Related papers: Logarithmic systolic growth for hyperbolic surface…
The purpose this article is to try to understand the mysterious coincidence between the asymptotic behavior of the volumes of the Moduli Space of closed hyperbolic surfaces of genus $g$ with respect to the Weil-Petersson metric and the…
In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to our knowledge its proof has never…
In this article we explore the relationship between the systole and the diameter of closed hyperbolic orientable surfaces. We show that they satisfy a certain inequality, which can be used to deduce that their ratio has a (genus dependent)…
We prove that the number of nodal domains of a density one subsequence of eigenfunctions grows at least logarithmically with the eigenvalue on negatively curved `real Riemann surfaces'. The geometric model is the same as in prior joint work…
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…
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface…
We show that any infinite-type surface without planar ends admits arbitrarily large families of length isospectral hyperbolic structures. If the surface has infinite genus and its space of ends is self-similar, we construct an uncountable…
We show that all hyperbolic surfaces admit an ideal triangulation with bounded shear parameters. This upper bound depends logarithmically on the topology of the surface.
Let $M$ be an orientable 3-manifold with $\partial M$ a single torus. We show that the number of boundary slopes of immersed essential surfaces with genus at most $g$ is bounded by a quadratic function of $g$. In the hyperbolic case, this…
The study of random surfaces, especially in the asymptotics of large genus, has been of increasing interest in recent years. Many geometrical questions have analogous formulations in the theory of random graphs with a large number of…
This paper is about closed hyperbolic surface amalgams with a focus on the growth of the number of closed geodesics. As in the case of surfaces, we show that topological and volume entropies coincide, but we show stark differences in how…
Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the…
Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the l-regular graphs X associated to regular tessellations of hyperbolic plane by m-gons, the denominators of the growth series…
We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that…
We present a new construction of embedded minimal surfaces in hyperbolic space with $3$ asymptotically totally geodesic ends and arbitrary finite genus.
In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by…
We obtain the exact values of the systoles of these hyperbolic surfaces of genus $g$ with cyclic symmetries of the maximum order and the next maximum order. Precisely: for genus $g$ hyperbolic surface with order $4g+2$ cyclic symmetry, the…
We study properties of typical closed geodesics on expander surfaces of high genus, i.e. closed hyperbolic surfaces with a uniform spectral gap of the Laplacian. Under an additional systole lower bound assumption, we show almost every…
Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on…
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.…