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CMC-1 surfaces via osculating Möbius transformations between circle patterns

Wai Yeung Lam

Abstract

Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induces a $PSL(2,\mathbb{C})$-valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same shear coordinates or the same intersection angles. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature $H\equiv 1$ in hyperbolic space. We further establish convergence on triangular lattices.

CMC-1 surfaces via osculating Möbius transformations between circle patterns

Abstract

Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induces a -valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same shear coordinates or the same intersection angles. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature in hyperbolic space. We further establish convergence on triangular lattices.

Paper Structure

This paper contains 17 sections, 24 theorems, 115 equations, 8 figures, 2 tables.

Table of Contents

  1. Background
  2. Möbius transformations
  3. Cross ratios
  4. Hyperbolic space
  5. Totally umbilical hypersurfaces
  6. Smooth osculating Möbius transformations
  7. Smooth CMC-1 surfaces in hyperbolic space
  8. Discrete osculating Möbius transformations
  9. Coherent lifting to $SL(2,\mathbb{C})$
  10. Horospheres at vertices
  11. Discrete CMC-1 surfaces in hyperbolic space
  12. Integrated mean curvature via parallel surfaces
  13. Weierstrass representation
  14. A 1-parameter family of cross ratio systems
  15. Convergence to smooth surfaces
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

We denote a triangulation by $(V,E,F)$ and its dual mesh by $(V^*,E^*,F^*)$. Given two Delaunay circle patterns $z,\tilde{z}:V\to \mathbb{C}$ with cross ratios $X,\tilde{X}$ such that $|X| \equiv |\tilde{X}|$. Let $A:V^{*} \to SL(2,\mathbb{C})$ be the osculating Möbius transformation from $z$ to $\t is a horospherical net with integrated mean curvature over faces satisfying In particular $f$ is a

Figures (8)

  • Figure 1: Two neighboring triangles sharing the edge $\{ij\}$ together with circumscribed circles.
  • Figure 2: The top row shows two circle patterns sharing the same modulus of cross ratios, where the triangle meshes are indicated by dotted lines. The osculating Möbius transformation induces a realization of the dual graph into hyperbolic space (bottom). It forms a surface consisting of pieces of horospheres. Over each face, the ratio of the integrated mean curvature (Definition \ref{['def:intmean']}) to the face area is constantly equal to $1$. The hyperbolic Gauss map is the vertices of the circle pattern on the top right. Such a correspondence is a discrete analogue of the Weierstrass representation for CMC-1 surfaces in hyperbolic space.
  • Figure 3: The horospheres $\{H_i\}$ on the left passes through a common vertex $O$, the center of the unit ball model. The osculating Möbius transformation induces horospheres $\{\tilde{H}_i\}$ adapted to another circle pattern. As seen in Section \ref{['sec:weier']}, the thick arcs form a cross section of a discrete CMC-1 surface.
  • Figure 4: On the left, the solid line and the circle indicates a vertical cross section of two horospheres intersect at angle $|\alpha|$ in the upper half space model, where one of the horospheres is the horizontal plane $x_3=1$. The right figure shows the horospherical face on the horizontal plane $x_3=1$. The rhombi vertices denote the projection of the tangency points of the neighboring horospheres with $\partial \mathbb{H}^3$. Each circular edge is generated by a rotation centered at a rhombus vertex with angle $\theta$ and radius $\tilde{r}$. The rotation is clockwise if $\theta>0$ while counterclockwise if $\theta<0$.
  • Figure 6: The area of the shaded region is bounded above by $C (\Delta r_{ij}^2+ \Delta r_{jk}^2)$ for some constant $C>0$ independent of $t$.
  • ...and 3 more figures

Theorems & Definitions (49)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem: Lam2019
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 39 more