Related papers: Timelike minimal Lagrangian surfaces in the indefi…

Let $S_g$ ($g\geq 2$) be a closed surface of genus $g$. Let $K$ be any real number field and $A$ be any quaternion algebra over $K$ such that $A\otimes_K\mathbb{R}\cong M_2(\mathbb{R})$. We show that there exists a hyperbolic structure on…

The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed in [Izumiya, S. and Romero Fuster, M. C. Selecta Mathematica (NS), 13 23--55 (2007)] which is a natural Lorentzian analogue of…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

Consider a graphed holomorphic surface $u=F(x,y)$ in $\mathbb{C}^3_{x,y,u}$ under the action of the affine transformation group $A(3)$. In 1999, Eastwood and Ezhov obtained a list of homogeneous models by determining possible tangential…

Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite…

We classify the minimal algebraic surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, so that if $S$ is such a surface then there exist two smooth…

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play…

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a…

In this paper, we show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic $3$-manifolds except some special cases.

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving…

In this work we introduce a new method for the construction of minimal submanifolds of codimension two in even dimensional spheres and hyperbolic spaces. This is based on the theory of complex-valued harmonic morphisms. This gives the first…

In the three-dimensional Heisenberg group equipped with a certain left invariant Lorentzian metric, timelike minimal surfaces which have the Abresch-Rosenberg differentials with vanishing multiplication of the coefficient function and its…

For any irreducible compact homogeneous K\"ahler manifold, we classify the compact tight Lagrangian submanifolds which have the Z_2-homology of a sphere.

Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4…

We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine…

In response to a question raised by Belolipetsky and the first author, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to…

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 suggest a method for constructing minimal Lagrangian immersions of $R^2$ in $CP^2$ with induced diagonal metric in terms of Baker-Akhiezer functions of algebraic curves.

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichm\"uller theory than arbitrary non-compact surfaces. We show that the…

In this paper we study the flat (n+1)-spacetimes admitting a Cauchy surface diffeomorphic to a compact hyperbolic n-manifold. We show how to construct a canonical future complete one among all such spacetimes sharing the same holonomy. We…