8 papers
A Lewy-type theorem for pluriharmonic mappings in $\C^2$ and bi-Lipschitz quasiconformal harmonic maps
David Kalaj
Lewy's theorem says that a one-to-one harmonic mapping between plane domains has nonvanishing Jacobian. In dimensions at least three this statement is false for general harmonic ho…
The Gromov--Ros conjecture for rank-one symmetric spaces of noncompact type
David Kalaj
We prove the Gromov--Ros conjecture for every rank-one symmetric space of noncompact type: finite-perimeter isoperimetric regions are precisely the geodesic balls, up to ambient is…
A Lewy theorem for harmonic quasiregular mappings in three-space
David Kalaj, Jian-Feng Zhu
Lewy's classical theorem asserts that a one-to-one planar harmonic mapping has nonvanishing Jacobian. We prove a three-dimensional bounded-distortion analogue: if \[ f:Ω\subset \ma…
The Schwarz lemma for holomorphic and minimal disks at the boundary
David Kalaj
We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in . Further by using a Schwarz lemma for minimal conformal disks of Forstneri\v c and…
The Nitsche--Hopf conjecture for minimal graphs
David Kalaj, Jian-Feng Zhu
We prove the Nitsche--Hopf conjecture for non-parametric minimal graphs over disks. If \(S\) is a minimal graph over a disk of radius \(R\), and if \(ξ\) is the point above the ce…
On an extremal problem for harmonic maps conformal at a point
Franc Forstneric, David Kalaj
Let \(\mathbb D\) denote the unit disc in \(\mathbb C\). For a domain \(D\subset\mathbb C\) and a point \(p\in D\), let \(M_D(p)\) denote the supremum of \(\|df_0\|\) over all harm…