Every noncompact surface is a leaf of a minimal foliation

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface Σ endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field α-attractor models, being parameterized by a positive constant α, by the choice of a finitely-generated surface group Γ⊂PSL(2,R) (which is isomorphic with the fundamental group of Σ) and by the choice of a scalar potential defined on Σ. The traditional two-field α-attractor models arise when Γ is the trivial group, in which case Σ is the Poincaré disk. We give a general prescription for the study of such models through uniformization in the so-called “non-elementary” case and discuss some of their qualitative features in the gradient flow approximation, which we relate to Morse theory. We also discuss some aspects of the SRST approximation in these models, showing that it is generally not well-suited for studying dynamics near cusp ends. When Σ is non-compact and the scalar potential is “well-behaved” at the ends, we show that, in the naive local one-field truncation, our generalized models have the same universal behavior as ordinary one-field α-attractors if inflation happens near any of the ends of Σ where the extended potential has a local maximum, for trajectories which are well approximated by non-canonically parameterized geodesics near the ends; we also discuss spiral trajectories near the ends. Generalized two field α-attractors illustrate interesting consequences of nonlinear sigma models whose scalar manifold is not simply connected. They provide a large class of tractable cosmological models with non-trivial topology of the scalar field space.

We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to $-1$ and are $C^0$, but are not necessarily $C^1$, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves as an obstruction to "higher order decay" of the Riemann curvature operator. Finally, we establish Fredholm results for geometric elliptic operators, extending the work of Rafe Mazzeo and John M. Lee to this setting. As an application, we show that any weakly asymptotically hyperbolic metric is conformally related to a weakly asymptotically hyperbolic metric of constant negative curvature.

We develop a new method to bound the hyperbolic and spherical Fourier coefficients of Maass forms defined with respect to arbitrary uniform lattices.

The classical uniformization theorem, interpreted differential geomet-rically, states that any Riemannian metric on a 2-dimensional surface ispointwise conformal to a constant curvature metric. Thus one can con-sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met-ric. The primary interest in this question is not so much to give a newproof of the uniformization theorem, but rather to understand nonlinearparabolic equations better, especially those arising in differential geome-try. A sufficiently deep understanding of parabolic equations should yieldimportant new results in Riemannian geometry.The question in the preceding paragraph has been studied by RichardHamilton [3] and Brad Osgood, Ralph Phillips and Peter Sarnak [6]. In [3],Hamilton studied the following equation, which we refer to as

Given an arbitrary Riemannian metric on a closed surface, we consider length-minimizing geodesics in the universal cover. Morse and Hedlund proved that such minimal geodesics lie in bounded distance of geodesics of a Riemannian metric of constant curvature. Knieper asked when two minimal geodesics in bounded distance of a single constant-curvature geodesic can intersect. In this paper we prove an asymptotic property of minimal rays, showing in particular that intersecting minimal geodesics as above can only occur as heteroclinic connections between pairs of homotopic closed minimal geodesics. A further application characterizes the boundary at infinity of the universal cover defined by Busemann functions. A third application shows that flat strips in the universal cover of a nonpositively curved surface are foliated by lifts of closed geodesics of a single homotopy class.

A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the pro- file of the constant curvature metric. The comparison theorem implies that the evolving metric is bounded below by the model comparison for all time and hence converges to the constant curvature profile. This yields a curvature bound and a bound on the isoperimetric constant, leading to a direct proof that the metric converges to the constant cur- vature metric.

We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi's representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation. Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every $C^k$-norm. In the case of $C^0$ and $C^1$-convergence, we achieve this by coupling two particles. To get $C^2$-convergence (in particular, convergence of the curvature), we use a coupling of three particles. This triple coupling is developed here only for the case of constant curvature metrics on surfaces, though we suspect that some variants of this idea are applicable in other situations and therefore be of independent interest. Finally, for $k\ge3$, the $C^k$-convergence follows relatively easily using induction and coupling of two particles. None of these techniques appear in the Ricci flow literature and thus provide an alternative approach to the field.

