Minimal Number Of Critical Points Research Articles

Maps of degree 1, Lusternik–Schnirelmann category, and critical points

Let Crit M denote the minimal number of critical points (not necessarily non-degenerate) on a closed smooth manifold M. We are interested in the evaluation of Crit. It is worth noting that we do not know yet whether Crit M is a homotopy invariant of M. This makes the research of Crit a challenging problem.In particular, we pose the following question: given a map f:M→N of degree 1 of closed manifolds, is it true that CritM≥CritN? We prove that this holds in dimension 3 or less. Some high dimension examples are considered. Note also that an affirmative answer to the question implies the homotopy invariance of Crit; this simple observation is a good motivation for the research.

The minimal number of critical points of a smooth function on a closed manifold and the ball category

Introduced by Seifert and Threlfall [Variationsrechnung im Grossen, Hamburger Math Einzelschr., Heft 24, Teubner, Leipzig-Berlin, 1938], cylindrical neighborhoods of isolated critical points of smooth functions is an essential tool in the Lusternik-Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate. If, contrary to the assertion of the conjecture, there are isolated critical points that do not admit cylindrical ball neighborhoods, then we say that such critical points are exotic. We prove a Lusternik-Schnirelmann type theorem asserting that the minimal number of critical points of smooth functions without exotic critical points on a closed manifold of dimension at least 6 6 is the same as the minimal number of elements in a Singhof-Takens filling of M M by smooth balls with corners.

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most$3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric$3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

A multiplicity theorem for Fréchet spaces

This note serves to announce a multiplicity result for Keller C1c -functionals on Fréchet spaces which are invariant under the action of a discrete subgroup. For such functionals, we evaluate the minimal number of critical points by applying the Lyusternik—Schnirelmann category.

Fibred algebraic surfaces and commutators in the Symplectic group

We describe the minimal number of critical points and the minimal number s of singular fibres for a non isotrivial fibration of a surface S over a curve B of genus 1, exhibiting several examples and in particular constructing a fibration with s=1 and irreducible singular fibre with 4 nodes.Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group Sp(2g,Z).

A $$C^0$$ C 0 counterexample to the Arnold conjecture

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold must have at least as many fixed points as the minimal number of critical points of a smooth function on the manifold. It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal of this paper is to provide a counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in dimensions four and higher. More precisely, we prove that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point.

Global classification of isolated singularities in dimensions (4,3) and (8,5)

We characterize those closed $2k$-manifolds admitting smooth maps into $(k+1)$-manifolds with only finitely many critical points, for $k\in\{2,4\}$. We compute then the minimal number of critical points of such smooth maps for $k=2$ and, under some fundamental group restrictions, also for $k=4$. The main ingredients are King's local classification of isolated singularities, decomposition theory, low dimensional cobordisms of spherical fibrations and 3-manifolds topology.

Functions on Tori with Minimal Number of Critical Points and Rotation Type Solutions of Spatially Periodic Hamiltonian Systems

Abstract In this paper we use functions on Tn−1 with n ≥ 2 possessing minimal number of critical points given by Ljusternik-Schnirelman theory to construct spatially n-periodic Hamiltonian systems which possess precisely n distinct rotation type solutions with prescribed energy and rotation type vector.

Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$

We consider manifolds $M^{2n}$ which admit smooth maps into a connected sum of $S^1\times S^n$ with only finitely many critical points, for $n\in\{2,4,8\}$, and compute the minimal number of critical points.

On the minimal number of critical points of smooth maps between closed manifolds

New information concerning the minimal number of critical points of smooth proper mappings between closed connected surfaces (possibly with boundary) without critical points on the boundary is presented.

Controlled connectivity of closed 1–forms

We discuss controlled connectivity properties of closed 1-forms and their cohomology classes and relate them to the simple homotopy type of the Novikov complex. The degree of controlled connectivity of a closed 1-form depends only on positive multiples of its cohomology class and is related to the Bieri-Neumann-Strebel-Renz invariant. It is also related to the Morse theory of closed 1-forms. Given a controlled 0-connected cohomology class on a manifold M with n = dim M > 4 we can realize it by a closed 1-form which is Morse without critical points of index 0, 1, n-1 and n. If n = dim M > 5 and the cohomology class is controlled 1-connected we can approximately realize any chain complex D_* with the simple homotopy type of the Novikov complex and with D_i=0 for i < 2 and i > n-2 as the Novikov complex of a closed 1-form. This reduces the problem of finding a closed 1-form with a minimal number of critical points to a purely algebraic problem.

On the minimal number of critical points of functions on $h$-cobordisms

Let (W,M,M'), dim W > 5, be a non-trivial h-cobordism (i.e., the Whitehead torsion of (W,V) is non-zero). We prove that every smooth function f: W --> [0,1], f(M)=0, f(M')=1 has at least 2 critical points. This estimate is sharp: W possesses a function as above with precisely two critical points.

Continuation to the minimal number of critical points in gradient flows

Continuation to the minimal number of critical points in gradient flows

Covering numbers of manifolds and critical points of a morse function

For a closed connected triangulatedn-manifoldM, we study some numerical invariants (namedcategory andcovering numbers) ofM which are strictly related to the topological structure ofM. We complete the classical results of 3-manifold topology and then we prove some characterization theorems in higher dimensions. Finally some applications are given about the minimal number of critical points (resp. values) of Morse functions defined on a closed connected smoothn-manifold.

Periodic maps in symplectic topology

Symplectic topology arose with [I], where a conjecture on fixed points of symplectomorphisms generalizing the "Poincare geometric theorem" [2, 3] was formulated. This conjecture of Arnol'd gives as a lower bound for the number of fixed points of symplectomorphis which is a homologicai identity of a closed symplectic manifold* the minimal number of critical points of a function on this manifold. The present paper is basically devoted to a discussion of the following conjecture.

New generalizations of Poincar�'s geometric theorem

In this work we show that the number of closed trajectories of the field of kernels of a closed, nondegenerate, and center of gravity preserving 2-form on the total space of an oriented fiber bundle with fiber the circle over an orientable compact two-dimensional base is not less than the minimal number of critical points of a smooth function on the basis, under the assumption that the field of kernels is C1-close to a vertical field. Counting multiplicities, this number is not smaller than the minimal number of critical points of a Morse function on the base. We also give a lower estimate for the number of closed trajectories in the case of a higher-dimensional base. A form preserves.the center of gravity if its cohomology class is the lift of a class from the base. As an application we prove that the number of closed trajectories of a particle on a surface under the action of a strong and little varying magnetic field perpendicular to the surface is not smaller than the minimal number of critical points of a function on the given surface. The author is grateful to V. I. Arnol'd for formulating the problem on the closed trajectories of fields of kernels of differential forms preserving the center of gravity and for helping in work. I. Condition of Preservation of the Center of Gravity. Let S I + M ~ B be an orientable bundle over a closed (compact and boundaryless) orientable manifold B.