The Topological Classification of Minimal Surfaces in R^3

Topological entropy of minimal geodesics and volume growth on surfaces

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of (M, g) generalizing work of Freire and Mane.

Existence of proper minimal surfaces of arbitrary topological type

Consider a domain D in R^3 which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f : M --> D. Moreover, if D is smooth and bounded, then we prove that the immersion f : M --> D can be chosen so that the limit sets of distinct ends of M are disjoint connected compact sets in the boundary of D.

Classification of conformal minimal immersions from $S^2$ to $G(2,N;\mathbb{C})$ with parallel second fundamental form

In this paper, we determine all conformal minimal immersions of 2-spheres in complex Grassmann manifold $G(2,N; \mathbb{C})$ with parallel second fundamental form.

利用旋量的稳定最小曲面在R^3中的分类证明

A spinor proof of the classification of stable minimal surfaces in $\mathbb{R}^3$

本文通过Dirac算子在扭曲旋量丛上的索引理论，证明了R^3中所有完整双侧稳定最小曲面均为平坦的。

$Nil^{4}$中Codazzi超曲面的分类及极小超曲面注记

Classification of Codazzi and note on minimal hypersurfaces in $Nil^{4}$

本文研究李群$(Nil^{4},\tilde{g})$中的超曲面问题，完成了Codazzi超曲面的分类，刻画了一类极小超曲面并给出实例，丰富了该空间的超曲面理论。

Minimal geodesic foliation on T^2 in case of vanishing topological entropy

On a Riemannian 2-torus $(T^2,g)$ we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all $r \in \mathbb{R} \cup \{\infty\}$ the universal cover $\Br^2$ is foliated by minimal geodesics of rotation number $r$. For irrational $r \in \mathbb{R}$ all geodesics are minimal, for rational $r \in \mathbb{R} \cup \{\i…

Classification of Stable Minimal Surfaces Bounded by Jordan Curves in Close Planes

We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifolds which can be enumerated explicitely. One consequence of this result is the uniqueness of the area minimizing examples. Another is the asymptotic nonexistence of stable compact embedded minimal surfaces of positive genus bounded by two convex curves in parallel planes.

极小曲面的拓扑控制

Topological control for min-max free boundary minimal surfaces

本文通过极小方法研究自由边界极小曲面的拓扑性质，证明了贝蒂数的下半连续性，并展示了在可定向情况下亏格与边界组件数量的关系，以及在欧几里得单位球中构造自由边界极小三叶形的应用。

Classification of minimal Lorentz surfaces in indefinite space forms with arbitrary codimension and arbitrary index

Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are known. Hence, in this paper we investigate minimal Lorentz surfaces in arbitrary indefinite space forms. As a consequence, we obtain several classification results for minimal Lorentz surfaces in indefinite space forms. In particular, we completely classify all minimal Lorentz surfac…

Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.

Monodromy and topological classification of germs of holomorphic foliations

We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representation whose topological invariance was conjectured in the eighties by Cerveau and Sad and proved here under mild hypotheses.