Showing papers in "Inventiones Mathematicae in 2020"

On Falconer's distance set problem in the plane

Abstract: If $$E \subset \mathbb {R}^2$$ is a compact set of Hausdorff dimension greater than 5 / 4, we prove that there is a point $$x \in E$$ so that the set of distances $$\{ |x-y| \}_{y \in E}$$ has positive Lebesgue measure.

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

Abstract: In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension $$n \ge 3$$ . For every bounded open subset $$\Omega \subset M$$ with smooth boundary, we prove that $$\begin{aligned} \int \limits _{\partial \Omega } \left| \frac{\mathrm{H}}{n-1}\right| ^{n-1} \!\!\!\!\!{\mathrm{d}}\sigma \,\,\ge \,\,{\mathrm{AVR}}(g)\,\big |\mathbb {S}^{n-1}\big |, \end{aligned}$$ where $${\mathrm{H}}$$ is the mean curvature of $$\partial \Omega $$ and $${\mathrm{AVR}}(g)$$ is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if $$(M{{\setminus }}\Omega , g)$$ is isometric to a truncated cone over $$\partial \Omega $$ . An optimal version of Huisken’s Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

Square-free Gröbner degenerations

Abstract: Let I be a homogeneous ideal of $$S=K[x_1,\ldots , x_n]$$ and let J be an initial ideal of I with respect to a term order. We prove that if J is radical then the Hilbert functions of the local cohomology modules supported at the homogeneous maximal ideal of S/I and S/J coincide. In particular, $${\text {depth}} (S/I)={\text {depth}} (S/J)$$ and $${\text {reg}} (S/I)={\text {reg}} (S/J)$$ .

Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras

Abstract: This paper concerns the cohomological aspects of Donaldson–Thomas theory for Jacobi algebras and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman. We prove the Hodge-theoretic categorification of the integrality conjecture and the wall crossing formula, and furthermore realise the isomorphism in both of these theorems as Poincare–Birkhoff–Witt isomorphisms for the associated cohomological Hall algebra. We do this by defining a perverse filtration on the cohomological Hall algebra, a result of the “hidden properness” of the semisimplification map from the moduli stack of semistable representations of the Jacobi algebra to the coarse moduli space of polystable representations. This enables us to construct a degeneration of the cohomological Hall algebra, for generic stability condition and fixed slope, to a free supercommutative algebra generated by a mixed Hodge structure categorifying the BPS invariants. As a corollary of this construction we furthermore obtain a Lie algebra structure on this mixed Hodge structure—the Lie algebra of BPS invariants—for which the entire cohomological Hall algebra can be seen as the positive part of a Yangian-type quantum group.

Reductivity of the automorphism group of K-polystable Fano varieties

Abstract: We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $$\Theta $$ -reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.

Diophantine problems and p-adic period mappings

Abstract: We give an alternative proof of Faltings’s theorem (Mordell’s conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings’s original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of p-adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems of p-adic Hodge theory, and explicit topological computations of monodromy. By the same methods we show that, in sufficiently large dimension and degree, the set of hypersurfaces in projective space, with good reduction away from a fixed set of primes, is contained in a proper Zariski-closed subset of the moduli space of all hypersurfaces. This uses in an essential way the Ax–Schanuel property for period mappings, recently established by Bakker and Tsimerman.

Singularity formation for the two-dimensional harmonic map flow into S-2

Abstract: We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $$S^2$$, $$\begin{aligned} u_t&= \Delta u + | abla u|^2 u \quad \text {in } \Omega \times (0,T)\\ u&= \varphi \quad \text {on } \partial \Omega \times (0,T)\\ u(\cdot ,0)&= u_0 \quad \text {in } \Omega , \end{aligned}$$where $$\Omega $$ is a bounded, smooth domain in $$\mathbb {R}^2$$, $$u: \Omega \times (0,T)\rightarrow S^2$$, $$u_0:\bar{\Omega } \rightarrow S^2$$ is smooth, and $$\varphi = u_0\big |_{\partial \Omega }$$. Given any k points $$q_1,\ldots , q_k$$ in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corotational harmonic map. We build a continuation after blow-up as a $$H^1$$-weak solution with a finite number of discontinuities in space–time by “reverse bubbling”, which preserves the homotopy class of the solution after blow-up. Furthermore, we prove the codimension one stability of the one point blow-up phenomenon.

Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

Abstract: We prove the Topological Mirror Symmetry Conjecture by Hausel–Thaddeus for smooth moduli spaces of Higgs bundles of type $$SL_n$$ and $$PGL_n$$ . More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore, we prove for d prime to n, that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d. This proves a conjecture by Mozgovoy–Schiffmann in the coprime case.

Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials

Abstract: We show that the Masur–Veech volumes and area Siegel–Veech constants can be obtained using intersection theory on strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel–Veech constants for all strata. We also show that the same results hold for the spin and hyperelliptic components of the strata.

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature

Abstract: The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron.

Embedding minimal dynamical systems into Hilbert cubes

Abstract: We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube $$\left( [0,1]^N\right) ^{\mathbb {Z}}$$. This problem is intimately related to the theory of mean dimension, which counts the average number of parameters for describing a dynamical system. Lindenstrauss proved that minimal systems of mean dimension less than cN for $$c=1/36$$ can be embedded in $$\left( [0,1]^N\right) ^{\mathbb {Z}}$$, and asked what is the optimal value for c. We solve this problem by showing embedding is possible when $$c=1/2$$. The value $$c=1/2$$ is optimal. The proof exhibits a new interaction between harmonic analysis and dynamical coding techniques.

The Fried conjecture in small dimensions

Abstract: We study the twisted Ruelle zeta function ζX (s) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove Fried conjecture, relating Reidemeister torsion and ζX (0). In higher dimensions, we show more generally that ζX (0) is locally constant with respect to the vector field X under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where Fried conjecture holds true.

Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures)

Abstract: Using our earlier results on polynomiality properties of plethystic logarithms of generating series of certain type, we show that Schiffmann’s formulas for various counts of Higgs bundles over finite fields can be reduced to much simpler formulas conjectured by Mozgovoy. In particular, our result implies the conjecture of Hausel and Rodriguez-Villegas on the Poincare polynomials of twisted character varieties and the conjecture of Hausel and Thaddeus on independence of E-polynomials on the degree.

On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4 + 1

Abstract: We prove that every bounded stable solution of $$\begin{aligned} (-\Delta )^{1/2} u + f(u) =0 \qquad \text{ in } \mathbb {R}^3 \end{aligned}$$is a 1D profile, i.e., $$u(x)= \phi (e\cdot x)$$ for some $$e\in {\mathbb {S}}^2$$, where $$\phi :\mathbb {R}\rightarrow \mathbb {R}$$ is a nondecreasing bounded stable solution in dimension one. Equivalently, stable critical points of boundary reaction problems in $$\mathbb {R}^{d+1}_+=\mathbb {R}^{d+1}\cap \{x_{d+1}\ge 0\}$$ of the form $$\begin{aligned} \int _{\{x_{d+1\ge 0}\}} \frac{1}{2} | abla U|^2 \,dx\, dx_{d+1} + \int _{\{x_{d+1}=0\}} F(U) \,dx \end{aligned}$$are 1D when $$d=3.$$ These equations have been studied since the 1940’s in crystal dislocations. Also, as it happens for the Allen–Cahn equation, the associated energies enjoy a $$\Gamma $$-convergence result to the perimeter functional. In particular, when $$f(u)=u^3-u$$ (or equivalently when $$F(U)=\frac{1}{4} (1-U^2)^2 $$), our result implies the analogue of the De Giorgi conjecture for the half-Laplacian in dimension 4, namely that monotone solutions are 1D. Note that our result is a PDE version of the fact that stable embedded minimal surfaces in $$\mathbb {R}^3$$ are planes. It is interesting to observe that the corresponding statement about stable solutions to the Allen–Cahn equation (namely, when the half-Laplacian is replaced by the classical Laplacian) is still unknown for $$d=3$$.

Shifts of finite type as fundamental objects in the theory of shadowing

Abstract: Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let X be a compact totally disconnected space and $$f:X\rightarrow X$$ a continuous map. We demonstrate that f has shadowing if and only if the system $$(f,X)$$ is (conjugate to) the inverse limit of a directed system satisfying the Mittag-Leffler condition and consisting of shifts of finite type. In particular, this implies that, in the case that X is the Cantor set, f has shadowing if and only if (f, X) is the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type. Moreover, in the general compact metric case, where X is not necessarily totally disconnected, we prove that f has shadowing if $$(f,X)$$ is a factor of the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo-orbits.

