Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume

Annals of Mathematics

Euclidean Structures on Simplicial Surfaces and Hyperbolic VolumeAuthor(s): Igor RivinSource: Annals of Mathematics, Second Series, Vol. 139, No. 3 (May, 1994), pp. 553-580Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2118572 .

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Annals of Mathematics, 139 (1994), 553-580

Euclidean structures on simplicial surfaces

and hyperbolic volume By IGOR RIVIN*

Introduction

In this paper I develop a deformation theory of singular Euclidean structures on finite 2-dimensional simplicial oriented complexes, homeomorphic to surfaces with or without boundary. For a given complex T the deformation space can be parametrized by dihedral angles (see Definition 3.4), which are simple linear functions of the angles of the faces of T.

This theory yields a number of results in, seemingly, rather different areas of 2- and 3-dimensional geometry and topology. A partial list of these results follows:

* As parametrized by the dihedral angles, the space of convex (with dihedral angles not exceeding ir) Euclidean structures with cone-like singularities on a complex T, as above, has the structure of a precompact convex polytope &con (T) in a finite-dimensional Euclidean space (Theorem 6.1). This is proved via a variational argument.

* A parametrization of similarity structures with cone-like singularities is given on an oriented surface S, and a canonical association of a singular Eu- clidean structure (determined up to scaling; this indeterminacy can be removed by normalizing the area) to a similarity structure, hence a fibering of the space of singular similarity structures over the space of singular Euclidean structures. See Section 10.

* If there exists an affine structure with prescribed convex dihedral angles on T, then there exists one with any prescribed holonomy representation of the fundamental group of T\V(T). See Sections 7 and 8.

*Previous versions of this paper have been circulated as a preprint titled "Some applications of the hyperbolic volume formula of Lobachevsky and Milnor". The author would like to thank the referee for his patience, and for suggestions, which considerably improved the exposition. The work presented in this paper has been partially supported by the Minnesota Geometry Center under NSF grant #DMS-8506637 and some was supported by the Institute for Advanced Study under NSF Grant #DMS-9304580.



554 IGOR RIVIN

* There exists a Euclidean structure on a surface with prescribed cone angles satisfying the Euler characteristic relation. This result was known before for closed surfaces. See Section 11.

* A convex subset of E(T) parametrizes the embeddings of T which are the Delaunay tessellations of their vertex sets, and also provides a new combinatorial characterization of possible combinatorial structures of the complexes for which such embeddings are possible. The previous results in this direction (also due to the author) are seemingly quite different in flavor. See Section 13.

* For E(T), or any convex subset C', we can define a canonical structure in C', which has the maximal volume (see results on hyperbolic polyhedra below). It is shown that such a structure is unique and maximally symmetric. In particular there always exists a maximally symmetric Delaunay embedding of a planar T. See Section 12.

* If there exists an affine structure with prescribed convex dihedral angles on T, then there exists one with any prescribed holonomy representation of the fundamental group of T\V(T). See Sections 7 and 8.

By associating an ideal simplex in H3 to each triangle of T, we obtain a number of results concerning ideal polyhedra in H3. The two classes of polyhedra considered in this paper are star-shaped (with respect to one of their vertices) polyhedra-this class will be denoted by Pstar, and convex polyhedra (class Pcon C P~t=). The dihedral angles mentioned above correspond precisely to the hyperbolic dihedral angles.

* A polyhedron in 'Pt is uniquely determined by its dihedral angles. This uniqueness result was announced in t4] (though only for convex polyhedra).

* The set of polyhedra with prescribed combinatorics T in Pstar is parametrized by the dihedral angles. The set Pcon C Pstar has the structure of a convex polytope. See Sections 1 and 14.

* There exists an effective combinatorial description of convex ideal polyhedra in H3-this is a paraphrase of the statement about Delaunay tessellations above.

* The volume of star-shaped (and thus also convex) polyhedra is a convex function of the dihedral angles. See Sections 2, 12, 14.

o There is a unique convex polyhedron combinatorially equivalent to T of maximal volume.

The maximal volume polyhedron is maximally symmetric. In particular, the ideal platonic solids are of maximal volume among convex polyhedra of their combinatorical class. This was previously known only for the simplex. See Section 14.



EUCLIDEAN STRUCTURES AND VOLUME 555

* In Section 14 I give another geometric description of maximal volume polyhedra. This uses a volume variation formula of Milnor (Theorem 14.5), generalizing a classical result of Schlifli. As I am unaware of any published proof of Milnor's formula, one is supplied here.

The results of Section 14 also give interesting existence and uniqueness results on tangential arrangements of spheres in E3 with prescribed combinatorics.

1. Geometry of ideal polyhedra in H3

In the upper halfspace model, H3 = {(x, y, z) E R3 I z > 0}. The sphere at infinity, S2 of H3, is represented as the Riemann sphere C, where the finite complex plane is the plane P = {(x, y, z) E R3 I z = 0}.

Geodesic planes in H3 are then hemispheres centered on the sphere at infinity, except that those planes passing through the point cc of S2 are vertical half-planes.

Consider an ideal polyhedron P, such that one vertex vC'> of P is at the point oo. Assume that P is star-shaped with respect to vc. Furthermore, assume that P has triangular faces-extra edges (with dihedral angles ir) may need to be added to achieve this. Finally, consider the image of P under the projection map II: H3 -A C, such that H(x, y, z) = x + iy. Edges of P not passing through vo,, then project down to straight line segments in C, and the faces of P not passing through vo, project down to triangles in C.

Remark 3.10 of [6} states the following:

THEOREM 1.1. Consider an immersed polygon Q in the plane, with the interior of Q triangulated in such a way that edges of Q are edges of the triangulation T(Q) (in other words, there are no vertices of the triangulation on the edges of Q). Then T(Q) is the projection of an ideal polyhedron Q onto the plane C at infinity of H3, such that: (1) Q has vertices at the vertices of T(Q), plus one vertex at the point cc

of C. (2) Q is similar to the link of the vertex VOO of Q at oo. (3) Q is star-shaped with respect to V.. (4) If v1, -.. ,Vk are the vertices of Q, then the dihedral angle of Q corre-

sponding to the edge VkVCX) is the Euclidean angle of Q at Vk. (5) The dihedral angle of Q corresponding to the boundary edge vivi+1 is

equal to the Euclidean angle at the third vertex w of the (unique) triangle wuvivi+i of T(Q) containing the edge vivi+i.



556 IGOR RIVIN

(6) The dihedral angle of Q corresponding to a non-boundary edge AB of T(Q) is equal to the sum of the angles at C and D of the two triangles ACB and ADB abutting along the edge AB.

Note. It is necessary to explain what is meant by immersed polygon Q in the statement of Theorem 1.1. Let R be a two-dimensional Riemannian manifold with boundary. Furthermore, assume that R is locally isometric to E2, and that the boundary of R is piecewise-geodesic. Then Q is the image of R under its developing map D into E2. This map can be explained in terms of a geodesic triangulation of R. Consider two adjacent triangles ABC and ABD in R. Note that D(ABC) is uniquely determined (up to ambient isometry of E2), and D(ABD) is determined by D(ABC) and the requirement that D be a local isometry of a neighborhood of the edge AB. In simpler terms, the requirement is that there should be no fold along the edge AB.

For general R, the developing map D will be multiple-valued, but in the special case where R is homeomorphic to the disk D2, D is single-valued, and is an immersion (that is, D is not necessarily globally 1-to-I). If, in addition, R is geodesically convex (or, at least, star-shaped with respect to some point p), then D is actually an embedding. This is so because in E2 there is a unique line between any pair of points.

The observations in Theorem 1.1 can all be proved from first principles using the conformality of the half-space model. A particularly easy way to see them is to use the following:

Fact. Consider an ideal simplex A with vertices A, B, C, oo. Then the dihedral angles at the edges Aoo, Boo and Coo are equal to the dihedral angles at the edges BC, AC, AB, respectively.

Now consider simplices ABCoo and ABDoo. It is a matter of arithmetic that the dihedral angle at AB is equal to the sum of angles at C and D, and the rest of Theorem 1.1 follows easily.

2. The ideal simplex

Consider an ideal simplex S in H3. In the upper halfspace model of H3, let the vertices of S be at points A, B, C, oo of C, and let the angles at A, B, and C be a,,f, and -y, respectively. According to Milnor's formula (see [51), the volume of S can be expressed in terms of the angles of ABC as follows:

(2.1) V(S) = V(a, /, 7) = L(a) + L(3) + L(?y),




where L(x) is the Lobachevsky function

L(x) = log(2 sin 0)dO. 0

Note. Milnor's integrand is log 12 sin ~1, but for 0 E [0, ir] the two forms agree.

The set of possible angles of nondegenerate Euclidean triangles is the following open triangle in R3:

(2.2) T={(x1,x2,x3) E R3 xl >O,x2 >O,X3 >O,xl+x2?+X3 =r}.

THEOREM 2.1. The function V is concave down on T.

LEMMA 2.2. Let a and 3 be such that 0 < a, 0 < 3 and a +L3 < ir. Then cot (a) + cot (3) > 0.

Proof. Construct a triangle ABC, with angles a, /3, and 7 = r - a - 3. If the length of the altitude from the vertex C onto AB is h, then the length of AB is h(cot a + cot/3). 1I

Proof of Theorem 2.1. By formula (2.1), the Hessian matrix of V at the point a,/3, -y has the form

-cota 0 0 (2.3) H(a7, 03Y)= 0 -cot/3 0

n O 0 - cot Y

The statement of the theorem is equivalent to the statement that H defines a negative-definite quadratic form on the tangent space of T at all (a,,3<y) E T. The vectors vj = (1, 0,-I) and v2 = (0,1 ,-1) form a basis of the tangent space of T (at every point), and it is easy to verify that

v Hvi = - (cot a + cot-y),

vjTHv2 = - (cot,3 + coty),

vT Hv2 =vTHv1 =-cot-y.

To show that the matrix

(a, 3, 7) =H I <vz,v2> (or, }, 7) ( cot a + cot ) -cot ) (-(ctcot'Y) - cot'y+ ot

is negative-definite on T, observe first that by Lemma 2.2, both the diagonal elements of M are negative, hence

(2.4) trM < 0.



558 IGOR RIVIN

On the other hand,

det M(a,,3, -y) = cot a cot,3 + coty(cot a + cot,3).

Since y = - a -/3,

cot a cot13 - 1 cot y -cot(a +13) -cot a + cot

Thus

(2.5) det Mt(a,,3, ay) = 1,

for (a,1, -y) E T. The conclusion of the theorem follows from equations 2.4 and 2.5. E1

3. Singular Euclidean structures on surfaces.

Let T be an oriented finite simplicial complex homeomorphic to a surface (possibly with boundary). As usual, the sets of faces, edges, and vertices of T will be denoted by F (T), E (T), V (T), respectively. Individual faces, edges, and vertices will be denoted by f, e, and v, respectively.

A natural way to think of T is to imagine that each face of T is a Euclidean triangle, and adjacent triangles are glued together by an isometric identifica- tion of corresponding edges. The shapes (that is, similarity classes) of triangles are determined by their angles, so the following is a natural question:

What are the assignments of angles which allow a gluing as described above?

Definition 3.1. A locally Euclidean structure A on T is the assignment of positive angles a(t), /3(t), -y(t) to each triangle t of T, such that a(t) + 3(t) + y(t) = W.

A locally Euclidean structure is not sufficient for gluing, as demonstrated in the following example:

Example. Consider the simple "wheel" complex W shown in Figure 1. Now consider the following angle assignment: In triangle ABE: A = i/2, B = ir/4, E = ir/4. In triangle BCE: B = r/2, C = ir/4, E = -r/4. In triangle CDE: C = w/2, D = ir/4, E = or/4. In triangle DAE: D = 7r/2, A = 7r/4, E = ir/4. Suppose we set the length of the side AE to be 1 unit. Then the length

of BE will be X/2 units, the length of CE will be 2 units, the length of DE 2V/ units and finally the length of AE will be 4 $& 1 units.




A

D E B

C

FIGURE 1

Thus, a locally Euclidean structure defines a holonomy representation of H: 7ri(T\V(T)) -* GL2(R), and this structure corresponds to a gluing if and only if the image of H actually lies in S02(R). The holonomy H is explicitly computed below in terms of the angles of faces of T.

Definition 3.2. The set of all locally Euclidean structures on T will be denoted by A(T).

Remark 3.3. Throughout this paper A(T) is viewed as a subset of the linear space R3IF(T}I of angles of T. When viewed thus, it the solution set of a system of linear equations and inequalities, hence a convex polytope. This polytope has dimension 21F(T) , since there is one linear equation for each face of T. Alternately A(T) is the product of IF(Tfl copies of the triangle T described in equation (2.2) of Section 2. Then R3IF(T)I is viewed as the product of [F(T)[ copies of R3. Since all of the coordinates of points of A(T) are between 0 and ir, its closure is compact.

A complex T has two kinds of edges-internal edges and boundary edges. It will be convenient to add an extra vertex to T-the point at infinity. This vertex will be denoted by the symbol oo and will be viewed as connected to all of the boundary vertices of G. The following definition is directly modeled on Theorem 1.1.

Definition 3.4. Let A be a locally Euclidean structure on G. To each edge AB of G associate a dihedral angle A(AB) as follows:

If AB is an interior edge of T, then there are faces ABC and ABD of T, and A(AB) is the sum of the angle D of ABD and the angle C of ABC.

If AB is a boundary edge of T, then there is a face ABC of T and Zi(AB) is the angle C of ABC.

