Related papers: Translation surfaces with large systoles
The flow in a fixed direction on a translation surface S determines a decomposition of S into closed invariant sets, each of which is either periodic or minimal. We study this decomposition for translation surfaces in the hyperelliptic…
Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length…
Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite…
In this paper, we study the $k$-Hessian curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We first prove the existence of translating solutions with given asymptotic behavior. Then, we prove that for strictly convex…
In this paper, we study translation surfaces in the Euclidean space endowed with a canonical semi-symmetric non-metric connection. We completely classify the translation surfaces of constant sectional curvature with respect to this…
A translational surface is a tensor product surface constructed from two space curves by translating one along the other. These surfaces are common within geometric modeling and, since their description is parametric, it is desirable to…
In this note we are interested in the dynamics of the linear flow on infinite periodic $\mathbb{Z}^d$-covers of Veech surfaces. An elementary remark allows us to show that the kernel of some natural representations of the Veech group acting…
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…
We consider translation surfaces with poles on surfaces. We shall prove that any finite group appears as the automorphism group of some translation surface with poles. As a direct consequence we obtain the existence of structures achieving…
In this note we study the systoles of convex hypersurfaces in $\mathbb{R}^{2n}$ invariant under an anti-symplectic involution. We investigate a uniform upper bound of the ratio between the systole and the symmetric systole of the…
In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure on the moduli space $\mathcal{M}_g$. We show that as $g$ goes to infinity, a generic surface…
We study the quantity $\mbox{KVol}$ defined as the supremum, over all pairs of closed curves, of their algebraic intersection, divided by the product of their lengths, times the area of the surface. The surfaces we consider live in the…
Dynamical systems on an infinite translation surface with the lattice property are studied. The geodesic flow on this surface is found to be recurrent in all but countably many rational directions. Hyperbolic elements of the affine…
We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.
We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation…
Let $(M,g)$ be a closed, oriented, Riemannian manifold of dimension $m$. We call a systole a shortest non-contractible loop in $(M,g)$ and denote by $sys(M,g)$ its length. Let $SR(M,g)=\frac{{sys(M,g)}^m}{vol(M,g)}$ be the systolic ratio of…
We review the different notions about translation surfaces which are necessary to understand McMullen's classification of $GL_2^+(\mathbb{R})$-orbit closures in genus two. In Section 2 we recall the different definitions of a translation…
Continuing the work in \cite{ergodic-infinite}, we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group…
In this paper, for each $k\in \mathbb{N}$, we define a complex $\Gamma_k(S)$ for an infinite-type surface $S$ with non-planar ends, which serves as an analog of the Hatcher-Thurston complex for the infinite-type setting. We show that…
The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus $g$ can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice…