arXiv research

Embedding calculus proves convergence for surfaces.

Study on embedding surfaces into 3-manifolds, focusing on equivariant cases.

The study embeds graphs on translation surfaces, proving essential-systolic embeddings and estimating surface genera.

Discrete maximal surfaces identified from s-embeddings.

given two minimal surfaces embedded in $\S3$ of genus $g$ we prove the existence of a sequence of non-congruent compact minimal surfaces embedded in $\S3$ of genus $g$ that converges in $C^{2,α}$ to a compact embedded minimal surface provided some conditions are satisfied. These conditions also imply that, if any of th…

This paper answers a question about discrete embeddings to maximal surfaces.

A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.

Geodesic surfaces embed into hyperbolic 3-manifolds for all finite group actions.

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

In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; the…

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

For leveled spatial graphs, we find a surface embedding that allows cellular embedding.

For a non-orientable closed surface standardly embedded in the 4-sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou-Marin quadratic form of this embedded surface.

We classify all rotational surfaces in Euclidean space whose principal curvatures $κ_1$ and $κ_2$ satisfy the linear relation $κ_1=aκ_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find clos…

Constructs a family of genus three minimal surfaces with parallel ends.

The paper proves an adjunction inequality for Real embedded surfaces in 4-manifolds.

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz reflection extends to a complete minimal surface. Studying the group of Schwarz refl…

Algorithm calculates genus of embedded graphs on surfaces.

Survey article on multibranched surfaces in 3-manifolds.

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed pi_1-injective surface of negative Euler characteristic. We also determine …

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, Σ)$ among all possible embedded closed/bordered surfaces with given geometric/algebraic g…

Embeddings preserve stable commutator length for surfaces.

Article provides conditions for embedding braid groups into mapping class groups.

Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic embeddings in $\mathbb{R}^{3}$ of compact Riemann surfaces with finitely many punctures…

Arnlind, Hoppe and Huisken showed how to express the Gauss and mean curvature of a surface embedded in a Riemannian manifold in terms of Poisson brackets of the embedding coordinates. We generalize these expressions to the pseudo-Riemannian setting and derive explicit formulas for the case of surfaces embedded in $\R^m…

An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic su…

The Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmoni…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…

We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.

Improved bound on first eigenvalue of minimal surfaces in $S^3$.

We prove the existence of embedded closed constant curvature curves on convex surfaces.

We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space $\mathbb{H}^3$, and we use it to prove that any open, connected, orientable surface can be properly embedded in $\mathbb{H}^3$ as an area-minimizing surface. Moreover, the embedding can be constructed in such a way that t…

Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.

Spaces of circle embeddings in curved surfaces indexed by trees.

Graphs from van der Corput sequence embed into Chamanara surface.

Study characterizes quasi-isometric embeddings of maps from cusped surfaces into moduli space.

New formulas for minimal surfaces with specific end conditions.

Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity, and we analyze what happens when one blows up the surfaces near a point where…

Classifies and constructs all extendable automorphisms of closed surfaces over the 3-sphere.

Proved contractibility of geodesic triangulation space on hyperbolic surfaces.

Maps on surfaces can be embedded into spheres with minimal dimensions.

We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuch…

Those maps of a closed surface to the three-dimensional torus that are homotopic to embeddings are characterized. Particular attention is paid to the somewhat intricate case when the surface is nonorientable.

Embeddings of surfaces in 5-manifolds are classified by an invariant.

Nonorientable surface mapping class group embeds quasi-isometrically in its orientable cover.

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…

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

Vortices on conical surfaces embedded in hyperbolic space.