  This paper concerns free analytic maps on noncommutative domains. These maps
are free analogs of classical holomorphic functions in several complex
variables, and are defined in terms of noncommuting variables amongst which
there are no relations - they are free variables. Free analytic maps include
vector-valued polynomials in free (noncommuting) variables and form a canonical
class of mappings from one noncommutative domain D in say g variables to
another noncommutative domain D' in g' variables.
  Motivated by determining the possibilities for mapping a nonconvex
noncommutative domain to a convex noncommutative domain, this article focuses
on rigidity results for free analytic maps. Those obtained to date, parallel
and are often stronger than those in several complex variables. For instance, a
proper free analytic map between noncommutative domains is one-one and, if
g=g', free biholomorphic. Making its debut here is a free version of a theorem
of Braun-Kaup-Upmeier: between two freely biholomorphic bounded circular
noncommutative domains there exists a linear biholomorphism. An immediate
consequence is the following nonconvexification result: if two bounded circular
noncommutative domains are freely biholomorphic, then they are either both
convex or both not convex. Because of their roles in systems engineering,
linear matrix inequalities (LMIs) and noncommutative domains defined by an LMI
(LMI domains) are of particular interest. As a refinement of above the
nonconvexification result, if a bounded circular noncommutative domain D is
freely biholomorphic to a bounded circular LMI domain, then D is itself an LMI
domain.
