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Gradient flows of Morse-Bott functions. (English) Zbl 1011.53059

Recently F. R. Harvey and J. B. Lawson jun. presented a new unifying approach to the Morse theory based on the consistent use of the geometric measure theory [Ann. Math. (2) 153, No. 1, 1-25 (2001; Zbl 1001.58005)]. They consider the following question. For a given smooth flow \(\phi_t:X\to X\) on a manifold \(X\), when does the limit \(P(\alpha)=lim_{t\to\infty}\phi_t^*(\alpha)\) exist for a given smooth differential form \(\alpha\)? For a so called finite-volume flow they show the existence of the limit for all \(\alpha\) and continuity of the corresponding linear operator \(P:{\mathcal E}^*(X)\to {\mathcal D'}^*(X)\) from smooth forms to generalized forms, i.e., currents. The operator \(P\) is chain homotopic to the inclusion \(I:{\mathcal E}^*(X)\to {\mathcal D'}^*(X)\) where the homotopy \(I-P=d\circ T + T\circ d\) is given with the help of a continuous operator \(T:{\mathcal E}^*(X)\to {\mathcal D'}^*(X)\) of degree \(-1\) constructed with the help of the graph of the flow.
In the paper by Harvey and Lawson the finite-volume flow is the gradient flow of a Morse function \(f:X\to R\). In the paper under review the author generalizes this paper to the context of Morse-Bott functions \(f:X\to R\). He proves the existence, obtains explicit formulas for \(P\) and a geometric model for a Morse complex associated to \(f\) which might be used to compute the homology of \(X\) with \(\mathbb{Z}\) or \(\mathbb{Z}_2\) coefficients.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
55N91 Equivariant homology and cohomology in algebraic topology
57R70 Critical points and critical submanifolds in differential topology
58A25 Currents in global analysis

Citations:

Zbl 1001.58005
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