If B = oo, then A(AB) is equal to the sum of the face angles A of the faces of T incident to A.



560 IGOR RIVIN

Definition 3.5. The set of locally Euclidean structures on T with no dihedral angles exceeding ir will be denoted by An (T). This set is a convex subset of A(T).

Definition 3.6. The set of locally Euclidean structures on T with prescribed dihedral angles A: E(T) -* (0, 2ir) will be denoted by A(T, A) c A(T).

4. Vertex singularities

Definition 4.1. The cone angle C(A, v) of A at a vertex v of T is defined to be the sum of the angles of A incident to v. The vertex v is considered non-singular if CQA, v) = 27r.

It turns out that there is an alternative way of computing the cone angles:

LEMMA 4.2.

(4.1) C(A,v) = - (-(e)) e incident to v

Proof. This is a simple matter of arithmetic: The sum S of all the angles in all the triangles incident to v is equal to n7r, where n is the valence of v. On the other hand, every angle in a triangle incident to v contributes to either a dihedral angle at an edge incident to v or to C(A, v), but not both. Li

Remark 4.3. For a boundary vertex V of T, the above argument shows that the dihedral angle A(Voo), which is defined to be the sum of the angles incident to V, can be linearly expressed in terms of the dihedral angles at edges of T incident to V, to wit:

(4.2) 7r + A(voo) = (r -(e)) e incident to v

This allows us to drop any mention of the vertex at infinity from future discussion.

5. The space of dihedral angle assignments

Consider the collection of possible assignments of dihedral angles to edges of T. This collection E(T) will henceforth be viewed as a subset of the linear space RIE(T)L. The map A constructed in Definitions 3.4 and 3.6 is the restric- tion of the linear map (abusing notation) A: R31F(T)l * RJE(T)t, described as follows:




If the basis of R3IF(T)i is e1,... , e3jF(T)j (where each basis element corresponds to an angle of a face of T), and the basis of RIE(T)1 is fl, ... , fIE(T)I, then A(ei) = fj, where the edge corresponding to fj is opposite to the angle corresponding to ei-that is, if ei corresponds to the angle A of ABC, fj corresponds to the edge BC.

With these conventions, the set ?(T) of possible dihedral angle assignments on E(T) is simply A(A(T)), and hence a convex polytope in RIE(T)I; thus, we have:

THEOREM 5.1. The set of ?(T) of dihedral angle assignments of T has the natural structure of the interior of a convex polytope (not of full dimension) in RE(T)

Definition 5.2. The subset of ?(T) corresponding to dihedral angle assignments such that all the dihedral angles are no greater than 7r will be denoted by &cn(T).

6. Volume and finding the complete structure

The main theorem of this section is the following (with notation as in the previous section):

THEOREM 6.1. Each dihedral angle assignment in ?(T) corresponds to exactly one singular Euclidean structure on T.

Proof. This is a convenient restatement of Theorem 6.9. 0l

Definition 6.2. The volume V(A) of the locally Euclidean structure A on T is the sum of hyperbolic volumes of ideal simplices corresponding to the triangles of T with the angles assigned to them by A.

PROPOSITION 6.3. Volume is a concave down function on A.

Proof. This is an immediate consequence of Theorem 2.1. El

Definition 6.4. Consider a triangle t = ABC, such that the edge AC is clockwise from the edge AB with respect to the orientation of T. The dilatation of t with respect to the vertex A is the quantity D(t, A) = log IACI - log IABI.

LEMMA 6.5. Let the angles of t = ABC be a,,3,y, respectively. Then D (tA) = log sin/3 - log sin a.

Definition 6.6. Let c* = {t1, t2,... tk = tl} be an ordered collection of triangles of T, such that ti and tj share an edge if [i -GI = 1 Let Ai = ti-1 n ti n ti+j. Then define the holonomy H(c*, A) = >kj 1

D(tiI Ai).



562 IGOR RIVIN

The notation c* is meant to emphasize that c* is Poincare-dual to a cycle.

Definition 6.7. A structure A is complete, if its holonomy vanishes on all dual cycles.

The reason for the above definition is the theorem below:

THEOREM 6.8. There exists a Euclidean metric MA on T (with cone-like singularities at vertices of T), with angles prescribed by A if and only if A is complete in the sense of Definition 6.7.

Proof. First, assume A is complete. Pick a base face to of T, and choose the size of to. For any other face t, pick a chain of faces a* = {to, ... ,tk-t , such that tj and tj+j share the edge ej. This will fix a size for t, and the vanishing of the holonomy will assure that this size will be independent of the choice of the chain a*. Conversely, if there is a Euclidean metric MA with cone-like singularities corresponding to A, the ratios of the lengths of edges are independent of the co-chain connecting them, and hence A is complete. E

The main result of this section is the following theorem.

THEOREM 6.9. Let A be such that A(T,A) C Acon is nonempty. Then there exists a unique complete AA E A(T,A).

Proof. The statement of the theorem is a combination of the statements of Theorem 6.16 and Theorem 6.13, which are proved below. D

THEOREM 6.10. If A(T,A) C A~on is nonempty, then the function V achieves its maximum on A(T,A).

Proof. Let C = A(T, A) be the closure of A(T, A). The polytope C is compact, and V is continuous on C. To show that V achieves its maximum on the interior of C requires two observations:

Observation 1. The support planes of the boundary of C are given by the intersections of the planes a = 0, with the linear subspace E(A) defined by the dihedral angle equations (where a is some angle of T). Denote the outward-pointing normal to that plane by nal!. L

Observation 2. The function log(2 sin x) is continuous in (0, 7r), and goes to -oo at both 0 and ir.

Observation 3. The volume V(a,3 j, y) of the ideal simplex decreases as (a,l3, -y) approaches (0, 0, 7r), or some permutation thereof. This can be shown by a simple computation.




Consider a point p E &C, and a point q in the interior of C. It will be shown that for sufficiently small t, V(p + t(q - p)) > V(p). In coordinates, p = (al(p),... an(P), while q = (aj(q),...,a1n(q)). Since p is a boundary point, possibly after we reorder the coordinates,

al (P) = =aio (P) = 0,

and also

aio+j(P) = **= (P) = ir.

As is seen by Lemma 6.11, and Observation 3 above, this last set of equalities is empty. Further,

0 < aio+l(p) . . .,an(P) <ir..

Let M(p) = 1, if io = n, and otherwise

M(p) = mi (min(ji (p), lr-&ai (p))). io<i<n

On the other hand,

(6.1) ? < aleq), *, antq) < try

Note in particular that ai(q) > ai(p), when i < io. Now let v = q - p, and let qt = p + tv. Furthermore, let f(t) = V(qt). When t > 0, it is seen that

'(t) = Ad 30 (&j(q) - ej(p)). i=1

a t

By the mean value theorem, V(qt) - V(p) = f(t) - f(0) = f'((), where o <j < t. Assume that t < to = M(p)/(211v11). Then,

f'(C) > Siav | (ai(q) - ai(P)) - (n - io) log 2 sin 2P)

From Observation 2 and the comment following equation 6.1 it follows that for 1 < i < io, a ti can always be picked so that if 0 < C < tjI

01'i'> (ai(q)- ai(p) >(n-io) log(2sin MP))

while T01 (ai(q) - ai(p)) > 0.

