Polyhedral 3-manifolds of non-negative Alexandrov curvature - - PDF document

Polyhedral 3-manifolds

• f non-negative Alexandrov curvature

Vsevolod Shevchishin joint work with Vladimir Matveev Discrete Differential Geometry, Berlin, 2007 theme of the talk Discrete = ⇒ Differential Geometry Definition / Notation. Polyhedral manifold M is glued from (convex) polyhedra in R3 by means of isometric identification

• f faces.

It has therefore induced metric (distance function) d. Non-negativity of the curvature of such a metric is understood in the sense of Alexandrov, details below. A Riemannian metric dg is a distance function associated with some Riemannian metric tensor g. Theorem 1. Any closed polyhedral manifold (M,d) of non- negative curvature admits a Riemannian metric dg of non- negative Ricci curvature. Moreover, such dg can be chosen on a uniformly bounded Lipschitz distance and arbitrary close in any H¨

• lder C0,s-distance with s < 1. In other words,

C−1 d(x,y)

dg(x,y) C with C depending only on (M,d), and

‚ ‚d(x,y) − dg(x,y) ‚ ‚ ε(d(x,y))s for a given s < 1 and ε > 0

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• Remarks. 1. Hamilton’s theorem: Ricci flow on a 3-manifold

(M,g) with Ricg 0 converges to a Riemann metric of constant non-negative sectional curvature. So a manifold M as in the theorem is S3, or S2×S1, or the 3-torus S1×S1×S1,

• r a finite non-ramified quotient of those.

2. (J. Sullivan) Let (M,Pj) be a triangulated 3-manifold, all Pj are tetrahedra. Make each Pj regular of edge length

• 1. Then the induced polyhedral metric on (M,Pj) has non-

negative curvature ⇔ every edge is contained in 5 simplices (tetrahedra).

• 3. The metric is understood in the sense of length function:

there are “enough many” curves γ(t) in (M,d) for which ℓ(γ) := R | ˙ γ|dt is well-defined. In particular, any 2 points are connected by a geodesic curve realizing the distance. For such curves ℓ([x,z]) = ℓ([x,y]) + ℓ([y,z]) for any inner point y ∈ [x,z]. Such metric spaces are called length spaces.

• 4. Definition.

In view of 3, one has well-defined notions of a triangle and a chord in a triangle. A length space (M,d) has Alexandrov curvature K a2 if for any triangle and any chord d in it one has d d∗ for the corresponding chord in a congruent triangle in the round sphere of the curvature a, that is, of radius R = a−1. If a2 = 0, then d∗ is the the corresponding chord in a congruent triangle in R2.

• 5. Theorem.

(Alexandrov-Pogorelov). Let (S2,d) be a 2- sphere with a metric of Alexandrov curvature K a2. Then there exists an isometric embedding of (S2,d) in S3

R (in R3

if a = 0) of the curvature a2. Moreover, the image of (S2,d)

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is the boundary of a convex ball B in S3

• R. Alexandrov case:

(S2,d) is a polyhedral 2-sphere of non-negative curvature, K 0.

a b’’ b’ c’’ c’ d* a b’’ c’’ c’ b’ d Alexandrov positivity: d* > d

Figure 1: Length comparison and Alexandrov curvature. Definition / Notation. Let (M,d) be a polyhedral 3- manifold. Then the singular locus of d is the set of points where d is not locally isometric to R3. This set is an embedded polyhedral graph of valency 3, possibly not connected and containing closed curves. I call the vertices and edges of Sing(M,d) essential vertices and edges of (M,d). Easy case: There are no essential vertices. Warning: There exist polyhedral structures on S3 with no essential vertices. On the other hand, if an essential vertex exists, then M is S3 or its finite quotient. The result follow from the proofs of Hamilton’s and our theorems. From now on: Assume that an essential vertex exists.

