arXiv research

Introduces fundamental heaps for surfaces, linking them to cocycle invariants.

In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In th…

We determine all the Q-fundamental surfaces in $(p,1)$-lens spaces and $(p,2)$-lens spaces with respect to natural triangulations with $p$ tetrahedra. For general $(p,q)$-lens spaces, we give an upper bound for elements of vectors which represent Q-fundamental surfaces with no quadrilateral normal disks disjoint from t…

This paper classifies fundamental groups of perforated surfaces.

The goal of these lectures is to give an introduction to the study of the fundamental group of a Klein surface. We start by reviewing the topological classification of Klein surfaces and by explaining the relation with real algebraic curves. Then we introduce the fundamental group of a Klein surface and present its mai…

Study properties of surfaces with nonvanishing third fundamental form.

Proves surface embedding theorem for 4-manifolds with good fundamental group.

Study mapping class group actions on surface fundamental groups.

Study surfaces with free product fundamental groups, proving existence and properties.

The theorem connects surface mapping groups to fundamental groupoids.

Constructs constant mean curvature surfaces using geometric flow.

We obtain an infinite family of complete non embedded rotational surfaces in $\mathbb R^3$ whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homo…

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in the fundamental group of M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a "sufficient" collection of surfaces in M, then the fundamental group of M is…

It is proved that every disconnected surface-link with meridian-based free fundamental group is a trivial (i.e., an unknotted-unlinked) surface-link. This result is a surface-link version of the author's recent announcement result on smooth unknotting of a surface-knot.

The paper proves fundamental theorems for timelike surfaces in Minkowski 4-space.

Study reveals vanishing Massey products on compact complex surfaces, impacting their fundamental group structure.

Study characterizes points on projective surfaces using a cubic form.

Study surfaces in 4-manifolds with cyclic fundamental group.

In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals. Also, we characterize in a simple way these singular surfaces and its fundament…

The study improves fundamental gap estimates for surfaces with non-constant positive curvature.

The paper provides infinite presentations for surface groups.

Study covers of surfaces, showing types and properties.

It is shown that with finitely many exceptions, the fundamental group obtained by Dehn surgery on a one cusped hyperbolic 3-manifold contains the fundamental group of a closed surface.

In this paper we study the second fundamental form of translation surfaces in E3. We give a non-existence result for polynomial translation surfaces in E3 with vanishing second Gaussian curvature KII. We classify those translation surfaces for which KII and H are proportional. Finally we obtain that there are no II-min…

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

No nontrivial automorphisms for cubic surfaces moduli space.

Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted. Using a twist-spinning construction from high-dimensional knot theory, we constr…

Proves incoherence of free-by-free and surface-by-free groups, solving two problems.

We prove that if S is a properly embedded incompressible surface in a compact 3-manifold M, then the fundamental group of S is separable in the fundamental group of M.

Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.

In this paper we study the topology of compact manifolds of positive isotropic curvature (PIC). There are many examples of non-simply connected compact manifolds with positive isotropic curvature. We prove that the fundamental group of a compact Riemannian manifold with PIC, of dimension greater than or equal to 5, doe…

New method for quandle presentations of surface knots in 4-manifolds.

Classifies $4$-manifolds with specific properties and fundamental group.

The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres.

Lower bound on crossing number of knots and links on a 2-sided surface in a 3-manifold

This note provides a formula for the character of the Lie algebra of the fundamental group of a surface, viewed as a module over the symplectic group.

Classification of groups as symmetries of infinite translation surfaces.

We give a diameter bound for fundamental domains for isometric actions of the fundamental group of a closed hyperbolic surface on a delta-hyperbolic space, where the bound depends on the hyperbolicity constant delta, the genus of the surface, and the injectivity radius of the action, which we assume to be strictly posi…

Formula found for surfaces in Sol_3, leading to gap results.

The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the…

Characterizes Stein surfaces with finite homotopy rank-sum.

We investigate some relations concerning the first and the second Beltrami operators corresponding to the fundamental forms I, II, III of a surface in the three-dimensional Euclidean space and we study surfaces which are of finite type in the sense of B.-Y. Chen with respect to the fundamental forms II and III.

The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.

Study of complex surfaces in a specific pseudo-Riemannian space.

There is a question asking whether a handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link. This question for the case of a trivial surface-link is affirmatively answered. That is, a handle-irreducible summand of every stably trivial surface-link is only a trivial 2-link. By com…

Study knotted surfaces in simply-connected 4-manifolds with trivial boundary.

We determine which amalgamated products of surface groups identified over multiples of simple closed curves are not fundamental groups of 3-manifolds. We prove each surface amalgam considered is virtually the fundamental group of a 3-manifold. We prove that each such surface group amalgam is abstractly commensurable to…

We show that for each aspherical compact complex surface $X$ whose fundamental group $π$ fits into a short exact sequence $1\to K \to π\to π_1(S) \to 1$ where $S$ is a compact hyperbolic Riemann surface and the group $K$ is finitely-presentable, there is a complex structure on $S$ and a nonsingular holomorphic fibr…