Polyhedral 3-manifolds of 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 R 3 by means of isometric identification of 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 d g 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 d g of non- negative Ricci curvature. Moreover, such d g can be chosen on a uniformly bounded Lipschitz distance and arbitrary close in older C 0 ,s -distance with s < 1 . In other words, any H¨ C − 1 � d ( x,y ) dg ( x,y ) � C with C depending only on ( M,d ) , and ‚ � ε ( d ( x,y )) s for a given s < 1 and ‚ ‚ ‚ d ( x,y ) − d g ( x,y ) ε > 0 1
Remarks. 1. Hamilton’s theorem: Ricci flow on a 3 -manifold ( M,g ) with Ric g � 0 converges to a Riemann metric of constant non-negative sectional curvature. So a manifold M as in the theorem is S 3 , or S 2 × S 1 , or the 3 -torus S 1 × S 1 × S 1 , or a finite non-ramified quotient of those. 2. (J. Sullivan) Let ( M,P j ) be a triangulated 3 -manifold, all P j are tetrahedra. Make each P j regular of edge length 1 . Then the induced polyhedral metric on ( M,P j ) 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 R ℓ ( γ ) := | ˙ 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 � a 2 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 , If a 2 = 0 , then d ∗ is the the that is, of radius R = a − 1 . corresponding chord in a congruent triangle in R 2 . Let ( S 2 ,d ) be a 2 - 5. Theorem. (Alexandrov-Pogorelov). sphere with a metric of Alexandrov curvature K � a 2 . Then there exists an isometric embedding of ( S 2 ,d ) in S 3 R (in R 3 if a = 0 ) of the curvature a 2 . Moreover, the image of ( S 2 ,d ) 2
is the boundary of a convex ball B in S 3 R . Alexandrov case: ( S 2 ,d ) is a polyhedral 2 -sphere of non-negative curvature, K � 0 . b’’ b’’ c’’ c’’ b’ d* b’ d c’ c’ a a 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 R 3 . 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 . There exist polyhedral structures on S 3 with no Warning: essential vertices. On the other hand, if an essential vertex exists, then M is S 3 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. 3
Step I of the proof. There exists a Lipschitz-small polyhedral deformation d ∗ of d and a triangulation of M = ∪ j P j such that all vertices and edges of any tetrahedron P j 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 = ∪ j P j 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 CP 2 . 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 T v M , glued from convex Euclidean cones. Take the unit sphere ST v M in T v M . Then ST v M has curvature K � 1 . By Alexandrov-Pogorelov theorem, ST v M is realized as the boundary of some convex ball B in S 3 . Embed S 3 in R 4 and span the cone with the vertex at origin 0 ∈ R 4 generated by B ⊂ S 3 . I call this cone the Alexandrov cone to M at v and denote AC v M . AC v M is a convex polyhedral cone in R 4 whose Fact. boundary is naturally isometric to the tangent cone T v M . Now take a hyperplane H in R 4 such that Q := H ∩ AC v M 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 4
and the new vertex v ′ . Let U be the union of side faces of V and U ′ the same for V ′ . Then U = T v M ∩ V is a polyhedral neighbourhood of v in T v M , and U ′ is a polyhedral ball, and ∂U and ∂U ′ are isometric. Assume that U is embeddable in ( M,d ) . Replace U by U ′ in ( M,d ) . Folding: 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 P j 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 of ( M,d + ) into convex polyhedra P j , such that d + -essential edges are edges of P j . Afterwards ( M,d + , { P j } ) 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 of the curvature a 2 with the same length of edges. Then P # j P # can be glued together yielding a spherical polyhedral j metric d # on M . For a 2 small enough d # has only “positive” essential edges and curvature K � a 2 . There exists a deformation of d # into a singular Claim. Riemannian metric d g of sectional curvature K g � a 2 − = a 2 − ε ( 0 < ε ≪ a 2 ) with singularities only at vertices v of P # j . Moreover, at each vertex v the corresponding Riemannian 5
metric g is the spherical cone of curvature a 2 − over the sphere ( S 2 ,g ′ ) of curvature K g ′ � 1 , i.e., sin 2( a − R ) g = dR 2 + g ′ . a 2 − I skip the proof and go to Step III of the proof. Smoothing vertices. Use Alexandrov- Pogorelov theorem and embed ( S 2 ,g ′ ) into the sphere S 3 of curvature 1 . Recall that there exists a (smooth) convex set B in S 3 such that ∂B is isometric to ( S 2 ,g ′ ) . In turn, embed S 3 as a round sphere in the 4 -sphere S 4 a Thus S 3 is a of constant curvature a 2 , i.e., of radius a − 1 . √ a − 2 − 1 in S 4 Let v ∈ S 4 geodesic sphere of radius a . a be the center of S 3 . Denote by C a ( v,B ) the spherical cone in S 4 a with the vertex v generated by the set B . The only non-smooth point of C a ( v,B ) is its vertex. To smooth it, take the standard isometric embedding of = R 4 be the tangent hyperplane T v S 4 Let W ∼ S 4 a in R 5 . a and π : S 4 a → W the stereographic projection from the origin 0 ∈ R 5 (defined only on the corresponding hemisphere in S 4 a ). Main property of π : S 4 a → W : geodesics are mapped into geodesics, convexity is preserved. So the construction succeeds as follows. (1) Project C a ( v,B ) onto C := π ( C a ( v,B )) . This is a = R 4 with the unique singular point smooth convex cone in W ∼ v . (2) Deform C into a (everywhere) smooth convex set C ′ , 6
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 ′ ) ⊂ S 4 a , this will be a smooth Take the induced Riemannian metric g ′′ on the convex set. boundary ∂C ′′ . 7
☞✌ ☎✆ ✡☛ ✟✠ ✝✞ �✁ ✂✄ Constructions used in Step II of the proof. Fix a “double cone” domain W in a spherical polyhedral manifold ( M,d ) of curvature K � a 2 as in the picture. The vertices v l , v r and the edge [ v l ,v r ] are essential. x v v r l D W Figure 3: Domain W . The shadowed disk D is equidistant with respect to v l and v r . Fix “left” and “right” polar coordinate r x 0 x R R r θ * l r θ θ l r v l v v r t Figure 4: The left and the right polar coordinates ( R l ,θ l ) and ( R r ,θ r ) and the coordinates ( t,r ) on ∆ . Then the metric has the form g = dR 2 + sin 2( aR ) ´ 2 dφ 2 ” ` α “ dθ 2 + sin 2 ( θ ) +Φ 2 ( R,θ ) dφ 2 a 2 2 π and the first 2 terms the metric on the triangle ∆ . It will be denoted by g ∆ . 8
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