Several Complex Variables and Banach Algebras pp 187-193 | Cite as

# Topology of Hulls

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## 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 ℝ^{s} . Let *ϕ* be a smooth function on *M*. A point *p* ∈ *M* is a critical point of *ϕ* if ∂*ϕ*/∂*x*_{k}(*p*) = 0 for 1 ≤ *k* ≤ *s*. A critical point *p* is nondegenerate if the (real) Hessian matrix (∂^{2}*ϕ*/∂*x*_{j}∂*x*_{k}(*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 *M*^{a} ≡ {*x* ∈ *M* : *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## Preview

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