Let t = min(to, . .. , tir). By the above argument, f'() < 0, when 0 < C < t, and thus V(qt) = f(t) > f(0) = V(p).



564 IGOR RIVIN

Denote the site of the maximum found above by

M = (a1 (M), aM2(M).... * Cf3IF(T)I (M))

LEMMA 6.11. Let A E Az0n(T) have dihedral angles A(A), such that A(TA(A)) has nonempty interior. Then, either no angle of any triangle of T equals ir, or there exists a triangle T, to which A assigns the angles 0, a, r- a, with a > 0.

Proof. The result is arrived at by contradiction. Suppose the triangle ABC has angle A equal to 7r. Then there are two possibilities:

Firstly, BC could be an edge of the boundary of T. That is impossible, since then the angle at A would have to be ir, contradicting the hypothesis that A(T, A(A)) has nonempty interior.

Secondly, BC is an interior edge of T, so there is a triangle BCD abutting ABC. The angle of BCD at D has to be 0, since D + A < r, hence BCD is the desired triangle T, or the angles of BCD at B and C must equal 0. Assuming T is never found, evidently a sequence S of triangles T1 = ABC, T2 = BCD, ... , Tk ... is obtained. Since, by the observation above, no Tj can be incident to the boundary of T, and since T is a finite complex, it must be true that S contains a minimal cycle (where, without loss of generality, T1 = T1). The sum of the angles of T1,... , T1 equals Iir, and since it is the sum of dihedral angles at the edges ej = Tjfl T(il)mod li it must then be true that every angle of T1 (to name one) must be opposite to some ek, since the sum of the angles not having that property must be 0. However, only el and el are incident to T1, so a contradiction is reached. E

Now, view A(T, A) as a (linear) surface in the space R3F(T) of angles of T. Recall that A(T, A) is defined (away from its frontier) by the equations Eqt: a(t) + ,3(t) + -y(t) = ir, where t ranges over the faces of T and also the equations Eqs: a(e) + ,3(e) = A(e), where e ranges over the edges of T, and a and 3 are the two angles opposite to e.

At the extremal point M of V on A(T, A), the gradient of V is orthogonal to A(T, A) (this is the principle of "Lagrange multipliers"), hence can be written as a linear combination of the normal vectors of the defining planes of A(T, A):

(6.2) gradV= E C(t) Eqi + E C(e) Eqej . tEF(T) eEE(T)

For an angle a, let t(a) be the face of T containing Oa and let e(a) be the edge of T opposite to (x. By equation (6.2):

(6.3) log(2 sin ai(M)) = A | = C(t(aj)) + C(e(ai))




LEMMA 6.12. At M the holonomy H(c*,A) vanishes for any dual cycle c* of T.

Proof. Let c* = {ti, t2, ... , tk}, as in Definition 6.6. Also as in Definition 6.6, let Ai = ti-1 n ti n ti+j. Then, if ei = ti n ti-1 and ei+l = ti n ti+1, it follows from equation 6.3 that D(ti, Ai) = C(ei) - C(ei+i). The statement of the lemma follows. f

THEOREM 6.13. There exists a complete structure on A(T,A) as long as A(T,A) is nonempty.

Proof. This follows from Theorem 6.10 and Lemma 6.12. E

In order to show that the complete structure is unique, it is sufficient to show that any complete structure is a critical point of the volume on A(T, A).

Definition 6.14. Let S map corners of triangles of T -k R. If t = ABC, and the edge AC is clockwise from the edge AB, let Ds(t, A) = S(B) - S(A). The holonomy Hs(c*) of S with respect to a dual cycle c* is defined as in Definition 6.6, with Ds in place of D.

LEMMA 6.15. Let S, as in Definition 6.14, be such that the holonomy Hs(c*) vanishes for every dual cycle c*. Then, there exist functions C1: F(T) -- R and C2: E(T) - R., such that (in a triangle t = ABC) S(A) =

Ci(t) + C2(BC).

Proof. It is sufficient to construct the map C1. Let ABC and ABD be two abutting triangles of T. It must be true that Cl(ABC) - Ci(ABD) = S(A) - S(D). It is clear that the holonomy condition on S allows one to construct C1 satisfying this relation. E

THEOREM 6.16. There exists at most one complete structure on A(T,A).

Proof. Let S(A) = log(2sina), and apply Lemma 6.15 to S. It follows that, by Lagrange multipliers, any complete structure corresponds to a critical point of V on A (T, A). Since V is convex on A(T) , there is at most one such critical point. El

7. Incomplete structures with prescribed holonomy

In this section, the methods of the previous sections are generalized some- what to study locally Euclidean structures with prescribed, not necessarily trivial, holonomy.



566 IGOR RIVIN

Definition 7.1. Let c* = {t1, t2,.. . , tk = t1} be an ordered collection of triangles of T, such that ti and tj share an edge if ji - ji = 1. Let Ai = ti-1 n ti f ti+1. Then define the holonomy Hs(-c*) = Ekj-1 Ds(ti, Ai).

Recall also the ordinary holonomy H(c*, A) of a locally Euclidean structure A around a cocycle c* is defined as in Definition 7.1, but with the function S, such that S(A) = log sin a, where A is a corner of a triangle and a is the angle at that corner.

The main result of this section will be the following:

THEOREM 7.2. Let A(T,L) C Acon(T) be the set of convex locally Eu- clidean structures on T with prescribed dihedral angles A, and assume that A(T,A) is nonempty. Furthermore, let S be a function as in Definition 6.14. Then, there exists a unique A E A(T-,A), such that Hs(c*) = H(c*,A) for all cocycles c* of T.

The proof of Theorem 7.2 is very similar to that of Theorem 6.9. First, define the modified volume functional Vs on A(T).

Definition 7.3. For a given function S mapping corners of triangles of T -+ R, define

Vs: A(T)-+ R = V- S(A)a(A), AE corners of triangles of T

where a(A) is the angle assigned to A by the structure in A(T).

Since Vs - V is a linear function of the angles, it follows that for any S, Vs is a strictly concave function (just like V), and, also like V, Vs restricted to any nonempty A(T, A) has a critical point As (A) (since the proof of Theorem 6.10 goes through essentially unchanged).

THEOREM 7.4. A structure AO E A(T,A) is a critical point of Vs if and only if H(c*,Ao) = Hs(c*) for every cocycle c* of T.

Proof. First, observe that

a (A) = log sin a(A) - S(A).

