Two Connections between Combinatorial and Differential Geometry John M. Sullivan Institut für Mathematik, Technische Universität Berlin Berlin Mathematical School DFG Research Group Polyhedral Surfaces DFG Research Center MATHEON Discrete Differential Geometry 2007 July 16 20 Berlin
Triangulations of the torus T 2 Average vertex degree 6 Exceptional vertices have d 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 5 2 7 2 triangulations (four exceptional vertices) Quotients of some such tori are 5,7 triangulations of Klein bottle John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 3 / 29
Two-vertex torus triangulations regular 4,8 3,9 2,10 1,11 John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 4 / 29
Refinement or subdivision schemes 3 fold 2 fold 7 fold 3 fold Exceptional vertices preserved Old vertex degrees fixed New vertices regular Lots more 4,8, 3,9, 2,10 and 1,11 triangulations John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 5 / 29
Is there a 5,7 triangulation of the torus? (any number of regular vertices allowed) John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 6 / 29
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) Günter Rote (FU Berlin) Boris Springborn (TU Berlin) John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 6 / 29
Combinatorics and topology Triangulation of any surface Double-counting edges gives: dv = 2E = 3F χ dv = χ 2E = χ 3F = 1 d 1 2 + 1 3 6χ = d (6 d)v d Notation d := average vertex degree v d := number of vertices of degree d John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 7 / 29
Eberhard s theorem Triangulation of S 2 12 = d (6 d)v d Theorem (Eberhard, 1891) Given any (v d ) satisfying this condition, there is a corresponding triangulation of S 2, after perhaps modifying v 6. Examples 5 12 triangulation exists for v 6 1 3 4 triangulation exists for v 6 2 and even John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 8 / 29
Torus triangulations The condition 0 = (6 d)v d is simply d = 6. Analog of Eberhard s Theorem would say 5,7 triangulation for some v 6 Instead, we show there are none John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 9 / 29
Euclidean cone metrics Euclidean cone metrics Definition Triangulation on M induces equilateral metric: each face an equilateral euclidean triangle. Exceptional vertices are cone points Definition Euclidean cone metric on M is locally euclidean away 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 John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 10 / 29
Euclidean cone metrics Regular triangulations on the torus Theorem (cf. Alt73, Neg83, Tho91, DU05, BK06) A triangulation of T 2 with no exceptional vertices is a quotient of the regular triangulation T 0 of the plane, or equivalently a finite cover of the 1-vertex triangulation. Proof. Equilateral metric is flat torus R 2 /Λ. The triangulation lifts to the cover, giving T 0. Thus Λ Λ 0, the triangular lattice. Corollary Any degree-regular triangulation has vertex-transitive symmetry. John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 11 / 29
Euclidean cone metrics Holonomy of a cone metric Definition M o := M cone points h : π 1 (M o ) SO 2 H := h(π 1 ) Lemma For a triangulation, H is a subgroup of C 6 := 2π/6. Proof. As we parallel transport a vector, look at the angle it makes with each edge of the triangulation. John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 12 / 29
Holonomy theorem Holonomy theorem Theorem A torus with two cone points p ± of curvature κ = ±2π/n has holonomy strictly bigger than C n. Corollary There is no 5,7 triangulation of the torus. Proof. Lemma says H contained in C 6 ; theorem says H strictly bigger. John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 13 / 29
Holonomy theorem Proof of Holonomy theorem. Shortest nontrivial geodesic γ avoids p +. If it hits p and splits excess angle 2π/n there, consider holonomy of a pertubation. Otherwise, γ avoids p or makes one angle π there, so slide it to foliate a euclidean cylinder. Complementary digon has two positive angles, so geodesic from p to p within the cylinder does split the excess 2π/n. π π p γ γ p + John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 14 / 29
Holonomy theorem Quadrangulations and hexangulations Theorem The torus T 2 has no 3,5 quadrangulation no bipartite 2,4 hexangulation 2,6 quad 3 2 5 2 quad 2,4 hex 1,5 hex bip 1,5 hex John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 15 / 29
Riemann surfaces Generalizing the holonomy theorem Question Given n > 0 and a euclidean cone metric on T 2 whose curvatures are multiples of 2π/n, when is its holonomy H contained in C n? Curvature as divisor Cone metric induces Riemann surface structure Cone point p i has curvature m i 2π/n Divisor D = m i p i has degree 0 Cone metric gives developing map from universal cover of M o to C. John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 16 / 29
Riemann surfaces Main theorem Theorem Proof. H < C n D principal Consider the n th power of the derivative of the developing map. This is well-defined on M iff H < C n. If so, its divisor is D. Conversely, if D is principal, corresponding meromorphic function is this n th power. Note The case n = 2 is the classical correspondance between meromorphic quadratic differentials and singular flat structrues. John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 17 / 29
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 4761 three-spheres plus 26 finite quotients [Matveev, Shevchishin]: Can smooth to get positive curvature John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 18 / 29
k-point metrics CMC Surfaces CMC Surfaces Definition A coplanar k-unduloid is an Alexandrov-embedded CMC (H 1) surface M with k ends and genus 0, contained in a slab in R 3. Note: each end asymptotic to unduloid John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 19 / 29
k-point metrics CMC Surfaces Classifying map M has mirror symmetry Upper half M + is a topological disk with k boundary curves in mirror plane Conjugate cousin M + is minimal in S 3 with k boundary Hopf great circles Hopf projection gives spherical metric on open disk with k completion boundary points John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 20 / 29
k-point metrics CMC Surfaces Classification Theorem (with Karsten Große-Brauckmann and Rob Kusner) Classifying map is homeomorphism from moduli space of coplanar k-unduloids to space D k of spherical k-point metrics, which is a connected (2k 3) manifold. New work (also with Nick Korevaar) D k = R 2k 3 John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 21 / 29
k-point metrics 2- and 3-point metrics 2-point metrics Universal cover of S 2 {p, q} Bi-infinite chain of slit spheres D 2 = (0, π] John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 22 / 29
k-point metrics 2- and 3-point metrics Triunduloids classified by spherical triples John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 23 / 29
k-point metrics 2- and 3-point metrics 3-point metrics Spherical triangle with three chains of slit spheres D 3 = T3 = B 3 = R 3 C 3 := D 3 /Möb = { } General case We show C k = C k 3, so D k = R 2k 3 John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 24 / 29
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 circles with 3 boundary points Möbius-invariant notion John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 25 / 29
k-point metrics Associahedron Metric trees Theorem The space of planar trees with k ends (labeled in order) length 0 on each midsegment is cone over dual associahedron, homeomorphic to R k 3. a k = 4 gives R a, b 0, at most one positive two rays join to form R a b b John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 26 / 29
k-point metrics Associahedron k = 5 a b a c e d b c d b e (a, b) a Combinatorial trees triangulations of pentagon Five quadrants glue together to form plane John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 27 / 29
k-point metrics Associahedron k = 6 14 generic trees 14 triangulations of hexagon 14 octants fit together to form R 3 John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 28 / 29
k-point metrics Associahedron Complexification C k = C k 3 Medial axis is a tree Midsegment is circles through pq ending 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 Möbius-invariant John M. Sullivan (TU Berlin) Combinatorial and Differential Geometry 2007 July 16 29 / 29