International Journal of Mathematics

Let $X$ be a compact Kähler manifold and let $T$ be a foliation cycle directed by a transversely Lipschitz lamination on $X$. We prove that the self-intersection of the cohomology class of $T$ vanishes as long as $T$ does not contain currents of integration along compact manifolds.

As a consequence, we prove that transversely Lipschitz laminations of low codimension in certain manifolds, e.g. projective spaces, do not carry any foliation cycles except those given by integration along compact leaves.

We show that under some topological assumptions, an exact Lagrangian cobordism $(W;L_0,L_1)$ of dimension $\mathrm{\text{dim}}(W)\ge 6$ is a Lagrangian pseudo-isotopy. This result is a weaker form of a conjecture proposed by Biran and Cornea, which states that any exact Lagrangian cobordism is Hamiltonian isotopic to a Lagrangian suspension.

Let $S$ be a surface with $p_g(S)=q(S)=0$ and endowed with a very ample line bundle ${{𝒪}}_S(h)$ such that $h^1(S,{{𝒪}}_S(h))=0$. We show that $S$ supports special (often stable) Ulrich bundles of rank $2$, extending a recent result by A. Beauville. Moreover, we show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$ except for very few cases. We also show that the same is true for each linearly normal non-special surface with $p_g(S)=q(S)=0$ in ${\mathbb{P}}^4$ of degree at least $4$, Enriques surface and anticanonical rational surface.

We prove that a complex curve charged by a local Ahlfors current is either a disk or an annulus.

In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note, we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Cartan–Remmert reduction of a holomorphically convex space. Moreover, in our classification theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where passing to a double cover cannot be avoided.

We consider a nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure and study non-invariant hypersurface of a nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure. We obtain some properties of nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure. Further, we find the necessary and sufficient conditions for totally umbilical non-invariant hypersurface with $(f,g,u,v,\lambda )$-structure of nearly hyperbolic Sasakian manifold to be totally geodesic. We also calculate the second fundamental form of a non-invariant hypersurface of a nearly hyperbolic Sasakian manifold with $(f,g,u,v,\lambda )$-structure under the condition when f is parallel.

We study the prescribed mean curvature equation with Neumann boundary conditions in domains of Riemannian manifold. The main goal is to establish the gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.

In this paper, we give an explicit formula of the determinant of a Heisenberg representation $\rho$ of a finite group $G$. Heisenberg representations are induced by 1-dimensional characters in multiple ways, but our formula will be independent of any particular choice of induction.