We construct for every integer $n)1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\mathbb{C}^n$ (i.e., a long $\mathbb{C}^n$), but it does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new biholomorphic invariants of a complex manifold $X$, the stable core and the strongly stable core, that are based on the long term behavior of hulls of compact sets with respect to an exhaustion of $X$. We show that every compact polynomially convex set $B \subset \mathbb{C}^n$ which is the closure of its interior is the strongly stable core of a long $\mathbb{C}^n$; in particular, biholomorphically nonequivalent sets give rise to nonequivalent long $\mathbb{C}^n$'s. Furthermore, for any open set $U \subset \mathbb{C}^n$ there exists a long $\mathbb{C}^n$ whose stable core is dense in $U$. It follows that for any $n)1$ there is a continuum of pairwise nonequivalent long $\mathbb{C}^n$'s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long standing open problems.
COBISS.SI-ID: 17834073
We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1 ( p ( \infty$, a Hörmander-type holomorphic functional calculus on $L^p$ in the sector of angle $\phi_p^\ast = \arcsin \vert 1-2/p \vert$. The obtained angle is optimal.
COBISS.SI-ID: 17897305
A complex manifold $X$ of dimension $n$ is said to be $q$-complete for some $q \in \{ 1, \dots ,n\}$ if it admits a smooth exhaustion function whose Levi form has at least $n-q+1$ positive eigenvalues at every point; thus, 1-complete manifolds are Stein manifolds. Such an $X$ is necessarily noncompact and its highest-dimensional a priori nontrivial cohomology group is $H^{n+q-1}(X; \mathbb{Z})$. In this paper we show that if $q ( n$, $n+q-1$ is even, and $X$ has finite topology, then every cohomology class in $H^{n+q-1}(X; \mathbb{Z})$ is Poincaré dual to an analytic cycle in $X$ consisting of proper holomorphic images of the ball. This holds in particular for the complement $X = \mathbb{CP}^n \setminus A$ of any complex projective manifold $A$ defined by $q ( n$ independent equations. If $X$ has infinite topology, then the same holds for elements of the group $\mathscr{H}^{n+q-1}(X; \mathbb{Z}) = \lim_j H^{n+q-1}(X; \mathbb{Z})$, where $\{M_j\}_{j \in \mathbb{N}}$ is an exhaustion of $X$ by compact smoothly bounded domains. Finally, we provide an example of a quasiprojective manifold with a cohomology class which is analytic but not algebraic.
COBISS.SI-ID: 17622361
In 1977, P. Yang asked whether there exist complete immersed complex submanifolds $\varphi \colon M^k \to \mathbb{C}^N$ with bounded image. A positive answer is known for holomorphic curves $(k=1)$ and partial answers are known for the case when $k)1$. The principal result of the present paper is a construction of a holomorphic function on the open unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ whose real part is unbounded on every path in $\mathbb{B}_N$ of finite length that ends on $b\mathbb{B}_N$. A consequence is the existence of a complete, closed complex hypersurface in $\mathbb{B}_N$. This gives a positive answer to Yang's question in all dimensions $k$, $N$, $1 \le k ( N$, by providing properly embedded complete complex manifolds.
COBISS.SI-ID: 17459545
In this paper we show that for every conformal minimal immersion $u \colon M \to \mathbb{R}^3$ from an open Riemann surface $M$ to $\mathbb{R}^3$ there exists a smooth isotopy $u_t \colon M \to \mathbb{R}^3$ $(t \in [0,1])$ of conformal minimal immersions, with $u_0 = u$, such that $u_1$ is the real part of a holomorphic null curve $M \to \mathbb{C}^3$ (i.e. $u_1$ has vanishing flux). Furthermore, if $u$ is nonflat then $u_1$ can be chosen to have any prescribed flux and to be complete.
COBISS.SI-ID: 17540953