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Searching normal forms for real analytic submanifolds of C^n involves convergence problems. In 1983, J.K. Moser and S.M. Webster provided examples of real analytic surfaces in C^2 having an isolated hyperbolic (in the sense of E. Bishop)…

Let $\Gamma \stackrel{i}{\hookrightarrow} L$ be a lattice in the real simple Lie group $L$. If $L$ is of rank at least 2 (respectively locally isomorphic to $Sp(n,1)$) any unbounded morphism $\rho: \Gamma \longrightarrow G$ into a simple…

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. The…

We consider a formally integrable, strictly pseudoconvex CR manifold $M$ of hypersurface type, of dimension $2n-1\geq7$. Local CR, i.e. holomorphic, embeddings of $M$ are known to exist from the works of Kuranishi and Akahori. We address…

We give necessary and sufficient conditions on a smooth local map of a Riemannian manifold $M^m$ into the sphere $S^m$ to be the Gauss map of an isometric immersion $u:M^m \to R^n$, $n=m+1$. We briefly discuss the case of general $n$ as…

The notion of ideal embeddings was introduced in [B.-Y. Chen, {Strings of Riemannian invariants, inequalities, ideal immersions and their applications.} The Third Pacific Rim Geometry Conference (Seoul, 1996), 7-60, Int. Press, Cambridge,…

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by $a$ and…

Suppose that $\Sigma=\partial\Omega$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(\Omega^{n+1},g)$ whose scalar curvature $R\ge…

We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of…

Let $f:{\mathbb B}^n \to {\mathbb B}^N$ be a holomorphic map. We study subgroups $\Gamma_f \subseteq {\rm Aut}({\mathbb B}^n)$ and $T_f \subseteq {\rm Aut}({\mathbb B}^N)$. When $f$ is proper, we show both these groups are Lie subgroups.…

A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow…

In \cite{Kramer11} Kramer proves for a large class of semisimple Lie groups that they admit just one locally compact $\sigma$-compact Hausdorff topology compatible with the group operations. We present two different methods of generalising…

Let $K$ be a complete discrete valued field with residue field $k$ and $F$ the function field of a curve over $K$. Let $A \in {}_2Br(F)$ be a central simple algebra with an involution $\sigma$ of any kind and $F_0 =F^{\sigma}$. Let $h$ be…

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map…

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold…

We show that a closed piecewise-linear hypersurface immersed in $R^n$ ($n\ge 3$) is the boundary of a convex body if and only if every point in the interior of each $(n-3)$-face has a neighborhood that lies on the boundary of some convex…

Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies…

We prove that for any open Riemann surface $N$ and finite subset $Z\subset \mathbb{S}^1=\{z\in\mathbb{C}\,|\;|z|=1\},$ there exist an infinite closed set $Z_N \subset \mathbb{S}^1$ containing $Z$ and a null holomorphic curve…

Consider a strictly convex bounded regular domain $C$ of $\R^3$. For any arbitrary finite topological type we find a compact Riemann surface $\mathcal{M}$, an open domain $M\subset \mathcal{M}$ with the fixed topological type, and a…

In this paper we investigate the global behavior of proper holomorphic maps f from the unit disc U={|z|<1} to C^2. The fact that U is transcendental imposes certain restrictions on the image f(U). For instance, f(U) cannot be contained in…