A Burns-Krantz type theorem for domains with corners

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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 vertex p. Such results have no analogues in one complex variable in contrast to the situation when a domain is preserved. And second, to extend the above results from boundaries of domains to submanifolds of higher codimension.