Define Hs(c*) = H(c*, A) - Hs(c*). The arguments used to prove Lemma 6.12 and Lemma 6.15 go through unchanged, to show that AO is a critical point of Vs on A(T, A) if and only if Hs is trivial, which is equivalent to the statement of the theorem. a




Proof of Theorem 7.2. It follows that the structure As(A) maximizing Vs on A(T, A) is the unique structure whose existence is postulated in the statement of Theorem 7.2, as shown by Theorem 7.4 and the preceding discussion. E

It should be noted that Theorem 7.2 gives A(T, A) a natural linear structure. Indeed, holonomies are elements of the dual C* of the space C* generated by cocycles, and the set 1- of possible holonomies is the image of the linear map f from the space S = {S: corners of triangles of T -+ R} to C*, where f(S)(c*) = Hs(c*).

8. The structure of A(T)

In this section it will be assumed for simplicity that T is a closed (and, as always, oriented) surface, though the extension to surfaces with boundary is quite straightforward.

Indeed, consider a structure AO E A(T), and consider the tangent space T = TA0(A(T)). The space T has dimension 21F(T)I, and splits as follows:

(8.1) T = TAO (A(T, Ao)) ?D TAo (e(T)),

where Ao = A (Ao). The space TA0(?(T)) can be viewed as a linear subspace of RIE(T)I. As

such, it has codimension at least one, since the sum of all dihedral angles is equal to the sum of the angles of all the faces, which, in turn, is equal to wrIF(T)I. Therefore,

(8.2) dimTA0(&(T)) < IE(T)I - 1.

On the other hand, by the results of Section 7, the space TAO(A(T, AO)) can be identified with a linear subspace of the tangent space of the space 1? of representations of wr, (T\V(T)) into R, which has dimension IV(T) I + 2g - 1, and so

(8.3) dimTAO(A(T, Ao)) < IV(T)I + 2g -1.

Combining the splitting (8.1) with inequalities (8.2), and (8.3), we see that

(8.4) 21F(T)l = dimT > IE(T)I + IV(T)I + 2g-2.

It is easily seen that IE(T)I = 'IF(T)I, and so by Euler's formula it follows that IV(T) = I F(T)I + 2- 2g, and thus equality holds in (8.4), and, consequently in (8.2) and (8.3) also. Therefore, the results of Section 7 and of the preceding discussion imply the following theorem:



568 IGOR RIVIN

THEOREM 8.1. Any representation R of wri(T\V(T)) into R can be re- alized as the holonomy of a set AR of locally Euclidean structures. The set AR is (IE(T)I - l)-dimensional. The subset A.R C AR of structures with all dihedral angles not exceeding wr can be naturally identified with the polytope Scon (f)

9. Symmetries of complete structures

Consider the group G = Aut(T) of combinatorial automorphisms of T. The group G can be seen to act on A(T) in the way described below.

An automorphism a E G is completely determined by its action on the corners of triangles of T, which is the same as saying that it is determined by its action on the 0-skeleton and the 2-skeleton of T. This observation induces a faithful representation a --+ of G into GL(R3IF(T)I), as follows: Let ABC be a face of T, and c(ABC) = DEF, so that C(A) = D, C(B) = E, a(C) = F. Then, 3(e(ABC,A)) = e(DEFE), and so on, where e(t~v) is the basis vector corresponding to the corner v of t. Defined thus, it is seen that C acts on A(T).

In the sequel, MA denotes the singular Euclidean metric on T corresponding to the (complete) structure A, as in Theorem 6.8.

THEOREM 9.1. Let A E Aco, (T) be a complete convex structure, and let H be a subgroup of G = Aut (T), such that H fixes A. Then H acts on MA by isometrics.

Proof. Let TH = H\T be the quotient of T by H, and let AH be the structure in A(TH) induced by A. There exists a complete structure AH E A(TH), with the same dihedral angles as AH, by Theorem 6.1. Also, AH lifts to a structure A on T. Now, note that V(AH) > V(Ah), and also note that V(A) = IHIV(AH). Consequently,

(9.1) V(A) = IHIV(AH) > IHIV(AH) = V(A).

Since the dihedral angles of A are the same as those of A, and A is complete, it follows from the results of Section 6 that A = A, and the statement of the theorem follows. L

10. Singular Euclidean and similarity structures

In this section, some of the preceding results, together with Theorem 10.1 (whose proof is not difficult, and is contained in [9]), will be used to study singular Euclidean and similarity structures on surfaces.




THEOREM 10.1. Let (S,g) be a surface with a singularity structure with cone-like singularities, with the set of marked points P = {P1, . ,Pk}, which includes the set of cone points of (S,g). Then there exists a geodesic triangulation T (a Delaunay triangulation) of S, such that the vertex set of T equals P, and whose dihedral angles do not exceed wr. If the edges of T(g) with angle wr are erased, then the Delaunay tessellation (denoted by T(g)) is obtained and T(g) is uniquely determined by (S,g) and P.

THEOREM 10.2. Let T' be any geodesic triangulation of (S,g) whose set of vertices contains the set of cone points of (S,g). The T' can be transformed into a Delaunay triangulation with the same holonomy by a sequence of flip moves using the following algorithm:

Whenever ABC and ABD -are abutting triangles of T', such that the dihedral angle at the edge AB is greater than ir, change the diagonal in the (metric) quadrilateral ABCD from AB to CD.

Theorem 10.2 results from the following:

LEMMA 10.3. Let T1 be a triangulation, and T2 be a triangulation resulting from T1 after a flip move as described in Theorem (10.2). Then V(T2) > V(T1).

Proof. Let ABC and ABD be two abutting triangles of triangulation T1 of (S, g), such that the dihedral angle at AB is greater than wr. Develop the quadrilateral ACBD into the plane at infinity of H3, and examine the two ideal solids S, = ABCoo U ABDoo and S2 = ACDoo U CBDoo. It is seen that Si C S2, hence V(Si) < V(S2). a

Proof of Theorem 10.2. There are only a finite number of combinatorial types of triangulations of S with k vertices. Since the volume is increased by flip moves, the algorithm described in the statement of the theorem must terminate. That the holonomy is the same follows from the observation that the flip moves preserve holonomy. L3

Remark 10.4. If two triangles ABC and ABD of a Delaunay triangulation are such that the dihedral angle at AB is equal to ir, the flip move in ABCD will transform one Delaunay triangulation into another.

COROLLARY 1. Among all the geodesic triangulations of (S,g) with the vertex set P, the triangulation T(g) satisfying the hypotheses of Theorem 10.1 have the greatest volume (as defined earlier). If T1(g) and T2(g) are two triangulations satisfying the hypotheses of Theorem 10.1, then their volumes are equal.



570 IGOR RIVIN

Proof of Corollary 1. This follows immediately from the proof of Theorem 10.2. ?

