Math 2.0

Global analysis, analysis on manifolds Definition

Lusternik–Schnirelmann theory of critical points

Builds on Smooth manifoldLusternik–Schnirelmann categoryCritical pointCritical point theory

Definition

Let MM be a compact connected smooth manifold without boundary and let f ⁣:M→ℝf \colon M \to \R be a C1C^1 function. For a closed subset A⊆MA \subseteq M, let cat⁡M(A)\operatorname{cat}_M(A) be the least m≥0m \ge 0 such that m+1m + 1 open subsets of MM, each contractible in MM, cover AA. Then cat⁡M(M)=cat⁡(M)\operatorname{cat}_M(M) = \operatorname{cat}(M), the normalized Lusternik–Schnirelmann category. For k=0,1,…,cat⁡(M)k = 0, 1, \dots, \operatorname{cat}(M), the Lusternik–Schnirelmann minimax values of ff are ck=inf⁡{max⁡x∈Af(x):A⊆M closed, A≠∅, cat⁡M(A)≥k}.c_k = \inf \set{ \max_{x \in A} f(x) : A \subseteq M \text{ closed}, \ A \ne \emptyset, \ \operatorname{cat}_M(A) \ge k }. Lusternik–Schnirelmann theory is the critical point theory built on these values. Its basic theorem states:

  1. each ckc_k is a critical value of ff, and c0≤c1≤⋯≤ccat⁡(M)c_0 \le c_1 \le \cdots \le c_{\operatorname{cat}(M)};
  2. if ck=ck+1c_k = c_{k+1} for some kk, then ff has infinitely many critical points at that level;
  3. hence ff has at least cat⁡(M)+1\operatorname{cat}(M) + 1 critical points.

Remarks

The proof uses a deformation along a pseudo-gradient flow. If cc is not a critical value, the sublevel set {f≤c+ε}\set{f \le c + \varepsilon} can be pushed into {f≤c−ε}\set{f \le c - \varepsilon}, and such deformations do not increase category. If the critical set at level ck=ck+1c_k = c_{k+1} were finite, it would have a neighbourhood of category 00, and subadditivity of category would give a contradiction. Since one-point sets are allowed for k=0k = 0, the first value is c0=min⁡fc_0 = \min f.

Example: for M=T2M = T^2, cat⁡(T2)=2\operatorname{cat}(T^2) = 2, so every C1C^1 function on the torus has at least three critical points. This bound is attained by suitable functions, while a Morse function on T2T^2 has at least four.

Non-example: the theory does not apply as stated to f(x)=arctan⁡xf(x) = \arctan x on the non-compact manifold ℝ\R. There cat⁡(ℝ)=0\operatorname{cat}(\R) = 0 would predict one critical point, but ff has none, since the infimum defining c0c_0 is not attained. In infinite dimensions compactness of MM is replaced by the Palais–Smale condition together with completeness and a lower bound on ff.

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Critical point, concept, no evidence levelCritical pointconceptCritical point theory, concept, no evidence levelCritical point theoryconceptLusternik–Schnirelmann category, concept, no evidence levelLusternik–Schnirelman…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptLusternik–Schnirelmann theory of critical points, definition, no evidence levelLusternik–Schnirelman…definition
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  1. Critical point concept
  2. Critical point theory concept
  3. Lusternik–Schnirelmann category concept
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  5. Lusternik–Schnirelmann theory of critical points definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Lusternik–Schnirelmann_category
About this page
TagGSV68DRC1D
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Kinddefinition
WritingW0 stub
MSC58E05, 55M30
DefinesLusternik–Schnirelmann theory of critical points
UsesSmooth manifold, Lusternik–Schnirelmann category, Critical point, Critical point theory

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