1. CJM Online first
 Roth, Oliver

Pontryagin's maximum principle for the Loewner equation in higher dimensions
In this paper we develop a variational method for the Loewner
equation in higher dimensions. As a result we obtain a version of Pontryagin's
maximum principle from optimal control theory for the Loewner equation in
several complex variables. Based on recent work of Arosio, Bracci and
Wold,
we then apply our version of the Pontryagin maximum
principle to obtain firstorder necessary conditions for the extremal
mappings for a wide class
of extremal problems over the set of normalized biholomorphic
mappings on the unit ball in $\mathbb{C}^n$.
Keywords:univalent function, Loewner's equation Categories:32H02, 30C55, 49K15 

2. CJM 2009 (vol 61 pp. 566)
 Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Pfaltzgraff, John A.

Convex Subordination Chains in Several Complex Variables
In this paper we study the notion of a convex subordination chain in several
complex variables. We obtain certain necessary and sufficient conditions for a
mapping to be a convex subordination chain, and we give various examples of
convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We
also obtain a sufficient condition for injectivity of $f(z/\z\,\z\)$
on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain
over $(0,1)$.
Keywords:biholomorphic mapping, convex mapping, convex subordination chain, Loewner chain, subordination Categories:32H02, 30C45 

3. CJM 2002 (vol 54 pp. 324)
 Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela

Parametric Representation of Univalent Mappings in Several Complex Variables
Let $B$ be the unit ball of $\bb{C}^n$ with respect to an arbitrary norm. We
prove that the analog of the Carath\'eodory set, {\it i.e.} the set of normalized
holomorphic mappings from $B$ into $\bb{C}^n$ of ``positive real part'', is
compact. This leads to improvements in the existence theorems for the Loewner
differential equation in several complex variables. We investigate a subset
of the normalized biholomorphic mappings of $B$ which arises in the study of
the Loewner equation, namely the set $S^0(B)$ of mappings which have
parametric representation. For the case of the unit polydisc these mappings
were studied by Poreda, and on the Euclidean unit ball they were studied by
Kohr. As in Kohr's work, we consider subsets of $S^0(B)$ obtained by placing
restrictions on the mapping from the Carath\'eodory set which occurs in the
Loewner equation. We obtain growth and covering theorems for these subsets of
$S^0(B)$ as well as coefficient estimates, and consider various examples.
Also we shall see that in higher dimensions there exist mappings in $S(B)$
which can be imbedded in Loewner chains, but which do not have parametric
representation.
Categories:32H02, 30C45 