Consider now the set Euc(S) of Euclidean structures with k cone points on the surface S. Euc(S) naturally decomposes into regions R(T), corresponding to the isotopy type of the Delaunay triangulation constructed in Theorem 10.1. Since each R(T) is exactly equal to ?gwn(T), by the results of Section 6, each R(T) has the natural structure of a convex polytope. The (codimension-one) intersection of R(T1) and R(T2) corresponds exactly to the same two triangles ABC and ABD of (say) T1, with the dihedral angle at AB equal to ir; hence, the intersection of R(T1) and R(T2) corresponds to the tessellation with ABC and ABD replaced by the concyclic quadrilateral ABCD. In T2 the diagonal goes the other way, so that the two triangles are ACD and CBD. The higher codimension intersections are described in a similar way.

Remark 10.5. Any isotopy class T of triangulations of S can arise as a Delaunay triangulation for some g. Indeed, represent the triangles of T by equilateral triangles of the same size. All the dihedral angles are then 2ir/3 < -r, so T is indeed the Delaunay triangulation.

In view of the remark above and the preceding discussion, it is quickly apparent that the cellulation of Euc(S) by the polytopes R(Ti) is combinatorially none other than the "Harer complex," which gives an equivariant cellulation of the Teichmiiller space of S. In the construction above, the equiv- ariance is immediately obvious. In addition, combined with the elementary proof by Hatcher ([3]) of the contractibility of the Harer complex, the above discussion tells us that the space of Euclidean structures on S is contractible (without recourse to Teichmiiller theory).

Now, the above discussion combines with the results of Sections 7 and 8, to give a nice picture of the space Sim(S) of similarity structures on S with cone singularities. To wit, it is seen that there is a canonical splitting

(10.1) Sim(S) = Euc(S) x R(S),

where R7(S) is the space of real representations of the fundamental group of S\P (recall that P is the set of cone points of S). The projection of each similarity cone structure is found in a canonical way, by constructing the Delaunay tessellation and constructing the Euclidean cone structure with the same isotopy class of the Delaunay triangulation, and the same cone angles. The resulting Euclidean structure is only determined up to scaling, though if desired, it can be normalized, by, for example, requiring it to have area 1. Further results on Euclidean structures are developed in the next section.




11. Euclidean structures with prescribed cone-angles

In this section some of the preceding results above will be applied to the study of Euclidean structures with cone-like singularities on surfaces.

Definition 11.1. Let Ts be the set of simplicial complexes T with k vertices, such that ITI is homeomorphic to S. Let

As= A() TETS

where the union is disjoint.

Remark 11.2. In a natural fashion As can be viewed as a finite (possibly empty) collection of compact convex polytopes.

Consider a Euclidean structure E on S with k cone points V1, ... Vk. To each cone point Vi is associated the cone angle 6i. The cone angles must a priori satisfy the following relations:

(1 1. 1) 0i> O. 1 <iz<k.

(11.2) E (ir - Oi) + E (2r - Oj) = 2irX(S). ViEas vjoas

Definition 11.3. Define

-k {(= 1 . . .,k) E Rk I 01..., ok satisfy equations 11.1 and 11.2 above}.

Let e: As -'S be the map that associates to each locally Euclidean structure its set of cone angles.

PROPOSITION 11.4. E(AS) = 4S or E(AS) is empty.

Proposition 11.4 will be established by demonstrating that a(AS) is open and closed in S. From now on it will be assumed that E3(As) is nonempty, which is equivalent to saying that there exists a triangulation of S with k vertices.

THEOREM 11.5. e(A(T)) is open in '"S for each T E Tks

Proof. The tangent space of 'S at each point is generated by the vectors vZ3, where v = 1, v3J -1 and v0J = 0, when k i {ijj}; consequently it will be enough to demonstrate that for any two vertices Vi and Vj of T it is possible to perturb the locally Euclidean structure A in such a way as to increase the cone angle C(A, Vi) by a (sufficiently small) ?, and simultaneously decrease the cone angle C(A, Vj) by that same e. Since 121 is connected, it is sufficient to show this when Vi and Vj are adjacent. But in that case, there



572 IGOR RIVIN

exists a triangle OViVj, and the required perturbation consists of increasing the angle at Vi slightly and decreasing the angle at Vj by the same amount. If the required triangulation is not Delaunay, it can be transformed into a Delaunay triangulation with the same holonomy by a sequence of flip moves (by Theorem 10.2). Then, by the results of Section 6, it is possible to construct a singular Euclidean structure with the prescribed dihedral (and hence cone) angles. ?-

Notation. In the proof of Theorem 11.6, A(ABC) will denote the angle of the triangle ABC at the vertex A (similarly B(ABC), C(ABC).

THEOREM 11.6. e(A(T)) is closed in _S

Proof. Let (, . , G, .., ( E @(AS) be a sequence of points in 'S converging to (. To establish Theorem 11.6 it is necessary to show that

E E (AS). For each (i let Ai E E-1(xi). Since As is compact, the sequence A1,..., Ai,... has an accumulation point A in As. If A is in As, the proof is finished, so assume that A E OAS, which means that there is a complex T, such that A E OA(T). This means that some angles of A are equal to zero. However, it turns out that it is possible to pick an A' with an underlying complex T', with all angles nonzero, and with O(A') = 0(A) using the following algorithm, each step of which transforms an Ai to an Ai+,, such that the number of zero angles of A' is strictly smaller than that of A. Thus, in a finite number of steps the algorithm will produce the desired A'.

Suppose there is a triangle XYZ of A with X(XYZ) = 0. In that case there is such a triangle with an abutting triangle XYW, such that the angle X(XYW) =A 0. That is so since the cone angle at the vertex X is not zero by assumption.

Now there are two cases to consider:

Transformation 1. The angle Y(XYZ) > 0. In this case, increase the angle X(XYZ) by a sufficiently small ?, decrease the angle Y(XYZ)by ?, decrease the angle X(XYW) by ?, and increase the angle Y(XYW) by ?. L

Transformation 2. The angle Y(XYZ) = 0. Thus the angle Z(XYZ) = 7r. In this case, replace the pair of triangles XYZ and XYW by the pair of triangles ZXW and ZYW, such that:

(i) X(ZXW) = X(XYW), (ii) Y(ZYW) = Y(XYW), (iii) W(ZXW) = W(ZYW) = W(XYW)/2, (iv) Z(ZXW) =7r - X(ZXW) - W(ZXW), (v) Z(ZYW) =X - Y(ZYW) - W(ZYW).




Clearly, neither of the two transformations above changes the cone angle at any vertex, and at the same time each decreases the number of zero angles, as desired. El

There is one further observation to be made. This is that some dihedral angles of the triangulation obtained as above might exceed 7r, as a result of transformations above. However, by Theorem 10.2 this triangulation can always be transformed into a Delaunay triangulation with the same holonomy. There is then a singular Euclidean structure with the same dihedral (and hence cone) angles.

The proof of Proposition 11.4 is now complete. El

PROPOSITION 11.7. For every e E Ok there is a Euclidean cone manifold structure on S, whose cone angles are equal to (i, ...

