arXiv research

Classifies $π_1$-injective maps between non-compact surfaces.

Non-proper surface group action on product of trees found.

We show that immersed minimal surfaces of $\mathbb{R}^{3}$ with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map is injective the whole surface is a wide component. Prior to these results it wa…

New minimal surfaces grow area very quickly.

We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of C^2, and a complete proper holomorphic embedding into a ball of C^3.

Every nonflat conformal minimal surface is homotopic to a proper one.

Maps between non-compact surfaces can have geometric kernels under certain conditions.

Study on descent properties of complex affine surfaces under proper morphisms.

The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.

Constructs exotic proper 2-knots from open 2-handles.

Ribbonness proven for slice knots, solving an old question.

Study shows mapping class group dimension for surfaces with punctures.

Proper superminimal surfaces in hyperbolic 4-space can be approximated by conformal immersions.

Finite-type surfaces have a topological Hopf property.

The study of Bonnet surfaces in 4D space forms reveals new conformally invariant properties and characterizes proper Bonnet surfaces.

The paper studies $f$-biharmonic maps and submersions in space forms.

We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of $S^2(1/\sqrt{2})$ in $S^3$. We also give complete c…

Proves open Riemann surfaces can be embedded into 4D space.

Paper classifies special slant surfaces with varying curvature.

Alternative proof for ribbon surfaces in 3D space.

Study on dimensions of mapping class groups of non-orientable surfaces.

The Mittag-Leffler theorem is extended to meromorphic curves and minimal surfaces.

Perfect mapping class groups of specific surfaces have no proper subgroups.

CMC surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h\in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in $\sn^5$ with $|H|=h$, while a necessar…

Strong rigidity proven for non-compact surfaces.

A $3$-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part,…

The main goal of this paper is to show a counterexample to the following conjecture: {\bf Conjecture} [Meeks, Sullivan]: If $f:M\to \mathbb{R}^3$ is a complete proper minimal immersion where $M$ is a Riemannian surface without boundary and with finite genus, then $M$ is parabolic. We have proved: {\bf Theorem:} There e…

CMC-1 surfaces found on compact Riemann surfaces with Cantor sets.

A simple surface amalgam is the union of a finite collection of surfaces with precisely one boundary component each and which have their boundary curves identified. We prove if two fundamental groups of simple surface amalgams act properly and cocompactly by isometries on the same proper geodesic metric space, then the…

We show that the mapping class group of a closed surface admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension.

Examples of complete minimal surfaces properly embedded in H^2 x R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces embedded in H^2 x R, not necessarily proper, which are invariant by a vertical tra…

Proves conditions for minimal surfaces in complex hyperbolic space.

For any open orientable surface $M$ and convex domain $Ω\subset \mathbb{C}^3,$ there exists a Riemann surface $N$ homeomorphic to $M$ and a complete proper null curve $F:N\toΩ.$ This result follows from a general existence theorem with many applications. Among them, the followings: For any convex domain $Ω$ in $\mathbb…

Complex equivalence classes found in graph homotopy.

We give necessary and sufficient conditions for an affine deformation of a Schottky subgroup of O(2,1) to act properly on affine space. There exists a real-valued biaffine map between the cohomology of the Schottky group and the space of geodesic currents on the corresponding hyperbolic surface S. For a fixed cohomolog…

The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We more…

Constructs continuous families of minimal surfaces and holomorphic immersions.

Let $Σ$ be a compact Riemann surface and $D_1,...,D_n$ a finite number of pairwise disjoint closed disks of $Σ$. We prove the existence of a proper harmonic map into the Euclidean plane from a hyperbolic domain $Ω$ containing $Σ\backslash\cup_{j=1}^n D_j$ and of its topological type. Here, $Ω$ can be chosen as close as…

Study of homeomorphisms on infinite type surfaces with a classification theorem.

Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.

New insights into surface group actions and entropy.

We give a complete classification of irreducible symmetric spaces for which there exist proper SL(2,R)-actions as isometries, using the criterion for proper actions by T. Kobayashi [Math. Ann. '89] and combinatorial techniques of nilpotent orbits. In particular, we classify irreducible symmetric spaces that admit surfa…

Metric WPD for pseudo-Anosov maps shows many unbounded quasi-morphisms.

We prove that for any open Riemann surface $N$ and finite subset $Z\subset \mathbb{S}^1=\{z\in\mathbb{C}\,|\;|z|=1\},$ there exist an infinite closed set $Z_N \subset \mathbb{S}^1$ containing $Z$ and a null holomorphic curve $F=(F_j)_{j=1,2,3}:N\to\mathbb{C}^3$ such that the map $Y:Z_N\times N\to \mathbb{R}^2,$ $Y(v,P)…

Analytic completeness criterion applied to constant mean curvature surfaces.

Generalizing the notion of domains of dependence in the Minkowski space, we define and study regular domains in the affine space with respect to a proper convex cone. In dimension three, we show that every proper regular domain is uniquely foliated by a particular kind of surfaces with constant affine Gaussian curvatur…

Study on exotic smooth embeddings of surfaces in 4-manifolds, revealing different properties and complexities.

Consider a domain D in R^3 which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f : M --> D. Moreover, if D is smooth and bounded, then we prove that the immersion f : M --> D can be chosen so that the limit sets of …