Unit tangent bundle of a surface carries various information of tangent vector fields on that surface. For 2-spheres (ie genus-zero closed surfaces), the unit tangent bundle is a closed 3-manifold that has non-trivial topology and cannot be embedded in R3. Therefore it cannot be constructed by existing mesh generation algorithms directly. This work aims at the first discrete construction of unit tangent bundles over 2-spheres using tetrahedral meshes. We propose a two-stage algorithm for the construction, which starts from constructing two local bundles and then combines them into a global bundle.

In this paper, we show that any compact Kähler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Kähler–Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold X homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure J compatible with the symplectic form, there is no non-constant J-holomorphic entire curve f:C→X.

In this paper, we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by their period. These surfaces are constructed by adding a handle to the toroidal half-plane layers defined by H. Karcher. The technics that we use are to solve a Jenkins-Serrin problem over a strip domain and to consider the conjugate minimal surface to the graph. To construct the Jenkins Serrin graph, we solve in fact the maximal surface equation and use an other conjugation technic.

Let $M = \sum_1 \times \sum_2 $ be the product of two compact Riemannian manifolds of dimension $n \ge 2$ and two, respectively. Let $\sum $ be the graph of a smooth map $f: \sum_1 \to \sum_2$, then $\sum$ is an n-dimensional submanifold of $M$. Let $\mathcal{G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of all n-dimensional subspaces of the tangent space of $M$. The Gauss map $\gamma :\sum \to \mathcal{G}$ assigns to each point $x \in \sum $ the tangent space of \sum$ at x. This article considers the mean curvature flow of $\sum $ in $M$. When $\sum_1|$ and $\sum_2|$ are of the same non-negative curvature, we show a sub-bundle \mathcal{S}$ of the Grassmannian bundle is preserved along the flow, i.e. if the Gauss map of the initial submanifold $\sum $ lies in \mathcal{S}$, then the Gauss map of $\sum_t$ at any later time $t$ remains in $\mathcal{S}$. We also show that under this initial condition, the mean curvature flow remains a graph, exists for all time and converges to the graph of a constant map at infinity . As an application, we show that if $f$ is any map from $S^n$ to $S^2$ and if at each point, the restriction of $df$ to any two dimensional subspace is area decreasing, then f is homotopic to a constant map.

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