arXiv subjects
Publications and source records attributed to Eran Makover.
In this paper we continue to investigate the systolic landscape of translation surfaces started in [CHMW]. We show that there is an infinite sequence of surfaces $(S_{g_k})_k$ of genus $g_k$, where $g_k \to \infty$ with large systoles. On the other hand we show that for hyperelliptic surfaces we can find a suitable homology basis, where a large number of loops that induce the basis are short.
We present four counterexamples in surface homology. The first example shows that even if the loops inducing a homology basis intersect each other at most once, they still may separate the surface into two parts. The other three examples show some difficulties in working with minimal homology bases.
Given a hyperelliptic hyperbolic surface $S$ of genus $g \geq 2$, we find bounds on the lengths of homologically independent loops on $S$. As a consequence, we show that for any $λ\in (0,1)$ there exists a constant $N(λ)$ such that every such surface has at least $\lceil λ\cdot \frac{2}{3} g \rceil$ homologically independent loops of length at most $N(λ)$, extending the result in [Mu] and [BPS]. This allows us to extend the constant upper bound obtained in [Mu] on the minimal length of non-zero period lattice vectors of hyperelliptic Riemann surfaces to almost $\frac{2}{3} g$ linearly independent vectors.
We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface $S$ where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian torus of $S$ develops into a torus that splits. If the geodesic is nonseparating then the Jacobian torus of $S$ degenerates. The aim of this work is to get insight into this process and give estimates in terms of geometric data of both the initial surface $S$ and the final surface, such as its injectivity radius and the lengths of geodesics that form a homology basis. As an invariant we introduce new families of symplectic matrices that compensate for the lack of full dimensional Gram-period matrices in the noncompact case.
We give a general method for constructing compact Kähler manifolds $X_1$ and $X_2$ whose intermediate Jacobians $J^k(X_1)$ and $J^k(X_2)$ are isogenous for each $k$, and we exhibit some examples. The method is based upon the algebraic transplantation formalism arising from Sunada's technique for constructing pairs of compact Riemannian manifolds whose Laplace spectra are the same. We also show that the method produces compact Riemannian manifolds whose Lazzeri Jacobians are isogenous.
In this paper we construct quasiconformal embeddings from Y-pieces that contain a short boundary geodesic into degenerate ones. These results are used in a companion paper to study the Jacobian tori of Riemann surfaces that contain small simple closed geodesics.
We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for small values of $n$. We also show that this function is greater than a function that grows logarithmically in function of the ratio $g/n$.
A result of Bangert states that the stable norm associated to any Riemannian metric on the $2$-torus $T^2$ is strictly convex. We demonstrate that the space of stable norms associated to metrics on $T^2$ forms a proper dense subset of the space of strictly convex norms on $\R^2$. In particular, given a strictly convex norm $\Norm_\infty$ on $\R^2$ we construct a sequence $<\Norm_j >_{j=1}^{\infty}$ of stable norms that converge to $\Norm_\infty$ in the topology of compact convergence and have the property that for each $r > 0$ there is an $N \equiv N(r)$ such that $\Norm_j$ agrees with $\Norm_\infty$ on $\Z^2 \cap \{(a,b) : a^2 + b^2 \leq r \}$ for all $j \geq N$. Using this result, we are able to derive results on multiplicities which arise in the minimum length spectrum of $2$-tori and in the simple length spectrum of hyperbolic tori.
We investigate the size of the embedded regular tree rooted at a vertex in a $d$ regular random graph. We show that almost always, the radius of this tree will be ${1/2}\log n$, where $n$ is the number of vertices in the graph. And we give an asymptotic estimate for Gauss' Hypergeometric Function.
We consider random Fibonacci sequences given by $x_{n+1}=\pm βx_{n}+x_{n-1}$. Viswanath (\cite{viswanath}), following Furstenberg (\cite{furst}) showed that when $β= 1$, $\lim_{n\to \infty}|x_{n}|^{1/n}=1.13...$, but his proof involves the use of floating point computer calculations. We give a completely elementary proof that $1.25577 \ge (E(|x_{n}|))^{1/n} \ge 1.12095$ where $E(|x_{n}|)$ is the expected value for the absolute value of the $n$th term in a random Fibonacci sequence. We compute this expected value using recurrence relations which bound the sum of all possible $n$th terms for such sequences. In addition, we give upper an lower
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 independent of the genus.
In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi surfaces. And in this construction we can control the geometry of the compact Riemann surface by the geometry of the graph. We show that almost all such surfaces have large first eigenvalue and large Cheeger constant.
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.