Given an smooth function $K <0$ we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genus $g>1$. We do so by minimizing an appropriate functional using elementary analysis. In particular for $K$ a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genus $g >1$.
Let $h$ be a complete metric of Gaussian curvature $K_0$ on a punctured Riemann surface of genus $g \geq 1$ (or the sphere with at least three punctures). Given a smooth negative function $K$ with $K=K_0$ in neighbourhoods of the punctures we prove that there exists a metric conformal to $h$ which attains this function as its Gaussian curvature for the punctured Riemann surface. We do so by minimizing an appropriate functional using elementary analysis.
We construct higher genus Riemann's minimal surfaces properly embedded in the Euclidean space. To do that we glue end by end a Costa-Hoffman-Meeks examples to two halves genus zero Riemann's minimal surfaces. In first we need to perform a deformation of a Costa-Hoffman-Meeks example to prescribe the flux vector along the catenoidal ends. Then we study the mapping property of the Jacobi operator on the half Riemann example as a perturbation analysis of a CMC-Delaunay half cylinder.
Involutions without fixed points on hyperbolic closed Riemann surface are discussed. For an orientable surface $X$ of even genus with an arbitrary Riemannian metric $d$ admitting an involution $τ$, it is known that $\min_{p\in X}d(p,τ(p))$ is bounded by a constant which depends on the genus of $X$. The equivalent result is proved to be false in odd genus, and the optimal constant for hyperbolic Riemann surfaces is calculated in genus 2.
We show the existence of 1-parameter families of non-periodic, complete, embedded minimal surfaces in euclidean space with infinitely many parallel planar ends. In particular we are able to produce finite genus examples and quasi-periodic examples of infinite genus.
Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (globally) maximal surface $S_{max}$ a compact Riemann surface of genus $g$ whose systole has length $\mathop{msys}(g)$. In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prov…
We give examples of isospectral non-isometric surfaces of genus 2 and 3 with variable curvatures and apply the result to construct isospectral potentials on Riemann surfaces of genus 2.
In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comparable to $\frac{1}{g^2}$ in $g$. In the proof of the upper bound, for any constant $ε>0$, we will construct a closed Riemann surface of genus $g$ in any $ε$-thick part of moduli space such that it admits a pants decomposition whose boundary curves all have length equal to $ε$, a…
Brooks and Makover introduced an approach to random Riemann surfaces based on associating a dense set of them - Belyi surfaces - with random cubic graphs. In this paper, using Bollobas model for random regular graphs, we examine the topological structure of these surfaces, obtaining in particular an estimate for the expected value of their genus.
Let $X$ be a compact connected Riemann surface of genus at least two. The main theorem of arxiv:1010.1488 says that for any positive integer $n \leq 2({\rm genus}(X)-1)$, the symmetric product $S^n(X)$ does not admit any Kähler metric satisfying the condition that all the holomorphic bisectional curvatures are nonnegative. Our aim here is to give a very simple and direct proof of this result of Bökstedt and Romão.
This article is an attempt to generalize Riemann's bilinear relations on compact Riemann surface of genus at least 2, which may lead to new structures in the theory of hyperbolic Riemann surfaces. No significant result is obtained, the article serves to bring the readers' attention to the observation made by [Bol-1949], and some easy consequences.
To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed, that the square of the first successive minimum, the squared norm of the shortest non-zero vector in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. We show, that in the case of a hyperelliptic surface this geometric invariant is bounded …
We will show that any open Riemann surface $M$ of finite genus is biholomorphic to an open set of a compact Riemann surface. Moreover, we will introduce a quotient space of forms in $M$ that determines if $M$ has finite genus and also the minimal genus where $M$ can be holomorphically embedded.
Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the inclusion maps converge in $C^0$ to a map from a surface of genus $g$ to $M$. We also show that, on passing to a further subsequence, the distance functions corresponding to pullback metrics converge to a pseudo-metric of fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and g…
The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface $S_{m,s}$ with $s$ ends and exactly $m$ of them with infinite genus, such that $m,s\in \mathbb{N}$ and $1<m\leq s$, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group $Γ_{m,s}$ such that the quotient space $\mathbb{H}/ Γ_{m, s}$ is homeomorphic to $S_{m,s}$ and has infinite area. For this construction, we exhibit a hyperbolic polygon with an infinite number of sides and giv…
In this article, we provide bounds on systoles associated to a holomorphic $1$-form $ω$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,ω)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $ω$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.
In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than $\frac{\mathcal{L}_1(X_g)}{g^2}$ up to a uniform positive constant multiplication. Where $\mathcal{L}_1(X_g)$ is the shortest length of multi closed curves separating $X_g$. Moreover,we also show that this new lower bound is optimal as $g \to \infty$.
We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation for the action functional in terms of the geometry of different fiber…
We consider stable minimal surfaces of genus 1 in Euclidean space and in Riemannian manifolds. Under the condition of covering stability (all finite covers are stable) we show that a genus 1 finite total curvature minimal surface in $\mathbb R^n$ lies in an even dimensional affine subspace and is holomorphic for some constant orthogonal complex structure. For stable minimal tori in Riemannian manifolds we give an explicit bound on the systole in terms of a positive lower bound on the isotropic curvature. As an application we estimate the systole of noncyclic abelian subgroups of the fundamenta…
A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater…
Let $S_g$ be a closed surface of genus $g$ and $\mathbb{M}_g$ be the moduli space of $S_g$ endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of $\mathbb{M}_g$ for large genus $g$. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectional curvatures at certain thick surfaces in $\mathbb{M}_g$ as $g \to \infty$. Then we prove two curvature properties on the whole space $\mathbb{M}_g$ as $g\to \infty$ in a probabilistic way.
Let $(X,g)$ be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure $J$ compatible with $g$, then the canonical bundle $K_X$ is not pseudo-effective and the Kodaira dimension $κ(X,J)=-\infty$. We also introduce the complex Yamabe number $λ_c(X)$ for compact complex manifold $X$, and show that if $λ_c(X)>0$, then $κ(X)=-\infty$; moreover, if $X$ is also spin, then the Hirzebruch $A$-hat genus $\hat A(X)=0$.
Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length $γ_1\le γ_2\le ... \le γ_i ,...$, then for fixed $k$, if one allows the genus to go to infinity, the length of $γ_{k}$ is inde…
In this paper we study the behavior of the spectrum of a compact, connected Riemannian manifold $(M,g)$ of dimension $d \ge 2$, when we add an increasing number of increasingly small handles. No assumptions on any of the curvatures are needed.
Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self-contained survey of their basic spectral properties, we study the zeta-regularized determinant of the Laplacian as a functional on the moduli space of these surfaces. An explicit formula for this determinant is obtained.
For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.
We relate the Donaldson invariants of two four-manifolds $X_i$ with embedded Riemann surfaces of genus 2 and self-intersection zero with the invariants of the manifold X which appears as a connected sum along the surfaces. When the original manifolds are of simple type with $b_1=0$ and $b^+>1$, X is of simple type with $b_1=0$ and $b^+>1$ as well, and the relationship between the invariants is expressed as constraints in the basic classes for X. Also we give some applications. For instance, if $X_i$ have both $b_1=0$ then X is of simple type with $b_1=0$, $b^+>1$, and has no basic classes eval…
On the thick part of the moduli space of Riemann surfaces, where there is a positive lower bound of the systole of the surface, we show that all Weil-Petersson Riemannian curvatures are bounded, independent of the genus of the surface.
The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number of appearances of fixed finite sets of combinatorial types of curves. Of any such set we determine the asymptotics of the probability distribution. It turns out that these distributions are independent of the genus in an appropriate sense. As an application of our results we stu…
