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We will usually identify the germζ f with one of its representatives—that is, a map f : U → C defined in a small neighborhood U 3 ζ and satisfyingf(U ∩ 0) ⊂ 0 ′. Several authors have studied this problem. Alexander [A] generalized Poincaré’s theorem to higher dimensions in 1974. A year later, Pinchuk [P1] proved that any germ of a biholomorphic mapping from a connected strictly pseudoconvex real-analytic hypersurface 0 ⊂ C to ∂B extends analytically along any path on0 as a locally biholomorphic map with the inclusion f(0) ⊂ ∂B. Recall that a strictly pseudoconvex real-analytic hypersurface 0 ⊂ C is called sphericalat a pointp ∈0 if there exists a germ of a biholomorphic map at p from 0 to ∂B. It follows from [P1] that, if a connected strictly pseudoconvex hypersurface is spherical at a point, then it is spherical at any point. Pinchuk’s result clearly holds if, in the target space, ∂B is replaced by an arbitrary simply connected compact strictly pseudoconvex spherical hypersurface 0 ′. Indeed, if0 ′ is spherical then a germ of a biholomorphic mapping g : 0 ′ → ∂B extends along any path on0 ′. Since0 ′ is simply connected, g extends to a global mapping from 0 ′ to ∂B. But then0 ′ is biholomorphically equivalent to ∂B. If 0 ′ is not simply connected, the result is no longer true. In fact, Burns and Shnider [BS] constructed some examples of compact and spherical but not simply connected hypersurfaces inC. For any such hypersurface 0 ′ ⊂ C, there exists a germ of a biholomorphic mappingf : ∂B → 0 ′ that does not extend holomorphically along some paths on ∂B.

Analytic continuation of germs of holomorphic mappings between real hypersurfaces in Cn.