For the moduli space of unmarked convex $\mathbb{RP}^2$ structures on the surface $S_{g,m}$ with negative Euler characteristic, we investigate the subsets of the moduli space defined by the notions like boundedness of projective invariants, area, Gromov hyperbolicity constant, quasisymmetricity constant etc. These subsets are comparable to each other. We show that the Goldman symplectic volume of the subset with certain projective invariants bounded above by $t$ and fixed boundary simple root lengths $\mathbf{L}$ is bounded above by a positive polynomial of $(t,\mathbf{L})$ and thus the volume of all the other subsets are finite. We show that the analog of Mumford's compactness theorem holds for the area bounded subset.
In this paper we survey a number of recent results concerning the existence and moduli spaces of solutions of various geometric problems on noncompact manifolds. The three problems which we discuss in detail are: I. Complete properly immersed minimal surfaces in $\RR^3$ with finite total curvature. II. Complete embedded surfaces of constant mean curvature in $\RR^3$ with finite topology. III. Complete conformal metrics of constant positive scalar curvature on $M^n \setminus Λ$, where $M^n$ is a compact Riemannian manifold, $n\geq3$ and $\Lam \subset M$ is closed. The existence results we discu…
In this short article we investigate the topology of the moduli space of two-convex embedded tori $S^{n-1}\times S^1\subset \mathbb{R}^{n+1}$. We prove that for $n \geq 3$ this moduli space is path-connected, and that for $n = 2$ the connected components of the moduli space are in bijective correspondence with the knot classes associated to the embeddings. Our proof uses a variant of mean curvature flow with surgery developed in our earlier article (arXiv:1607.05604) where neck regions are deformed to tiny strings instead of being cut out completely, an approach which preserves the global topo…
Let S be an orientable, finite type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first author. In this article, we give an explicit description of this topology using explicit coordinates. More precisely, given every point in this augmented moduli space, we find explicit continuous coordinates on the quotient of a suitable open neighborhood about this point by a suitable subgroup of the mapping class group of S. Using this, we give a simpler proof of the fact that the augmented moduli s…
We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifolds.
There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to t…
Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of punctured surfaces. We extend this cell decomposition to the moduli space of decorated strictly convex projective structures of finite volume on punctured surfaces. The proof uses Fock and Goncharov's $\mathcal{A}$-coordinates for doubly decorated structures. In addition, we describe a s…
Let S be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural correspondence between convex RP^2 structures on S and pairs (Σ,U) consisting of a conformal structure Σon S and a holomorphic cubic differential U over Σ. The pairs (Σ,U$, for Σvarying in moduli space, allow us to define natural holomorphic coordinates on the moduli space of convex RP^2 structures. We consider geometric limits of convex RP^2 structures on S in which the RP^2 structure degenerates only along a set of simple, non-int…
We define convex projective structures on 2D surfaces with holes and investigate their moduli space. We prove that this moduli space is canonically identified with the higher Teichmuller space for the group PSL_3 defined in our paper math/0311149. We define the quantum version of the moduli space of convex projective structures on surfaces with holes. The present paper can serve as an introduction to math/0311149. In the Appendix we show that the space of configurations of 5 flags in the projective plane is of cluster type E_7.
We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS13], we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary.
We construct a compact convex generating set $\mathcal{C}_n$ of the moduli set of closed connected projective special real manifolds of fixed dimension $n$. We show that a closed connected projective special real manifold corresponds to an inner point of $\mathcal{C}_n$ if and only if it has regular boundary behaviour. Our results can be used to describe deformations of 5d supergravity theories with complete scalar geometries.