Proof. By the results of Section 3, for every a E As, there exists a unique complete structure with the same dihedral angles, and hence, by Lemma 4.2, the same cone angles, and so the result follows from Proposition 11.4, since in case there is no triangulation of S with k vertices, there is surely one with more than k vertices, all but k of which can be assigned cone angle 27r, or 7r if they happen to be on the boundary. O

Remark 11.8. Proposition 11.7 was proved by M. Troyanov in [111, for S a closed surface, by a nice analytic argument involving quadratic differentials. Troyanov also shows that there is a unique Euclidean structure with prescribed cone angles in each conformal class.

12. Properties of volume on the space of Euclidean structures ?(T)

THEOREM 12.1. Let V(A) be the volume of the singular structure in E(T) with dihedral angles A. Then V is a concave down function on E(f).

Proof. Let A1 and A2 be two nearby dihedral angle assignments. Let A3 = AA1 + (1 - A)A2. Let AA1 and AA2 be the structures corresponding to A1 and A2. Clearly A' = AAA, + (1 - A)AA2 (where the structures A~j are viewed as vectors of angles) is in AA3. Furthermore, V(A') > AV(AA1) + (1 - A)V(AA2), by Proposition 6.3. On the other hand, by the proof of Theorem 6.10, the volume of the complete structure AA3 is greater yet than the volume of A'. El

In the sequel, ?' is a convex subset of ?.



574 IGOR RIVIN

COROLLARY 1. V achieves a unique maximum on ?'. If ?' is the intersection of ? with an affine subspace, than the maximum lies in the interior of ?,.

Proof. The first statement follows from Theorem 12.1, the second follows from Theorem 12.1 and the argument used to prove Theorem 6.10, after pulling back to A(T). C

Definition 12.2. Denote the structure in ?' where V achieves its maximum by MV(C').

THEOREM 12.3. Let H be a subgroup of Aut (T), such that H preserves ?'. Then: (1) MV(C') is fixed by H. (2) H is isomorphic to a subgroup of Isom (MMv(e')).

Proof. The first assertion follows from the uniqueness of MV(C'). Now, Theorem 9.1 applies, with A = MV(C'). O

Theorem 12.3 is not very interesting for ?' = ?, since MV(C) is very easy to describe: All of its faces are equally sized equilateral triangles. Other choices for ?' are much more interesting. All of the examples below are invariant under all of Aut(T).

Example 12.4. go: Structures with all cone angles equal to 2ir. These are the nonsingular Euclidean structures on T.

Example 12.5. El: Structures with all cone angles less than or equal to 2ir. This space is empty if T is a closed surface of negative Euler characteristic. If T is homeomorphic to S2, then the El correspond to convex polyhedra in R3.

Example 12.6. 63: Structures with all dihedral angles no greater than ir. These correspond to convex ideal polyhedra; see Section 14.

13. Planar Delaunay tessellations

Observation. A triangulation is the Delaunay triangulations of its vertex set precisely when all of the dihedral angles are smaller than ir. If a dihedral angle at an edge e is precisely equal to ir, then the two triangles abutting e are concyclic and the Delaunay triangulation is degenerate.

Hence, Theorem 6.1 gives a linear characterization of the embeddings of a fixed graph G (not necessarily a triangulation) which are the Delaunay




tessellations of the vertex set. In particular, determining whether there is any such embedding is a feasibility problem for a simple linear program whose coefficients are 0 or 1 (this is obtained by dividing all of the constraints by ir). This program can be solved in polynomial time, using standard technology. This characterization can also be derived directly from another result of the author (see [4],[8],[7]).

The vertex oo is covered by the statement of Theorem 13.1.

THEOREM 13.1. The exterior dihedral angles of a Delaunay tessellation D satisfy the following constraints:

(1) All dihedral angles are greater than 0 and smaller than ir. (2) The sum of the exterior dihedral angles of edges incident to a vertex is

equal to 2X. (3) The sum of the exterior dihedral angles of edges forming a cutset of D,

but not all incident to the same vertex is- greater than 27r. Furthermore, every system of weights on the edges of D satisfying the

above constraints constitutes a system of dihedral angles for a Delaunay triangulation with combinatorics D.

The above characterization has an exponential number of constraints, but can still be checked in polynomial time using standard methods of convex programming, as observed by Warren Smith (see [41).

14. Hyperbolic ideal polyhedra, revisited

The theory developed above can now be applied to the study of (possibly immersed) hyperbolic ideal polyhedra, star-shaped with respect to one of their vertices (from now on simply "star-shaped"), or convex. Since the abstract dihedral angles correspond to the hyperbolic dihedral angles, the following results are immediate corollaries of the results of Sections 3 and 12. Although the results apply to any hyperbolic cone manifold formed of a collection of ideal simplices sharing a vertex oo, the results for polyhedra are particularly easy to state, and in some cases are the most aesthetically pleasing.

THEOREM 14. 1. A star-shaped (and in particular convex) hyperbolic ideal polyhedron is uniquely determined by its dihedral angles.

Proof. This is a restatement of Theorem 6.16. 0

Note. Theorem 14.1 for convex polyhedra was announced as Theorem 2 in [4].

THEOREM 14.2. The set of dihedral angles of convex polyhedra forms a convex polytope.



576 IGOR RIVIN

Proof. The set is ?C(T), described in Theorems 5.1 and 6.1. Convex polyhedra are those whose dihedral angles are not greater than ir, and are thus the intersection of ?(T) with a collection of linear half-spaces. [

Note. This is a different description of the polytope of dihedral angles of convex ideal polyhedra from that announced in [4] and proved in [8],[7].

THEOREM 14.3. There exists a unique convex ideal polyhedron of a fixed combinatorial type of maximal volume. This polyhedron is maximally symmetric.

Proof. This follows from Theorem 14.2 and Corollary 12.1 to Theorem 12.1. [

COROLLARY 1. Regular ideal Platonic solids are of maximal volume among convex ideal polyhedra in their respective combinatorial classes.

THEOREM 14.4. There exists a unique convex polyhedron of a fixed combinatorial type of maximal volume. This polyhedron is maximally symmetric.

In the last part of this section I will use the following generalization of Schliifli's formula 14.2, due to Milnor:

THEOREM 14.5. Let P be a polyhedron with vertices vi, ... ,van where vi is ideal for i > m. Let Hm+i, ... ,Hn be a collection of horospheres, such that Hi is centered on vi. Further, if there is an edge eij in P connecting vi and vj, then dij is the distance between vi and v;, if ij < m, the signed distance between Hi and Hj (negative if the corresponding horoballs intersect) if ij > m, and the signed distance between Hi and vj if i < m, j > m.

(14.1) dV(P) =- dij dij vi connected to v;

where caij is the dihedral angle at the edge eij.

