Related papers: Translation surfaces with no convex presentation
We prove the existence and uniqueness of a $C^{1,1}$ solution of the $Q_k$ flow in the viscosity sense for compact convex hypersurfaces $\Sigma_t$ embedded in $R^{n+1}$ ($n \geq 2$) . In particular, for compact convex hypersurfaces with…
We classify all surfaces with constant Gaussian curvature $K$ in Euclidean $3$-space that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable. If $K=0$, we prove…
We introduce and study tame homeomorphisms of surfaces of infinite type. These are maps for which curves under iterations do not accumulate onto geodesic laminations with non-proper leaves, but rather just a union of possibly intersecting…
This book explores infinite-type translation surfaces and is intended as an introductory text for graduate and PhD students, as well as a reference for more advanced researchers. Chapter 1 introduces the three definitions of translation…
In this note we present the classification of non-symplectic automorphisms of prime order on K3 surfaces, i.e.we describe the topological structure of their fixed locus and determine the invariant lattice in cohomology. We provide new…
In this paper, we first investigate the flow of convex surfaces in the space form $\mathbb{R}^3(\kappa)~(\kappa=0,1,-1)$ expanding by $F^{-\alpha}$, where $F$ is a smooth, symmetric, increasing and homogeneous of degree one function of the…
It is conjectured that for every convex disks K, the translative covering density of K and the lattice covering density of K are identical. It is well known that this conjecture is true for every centrally symmetric convex disks. For the…
In 1949 Siegel gave an example of a complex two-torus with no nonconstant meromorphic functions. In 1964 Kodaira showed that compact complex surfaces with no nonconstant meromorphic must be of the following three types: tori, Hopf type…
We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that…
For each member of an infinite family of homology classes in the K3-surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also…
We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of…
In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More…
We classify the non-degenerate homogeneous hypersurfaces in real and complex affine four-space whose symmetry group is at least four-dimensional.
We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive…
A convex surface contracting by a strictly monotone, homogeneous degree one function of curvature remains smooth until it contracts to a point in finite time, and is asymptotically spherical in shape. No assumptions are made on the…
We study the geometry of hyperconvex representations of surface groups in ${\rm PSL}(d,\mathbb{C})$ and their deformation spaces: We produce a natural holomorphic extension of the classical Ahlfors--Bers map to a product of Teichm\"uller…
We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichm\"uller curves. Using an algorithm…
We show that in any non-arithmetic rank 1 orbit closure of translation surfaces, there are only finitely many Teichm\"uller curves. We also show that in any non-arithmetic rank 1 orbit closure, any completely parabolic surface is Veech.
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…
We announce a classification of genus 2 Veech surfaces in the stratum with a single double zero. Furthermore, we classify all completely periodic translation surfaces in genus 2.