A New Gap Theorem Result for Proper Holomorphic Mappings Between Complex Balls



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In this dissertation, we study how rigidity properties of proper holomorphic mappings from complex balls of dimension n to complex balls of dimension N allow us to classify all such maps for particular values of n and N, up to equivalence by automorphism on the boundary. For maps F:BnBN, when N takes values in certain intervals, all maps are equivalent to (G,0), where G is a proper holomorphic map from Bn to BM, with M<N. These intervals are established by the First, Second, and Third Gap Theorems. The cases where N lies on the upper boundary of one of these gaps are more difficult than cases where N is smaller. We review the cases where N<3n−3 and where 3n<N<4n−6, and then we prove a classification theorem for the case where N=3n−3.

In Chapter 10, we preemptively simplify our calculation by examining the coefficients of the (1,1) terms of the Taylor expansion of the codimension components ϕk of a normalized map. Given any p, our map F is linear fractional along some affine subspace through p, which we use to show that all but one of the coefficients we investigate in this chapter are 0.

We assume the geometric rank κ0=2, because the smaller cases are covered in previous papers, and larger κ0 has been shown to be impossible (cf. [HJX06, Corollary 1.3]). In Chapter 11, we perform a long calculation to show that deg⁡(F)≤2 along a certain Segre variety. This takes two steps. First, we show that deg⁡(F)≤3, and then we use an explicit statement of F as a rational map to show deg⁡(F)≤2. Our calculation allows us to apply two theorems, [HJX06,Theorem 5.4] and [Leb11, Theorem 1.5], to prove that F must be equivalent to the Generalized Whitney Map.



CR geometry, Complex analysis