By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with $n$ cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a natural decomposition into real hyperbolic convex polyhedra of dimensions $n-3$ and $\leq \frac{1}{2}(n-1)$. By a result of W.~Veech, the moduli space of flat metrics on a compact surface with cone singularities of prescribed negative curvatures has a foliation whose leaves have a…
We introduce the notion of affine Legendrian submanifolds in Sasakian manifolds and define a canonical volume called the $ϕ$-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the $ϕ$-volume to obtain the stability result in some $η$-Einstein Sasakian manifolds. It also implies the convexity of the $ϕ$-volume functional on the space of affine Legendrian submanifolds. Next, we introduce the notion of special affine Legendrian submanifolds in Sasaki-Einstein manifolds as a generalization of that of speci…
We introduce the moduli space of spectral curves of constant mean curvature (\cmc\hspace{-5pt}) cylinders of finite type in the round unit 3-sphere. The subset of spectral curves of mean-convex Alexandrov embedded cylinders is explicitly determined using a combination of integrable systems and geometric analysis techniques. We prove that these cylinders are surfaces of revolution. As a consequence all mean-convex Alexandrov embedded {\sc{cmc}} tori in the 3-sphere are surfaces of revolution.
We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie-Loftin parameterization of convex RP^2 structures on a compact surface by the bundle of holomorphic cubic differentials over Teichmuller space.
The purpose of this note is introduce a new axiom (called the Descent Axiom) in the theory of $r$-spin cohomological field theories. This axiom explains the origin of gravitational descendants in this theory. Furthermore, the Descent Axiom immediately implies the Vanishing Axiom, explicating the latter (which has no a priori analog in the theory of Gromov-Witten invariants), in terms of the multiplicativity of the virtual class. We prove that the Descent Axiom holds in the convex case, and consequently in genus zero.
A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space $V$, then apply this framework to show that the moduli space $\mathcal{M}(P)$ of flat connections on a principal bundle $P$ over a compact manifold $M$ is a polysymplectic reduction of the space $\mathcal{A}(P)$ of all connections on $P$ by the action of the gauge group $\mathcal{G}$ with respect to a natural $Ω^2(M)/B^2(M)$-valued symplectic structure on $\mathcal{A}(P)…
We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami and Rivin. It is proved by A. Kitaev and the author that any closed 3-manifold has a triangulation supporting an angle structure. The moduli space of all angle structures on a triangulated 3-manifold is a bounded open convex polytope in a Euclidean space. The volume of an angle s…
It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed and endowed with a suitable area, is isometric to a spherical polyhedron. These polygons have an infinite number of vertices, are space-like, contained in the future cone of the origin, and setwise invariant under the action of a linear isometry.
A protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit PL (piecewise linear) convex function from the interval with at most 18 non-linear points. Forgetting its actual function behaviour, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples. From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyperKahler metrics to conjectural segments, as in the work of Honda-Sun-Zhang. On t…
Let $X$ be a closed $3$-manifold, $\mathcal{M}_{R>0}$ the space of metrics on $X$ with positive scalar curvature, and $\text{Diff}(X)$ the group of diffeomorphisms of $X$. Marques proves the fundamental result that $\mathcal{M}_{R>0}/ \text{Diff}(X)$ is path connected. Using this and the theorem of Cerf in differential topology, Marques shows that the space of asymptotically flat metrics with nonnegative scalar curvature on $\mathbb{R}^3$ is path connected. Based on Carlotto-Li's generalization of Marques' result to the case of compact manifold with boundary, we show that the space of asymptot…
Using global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on $S\times\mathbb{R}$ is the tangent bundle of the Teichmüller space of $S$, if $S$ is a closed surface. One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relation between the Mess parameters and the embedding data of any Cauchy surface. This relation is pointed out by using some specific properties of Codazzi tensors on hyperbolic surfaces. As a by-product we get a new Lorentzian proof of Goldman's celebrated result about the coinciden…
Denote the free group on two letters by F2 and the SL(3,C)-representation variety of F2 by R = Hom(F2, SL(3, C)). There is a SL(3,C)-action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine variety X. We determine explicit minimal generators and defining relations for the subring of invariants and show X is a hyper-surface in C9. Our choice of generators exhibit Out(F2) symmetries which allow for a succinct expression of the defining relations. We then show C[X] is a Poisson algebra with respect to a presentation of F2 imposed by a punctured surfac…