Since I am not aware of any existing proof of Theorem 14.5 (a version was used, without a complete proof, by Craig Hodgson), I supply one here. A few ingredients are needed; the first is Schlkfli's variation formula for a compact simplex T:

(14.2) dV(T) =-2 E lij daij, X2

where lithjth is the length of the edge of T connecting the ith and jth vertices. A version of equation 14.2 for spherical simplices of arbitrary dimension was proved in [10], although the 3-dimensional hyperbolic version of equation 14.2




was apparently already known to Lobachevsky. For a nice modern proof of equation 14.2 in all constant curvature space forms of arbitrary dimension, see [11.

LEMMA 14.6. The right-hand side of equation (14.1) is independent of the choice of horospheres Hi.

Proof. Without loss of generality, we may replace the horosphere Hm+i by the horosphere H' +1, such that d(Hm+l, H, +1) = dm+i. Then l(m+1)j -

i(m+)j = dm+i. The right-hand side of equation (14.1) thereby changes by - dl Ej dalj. Since the link of the vertex vm+l is Euclidean, the last quantity is equal to 0. L

LEMMA 14.7. If Theorem 14.5 is true for a simplex with a single ideal vertex, it is true for any ideal polyhedron.

Proof. Any ideal polyhedron P can be decomposed into a union T of compact simplices and simplices with a single ideal vertex, and so the variation of volume of P will be the sum of terms of the form 1(e) E dai&(e), where cxi are the dihedral angles of the simplices of T incident to e, and 1(e) is the generalized length of e, as defined in the statement of Theorem 14.5. If e is not a part of an edge of P, the total angle at e will remain constant, and hence the term corresponding to e will contribute 0 to the sum. Otherwise, the contribution of all of the pieces of an edge E of P is seen to be exactly as needed. ?

Proof of Theorem 14.5. By Lemma 14.6 and Lemma 14.7 it is sufficient to show Theorem 14.5 for the special case where P is a simplex with a single ideal vertex V, and H is a horosphere which only intersects edges of P incident to V. The argument uses a construction reminiscent of dicing of a carrot: Consider the infinite family of decreasing concentric horospheres HI = H, H2, H3, such that d(Hi, Hi+1) = 1. Furthermore, if the edges of P incident to V are el, e2, and e3, then let the corresponding vertices of P be x0, yo, and zo, and let Hi n el = xi, Hi A e2 = yi, Hi n e3 = zi. Finally, let Pi be the triangular prism with vertices xi, yi, z xi+1 Yi+11 Zi+

Since P = Po U PU U U Pi U , it follows that

00

(14.3) V(P) = ZV(Pi) i=O



578 IGOR RIVIN

Each Pi has nine edges, and since it is compact, its volume variation can be computed using equation (14.2):

dV(Pi) =da.,iyil(xiyi) + dayjizil(yizi) + datzixl(zixj)

+ da1i+ yi+ll(Xi+jYi+1) + daxi+1Zi+1l(Yi+1Zi+1) + daZi+ 1i+1l(Zi+1xi+1)

+ dacxjx+jl(xixi+1) + daYiYi+ll(yiyi+,) + dazjzj+j(zjzi+j)).

Since l(xixi+l) = 1(yiyi+j) = l(zjzj+j), and the link of V stays Euclidean, the last three terms drop out when i > 0. Also since l(xnyn) = O(exp(-n)) (and likewise for the other "horizontal" edges), and the sum in equation (14.3) converges absolutely (all of the terms are positive), the sum in equation (14.3) can be differentiated under the summation. Since the total angles at the edges xiyi, etc, remain constant at 7r/2 the sum telescopes, and we remain with exactly the terms specified by Theorem 14.5. El

The following observation is due to Thurston:

THEOREM 14.8. If P is a critical point of volume, then there exist horospheres H1,...N Hn centered on the vertices v1,.. ,vn, respectively, such that Hi and Hj are tangent, whenever there is an edge in P joining vi and v;.

Proof. Select the Hi arbitrarily. The exterior dihedral angles incident to any vertex vi of P sum to 27r,

(14.4) (r - aij) = 2Xr.

In the space of dihedral angles RIE(P)I, the normal vector to the hyperplane defined by equation (14.4) is

(14.5) 4=Zeij,

where eij are the basis vectors of RE(P). Theorem 14.1, together with a dimension count, implies that equations (14.4) are the only linear constraints on the dihedral angles. By the principle of Lagrange multipliers, if P is a critical point of volume, then

(14.6) Zdijeij = grad V(P) = CiVi ii i

where Ci are arbitrary real numbers. Expanding out and collecting terms, we finally obtain

(14.7) dij = Ci + C3.




Thus, if instead of the horospheres Hi, horospheres H' were picked, such that Hi is Ci farther out from vi than Hi, then the corresponding d'j are seen to be all 0. L

THEOREM 14.9. If the polyhedron P, whose existence is guaranteed by Theorem 14.4 is strictly convex, then there exist horospheres centered at the vertices of P, such that Hv and Hw are tangent if v and w are adjacent in P.

15. Further work

The results and ideas described in the current paper can be developed in various directions, some of which are sketched below:

Deformation theory of singular hyperbolic structures on 3-manifolds. In Thurston's theory of geometric structures on 3-manifolds, in particular in the study of Dehn surgery spaces, a prominent role is played by a hyperbolic cone- manifold made of ideal hyperbolic simplices. Using the ideas of the current paper it can be shown, for example, that the cone-manifold structure is determined uniquely by the cone angles. Uniqueness in the case of hyperbolic manifolds, where all of the cone angles are equal to 27r, follows from Mostow rigidity, but that approach does not extend to general cone-manifolds.

Geometry of planar graphs. The volume function seems to be a canonical "goodness" measure of the layout (drawing) of a graph. What is more, it turns out that if all of the dihedral angles are smaller than 7r, the drawing with those dihedral angles can be constructed in polynomial time in the size of the graph and the number of digits of precision required. These results also can be extended to layout on surfaces of high genus.

Infinite complexes. The results of this paper can be used to understand the geometry of infinite triangulations in the plane, and of ideal polyhedra with an infinite number of vertices. One interesting result that comes out of this is that an assignment of dihedral angles to a convex ideal polyhedron is feasible if and only if it is feasible for every finite sub-complex.

Spectral geometry. The Hessian of hyperbolic volume of a simplex (see Section 2) is closely related to the finite-element approximation to the Lapla- cian (which can be viewed as the explanation of Theorem 2.1). It is interesting to explore further relationships between current work and spectral and conformal geometry.

PRINCETON UNIVERSITY, PRINCETON, NJ NEC RESEARCH INSTITUTE, PRINCETON, NJ INSTITUTE FOR ADVANCED STUDY, PRINCETON, NJ



580 IGOR RIVIN

REFERENCES

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[2] BRIAN BOWDITCH, Singular Euclidean structures on surfaces, J. London Math. Soc. 44 (1991), 553-565.

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(Received May 13, 1992)



Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume

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