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Topology of Hulls

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

Abstract

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.

Keywords

Morse Theory Morse Function Plurisubharmonic Function Maximal Ideal Space Polynomially Convex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1998

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