -10\pi$. The results highlight deep connections between the complex Weierstrass data, end behavior, and global curvature bounds, contributing to the affine Front theory and offering concrete surfaces with embedded ends and extremal curvature properties.

Abstract

We provide a classification of complete improper affine spheres with singularities (say \emph{improper affine fronts}) in unimodular affine three-space $\boldsymbol{R}^3$ whose total curvature is greater than or equal to $-6π$, and a partial classification in the case of total curvature $-8π$. For the case of total curvature $-8π$, we give a complete classification for genus $0$ case and show the existence of an example and a one parameter family with genus $1$. We also study the asymptotic behavior of embedded ends of complete improper affine fronts. Moreover, we give new examples for this class of surfaces, including one which satisfies the equality condition of an Osserman-type inequality and is of positive genus.

The classification of complete improper affine spheres with singularities of low total curvature and new examples

TL;DR

This work advances the global classification of complete improper affine fronts in by total curvature, revealing that TC forces genus zero, and that TC can admit genus 1, with partial results and explicit Weierstrass data for these low-curvature cases. The authors develop an end-by-end analysis, linking embedded ends to equality in an Osserman-type inequality and identifying three asymptotic end-types (type-P, type-R, type-NR). They construct new embedded-end examples, including multi-end configurations, and provide a detailed taxonomy for TC with explicit data, culminating in a genus-1 existence result for TC and a new genus-1 example with TC . The results highlight deep connections between the complex Weierstrass data, end behavior, and global curvature bounds, contributing to the affine Front theory and offering concrete surfaces with embedded ends and extremal curvature properties.

Abstract

We provide a classification of complete improper affine spheres with singularities (say \emph{improper affine fronts}) in unimodular affine three-space whose total curvature is greater than or equal to , and a partial classification in the case of total curvature . For the case of total curvature , we give a complete classification for genus case and show the existence of an example and a one parameter family with genus . We also study the asymptotic behavior of embedded ends of complete improper affine fronts. Moreover, we give new examples for this class of surfaces, including one which satisfies the equality condition of an Osserman-type inequality and is of positive genus.
Paper Structure (10 sections, 10 theorems, 108 equations, 12 figures)

This paper contains 10 sections, 10 theorems, 108 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic behavior of complete embedded ends
  4. Asymptotic behavior of the complete embedded ends
  5. New examples with embedded ends
  6. Classification of complete improper affine fronts of total curvature $-2m\pi$
  7. The case of total curvature $-2\pi$
  8. The case of total curvature $-4\pi$
  9. The case of total curvature $-6\pi$
  10. The case of total curvature $-8\pi$

Key Result

Theorem 1.1

Figures (12)

  • Figure 1: Complete improper affine fronts with only embedded ends
  • Figure 2: New examples with $2n$ embedded ends
  • Figure 3: Complete improper affine front with total curvature $-2\pi$\ref{['-2pi']}
  • Figure 4: Complete improper affine front with total curvature $-4\pi$
  • Figure 5: Complete improper affine front with total curvature $-6\pi$, and one end
  • ...and 7 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • Definition 3.1
  • Theorem 3.2
  • ...and 20 more