Two Connections between Combinatorial and Differential ... · Two Connections between Combinatorial and Differential Geometry John M. Sullivan Institut fur Mathematik, Technische

Two Connections betweenCombinatorial and Differential Geometry

John M. Sullivan

Institut fur Mathematik, Technische Universitat BerlinBerlin Mathematical School

DFG Research Group Polyhedral SurfacesDFG Research Center MATHEON

Discrete Differential Geometry2007 July 16–20 Berlin

The 5,7–triangulation problem

Triangulations of the torus T2

Average vertex degree 6

Exceptional vertices have d 6= 6

Regular triangulations have d ≡ 6

John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 2 / 29


Edge flips give new triangulations

Flip changes four vertex degrees

Can produce 5272–triangulations(four exceptional vertices)

Quotients of some such tori are5,7–triangulations of Klein bottle



Two-vertex torus triangulations

regular 4,8 3,9 2,10 1,11



Refinement or subdivision schemes

√3–fold 2–fold

√7–fold 3–fold

Exceptional vertices preservedOld vertex degrees fixed

New vertices regular

Lots more 4,8–, 3,9–, 2,10– and 1,11–triangulations



Is there a 5,7–triangulation of the torus?

(any number of regular vertices allowed)

No!We prove this combinatorial statement geometrically

using curvature and holonomy

or complex function theory

Joint work with

Ivan Izmestiev (TU Berlin)

Rob Kusner (UMass/Amherst)

Gunter Rote (FU Berlin)

Boris Springborn (TU Berlin)



Is there a 5,7–triangulation of the torus?

(any number of regular vertices allowed)

No!We prove this combinatorial statement geometrically

using curvature and holonomy

or complex function theory

Joint work with

Ivan Izmestiev (TU Berlin)

Rob Kusner (UMass/Amherst)

Gunter Rote (FU Berlin)

Boris Springborn (TU Berlin)



Combinatorics and topology

Triangulation of any surfaceDouble-counting edges gives:

dV = 2E = 3F

χ

dV=

χ

2E=

χ

3F=

1

d− 1

2+

13

6χ =∑

d

(6− d)vd

Notation

d := average vertex degree

vd := number of vertices of degree d



Eberhard’s theorem

Triangulation of S2

12 =∑

d

(6− d)vd

Theorem (Eberhard, 1891)Given any (vd) satisfying this condition, there is a correspondingtriangulation of S2, after perhaps modifying v6.

Examples

512–triangulation exists for v6 6= 1

34–triangulation exists for v6 6= 2 and even



Torus triangulations

The condition 0 =∑

(6− d)vd is simply d = 6.

Analog of Eberhard’s Theorem would say∃ 5,7–triangulation for some v6

Instead, we show there are none


The 5,7–triangulation problem Euclidean cone metrics

Euclidean cone metrics

DefinitionTriangulation on M induces equilateral metric:each face an equilateral euclidean triangle.

Exceptional vertices are cone points

DefinitionEuclidean cone metric on M is locally euclideanaway from discrete set of cone points.

Cone of angle ω > 0 has curvature κ := 2π − ω.

Vertex of degree d has curvature (6− d)π/3



Regular triangulations on the torus

Theorem (cf. Alt73, Neg83, Tho91, DU05, BK06)

A triangulation of T2 with no exceptional vertices is a quotient of theregular triangulation T0 of the plane, or equivalently a finite cover ofthe 1-vertex triangulation.

Proof.

Equilateral metric is flat torus R2/Λ. The triangulation lifts to the cover,giving T0. Thus Λ ⊂ Λ0, the triangular lattice.

CorollaryAny degree-regular triangulation has vertex-transitive symmetry.



Holonomy of a cone metric

DefinitionMo := M r cone points

h : π1(Mo)→ SO2

H := h(π1)

LemmaFor a triangulation, H is a subgroup of C6 := 〈2π/6〉.

Proof.As we parallel transport a vector, look at the angle it makes with eachedge of the triangulation.


The 5,7–triangulation problem Holonomy theorem

Holonomy theorem

TheoremA torus with two cone points p± of curvature κ = ±2π/nhas holonomy strictly bigger than Cn.

CorollaryThere is no 5,7–triangulation of the torus.

Proof.Lemma says H contained in C6; theorem says H strictly bigger.



