Bjoern Muetzel

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.

In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic ratio of origamis in the stratum . We compute the maximal systolic ratio of all origamis in with up to 67 squares. These computations support a conjecture of Judge and Parlier about the maximal systolic ratio in .

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 variety of S develops into a variety that splits. If the geodesic is nonseparating then the Jacobian 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. The Jacobians in this paper are represented by Gram period matrices. 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.

In Sect. 1 we reformulate a theorem of Blichfeldt in the framework of manifolds of non-positive curvature. As a result we obtain a lower bound on the number of homotopically distinct geodesic loops emanating from a common point q whose length is smaller than a fixed constant. This bound depends only on the volume growth of balls in the universal covering and the volume of the manifold itself. We compare the result with known results about the asymptotic growth rate of closed geodesics and loops in Sect. 2.

We give a general method for constructing compact K\"ahler 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.

Let S be a compact hyperbolic Riemann surface of genus $${g \geq 2}$$ g ≥ 2 . We call a systole a shortest simple closed geodesic in S and denote by $${{\rm sys}(S)}$$ sys ( S ) its length. Let $${{\rm msys}(g)}$$ msys ( g ) be the maximal value that $${{\rm sys}(\cdot)}$$ sys ( · ) 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 $${{\rm msys}(g)}$$ 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 prove several inequalities relating $${{\rm msys}(\cdot)}$$ msys ( · ) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.