Abstract
We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph \(\Gamma \) is realized as the 1-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if \(\Gamma \) is realized as the 1-skeleton of a polyhedron inscribed in the sphere and \(\Gamma \) admits a Hamiltonian cycle. This answers a question asked by Steiner in 1832. Rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space. We study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti-de Sitter geometry and in half-pipe geometry. Just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid, half-pipe geometry is naturally a limit of both hyperbolic and anti-de Sitter geometry. We promote a unified point of view to the study of the three cases throughout.
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Acknowledgements
Some of this work was completed while we were in residence together at the 2012 special program on Geometry and analysis of surface group representations at the Institut Henri Poincaré; we are grateful for the opportunity to work in such a stimulating environment. Our collaboration was greatly facilitated by support from the GEAR network (U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties”). We are grateful to Arnau Padrol for informing us that Theorem 1.2 answers a question asked by Steiner in [37], and to the anonymous referees for many constructive remarks. We also thank Qiyu Chen for pointing out a sign mistake.
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Jeffrey Danciger was partially supported by the National Science Foundation Grants DMS 1103939, DMS 1510254, and DMS 1812216 and by an Alfred P. Sloan Foundation fellowship. Sara Maloni was partially supported by the National Science Foundation Grants DMS 1506920, DMS 1650811, DMS 1839968 and DMS 1848346. Jean-Marc Schlenker was partially supported by FNR Grants INTER/ANR/15/11211745 and OPEN/16/11405402. The authors acknowledge support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 RNMS: “Geometric Structures and Representation Varieties” (the GEAR Network).
Appendix A. Ideal polyhedra with dihedral angles going to zero
Appendix A. Ideal polyhedra with dihedral angles going to zero
We outline an alternative proof of Proposition 1.16 using transitional geometry ideas. The argument uses Lemmas 1.14 and 1.15 to produce deformation paths of polyhedra with dihedral angles going to zero in a prescribed manner. Here is the basic idea: starting from an ideal polyhedron \(P \in \mathsf {AdSPolyh}\) with dihedral angles \(\theta \), we deform P so that the dihedral angles are proportional to \(\theta \) and decrease toward zero. An appropriate rescaled limit of these collapsing polyhedra yields an ideal polyhedron \(P_\infty '\) in \({\mathbb {HP}}^3\) whose (infinitesimal) dihedral angles are precisely \(\theta \); we then conclude, via Proposition 1.10, that \(\theta \) was in \(\mathcal {A}\) to begin with.
The main ingredient is the following proposition. Recall the projective transformations \(\mathfrak {a}_t\) of Sect. 2.6, which when applied to (the projective model of) \(\mathbb {A}\mathrm {d}\mathbb {S}^3\) yield \({\mathbb {HP}}^3\) in the limit as \(t \rightarrow 0\).
Proposition A.1
Let \(\Gamma \in \mathsf {Graph}(\Sigma _{0,N}, \gamma )\) and consider weights \(\theta \in \mathbb {R}^{E(\Gamma )}\) that satisfy conditions (i), (ii) of Definition 1.3, and the following weaker version of (iii):
- (iii’)::
If \(e_1^*, \ldots , e_n^*\) form a simple circuit that does not bound a face of \(\Gamma ^*\), and such that exactly two of the edges are dual to edges of the equator, then \(\theta (e_1) +\cdots + \theta (e_n) \ne 0\).
Let \(P_k\) be a sequence in \(\mathsf {AdSPolyh}_{\Gamma }\) with dihedral angles \(t_k \theta \) such that \(t_k \rightarrow 0\). Then:
- (1)
\(P_k\) converges to an ideal N-gon \(P_\infty \) in the hyperbolic plane.
- (2)
\(\mathfrak {a}_{t_k} P_k\) converges to an ideal polyhedron \(P_\infty '\) in \({\mathbb {HP}}^3\) with 1-skeleton \(\Gamma \) and infinitesimal dihedral angles \(\theta \).
Proposition A.1 will be applied in the alternative proof of Proposition 1.16 below to show by contradiction that the dihedral angles \(\theta \) of an ideal polyhedron in \(\mathsf {AdSPolyh}\) must satisfy Condition (iii). In particular, if \(\theta \) satisfies (iii’) but not (iii), then Proposition A.1 produces an ideal polyhedron in \({\mathbb {HP}}^3\) with the same dihedral angles, contradicting the already proved HP geometry version of Proposition 1.16, namely Proposition 1.10.
