Morse Inequalities Research Articles

Remark on the Off-Diagonal Expansion of the Bergman Kernel on Compact Kähler Manifolds

In this short note, we compare our previous works on the off-diagonal expansion of the Bergman kernel and the recent preprint of Lu-Shiffman (arxiv.1301.2166). In particular, we note that the vanishing of the coefficient of p^{-1/2} is implicitly contained in Dai-Liu-Ma's work (J. Differential Geom. 72 (2006), no. 1, 1-41) and was explicitly stated in our book (Holomorphic Morse inequalities and Bergman kernels, Progress in Math., vol. 254, Birkh\"auser, 2007).

On the homotopy type of spaces of Morse functions on surfaces

Let be a smooth closed orientable surface. Let be the space of Morse functions on with a fixed number of critical points of each index such that at least critical points are labelled by different labels (numbered). The notion of a skew cylindric-polyhedral complex is introduced, which generalizes the notion of a polyhedral complex. The skew cylindric-polyhedral complex (“the complex of framed Morse functions”) associated with the space is defined. In the case the polytope is finite; its Euler characteristic is calculated and the Morse inequalities for its Betti numbers are obtained. The relation between the homotopy types of the polytope and the space of Morse functions equipped with the -topology is indicated.Bibliography: 51 titles.

Morse theory on transnormal embeddings of S^m

In this work we use Morse theory to study the generating frames of transnormal embeddings of S m in R n . In general we start with M as a smooth compact connected m-manifold without boundary and f : M −→ R as a function whose critical points are non-degenerate. Using Morse Inequalities, we show that the generating frame of a transnormal embedding of S 1 splits into two transitive sets. Also we give a detailed proof for Robertson’s conclusion [13] regarding r-transnormal embeddings of S m , m> 1.

Morse equation of attractors for nonsmooth dynamical systems

This paper is concerned with a Morse theory of attractors for finite-dimensional nonsmooth dynamical systems described by differential inclusions with upper semi-continuous righthand sides. We first show that all open attractor neighborhoods of an attractor share the same homotopy type. Then based on this basic fact we introduce the concept of homology index for Morse sets and establish Morse inequalities and Morse equation by using smooth Morse–Lyapunov functions.

The Witten deformation for even dimensional conformally conic manifolds

The goal of this article is to generalise the Witten deformation to even dimensional conformally conic manifolds X X and a class of functions f : X → R f: X \to \mathbb R called admissible Morse functions. We get Morse inequalities relating the L 2 \mathrm {L}^2 -Betti numbers of X X with the number of critical points of the function f f . Hereby the contribution of a singular point p p of X X to the Morse inequalities can be expressed in terms of the intersection cohomology of the local Morse data of f f at p p . The definition of an admissible Morse function is inspired by stratified Morse theory as developed by Goresky and MacPherson.

Discrete Morse theory for manifolds with boundary

We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain Relative Morse Inequalities relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman's Sphere Theorem. The main corollaries of our work are: (1) For each d ≥ 3 and for each k ≥ 0, there is a PL d-sphere on which any discrete Morse function has more than k critical (d -1)-cells. (This solves a problem by Chari.) (2) For fixed d and k, there are exponentially many combinatorial types of simplicial d-manifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most k critical interior (d -1)-cells. (This connects discrete Morse theory to enumerative combinatorics/ discrete quantum gravity.) (3) The barycentric subdivision of any simplicial constructible d-ball is collapsible. (This almost solves a problem by Hachimori.) (4) Every constructible ball collapses onto its boundary minus a facet. (This improves a result by the author and Ziegler.) (5) Any 3-ball with a knotted spanning edge cannot collapse onto its boundary minus a facet. (This strengthens a classical result by Bing and a recent result by the author and Ziegler.)

Conditional extremals in complete Riemannian manifolds

Conditional extremal curves in a complete Riemannian manifold M are defined as the critical points of the squared L 2 distance between the tangent vector field of a curve and a so-called prior vector field. We prove that this L 2 distance satisfies the Palais–Smale condition on the space of absolutely continuous curves joining two submanifolds of M , and thus establish the existence of critical points. We also prove a Morse index theorem in the case where the two submanifolds are single points, and use the Morse inequalities to place lower bounds on the number of critical points of each index.

Remark on dynamical Morse inequality

We solve a transversality problem relating to Bertelson-Gromov’s “dynamical Morse inequality”.

Delocalized Betti numbers and Morse type inequalities

In this paper we state and prove Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle.