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Step I of the proof. There exists a Lipschitz-small polyhedral deformation d∗ of d and a triangulation of M = ∪jPj such that all vertices and edges of any tetrahedron Pj are essential. Remark 6. The claim is false in higher dimension: Fact 1. (Theorem of Cheeger): If a manifold M of any dimension n admits a triangulation of M = ∪jPj and a polyhedral metric d of non-negative Alexandrov curvature, such that singular locus is n − 2 skeleton (all faces of codim 2 are essential), then M is a rational homology sphere with finite fundamental group. Fact 2. ([Banchoff-K¨ uhnel]) There exists a polyhedral metric d of non-negative Alexandrov curvature on CP2. Construction I. Convex Folding. Take a polyhedral 3- manifold (M,d) of non-negative curvature and an essential vertex v. Then we have a well-defined tangent cone TvM, glued from convex Euclidean cones. Take the unit sphere STvM in TvM. Then STvM has curvature K 1. By Alexandrov-Pogorelov theorem, STvM is realized as the boundary of some convex ball B in S3. Embed S3 in R4 and span the cone with the vertex at origin 0 ∈ R4 generated by B ⊂ S3. I call this cone the Alexandrov cone to M at v and denote ACvM. Fact. ACvM is a convex polyhedral cone in R4 whose boundary is naturally isometric to the tangent cone TvM. Now take a hyperplane H in R4 such that Q := H ∩ACvM is a (convex) bounded set. Consider the pyramid V with the base Q and the vertex v. Move slightly v inside V . Let v′ be the new position of v, and V ′ the pyramid with the same base Q

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and the new vertex v′. Let U be the union of side faces of V and U ′ the same for V ′. Then U = TvM ∩ V is a polyhedral neighbourhood of v in TvM, and U ′ is a polyhedral ball, and ∂U and ∂U ′ are isometric. Assume that U is embeddable in (M,d). Folding: Replace U by U ′ in (M,d). This gives us a new polyhedral metric d′ on M of non-negative Alexandrov curvature, such that all edges of V ′ are essential. We obtain: (1) new essential edges; (2) convex polyhedra Pj in M whose all edges are d′-essential.

• Claim. Repeating the folding construction, we can Lipschitz-

small deform d into a new polyhedral metric d+ of non- negative Alexandrov curvature and construct a decomposition

• f (M,d+) into convex polyhedra Pj, such that d+-essential

edges are edges of Pj. Afterwards (M,d+,{Pj}) can be refined to (M,d∗,{P ∗

j })

with the desired properties. In particular, all P ∗

j are tetrahedra.

Step II of the proof. Smoothing edges. Fix very small a > 0. Replace each flat tetrahedron P ∗

j by a spherical tetrahedron

P #

j

• f the curvature a2 with the same length of edges. Then

P #

j

can be glued together yielding a spherical polyhedral metric d# on M. For a2 small enough d# has only “positive” essential edges and curvature K a2. Claim. There exists a deformation of d# into a singular Riemannian metric dg of sectional curvature Kg a2

− = a2−ε

(0 < ε ≪ a2) with singularities only at vertices v of P #

j .

Moreover, at each vertex v the corresponding Riemannian

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metric g is the spherical cone of curvature a2

− over the sphere

(S2,g′) of curvature Kg′ 1, i.e., g = dR2 +

sin2(a−R) a2 −

g′. I skip the proof and go to Step III of the proof. Smoothing vertices. Use Alexandrov- Pogorelov theorem and embed (S2,g′) into the sphere S3 of curvature 1. Recall that there exists a (smooth) convex set B in S3 such that ∂B is isometric to (S2,g′). In turn, embed S3 as a round sphere in the 4-sphere S4

a

• f constant curvature a2, i.e., of radius a−1.

Thus S3 is a geodesic sphere of radius √ a−2 − 1 in S4

a.

Let v ∈ S4

a be

the center of S3. Denote by Ca(v,B) the spherical cone in S4

a with the vertex v generated by the set B.

The only non-smooth point of Ca(v,B) is its vertex. To smooth it, take the standard isometric embedding of S4

a in R5.

Let W ∼ = R4 be the tangent hyperplane TvS4

a

and π : S4

a → W the stereographic projection from the origin

0 ∈ R5 (defined only on the corresponding hemisphere in S4

a).

Main property of π : S4

a → W : geodesics are mapped into

geodesics, convexity is preserved. So the construction succeeds as follows. (1) Project Ca(v,B) onto C := π(Ca(v,B)). This is a smooth convex cone in W ∼ = R4 with the unique singular point v. (2) Deform C into a (everywhere) smooth convex set C′,

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such that C and C′ coincide outside a sufficiently small neighbourhood of v. Figure 2: Smoothing the metric of a cone (replacing the cone (left) by a smooth ”cap”) (3) Take C′′ := π−1(C′) ⊂ S4

a, this will be a smooth

convex set. Take the induced Riemannian metric g′′ on the boundary ∂C′′.

