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Extra info for Hermitian Forms Meet Several Complex Variables: Minicourse on CR Geometry Using Hermitian Forms

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The diagonal entries in A f and Ag are the absolute values squared of the coefficients. We are done. Let us state the result in inhomogeneous coordinates. 3. Let f : S2n−1 → S2N−1 , n ≥ 2, be a nonconstant rational CR map of degree 2 or less. Then f is spherically equivalent to a monomial map. In particular, f is equivalent to a map taking (z1 , . . , zn ) to √ √ √ t1 z1 , t2 z2 , . . , tn zn , 1 − t1 z21 , 1 − t2 z22 , . . , 1 − tn z2n , 2 − t1 − t2 z1 z2 , 2 − t1 − t3 z1 z3 , . . , 2 − tn−1 − tn zn−1 zn , 0 ≤ t1 ≤ t2 ≤ .

If F : Bn → BN is a proper holomorphic mapping that is smooth (C∞ ) up to the boundary, then F is rational. In fact, the degree of F is bounded by some number D(n, N). The degree of a rational mapping is the maximum of the degree of the numerator and the denominator when written in lowest terms. Note that the requirement that the mapping extends to the boundary is necessary if N > n. There exist non-rational maps from Bn to BN that are only continuous up to the boundary. In fact such maps already exist when N = n + 1, see [D].

Let F : V → 2 and G : V → 2 be holomorphic maps such that F(z) 2 = G(z) 2 for all z ∈ V . Write F(z) = ∑ fα (z − p)α and G(z) = ∑ fα (z − p)α (where fα and gα are in 2 ). Then there exists an isometry U : span{ fα } → span{gα } such that F(z) = UG(z). Proof. As before translate and scale coordinates such that F is holomorphic in a neighborhood of the closed unit polydisc ∆ and let p = 0. Then the series of F(z) 2 complexifies to ∆ × ∆∗ . Let C be the matrix of coefficients of the function F(z) 2 , and as before C defines a bounded operator.

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