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We show that, for a closed orientable n -manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n − 1) -space ensures the existence of a totally real embedding into complex n -space. This implies that a closed orientable (4 k + 1) -manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4 k -space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we study the rotationally invariant minimal surfaces in the Bao-Shen's spheres, which are a class of 3-spheres endowed with Randers metrics of constant flag curvature K = 1, where are Berger metrics, are one-forms and k > 1 is an arbitrary real number. We obtain a class of nontrivial minimal surfaces isometrically immersed in the Bao-Shen's spheres, which is the first class of nontrivial minimal surfaces with respect to the Busemann-Hausdorff measure in Finsler spheres. Moreover, we also obtain a new class of explicit minimal surfaces in the classical Berger spheres , which was expected to get in [F. Torralbo, Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds, Differential Geom. Appl. 28(5) (2010) 593-607]. [ABSTRACT FROM AUTHOR]

We obtain all linear Chern number inequalities satisfied by any smooth complete intersection threefold with ample canonical bundle. [ABSTRACT FROM AUTHOR]

Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note, we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpretations of the conjecture when the Euclidean background space is replaced by an arbitrary Riemannian manifold. Introducing the notion of ɛ-superbiharmonic submanifolds, which contains each of the previous notions as special cases, we prove that ɛ-superbiharmonic submanifolds of a complete Riemannian manifold which satisfy a growth condition at infinity are minimal. [ABSTRACT FROM AUTHOR]

Let M = Γ\G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension. [ABSTRACT FROM AUTHOR]