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Constructions used in Step II of the proof. Fix a “double cone” domain W in a spherical polyhedral manifold (M,d) of curvature K a2 as in the picture. The vertices vl, vr and the edge [vl,vr] are essential.

W

D x v

l

v

r

Figure 3: Domain W . The shadowed disk D is equidistant with respect to vl and vr. Fix “left” and “right” polar coordinate

x x0

vl v

r

vr

t R R

r

θ

l

r

θl

r

θ*

Figure 4: The left and the right polar coordinates (Rl,θl) and (Rr,θr) and the coordinates (t,r) on ∆. Then the metric has the form g = dR2+sin2(aR)

a2

“ dθ2 + sin 2(θ) ` α

2π

´2 dφ2” +Φ2(R,θ)dφ2 and the first 2 terms the metric on the triangle ∆. It will be denoted by g∆.

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Substep II.1. Adjusting the metric g∆. Let θ∗ be the side angle of ∆ and r∗ its height. Fix sufficiently small ε > 0 and find functions ˜ θ(θ) χ(t) as on the figure below.

’

/3

*

θ /3

*

θ /3

*

θ /3

*

θ

ε

1

θ θ θ(θ) ~ 2 2

*

θ

*

θ θ(θ) ~

χ

1

* *

/10 /10

r r −

Figure 5: The functions ˜ θε(θ), ˜ θ′

ε(θ), χ(t).

Stretch left and right corners by diffeomorphisms ml, mr.

θ ε θ ε θ ε vl vl vr vr

*

x θ ε x θ

* * * *

/3− /3+

/3−

/3 /3

m l

Figure 6: The diffeomorphims ml is identity in the sector θ 2θ∗/3, stretches the θ-coordinate in the sector 2θ∗/3 θ θ∗/3, and rotate by ε the sector θ θ∗/3. Then match the obtained left and right metrics gε,l,gε,r on ∆ using χ(t) introduced above. Main property of the obtained

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metric gε on ∆: Positive curvature Kgε a2

− with a− ≈ a,

and gε is still metric of constant curvature a2 near [vl,vr]. Conditions on Φ(R,θ). The curvature condition Kg a2 for g of the form g = g∆(R,θ) + Φ2(R,θ)dφ2 are: Kg∆ a2 and ∇2Φ + a2 Φg∆ 0 As above, we construct “left” and “right” pieces of new Φ and then match them together. For local pieces we use Ψ(θ) (independent of R) related to Φ by Φ(R,θ) = sin(aR) a · Ψ(θ). To construct Ψ(θ) we start with Ψ1(θ) = sin(θ) Ψ2(θ) :=

α∗ 2πsin(θ + ε) and bend one into another near the intersection

point of graphs. Moreover, we have the following convex integration property: the derivative of Ψǫ is always between derivatives of Ψ1 and Ψ2.

θ

Ψ Ψ (θ) Ψ (θ)

1 2

ε ϑ(ε) 2ϑ(ε)

Figure 7: Construction of Ψε(θ). The function Ψε(θ) coincides with Ψ1(θ) for θ < ϑ(ε)−Cε2 and with Ψ2(θ) for θ > ϑ(ε) + Cε2.

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Having constructed Ψǫ, we obtain “left” and “right” pieces Φl(Rl,θl) and Φr(Rr,θr) substituting the “left” and “right” coordinate systems (Rl,θl) and (Rr,θr). Both live on ∆ and satisfy the concavity condition ∇2Φ + a2 Φgε 0. Moreover, they coincide in a neighbourhood of the edge [vl,vr]. Final argument: The function Φmin(x) := min{Φl(x),Φr(x)} satisfy the same concavity condition in a weak sense and is smooth in the same neighbourhood of the edge [vl,vr]. We show explicitly that it can be smoothed into a new function Φδ preserving the needed concavity condition. The main idea is essentially the convex integration as above: The derivative ∇Φδ(x) lies in C · δ-neighbourhood of the interval connecting the derivatives of ∇Φl(x) and ∇Φr(x).

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