A Burns-Krantz type theorem for domains with corners

Abstract

In their pioneering paper [BK94],Burns andKrantzhave established a boundary version of the classical Cartan’s uniqueness theorem: If f is holomorphic self-map of a smoothly bounded strongly pseudoconvex domain D ⊂ C such that f (z) = z + o(|z − p|3) as z → p for some p ∈ ∂D, then f (z) ≡ z. Furthermore, they give an example showing that the exponent 3 in the above statement cannot be decreased. The goal of this paper is twofold. First, to give purely local boundary uniqueness results for maps defined only on one side as germs at a boundary point and hence not necessarily sending any domain to itself (and also under the weaker assumption that f (z) = z + o(|z − p|3) holds only for z in a proper cone in D with vertexp). Such results have no analogues in one complex variable in contrast with the situation when a domain is preserved. Our second goal is to extend the above results from boundaries of domains to real submanifolds M ⊂ C of higher codimension. Recall that M is called generic if its tangent space TpM at every point p ∈ M spans TpC over C (i.e. TpC N = TpM + iTpM). Here the usual replacement for a one-sided neighborhood of a boundary point is a wedge with edgeM at a pointp ∈ M in the direction of an open cone ⊂ NpM := TpC/TpM , i.e. a domain W ⊂ C such that for any pair of open cones ′, ′′ ⊂ NpM with ′ {0} ⊂ and {0} ⊂ ′′, there exists a neighborhood U of p in C such that (M ∩ U) + ( ′ ∩ U) ⊂ W and (M ∩U)+ ( ′′ ∩U) contains a neighborhood of p in W (where we identify NpM with a fixed complement of TpM in TpC ∼= C ). Furthermore, we extend the notion of strong pseudoconvexity to wedges as follows. We call a wedge W with edge M at p in the direction of strongly