相关论文: Simple closed geodesics on hyperelliptic translati…

In this article, we consider a closed rank one Riemannian manifold $M$ without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on $M$ with length at most $t$, and $\# P(t)$ its cardinality. We…

This purpose of this paper is to prove the following result: let phi be a strictly convex, smooth, convex body in the Euclidean plane, if the intersection of n translates of phi has a non-empty interior, and all of the translates contribute…

Consider a compact surface of genus $\geq 2$ equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than $2\pi$. Following the technique in the work of Burns, Climenhaga, Fisher, and Thompson,…

We show that the number of simple closed geodesics of length bounded by L on a hyperbolic surface of genus g with c cusps and b boundary components grows roughly like L^{6g+2b+2c-6}. This has been conjectured for some time.

In this note, we consider the entropy of unit area translation surfaces in the $SL(2, \mathbb R)$ orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a…

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 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 study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface,…

Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces…

We study translation minimal hypersurfaces and separable minimal hypersurfaces in the ($n+1$)-space with $2m$-norm.

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.

We prove that any ergodic $SL_2(R)$-invariant probability measure on a stratum of translation surfaces satisfies strong regularity: the measure of the set of surfaces with two non-parallel saddle connections of length at most $\epsilon_1,…

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this…

In the closed, non-Haken, hyperbolic class of examples generated by (2p,q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio p/q and determined to be unique in all…

The Gauss-Bonnet formula for classical translation surfaces relates the cone angle of the singularities (geometry) to the genus of the surface (topology). When considering more general translation surfaces, we observe so-called wild…

Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such…

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension…

A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with…

We prove that if a geodesic flow on a closed orientable $C^\infty$ surface is transitive and has positive topological entropy, then it has a unique measure of maximal entropy. This covers all previous results of the literature on the…

We compute the maximal ratio of the algebraic intersection of two closed curves on two families of translation surfaces with multiple singularities. This ratio, called the interaction strength, is difficult to compute for translation…