相关论文: Compact complete proper minimal immersions in stri…

In this paper, we prove a uniform approximation theorem with interpolation for complete conformal minimal surfaces with finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. As application, we obtain a Mittag-Leffler type…

In this paper we study the local regularity of closed surfaces immersed in a Riemannian 3-manifold flowing by Willmore flow. We establish a pair of concentration-compactness alternatives for the flow, giving a lower bound on the maximal…

We prove that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits a proper conformal constant mean curvature one ($\mathrm{CMC\text{-}1}$) immersion into hyperbolic $3$-space…

Let $M$ be a compact Riemannian manifold not containing any totally geodesic surface. Our main result shows that then the area of any complete surface immersed into $M$ is bounded by a multiple of its extrinsic curvature energy, i.e. by a…

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total…

We show that any isometric immersion of a flat plane domain into $\mathbb R^3$ is developable provided it enjoys the little H\"older regulairty $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,\alpha}$ regularity with $\alpha…

Any open Riemann surface $R_0$ of finite genus $g$ can be conformally embedded into a closed Riemann surface of the same genus, that is, $R_0$ is realized as a subdomain of a closed Riemann surface of genus $g$. We are concerned with the…

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary…

In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $\gamma$ contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of…

Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane $\R P^2$. We show that…

We prove that given any compact Riemannian 3-manifold with boundary M, there exists a smooth properly embedded one-manifold G, included in M, each of whose components is a simple closed curve and such that the domain D=Int(M)-G does not…

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by…

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at…

This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff…

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…

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the…

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine…

In this short note we consider very general bounded minimal homogeneous domains. Under certain natural additional conditions new sharp results on Bergman type analytic spaces in minimal bounded homogeneous domains are obtained. Domains we…

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured…