Topology of Hulls

Part of the Graduate Texts in Mathematics book series (GTM, volume 35)


We begin by stating some basic results about Morse theory. A nice description of what we need about Morse theory is given in Part I of Milnor’s text [Mi2]. We will restrict our attention to open subsets M of ℝs . Let ϕ be a smooth function on M. A point pM is a critical point of ϕ if ∂ϕ/∂xk(p) = 0 for 1 ≤ ks. A critical point p is nondegenerate if the (real) Hessian matrix (∂2ϕ/∂xjxk(p)) is nonsingular. Then one defines the index of ϕ at p to be the number of negative eigenvalues of this matrix. Nondegenerate critical points are necessarily isolated. A Morse function ρ on M is a smooth real function such that Ma ≡ {xM : ρ(x) ≤ a} is compact for all a, all critical points of ρ are nondegenerate, and ρ(p1) ≠ ρ(p2) for critical points p1 ≠ p2. The following lemma produces Morse functions.


Morse Theory Morse Function Plurisubharmonic Function Maximal Ideal Space Polynomially Convex 
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© Springer-Verlag New York, Inc. 1998

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