Filling Area I

Filling Area I M. Heuer ([email protected]) June 14, 2011 In this lecture, we will be learning about the filling volume of a Rieman- nian manifold. For a compact surface embedded in R3, this can easily be imagined as filling up the inside of it with water and hence calculating the volume of the resulting 3-dimensional manifold (but with a different metric than the usual one). However, we will use a more general approach by considering n-dimensional manifolds and a so-called filling by a manifold of higher dimension having the surface as its boundary. In particular, we will learn how to calculate the filling volume for spheres. Moreover, we will get to know an estimate of the displacement of a point in relation to the area of the hyperelliptic surface. 1 Preliminaries { As we have seen in the preceding lectures, for a Riemann surface Σ and a hyperelliptic involution J, we have a conformal branched 2-fold covering Q :Σ ! S2 of the sphere S2 2 π { Pu's inequality: systπ1(G) ≤ 2 area(G) { A closed Riemannian surface Σ is called ovalless real if it possesses a fixed point free, antiholomorphic involution τ. 2 Orbifolds Definition 2.1. { An action of a group G on a topological space X is called faithful if there exists an x 2 X for all g 2 G with g 6= e, e being the neutral element, such that g · x 6= x. { A group action of a group G on a topological space X is cocompact if X=G is compact. Seminar \Systolic Geometry\, SS 2011, University of Göttingen 1 2 2 Orbifolds We will now get to know a more general definition of a manifold. Definition 2.2. A smooth n-dimensional orbifold is a pair (X; A) corresponding to a Hausdorff paracompact topological space X called the underlying space and an atlas A which consists of { a covering fUi ⊂ Xji 2 Ig of X such that the open sets Ui are closed under finite intersection; n { a collection of open sets fVi ⊂ R ji 2 Ig; { finite groups of diffeomorphisms fΓiji 2 Ig acting on Vi whereas the Vi are invariant under a faithful action of Γi; { a collection of homeomorphisms 'i : Ui ! Vi=Γi. Furthermore, A has the following properties: { For every inclusion Ui ⊂ Uj there is a smooth embedding ij : Vi ! Vj called a gluing map and a monomorphism fij :Γi ! Γj such that ij is Γi-equivariant, i.e. it commutes with the action of Γi. −1 { The gluing maps are compatible with the atlas, i.e. ij = 'j ◦ 'i. An orbifold is called good if the underlying space X can be obtained by the quotient of a manifold M and a group Γ where the action of Γ is discrete and cocompact. An orbifold is bad if this is not the case. Remark 2.3. If the group Γ is a symmetry operation which does not possess a fixed points, then X is a manifold. If there are one or more fixed point, the result is an orbifold with singular points. Example 2.4. { a manifold with boundary, where the Γi are the trivial groups; 2 { Consider the upper hemisphere of the sphere S . By choosing a point w0 and its corresponding antipodal point, we can fold the boundary together by keeping these two points fixed and gluing the opposite semicircles together. The result is an orbifold which looks like an American football (cf. fig. 1). We will consider the last example in more detail later on. Definition 2.5. A Riemannian metric G on an orbifold (X; A) is the usual metric on XnΣ0 where Σ0 denotes the set of singular points of the orbifold. When G is lifted to the neighborhoods Vi, we obtain a Γi-invariant metric for all Vi; i 2 I and thus for the orbifold (X; A). Definition 2.6. A pullback metric on an orbifold X corresponding to a differentiable map Q : X ! Σ where (Σ; G) is a Riemannian manifold, is defined as ∗ (Q G)(v; w) = G(dQ(v); dQ(w)) = G(Q∗(v);Q∗(w)) 8v; w 2 TpX: When we have a metric H = Q∗G, then Q is called a local isometry. 3 To fill a circle 3 Figure 1: The football as an orbifold 3 To fill a circle If we have a Riemannian manifold N n, we can fill it in by using a manifold Xn+1 which has N as its boundary. The filling volume of N is then defined as the volume of X. Definition 3.1. Let X be a Riemannian manifold with metric G. 1. G is called a complete Riemannian metric if (X; G) is complete. 