Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions

Jonas Hirsch, Rob Kusner, Elena Mäder-Baumdicker

Abstract

We study complete minimal surfaces in $\mathbb{R}^n$ with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy $\mathcal{W}: =\frac{1}{4} \int|\vec H|^2$. In codimension one, we prove that the $\mathcal{W}$-Morse index for any inverted minimal sphere or real projective plane with $m$ such ends is exactly $m-3=\frac{\mathcal{W}}{4π}-3$. We also consider several geometric properties -- for example, the property that all $m$ asymptotic planes meet at a single point -- of these minimal surfaces and explore their relation to the $\mathcal{W}$-Morse index of their inverted surfaces.

Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions

Abstract

We study complete minimal surfaces in with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy . In codimension one, we prove that the -Morse index for any inverted minimal sphere or real projective plane with such ends is exactly . We also consider several geometric properties -- for example, the property that all asymptotic planes meet at a single point -- of these minimal surfaces and explore their relation to the -Morse index of their inverted surfaces.

Paper Structure

This paper contains 8 sections, 27 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Dimension of the span of the normals at infinity
  3. Asymptotic tangent planes
  4. The density at infinity
  5. The equivariant index and the sign of eigenfunctions of the Jacobi operator
  6. Real projective planes with embedded planar ends
  7. Lower semi-continuity of the index
  8. Self-adjointness of the Willmore-Jacobi operator

Key Result

Theorem 1.1

Let $X: \Sigma\setminus\{p_1,...,p_m\} \to\mathbb{R}^3$ be a complete conformal minimal immersion with embedded planar ends at $\{p_1,...,p_m\}\subset\Sigma$ with $m>1$. If $\Sigma$ is a sphere or a real projective plane, then we denote by $\nu:\mathcal{S}^2 \to \mathcal{S}^2\subset\mathbb{R}^3$ the As a consequence, the Morse index of the inverted surface $f:\Sigma \to\mathbb{R}^3$ is

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Corollary 2.4
  • ...and 41 more