Functions with minimal number of critical points (Q2771515)

The thesis studies the problem of estimating the minimal number of critical points, \(\text{crit}(M)\), which have smooth functions on a given closed smooth manifold \(M\). I will mention the main results presented in the thesis.NEWLINENEWLINENEWLINETheorem 2. Let \(M\) be a closed manifold and \(f:M \to\mathbb{R}\) be a smooth function. Then \(f\) has at least \(\text{bcat}(M)\) critical points.NEWLINENEWLINENEWLINEHere \(\text{bcat}(M)\) denotes the ball category of \(M\), i.e. the minimal number \(k\) such that \(M\) admits an open cover \(M=U_1\cup U_2\cup \cdots\cup U_k\) with each \(U_j\) homeomorphic to \(\mathbb{R}^n\), \(n=\dim M\). Clearly \(\text{bcat}(M)\geq\text{cat}(M)\). The classical Theorem of Lusternik and Schnirelman claims that the number of critical points of any function on \(M\) is at least \(\text{cat}(M)\).NEWLINENEWLINENEWLINETheorem 9. Let \(M\) be a closed manifold of dimension \(\leq 7\). Then \(\text{crit} (M\times S^n)\leq \text{crit}(M)+1\).NEWLINENEWLINENEWLINEFor high-dimensional manifolds the author proves: NEWLINENEWLINENEWLINETheorem 8. Let \(M\) be a closed manifold admitting a smooth function \(f:M\to \mathbb{R}\) with \(\text{crit} (M)\) critical points, among them exactly one minimum. Then \(\text{crit} (M\times S^{p_1}\times S^{p_2}\times \cdots\times S^{p_k})\leq \text{crit} (M)+k\).NEWLINENEWLINENEWLINEThe proofs of the above Theorems use the following claim and several of its modifications:NEWLINENEWLINENEWLINEFusing Lemma. Let \((W;V_0,V_1)\) be a compact connected triad of manifolds and let \(f:W\to [0,1]\) be a smooth function with \(f^{-1}(0)= V_0\), \(f^{-1}(1)=V_1\). Suppose that \(f\) has exactly two critical points \(x_0\) and \(x_1\) lying in the interior of \(W\) and there exists a pseudo-gradient vector field for \(f\) such that there are no connecting flow lines between \(x_0\) and \(x_1\). Then \(x_0\) and \(x_1\) fuse, i.e. one may find a smooth function \(g:W\to [0,1]\), such that \(g^{-1}(0) =V_0\), \(g^{-1}(1) =V_1\) and \(g\) has a single critical point lying in the interior of \(W\).NEWLINENEWLINENEWLINEThe proof of the Fusing Lemma is based on a well-known Theorem of \textit{F. Takens} [Invent. Math. 6, 197-244 (1968; Zbl 0198.56603)] which describing conditions for colliding several critical points of a smooth function into a single critical point.