2.A distance function D is defined by D(p; q) = inf lengthfpiecewise smooth paths between p and qg: If G is complete, then there always exists a geodesic between p and q for all p; q 2 X. Def. and prop. 3.2. Let N be a compact, connected, orientable manifold of dimension n ≥ 1 with a metric d which is not necessarily Riemannian. For n ≥ 2 we have n n+1 FillVol(N ; d) = infgVol(X ; G); where X is an arbitrary fixed manifold satisfying @X = N and G is a complete Rie- mannian metric such that its induced distance function D restricted to the boundary @X is bounded below by d, i.e. Dj@X (p; q) ≤ d(p; q) 8p; q 2 N. Remark 3.3. In particular, for n ≥ 2 the filling volume does not depend on the topology of X with @X = N. Therefore, we can consider the cylinder X = N ×[0; 1), since its boundary is N. Idea for a proof. To show the independence of the filling manifold, we have to take a n long journey. First, we consider a singular n−simplex σn : ∆ ! X. From this we construct a singular complex and consider a singular cycle which we “fill in" with a singular chain for which the boundary is the cycle. We then calculate the volume of the chain. Then this has to be generalized for arbitrary manifolds. For the proof in detail, cf. [Gro1983]. 4 3 To fill a circle The filling volume cannot easily be calculated, hence it is not known for any Rie- mannian metric. However, we have an interesting Conjecture 3.4. 1 FillVol(Sn; can) = Vol(Sn+1; can); 2 where can is the canonical metric on the spheres of curvature +1. We will prove later in corollary 3.8 that the conjecture holds for orientable fillings of genus 0 and 1. Conjecture 3.5. Let Σ be an orientable surface of even genus and G a Riemannian metric on Σ which allows an isometric involution τ that reverses orientation. Then there exists a point p 2 Σ with dist (p; τ(p))2 π G ≤ : area(G) 4 Example 3.6. For the sphere Σ = S2 with the canonical round metric G we have area(G) = 4π (proven in the lecture about the Gauss-Bonnet theorem) and τ = −id, we have distG(x; −x) = π. Theorem 3.7. (Optimal displacement bound). Let (Σ; τ; J) an ovalless real hyperelliptic surface of even genus g and G a Riemannian metric conformal to the original metric. 1. Then there is a point p 2 Σ which satisfies dist (p; τ(p))2 π G ≤ area(G) 4 2. In particular, there exists a curve joining p and τ(p) whose length is at most π 1=2 ( 4 area(G)) . This curve consists of at most g + 1 smooth curves γi. 3. Every γi is a geodesic with respect to the singular constant curvature metric τ◦J τ◦J AF(Σ ; wi), where wi is a Weierstrass point and the set of fixed points Σ of the involution τ ◦ J is a circle. The proof will be discussed in section 5. Corollary 3.8. The filling area conjecture, i.e. conjecture 3.4 in the case n = 1 is satisfied by all orientable fillings of the circle with genus 0 or 1. Proof. In the lecture about the Gauss-Bonnet theorem, we already calculated the area of the 2-sphere, which is 4π. Hence, it has to be shown that FillVol(S1;X)=2π. Firstly, we study the case of a genus 0 surface. We consider the 2-dimensional disc such that points at the boundary, i.e. on the sphere, cannot be connected by a curve of smaller length than in the metric on S1. 3 To fill a circle 5 Figure 2: Fillings of the circle Let's assume the conjecture is false. Then there exists a metric G on D2 with 2 areaG(D ) < 2π. Furthermore, we want that G factorizes to a metric H on RP2 in order to be able to 2 ∼ 2 use Pu's inequality. We have RP = D =∼ where \∼\ identifies antipodal points on the boundary S1, i.e. (x; y) ∼ (−x; −y) 8 (x; y) 2 @D2. Since S1 is one-dimensional, 2 2 2 it does not have any influence on the area of D , hence areaG(D ) = areaH(RP ), where H denotes the quotient metric on RP2. We assume here that the metric G is compatible with the quotient metric (that this is well-defined). 2 2 We regard a loop in RP satisfying length(b) = sysπ1(RP ; H). Let a be a lift of the loop to D2 with length(a) = length(b). Let's analyse at first the special case that a is a lift to the boundary S1 and let G0 denote its induced metric.

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