We briefly mention an analogue of Proposition A.1 in the setting of quasifuchsian hyperbolic three-manifolds. The first conclusion of the proposition can be seen as an analogue of Series’ theorem [35], which states that when the bending data of a sequence of quasifuchsian representations goes to zero in a controlled manner, the convex cores collapse to a Fuchsian surface. The second part is the analogue of work of Danciger–Kerckhoff [17] showing that after application of appropriate projective transformations (in our notation, the \(\mathfrak {a}_t\)), the collapsing convex cores of such quasifuchsian representations converge to a convex core in half-pipe geometry.
Proof
We adapt the proof of Lemma 1.14 (properness of the map \(\Psi ^{\mathsf {AdS}}\)). As in that proof, we may again assume that the ideal vertices \((v^L_{1,k}, v^R_{1,k}), \ldots , (v^L_{N,k}, v^R_{N,k})\) of \(P_k\) satisfy that:
\(v^L_{1,k} = v^R_{1,k} = 0\), \(v^L_{2,k} = v^R_{2,k} = 1\), \(v^L_{3,k} = v^R_{3,k} = \infty \);
For each \(i \in \{1,\ldots N\}\), \(v^L_{i,k} \rightarrow v^L_{i,\infty }\) and \(v^R_{i,k} \rightarrow v^R_{i,\infty }\); and
\(v^L_{i, \infty } = v^L_{i+1, \infty }\) if and only if \(v^R_{i, \infty } = v^R_{i+1, \infty }\).
Therefore, we again find that the limit \(P_\infty \) of \(P_k\) (in this normalization) is a convex ideal polyhedron in \(\mathbb {A}\mathrm {d}\mathbb {S}^3\), possibly of fewer vertices, and possibly degenerate (i.e. lying in a two-plane). The dihedral angle at an edge e of \(P_\infty \) is again the sum of \(\theta _\infty (e')\) over all edges \(e'\) of \(\Gamma \) which collapse to e, where in this case \(\theta _\infty = 0\). Therefore all dihedral angles of \(P_\infty \) are zero and we have that \(P_\infty \) is an ideal polygon lying in the hyperbolic plane \(\mathscr {P}\) containing the ideal triangle \(\Delta _0\) spanned by (0, 0), (1, 1), and \((\infty , \infty )\). To prevent collapse, we apply the appropriate projective transformations \(\mathfrak {a}_{t_k}\) to the \(P_k\).
Claim A.2
Up to taking a subsequence (in fact not necessary), the vertices \(\mathfrak {a}_{t_k} v_{i,k}\) converge to points \(v_{i,\infty }'\) in the projective boundary \(\partial {\mathbb {HP}}^3\).
Proof
This can be seen from the following simple compactness statement, which may be verified by induction: Given \(M \ge 1\) and \(\Theta > 0\), there exists two smooth families of space-like bounding planes \(\mathcal {Q}_+(t)\) and \(\mathcal {Q}_-(t)\), defined for \(t \ge 0\), such that
\(\mathcal {Q}_+(0) = \mathcal {Q}_-(0) = \mathscr {P}\).
\(\mathcal {Q}_+(t)\) and \(\mathcal {Q}_-(t)\) are disjoint for \(t > 0\) and their common perpendicular is a fixed time-like line \(\alpha \) (independent of t).
The time-like distance (along \(\alpha \)) between \(\mathcal {Q}_+(t)\) and \(\mathcal {Q}_-(t)\) is \({\text {O}}(t)\).
Any space-like convex connected ideal polygonal surface in \(\mathbb {A}\mathrm {d}\mathbb {S}^3\) for which \(\Delta _0\) is (contained in) a face, which has at most M faces, and all of whose dihedral angles are bounded by \(t \Theta \) lies to the past of \(\mathcal {Q}_+(t)\) and to the future of \(\mathcal {Q}_-(t)\).
The first three conditions above imply that the limit of \(\mathfrak {a}_t \mathcal {Q}_+(t)\) and \(\mathfrak {a}_t \mathcal {Q}_-(t)\) as \(t \rightarrow 0\) are two disjoint non-degenerate planes \(\mathcal {Q}_+'\) and \(\mathcal {Q}_-'\) in \({\mathbb {HP}}^3\). Therefore, the limit of \(\mathfrak {a}_{t_k} P_k\) must, after extracting a subsequence if necessary, converge to some polyhedron in \({\mathbb {HP}}^3 \cup \partial {\mathbb {HP}}^3\) lying below \(\mathcal {Q}_+'\) and above \(\mathcal {Q}_-'\). \(\square \)
As in the proof of Lemma 1.14, the limit of \(\mathfrak {a}_{t_k} P_k\) is the convex hull \(P_\infty '\) of \(v_{1,\infty }', \ldots , v_{n, \infty }'\) in \({\mathbb {HP}}^3\). The 1-skeleton \(\Gamma '\) of \(P_\infty '\) is obtained from the original 1-skeleton \(\Gamma \) by collapsing some edges to vertices and some faces to edges or vertices.