Invariance of white noise for KdV on the line

Abstract: We consider the Korteweg–de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting.

The Bieri–Neumann–Strebel invariants via Newton polytopes

Abstract: We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri–Neumann–Strebel (BNS) via a theorem of Sikorav. We offer several applications: we reprove Thurston’s theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincare duality groups of type $$\mathtt {F}_{}$$ in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl. We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the $$L^2$$-torsion polytope of Friedl–Luck is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl–Luck–Tillmann.

A uniqueness result for the decomposition of vector fields in $$\mathbb {R}^{{d}}$$Rd

Abstract: Given a vector field $$\rho (1,\mathbf {b}) \in L^1_\mathrm{loc}(\mathbb {R}^+\times \mathbb {R}^{d},\mathbb {R}^{d+1})$$ such that $${{\,\mathrm{div}\,}}_{t,x} (\rho (1,\mathbf {b}))$$ is a measure, we consider the problem of uniqueness of the representation $$\eta $$ of $$\rho (1,\mathbf {b}) {\mathcal {L}}^{d+1}$$ as a superposition of characteristics $$\gamma : (t^-_\gamma ,t^+_\gamma ) \rightarrow \mathbb {R}^d$$, $$\dot{\gamma } (t)= \mathbf {b}(t,\gamma (t))$$. We give conditions in terms of a local structure of the representation $$\eta $$ on suitable sets in order to prove that there is a partition of $$\mathbb {R}^{d+1}$$ into disjoint trajectories $$\wp _\mathfrak {a}$$, $$\mathfrak {a}\in \mathfrak {A}$$, such that the PDE $$\begin{aligned} {{\,\mathrm{div}\,}}_{t,x} \big ( u \rho (1,\mathbf {b}) \big ) \in {\mathcal {M}}(\mathbb {R}^{d+1}), \quad u \in L^\infty (\mathbb {R}^+\times \mathbb {R}^{d}), \end{aligned}$$can be disintegrated into a family of ODEs along $$\wp _\mathfrak {a}$$ with measure r.h.s. The decomposition $$\wp _\mathfrak {a}$$ is essentially unique. We finally show that $$\mathbf {b}\in L^1_t({{\,\mathrm{BV}\,}}_x)_\mathrm{loc}$$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible $${{\,\mathrm{BV}\,}}$$ vector fields.

Zimmer's conjecture for actions of $\mathrm{SL}(m,\mathbb{Z})$

Abstract: We prove Zimmer’s conjecture for $$C^2$$ actions by finite-index subgroups of $$\mathrm {SL}(m,{\mathbb {Z}})$$ provided $$m>3$$ . The method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in $$\mathrm {SL}(m,{\mathbb {R}})$$ (Brown et al. in Zimmer’s conjecture: subexponential growth, measure rigidity, and strong property (T), 2016. arXiv:1608.04995 ) but new ideas are needed to overcome the lack of compactness of the space $$(G \times M)/\Gamma $$ (admitting the induced G-action). Non-compactness allows both measures and Lyapunov exponents to escape to infinity under averaging and a number of algebraic, geometric, and dynamical tools are used control this escape. New ideas are provided by the work of Lubotzky, Mozes, and Raghunathan on the structure of nonuniform lattices and, in particular, of $$\mathrm {SL}(m,{\mathbb {Z}})$$ providing a geometric decomposition of the cusp into rank one directions, whose geometry is more easily controlled. The proof also makes use of a precise quantitative form of non-divergence of unipotent orbits by Kleinbock and Margulis, and an extension by de la Salle of strong property (T) to representations of nonuniform lattices.

Pathologies on the Hilbert scheme of points

Abstract: We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil’s Murphy’s Law holds up to retraction for this scheme. Our main tool is a generalized version of the Bialynicki-Birula decomposition.

Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice

Abstract: We consider a Hamiltonian given by the Laplacian plus a Bernoulli potential on the two dimensional lattice. We prove that, for energies sufficiently close to the edge of the spectrum, the resolvent on a large square is likely to decay exponentially. This implies almost sure Anderson localization for energies sufficiently close to the edge of the spectrum. Our proof follows the program of Bourgain–Kenig, using a new unique continuation result inspired by a Liouville theorem of Buhovsky–Logunov–Malinnikova–Sodin.

Real orientations of Lubin–Tate spectra

Abstract: We show that Lubin–Tate spectra at the prime 2 are Real oriented and Real Landweber exact. The proof is by application of the Goerss–Hopkins–Miller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for $$E_n$$ with its $$C_2$$ -action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $$C_2$$ -fixed points.

Arctic boundaries of the ice model on three-bundle domains

Abstract: In this paper we consider the six-vertex model at ice point on an arbitrary three-bundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon, whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo and Sportiello (J Stat Phys 164:1488–1523, 2016) as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case, which might be of independent interest.

Growth of periodic Grigorchuk groups

Abstract: On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group G we show that its growth $$v_{G,S}(n)$$ satisfies $$\lim _{n\rightarrow \infty }\log \log v_{G,S}(n)/\log n=\alpha _{0}$$, where $$\alpha _{0}=\frac{\log 2}{\log \lambda _{0}}\approx 0.7674$$, $$\lambda _{0}$$ is the positive root of the polynomial $$X^{3}-X^{2}-2X-4$$.

COHOMOLOGY OF p-ADIC STEIN SPACES

Abstract: We compute p-adic etale and pro-etale cohomologies of Drinfeld half-spaces. In the pro-etale case, the main input is a comparison theorem for p-adic Stein spaces; the cohomology groups involved here are much bigger than in the case of etale cohomology of algebraic varieties or proper analytic spaces considered in all previous works. In the etale case, the classical p-adic comparison theorems allow us to pass to a computation of integral differential forms cohomologies which can be done because the standard formal models of Drinfeld half-spaces are pro-ordinary and their differential forms are acyclic.

p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin

Abstract: Let E be a CM elliptic curve over the rationals and $$p>3$$ a good ordinary prime for E. We show that $$\begin{aligned} {\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1 \implies {\mathrm {ord}}_{s=1}L(s,E_{/{\mathbb {Q}}})=1 \end{aligned}$$for the $$p^{\infty }$$-Selmer group $${\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})$$ and the complex L-function $$L(s,E_{/{\mathbb {Q}}})$$. In particular, the Tate–Shafarevich group $$\hbox {X}(E_{/{\mathbb {Q}}})$$ is finite whenever $${\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1$$. We also prove an analogous p-converse for CM abelian varieties arising from weight two elliptic CM modular forms with trivial central character. For non-CM elliptic curves over the rationals, first general results towards such a p-converse theorem are independently due to Skinner (A converse to a theorem of Gross, Zagier and Kolyvagin, arXiv:1405.7294, 2014) and Zhang (Camb J Math 2(2):191–253, 2014).

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

Abstract: It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- $$A_\infty $$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $$ \Omega \subset \mathbb {R}^{n+1}$$ with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in $$\Omega $$ , with data in $$L^p(\partial \Omega )$$ for some $$p<\infty $$ . In this paper, we give a geometric characterization of the weak- $$A_\infty $$ property, of harmonic measure, and hence of solvability of the $$L^p$$ Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors–David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors–David bounds); moreover, the examples show that the upper and lower Ahlfors–David bounds are each quantitatively sharp.

Every classifiable simple C*-algebra has a Cartan subalgebra

Abstract: We construct Cartan subalgebras in all classifiable stably finite C*-algebras. Together with known constructions of Cartan subalgebras in all UCT Kirchberg algebras, this shows that every classifiable simple C*-algebra has a Cartan subalgebra.

An effective Chebotarev density theorem for families of number fields, with an application to $\ell$-torsion in class groups

Abstract: We prove a new effective Chebotarev density theorem for Galois extensions $$L/\mathbb {Q}$$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of L); this theorem holds for the Galois closures of “almost all” number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of L, without assuming GRH. We give many different “appropriate families,” including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidalL-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidalL-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of L-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $$\ell $$-torsion in class groups, for all integers $$\ell \ge 1$$, applicable to infinite families of fields of arbitrarily large degree.

Polyhedra inscribed in a quadric

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.