In this paper we define the Hardy space $H^1_{\Cal F}(\Bbb R^n)$ associated with a family $\Cal {F}$ of sections and a doubling measure $\mu$, where $\Cal {F}$ is closely related to the Monge-Ampère equation. Furthermore, we show that the dual space of $H^1_{\Cal F}(\Bbb R^n)$ is just the space $B\!M\!O_{\Cal F}(\Bbb R^n)$, which was first defined by Caffarelli and Gutiérrez. We also prove that the Monge-Ampère singular integral operator is bounded from $H^1_{\Cal F}(\Bbb R^n)$ to $L^1(\Bbb R^n,d\mu)$.

The classical Cohn-Vossen inequality states that for any complete 2-dimensional Riemannian manifold the difference between the Euler characteristic and the normalized total Gaussian curvature is always nonnegative. For complete open surfaces in Euclidean 3-space this curvature defect can be interpreted in terms of the length of the curve "at infinity''. The goal of this paper is to investigate higher dimensional analogues for open submanifolds of Euclidean space with cone-like ends. This is based on the extrinsic Gauss-Bonnet formula for compact submanifolds with boundary and its extension "to infinity''. It turns out that the curvature defect can be positive, zero, or negative, depending on the shape of the ends "at infinity''. We give an explicit example of a 4-dimensional hypersurface in Euclidean 5-space where the curvature defect is negative, so that the direct analogue of the Cohn-Vossen inequality does not hold. Furthermore we study the variational problem for the total curvature of hypersurfaces where the ends are not fixed. It turns out that for open hypersurfaces with cone-like ends the total curvature is stationary if and only if each end has vanishing Gauss-Kronecker curvature in the sphere "at infinity''. For this case of stationary total curvature we prove a result on the quantization of the total curvature.

We study two kinds of transformation groups of a compact locally conformally Kähler (l.c.K.) manifold. First, we study compact l.c.K. manifolds by means of the existence of holomorphic l.c.K. flow (i.e., a conformal, holomorphic flow with respect to the Hermitian metric.) We characterize the structure of the compact l.c.K. manifolds with parallel Lee form. Next, we introduce the Lee-Cauchy-Riemann ($\mathrm{LCR}$) transformations as a class of diffeomorphisms preserving the specific $G$-structure of l.c.K. manifolds. We show that compact l.c.K. manifolds with parallel Lee form admitting a non-compact holomorphic flow of $\mathrm{LCR}$ transformations are rigid: such a manifold is holomorphically isometric to a Hopf manifold with parallel Lee form.

In this paper we classify real hypersurfaces all of whose geodesics orthogonal to the characteristic vector field are plane curves in complex projective or complex hyperbolic spaces.

The transplantation operators for the Hankel transform are considered and their boundedness on the real Hardy space is established. As its application, we obtain the Hörmander-Mihlin type multiplier theorem for the Hankel transform on the real Hardy space.

We introduce a new geometric structure on differentiable manifolds. A Contact Pair on a $2h+2k+2$-dimensional manifold $M$ is a pair $(\alpha,\eta) $ of Pfaffian forms of constant classes $2k+1$ and $2h+1$, respectively, whose characteristic foliations are transverse and complementary and such that $\alpha$ and $\eta$ restrict to contact forms on the leaves of the characteristic foliations of $\eta$ and $\alpha$, respectively. Further differential objects are associated to Contact Pairs: two commuting Reeb vector fields, Legendrian curves on $M$ and two Lie brackets on the set of differentiable functions on $M$. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds, bundles over the circle and principal torus bundles.

We characterize the invariant $f$-structures $\mathcal{F}$ on the classical maximal flag manifold $\mathbb F(n)$ which admit (1,2)-symplectic metrics. This provides a sufficient condition for the existence of $\mathcal{F}$-harmonic maps from any cosymplectic Riemannian manifold onto $\mathbb F(n)$. In the special case of almost complex structures, our analysis extends and unifies two previous approaches: a paper of Brouwer in 1980 on locally transitive digraphs, involving unpublished work by Cameron; and work by Mo, Paredes, Negreiros, Cohen and San Martin on cone-free digraphs. We also discuss the construction of (1,2)-symplectic metrics and calculate their dimension. Our approach is graph theoretic.

The proof of the Combinatorial Hard Lefschetz Theorem for the "virtual'' intersection cohomology of a not necessarily rational polytopal fan as presented by Karu completely establishes Stanley's conjectures for the generalized $h$-vector of an arbitrary polytope. The main ingredients, Poincaré Duality and the Hard Lefschetz Theorem, rely on an intersection product. In its original constructions, given independently by Bressler and Lunts on the one hand, and by the authors of the present article on the other, there remained an apparent ambiguity. The recent solution of this problem by Bressler and Lunts uses the formalism of derived categories. The present article instead gives a straightforward approach to combinatorial duality and a natural intersection product, completely within the framework of elementary sheaf theory and commutative algebra, thus avoiding derived categories.

We derive variational formulas of natural first order functionals and obtain criteria for stability in particular at Riemannian subimmersions.