Lemma A.3
Given \(e' \in \Gamma '\), the infinitesimal dihedral angle \(\theta _\infty '(e')\) of \(P_\infty '\) at \(e'\) is the sum of \(\theta (e) = \frac{d}{dt} t \theta (e)\big |_{t=0}\) over all edges e which collapse to \(e'\).
Proof
The sequence of edges \(e_1, \ldots , e_\ell \in E(\Gamma )\) which collapse to \(e'\) may be arranged in order so that the dual edges in \(\Gamma ^*\) form a simple path. The sequence of adjacent faces \(f_0, \ldots , f_\ell \) of \(\Gamma \) are such that in \(P_k\), each of these faces collapses to \(e'\) except the first \(f_0\) and last \(f_\ell \). After applying \(\mathfrak {a}_{t_k}\), the planes \(W_1, \ldots , W_\ell \) containing these faces (we are suppressing the \(t_k\) dependence here) converge to planes \(W_1', \ldots , W_\ell '\) arranged in order around the common edge realizing \(e'\) in \({\mathbb {HP}}^3\). The angle \(\theta _\infty '(e')\), which is the dihedral angle between \(W_1'\) and \(W_\ell '\), is seen to be the sum over i of the dihedral angle between \(W_i'\) and \(W_{i+1}'\), which is precisely \(\theta (e_i)\). \(\square \)
Next, consider the projection \(\varpi : {\text {HP}}^3 \rightarrow \mathscr {P}\). Note that \(\varpi (v_{i,\infty }') = v_{i,\infty }\). Let \(\mathcal {H}\) denote the HP horo-cylinder which is the inverse image under \(\varpi \) of a small horocycle in \(\mathscr {P}\) centered at a vertex \(v_{i, \infty }\) of \(P_\infty \). The metric on \(\mathcal {H}\) inherited from \({\mathbb {HP}}^3\) is flat and degenerate; it is the pull-back under \(\varpi \) of the metric on a horocycle. The intersection of \(\mathcal {H}\) with \(P_\infty '\) is a convex polygon q in \(\mathcal {H}\). The infinitesimal angles at vertices of q are the same as the infinitesimal dihedral angles of the corresponding edges of \(P_\infty '\). Note that the vertices of q are the intersection with \(\mathcal {H}\) of all edges emanating from the ideal points \(v_{j, \infty }'\) such that \(v_{j, \infty } = v_{i,\infty }\). The following lemma is just the fact that the exterior angles of a convex polygon in the Euclidean plane sum up to a constant \(2\pi \), interpreted in the setting that the polygon is infinitesimally thin.
Lemma A.4
The infinitesimal angles of q sum to zero.
Now, suppose, for contradiction, that \(v_{i+1, \infty } = v_{i, \infty }\). Then, the vertices of q correspond to a path \(c'\) of edges of \(\Gamma '\) whose inverse image under the collapse is a path c of edges in \(\Gamma \) which do not bound a face of \(\Gamma ^*\). It follows from the above that the sum of \(\theta (e)\) over the edges e of the path c is zero, contradicting the condition (iii’). \(\square \)
Remark A.5
This argument also works in the context of hyperbolic ideal polyhedra with dihedral angles going to zero and \(\pi \) at a controlled rate.
Remark A.6
Assuming the stronger condition (iii) on \(\theta \), the limiting ideal polygon \(P_\infty \) must be the unique minimum of the length function \(\ell _\theta \) over the space \(\mathsf {polyg}_N\) of marked ideal polygons. See the proof of Theorem 1.9.