Proof of Holonomy theorem.Shortest nontrivial geodesic γ avoids p+. If it hits p− and splits excessangle 2π/n there, consider holonomy of a pertubation. Otherwise, γ

avoids p− or makes one angle π there, so slide it to foliate a euclideancylinder. Complementary digon has two positive angles, so geodesicfrom p− to p− within the cylinder does split the excess 2π/n.

π πp−

p+

γ γ′



Quadrangulations and hexangulations

Theorem

The torus T2 has

no 3,5–quadrangulation

no bipartite 2,4–hexangulation

2,6–quad 3252–quad 2,4–hex 1,5–hex bip 1,5–hex


The 5,7–triangulation problem Riemann surfaces

Generalizing the holonomy theorem

Question

Given n > 0 and a euclidean cone metric on T2 whose curvatures aremultiples of 2π/n, when is its holonomy H contained in Cn?

Curvature as divisorCone metric induces Riemann surface structure

Cone point pi has curvature mi2π/n

Divisor D =∑

mipi has degree 0

Cone metric gives developing map from universal cover of Mo to C.


The 5,7–triangulation problem Riemann surfaces

Main theorem

Theorem

H < Cn⇐⇒ D principal

Proof.

Consider the nth power of the derivative of the developing map. This iswell-defined on M iff H < Cn. If so, its divisor is D. Conversely, if D isprincipal, corresponding meromorphic function is this nth power.

NoteThe case n = 2 is the classical correspondance between meromorphicquadratic differentials and “singular flat structrues”.


The 5,7–triangulation problem Three dimensions

Combinatorics −→ geometry in three dimensions

Triangulated 3-manifold:make each tetrahedron regular euclidean

Edge valence ≤ 5⇐⇒ curvature bounded below by 0

Enumeration (with Frank Lutz, TU Berlin)Enumerate simplicial 3-manifolds with edge valence ≤ 5

Exactly 4761three-spheres plus 26 finite quotients

[Matveev, Shevchishin]:Can smooth to get positive curvature


k-point metrics CMC Surfaces

CMC Surfaces

DefinitionA coplanar k-unduloid is an Alexandrov-embedded CMC (H ≡ 1)surface M with k ends and genus 0, contained in a slab in R3.

Note: each endasymptotic to unduloid



Classifying map

M has mirror symmetry

Upper half M+ is a topological diskwith k boundary curves in mirror plane

Conjugate cousin M+ is minimal in S3

with k boundary Hopf great circles

Hopf projection gives spherical metric on open diskwith k completion boundary points



Classification

Theorem (with Karsten Große-Brauckmann and Rob Kusner)Classifying map is homeomorphism from moduli space of coplanark-unduloids to space Dk of spherical k-point metrics, which is aconnected (2k− 3)–manifold.

New work (also with Nick Korevaar)

Dk∼= R2k−3


k-point metrics 2- and 3-point metrics

2-point metrics

Universal cover of S2 r {p, q}Bi-infinite chain of slit spheres

D2∼= (0, π]



Triunduloids classified by spherical triples



3-point metrics

Spherical trianglewith three chains of slit spheres

D3∼= T3

∼= B3 ∼= R3

C3 := D3/Mob∼= {∗}

General case

We show Ck∼= Ck−3, so Dk

∼= R2k−3


k-point metrics Medial axis

Medial axis

Maximal balls within D

Touch ≥ 2 boundary points

Medial axis is tree (retract of D)

k ends are infinite rays

nodes are maximal circleswith ≥ 3 boundary points

Mobius-invariant notion


k-point metrics Associahedron

Metric trees

TheoremThe space of planar trees with

k ends (labeled in order)

length ≥ 0 on each midsegment

is cone over dual associahedron,homeomorphic to Rk−3.

k = 4 gives Ra, b≥ 0, at most one positive

two rays join to form R 3 a− b

a

b



k = 5

a b a b

c

d

e

a

bc

d

e

(a, b)

Combinatorial trees←→ triangulations of pentagon

Five quadrants glue together to form plane



k = 6

14 generic trees

14 triangulations of hexagon

14 octants fit together to form R3



Complexification

Ck∼= Ck−3

Medial axis is a tree

Midsegment is circles through pqending with ones touching r, s

Length is angle (in R≥0, not mod 2π) between end circles

But also have a “twist”

Together these are log of cross-ratio(p, q, r, s)

All these notions are Mobius-invariant


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Two Connections between Combinatorial and Differential ... · Two Connections between Combinatorial and Differential Geometry John M. Sullivan Institut fur Mathematik, Technische