Smooth scalar curvature decrease of big scale on a sphere

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to obtain the local classification. For the global one, a suitable result on completeness for WRS yields the complete simply connected models. In particular, any of the local models in the Randers classification does admit an extension to a unique model of wind Riemannian structure, even if it cannot be extended as a complete Finslerian manifold. Thus, WRS's emerge as the natural framework for the analysis of Randers spaceforms and, prospectively, wind Finslerian structures would become important for other global problems too. For the sake of completeness, a brief overview about WRS (including a useful link with the conformal geometry of a class of relativistic spacetimes) is also provided.

We show that for given four points on the sphere and prescribed angles at these points which are not multiples of 2 π 2\pi , the number of metrics of curvature 1 1 having conic singularities with these angles at these points is finite.

In this paper, we consider a hyperbolic structure on a manifold to be a Riemannian metric of constant sectional curvature −1 which is not necessarily complete. A hyperbolic 3-manifold is an orientable 3-manifold with a hyperbolic structure. It is well known that the complement of the figure-eight knot K in the 3-dimensional sphere S admits a complete, finite volume hyperbolic structure σ∞. By Mostow-Prasad rigidity, such a hyperbolic structure on S − K is unique. Incomplete hyperbolic structures are also of interest. Indeed, Thurston [8] analyzed the flexibility of hyperbolic structures on S − K by allowing incomplete hyperbolic structures. In fact, he showed that there are deformations of σ∞ on S3−K that are not complete hyperbolic structures. Such deformations are holomorphically parametrized by points in an open subset U of a complex affine plane curve C. The curve C and this subset U are given in (1) and (2) below. When the parameter of deformation becomes close to the boundary ∂U or enters C−U , degeneration of hyperbolic structures on S3−K occurs. There are cases in which such degeneration results in closed 3-manifolds obtained by Dehn fillings of S −K with other types of geometric structure. In fact, twenty such cases have been identified. In each case, this resultant closed 3-manifold is one of the following types: a Sol-manifold, a PSL2(R)-manifold, a Haken manifold which is decomposed into a PSL2(R)-manifold and a Euclidean manifold along an embedded torus, or a Euclidean orbifold. The genus of the curve C is one, because C is of degree four and has two ordinary double points. In fact, we give an explicit form of a birational map from C to a non-singular plane cubic curve in Weierstrass form E, which is given in (4) below. The curve E is an example of what is called an ‘elliptic curve defined over Q’. It is well known that any elliptic curve is an abelian group under an addition law. In this paper, we see that there is a concrete

Let $F=\alpha\phi(s)$, $s:=\beta/\alpha$, be a generalized Berwald $(\alpha,\beta)$-metric on a $2$-dimensional manifold. Suppose that $F$ has vanishing $S$-curvature and $\phi'(0)\neq 0$. We show that if $F$ is regular, then it is a locally Minkowskian metric. If $F$ is an almost regular and non-locally Minkowskian metric, then we explicitly determine the function $\phi$ which results in generalized Berwald metrics not belonging to the classes of Berwald, Landsberg or Douglas metrics. Furthermore, we prove that a left-invariant Finsler metric on a $2$-dimensional Lie group has vanishing $S$-curvature if and only if it is a Riemannian metric of constant Gaussian curvature. Finally, we construct a family of Randers-type generalized Berwald metrics on an arbitrary odd-dimensional manifold.

The Uniformization Theorem due to Koebe and Poincare implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

We consider canonical determinantal random point processes with N particles on a compact Riemann surface X defined with respect to the constant curvature metric. We establish strong exponential concentration of measure type properties involving Dirichlet norms of linear statistics. This gives an optimal central limit theorem (CLT), saying that the fluctuations of the corresponding empirical measures converge, in the large N limit, towards the Laplacian of the Gaussian free field on X in the strongest possible sense. The CLT is also shown to be equivalent to a new sharp strong Szego-type theorem for Toeplitz determinants in this context. One of the ingredients in the proofs are new Bergman kernel asymptotics providing exponentially small error terms in a constant curvature setting.

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a discrete Yamabe flow with surgery.