arXiv research

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

We construct a sequence of embedded minimal disks in a ball where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper.

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

We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $\mathcal{C}^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.

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.

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…

Survey on Stein manifolds embeddings and immersions, including a new result.

Ribbonness proven for slice knots, solving an old question.

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

Study proper holomorphic maps between symmetric domains, proving rigidity and isometric properties.

New minimal surfaces grow area very quickly.

For any $H$ in (0,1/2), we construct complete, non-proper, stable, simply-connected surfaces embedded in $H^2xR$ with constant mean curvature $H$.

For any H in [0,1), we construct complete, non-proper, stable, simply-connected surfaces with constant mean curvature H embedded in hyperbolic 3-space.

Alternative proof for ribbon surfaces in 3D space.

Generalizes Pontryagin's construction for proper maps in stable dimensions.

We prove that every proper $n$-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space $\mathbb{R}^{3n+6,1}$. By an "approximate isometric embedding" we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.

The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.

In this paper we prove that every bordered Riemann surface M admits a complete proper null holomorphic embedding into a ball of the complex Euclidean $3$-space $\mathbb{C}^3$. The real part of such an embedding is a complete conformal minimal immersion $M\to \mathbb{R}^3$ with bounded image. For any such $M$ we also co…

We construct knotted proper holomorphic embeddings of the unit disc in C^2.

Groups can embed uniformly but not act properly on contractible manifolds.

Given a closed complex hypersurface $Z\subset \mathbb{C}^{N+1}$ $(N\in\mathbb{N})$ and a compact subset $K\subset Z$, we prove the existence of a pseudoconvex Runge domain $D$ in $Z$ such that $K\subset D$ and there is a complete proper holomorphic embedding from $D$ into the unit ball of $\mathbb{C}^{N+1}$. For $N=1$,…

We prove the three embeddedness results as follows. $({\rm i})$ Let $Γ_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of $Γ_{2m+1}<2mπ$. In particular, the total curvature of $Γ_5<4π$ and thus any minimal surface $Σ\subset \…

Embeds Higson compactification into adelic solenoids.

We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.

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

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_ρ$ on Teichmüller space $\Teich$ which is a qualitative invariant of the holonomy representation $ρ$ of $π_1(S)$. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that…

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

New embeddings for manifolds using heat kernels.

The ball in complex 2-space can contain curves of any shape.

This paper extends boundary embedding results to coarsely convex spaces.

We prove that given an open Riemann surface $N,$ there exists an open domain $M\subset N$ homeomorphic to $N$ which properly holomorphically embeds in $\mathbb{C}^2.$ Furthermore, $M$ can be chosen with hyperbolic conformal type. In particular, any open orientable surface admits a complex structure properly holomorphic…

In this paper, we prove that every confomal minimal immersion of an open Riemann surface into $\mathbb{R}^n$ for $n\ge 5$ can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into $\mathbb{R}^5$. One …

If a class of finitely generated groups Curly(G) is closed under isometric amalgamations along free subgroups, then every G in Curly(G) can be quasi-isometrically embedded in a group Hat(G) in Curly(G) that has no proper subgroups of finite index. Every compact, connected, non-positively curved space X admits an isomet…

Curve shortening flow is not unique on certain metrics.

Constructs improperly embedded minimal disks with curvature issues.

Study symplectic embeddings of 4-manifolds using Lefschetz fibrations.

Normal forms and isotropic embeddings via Euler-like vector fields.

Improves two-sample hypothesis testing using kernel divergences and scoring rules.

Embeds CR manifolds into complex spaces using equivariant actions.

Sharpness of actions on reductive homogeneous spaces proven for various groups.

The paper develops theory for holomorphic null curves in SL2(C).

A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over…

Harmonic maps intersect all minimal surfaces with bounded curvature.

In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in n-manifolds. More precisely, we give conditions that imply properness of such surfaces and prove the existence of fixed size one-sided regular neighborhoods for certain constant mean curvature hypersurfaces in cer…

We use the solution set of a real ordinary differential equation which has order n which is at least 2 to construct a smooth curve C in R^n. We describe when C is a proper embedding of infinite length with finite total first curvature.

Unified theory of orbifolds and cohomology.

GC Stein manifolds characterized with embeddings and functions.

We show that the theory of stable complex $G$-cobordisms, for a torus $G$, is embedded into the theory of stable complex $G$-cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex $G$-cobordism theory does not…