Related papers: Curve complexes are rigid
A simple closed curve in the boundary surface of a handlebody is called primitive if there exists an essential disk in the handlebody whose boundary circle intersects the curve transversely in a single point. The primitive curve complex is…
Recently Bowden, Hensel and Webb defined the fine curve graph for surfaces, extending the notion of curve graphs for the study of homeomorphism or diffeomorphism groups of surfaces. Later Long, Margalit, Pham, Verberne and Yao proved that…
We show that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case(the double local case). Especially we give the normal form of transversely jointed…
For a fixed radius $r$ and a point $o$ in the curve complex of a surface, we define the sphere of radius $r$ to be the induced subgraph on the set of vertices of distance $r$ from $o$. We show that these spheres are almost simply connected…
Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components with $g \geq 5$, $n \geq 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $\lambda :\mathcal{T}(N) \rightarrow…
A classical result in complex geometry says that the automorphism group of a manifold of general type is discrete. It is more generally true that there are only finitely many surjective morphisms between two fixed projective manifolds of…
In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of…
We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…
We give a combinatorial characterization of the group of quasiconformal homeomorphisms of a closed, oriented surface $S$ of genus at least $2$. In particular, we prove they are exactly the automorphisms of a graph of essential quasicircles…
Suppose $S_{1}$ and $S_{2}$ are orientable surfaces of finite topological type such that $S_{1}$ has genus at least $3$ and the complexity of $S_{1}$ is an upper bound of the complexity of $S_{2}$. Let $\varphi : \mathcal{C}(S_{1})…
Consider a Hamiltonian diffeomorphism $g$ on a surface. We describe several scenarios where a curve $L$ and its image $g(L)$ provide a simple evidence that $g$ is not autonomous.
We prove that each injective simplicial map from the arc complex of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface. We also prove that the automorphism group of the…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
An integral curve is a closed piecewise linear curve comprised of unit intervals. A dome is a polyhedral surface whose faces are equilateral triangles and whose boundary is an integral curve. Glazyrin and Pak showed that not every integral…
A taming symplectic structure provides an upper bound on the area of an approximately pseudoholomorphic curve in terms of its homology class. We prove that, conversely, an almost complex manifold with such an area bound admits a taming…
A subcomplex $X\leq \mathcal{C}$ of a simplicial complex is strongly rigid if every locally injective, simplicial map $X\to\mathcal{C}$ is the restriction of a unique automorphism of $\mathcal{C}$. Aramayona and the second author proved…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of…
We determine all complex hyperelliptic curves with many automorphisms and decide which of their jacobians have complex multiplication.
Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.