Outline of alternative proof of Proposition 1.16
Let \(\Gamma \in \mathsf {Graph}(\Sigma _{0,N}, \gamma )\) and suppose \(P \in \mathsf {AdSPolyh}_\Gamma \) such that the dihedral angles \(\theta = \Psi ^{\mathsf {AdS}}(P) \in \mathbb {R}^{E(\Gamma )}\) violate condition (iii) in the definition of \(\mathcal {A}_\Gamma \). We argue by contradiction. First we show that there are nearby weights \(\theta '\) satisfying conditions (i), (ii), as well as condition (iii’) of Proposition A.1 above and so that at least one of the angle sum expressions of (iii’) is strictly negative. This may already be the case for \(\theta \). If not, then there is at least one angle sum expression as in (iii) which evaluates to zero, and we will perturb. In the case that \(\Gamma \) is a triangulation, it is simple to verify that none of the angle sum expressions in condition (iii) is locally constant when the equations of condition (ii) are satisfied, and therefore a nearby \(\theta '\) exists as desired, since (iii’) consists of only finitely many conditions . If \(\Gamma \) is not a triangulation, then it actually could be the case that an angle sum expression as in condition (iii) is constant equal to zero on the entire space of weights satisfying (ii). However, it is always possible to add a small number of edges (at most one for each angle sum expression of (iii) which evaluates to zero for \(\theta \)) with very small positive weights, while perturbing the other weights slightly, to produce \(\theta '\) as desired. Let us briefly explain.
Suppose \(e_1^*, \ldots , e_n^*\) form a simple circuit that does not bound a face of \(\Gamma ^*\), such that exactly two of the edges are dual to edges of the equator, and such that \(\theta (e_1) + \cdots + \theta (e_n) = 0\) for all angle assignments \(\theta \) satisfying (ii). Then, in the algebra of functions on the edges of \(\Gamma \), the sum \(\theta (e_1) + \cdots + \theta (e_n)\) is a linear combination of the vertex relations of (ii). More specifically, the angle sum \(\theta (e_1) + \cdots + \theta (e_n)\) is equal to the alternating sum of vertex relations for the ordered collection of vertices (necessarily odd in number) lying on one side of the simple circuit. See Fig. 10. Consider a face f of \(\Gamma \) containing two such adjacent vertices \(v_+\) and \(v_-\) which appear respectively with a \(+\) sign and a − sign in the alternating sum. The only non-equatorial edges emanating from \(v_-\) end at a vertex represented in the alternating sum with a positive sign. Hence, we may add a diagonal edge \(e'\) within f to \(\Gamma \) connecting \(v_+\) to another vertex represented in the sum with a positive sign. In this new combinatorics we may deform the angle sum \(\theta (e_1) + \cdots + \theta (e_n) = -2\theta (e')\) to negative values by assigning a small positive weight to \(e'\). If after this adjustment, there remains other simple circuits of the type considered in (iii’) with angle sum identically zero, we perform a similar adjustment for that simple circuit. As there are only finitely many simple circuits to consider, this process terminates in finitely many steps yielding an angle assignment on some supergraph of \(\Gamma \) satisfying (iii’).
Next, by Lemma 7.2 (which was a simple consequence of Lemma 1.15, independent of Proposition 1.16), there is an ideal polyhedron \(P' \in \mathsf {AdSPolyh}\), close to P, so that \(\Psi ^{\mathsf {AdS}}(P') = \theta '\). Now, consider the path of weights \(t\theta '\), defined for \(t > 0\). Lemma 7.2 implies that there exists a path \(P_t\) in \(\mathsf {AdSPolyh}\) such that \(\Psi ^{\mathsf {AdS}}(P_t) = t \theta '\), defined at least for t close to one. In fact, the path \(P_t\) may be defined for all \(1 \ge t > 0\). Indeed if the limit as \(t \rightarrow T > 0\) of \(P_t\) failed to exist, then the proof of Lemma 1.14 would imply that \(\Psi ^{\mathsf {AdS}}(P_t)\) either goes to infinity or limits to an element of \(\mathbb {R}^{E(\Gamma )}\) for which some angle sum expression as in (iii) is exactly zero, impossible since the limit as \(t \rightarrow T\) of \(\Psi ^{\mathsf {AdS}}(P_t)\) is, of course, equal to \(T \theta '\). Hence, we may apply Proposition A.1 to the path \(P_t\). The result is an ideal polyhedron \(P_\infty ' \in \mathsf {HPPolyh}\) whose infinitesimal dihedral angles are precisely \(\theta '\). This contradicts Proposition 1.10 since \(\theta '\) does not satisfy (iii). \(\square \)
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Danciger, J., Maloni, S. & Schlenker, JM. Polyhedra inscribed in a quadric. Invent. math. 221, 237–300 (2020). https://doi.org/10.1007/s00222-020-00948-9
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DOI: https://doi.org/10.1007/s00222-020-00948-9