On a bi-harmonic equation involving critical exponent: Existence and multiplicity results

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: Δ 2u = K( x) u p , u > 0 in Ω, Δ u = u = 0 on ∂Ω, where Ω is a smooth domain in ℝ n , n ≥ 5 , and p + 1 = 2 n n − 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension \({2n-1, n\,\geqslant\,2}\), and let L k be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in L k, for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex–concave manifolds.

The Arrhenius formula in kinetic theory and Witten's spectral asymptotics

A new approach to the proof of the Arrhenius formula of kinetic theory is proposed. We prove this formula by starting from the equation of diffusion in a potential. We put this diffusion equation in the form of evolutionary equation generated by some Schrödinger operator. We show that the Arrhenius formula for the rate of the over-barrier transitions follows from the formula for the rate of quantum tunnel transitions for the considered Schrödinger operator. The relation of the proposed approach and the Witten method of the proof of the Morse inequalities is discussed. In our approach, the Witten spectral asymptotics takes the form of the low-temperature limit and the Arrhenius formula is a correction to the Witten asymptotics.

The Witten complex for singular spaces of dimension two with cone‐like singularities

AbstractThe Witten deformation is a method proposed by Witten which, given a function \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$f:M \longrightarrow \mathbb {R}$\end{document} on a smooth compact manifold M, allows to prove the Morse inequalities. Witten’s proof of the Morse inequalities is analytical and can thus be applied to situations where the Thom‐Smale method is not accessible. In these notes we generalise the Witten deformation to certain singular Riemannian manifolds X which are metric models for singular algebraic curves, and functions on X which we call admissible Morse functions. They are particular examples of stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Existence and Multiplicity of Solutions for Discrete Nonlinear Two‐Point Boundary Value Problems

By using Morse theory, the critical point theory, and the character of K1/2, we consider the existence and multiplicity results of solutions to the following discrete nonlinear two‐point boundary value problem −Δ2x(k − 1) = f(k, x(k)), k ∈ ℤ(1, T) subject to x(0) = 0 = Δx(T), where T is a positive integer, ℤ(1, T) = {1, 2, …, T}, Δ is the forward difference operator defined by Δx(k) = x(k + 1) − x(k), and f : ℤ(1, T) × ℝ → ℝ is continuous. In argument, Morse inequalities play an important role.

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems

Let I be an open interval, M be a real manifold, T*M its cotangent bundle and \Phi={\phi_t}, t in I, a homogeneous Hamiltonian isotopy of T*M defined outside the zero-section. Let \Lambda be the conic Lagrangian submanifold associated with \Phi (\Lambda is a subset of T*M x T*M x T*I). We prove the existence and unicity of a sheaf K on MxMxI whose microsupport is contained in the union of \Lambda and the zero-section and whose restriction to t=0 is the constant sheaf on the diagonal of MxM. We give applications of this result to problems of non displaceability in contact and symplectic topology. In particular we prove that some strong Morse inequalities are stable by Hamiltonian isotopies and we also give results of non displaceability for positive isotopies in the contact setting. In this new version we suppress one hypothesis in the main theorem and we extend the result of non displaceability for positive isotopies.

PRESCRIBING MEAN CURVATURE ON 𝔹n

In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball 𝔹n, n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β∈(n-2, n-1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler–Hopf type formula.

The Witten deformation for even dimensional spaces with cone-like singularities and admissible Morse functions

In this Note we generalise the Witten deformation to even dimensional Riemannian manifolds with cone-like singularities X and certain functions f, which we call admissible Morse functions. As a corollary we get Morse inequalities for the L 2-Betti numbers of X. The contribution of a singular point p of X to the Morse inequalities can be expressed in terms of the intersection cohomology of the local Morse datum of f at p. The definition of the class of functions which we study here is inspired by stratified Morse theory as developed by Goresky and MacPherson. However the setting here is different since the spaces considered here are manifolds with cone-like singularities instead of Whitney stratified spaces.

Generic multiplicity for a scalar field equation on compact surfaces

We prove generic multiplicity of solutions for a scalar field equation on compact surfaces via Morse inequalities. In particular our result improves significantly the multiplicity estimate which can be deduced from the degree-counting formula in Chen and Lin (2003) [12]. Related results are derived for the prescribed Q-curvature equation.

Divisorial Zariski decomposition and algebraic Morse inequalities

In this paper we use the divisorial Zariski decomposition to give a more precise version of the algebraic Morse inequalities.