相关论文: Totally real immersions of surfaces

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a…

In this note we consider the Liouville type theorem for a properly immersed submanifold $M$ in a complete Riemmanian manifold $N$. Assume that the sectional curvature $K^N$ of $N$ satisfies…

A perforated surface is the complement $\mathring\Sigma:=\Sigma\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $\Sigma$. It is known that the topological type of $\Sigma\setminus A$ is independent of the…

This article is concerned with locally flatly immersed surfaces in simply-connected $4$-manifolds where the complement of the surface has fundamental group $\mathbb{Z}$. Once the genus and number of double points are fixed, we classify such…

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…

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an…

We construct three kinds of complete embedded minimal surfaces in $\Bbb H^2\times \Bbb R$. The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These…

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…

The space of orientation-compatible almost complex structures on the six-dimensional sphere naturally contains a copy of seven-dimensional real projective space. We show that the inclusion induces an isomorphism on fundamental groups and…

We prove that every immersed $C^2$-curve $\gamma$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_{\gamma}$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_{\gamma}$. The approximating curves satisfy a…

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this…

In an earlier paper (math.SG/0110169), we introduced absolute gradings on the three-manifold invariants developed in math.SG/0101206 and math.SG/0105202. Coupled with the surgery long exact sequences, we obtain a number of three- and…

We introduce a family of variational functionals for spinor fields on a compact Riemann surface $M$ that can be used to find close-to-conformal immersions of $M$ into $\mathbb{R}^3$ in a prescribed regular homotopy class. Numerical…

The work of Mann and Rafi gives a classification surfaces $\Sigma$ when $\textrm{Map}(\Sigma)$ is globally CB, locally CB, and CB generated under the technical assumption of tameness. In this article, we restrict our study to the pure…

In this paper we define two regular homotopy invariants c and i for immersions of oriented 3-manifolds into R^5 in a geometric manner. The pair (c(f),i(f)) completely describes the regular homotopy class of the immersion f. The invariant i…

Let (M,\omega) be a symplectic manifold, and (\Sigma,\sigma) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {\omega^{D}}_{(\Sigma, \sigma)} on the space of immersions \Sigma \to M that is a special case of…

A holomorphic curve in moduli spaces is the image of a non-constant holomorphic map from a hyperbolic surface $B$ of type $(g,n)$ to the moduli space $\mathcal{M}_h$ of closed Riemann surfaces of genus $h$. We show that, when all peripheral…

We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These…

We view all smooth metrics $g$ on a closed surface $\Sigma$ through their Nash isometric embeddings $f_g: (\Sigma,g) \rightarrow (\mathbb{S}^{\tilde{n}}, \tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We…

In this paper we extend Efimov's Theorem by proving that any complete surface in $\mathbb{R}^3$ with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly…