Morse Inequalities Research Articles

Computational complexity, torsion-freeness of homoclinic Floer homology and homoclinic Morse inequalities

Floer theory was originally devised to estimate the number of 1-periodic orbits of Hamiltonian systems. In earlier works, we constructed Floer homology for homoclinic orbits on two dimensional manifolds using combinatorial techniques. In the present paper, we study theoretic aspects of computational complexity of homoclinic Floer homology. More precisely, for finding the homoclinic points and immersions that generate the homology and its boundary operator, we establish sharp upper bounds in terms of iterations of the underlying symplectomorphism. This prepares the ground for future numerical works.Although originally aimed at numerics, the above bounds provide also purely algebraic applications, namely(1) torsion-freeness of primary homoclinic Floer homology,(2) Morse type inequalities for primary homoclinic orbits.

Transcendental Morse inequality and generalised Okounkov bodies

The main goal of this article is to construct arithmetic Okounkov for an arbitrary pseudo-effective (1,1)-class $\alpha$ on a Kahler manifold. Firstly, using Boucksom's divisorial Zariski decompositions for pseudo-effective (1,1)-classes on compact Kahler manifolds, we prove the differentiability of volumes of big classes for Kahler manifolds on which modified nef cones and nef cones coincide; this includes Kahler surfaces. We then apply our differentiability results to prove Demailly's transcendental Morse inequality for these particular classes of Kahler manifolds. In the second part, we construct the convex body $\Delta(\alpha)$ for any big class $\alpha$ with respect to a fixed flag by using positive currents, and prove that this newly defined convex body coincides with the Okounkov body when $\alpha\in {\rm NS}_{\mathbb{R}}(X)$; such convex sets $\Delta(\alpha)$ will be called generalized Okounkov bodies. As an application we prove that any rational point in the interior of Okounkov bodies is valuative. Next we give a complete characterisation of generalized Okounkov bodies on surfaces, and show that the generalized Okounkov bodies behave very similarly to original Okounkov bodies. By the differentiability formula, we can relate the standard Euclidean volume of $\Delta(\alpha)$ in $\mathbb{R}^2$ to the volume of a big class $\alpha$, as defined by Boucksom; this solves a problem raised by Lazarsfeld in the case of surfaces. Finally, we study the behavior of the generalized Okounkov bodies on the boundary of the big cone, which are characterized by numerical dimension.

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the M\"obius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory; to this aim we establish new geometric expansions of the derivative of the Willmore functional on exponentiated small Clifford tori degenerating, under the action of the M\"obius group, to small geodesic spheres with a small handle. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.

A Thom–Smale–Witten theorem on manifolds with boundary

Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization techniques. As a by-product, we obtain Morse inequalities for manifolds with boundary.

Witten’s perturbation on strata

The main result is a version of Morse inequalities for the mini- mum and maximum ideal boundary conditions of the de Rham complex on strata of compact Thom-Mather stratifications, endowed with adapted met- rics. An adaptation of the analytic method of Witten is used in the proof, as well as certain perturbation of the harmonic oscillator related with the Dunkl harmonic oscillator.

Conley\u2013Morse\u2013Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes

We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.

Existence of Holomorphic Sections and Perturbation of Positive Line Bundles over q–Concave Manifolds

By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on the manifold. The bound is given explicitely in terms of the volume of the domain in this metric and a certain integral on the boundary involving the defining function and its Levi form. As application we study the perturbattion of the complex structure of a q-concave manifold.

Morse theory for elastica

In Riemannian manifolds the elastica are critical points of the restriction of total squared geodesic curvature to curves with fixed length which satisfy first order boundary conditions. We verify that the Palais-Smale condition holds for this variational problem, and also the related problems where the admissible curves are required to satisfy zeroth order boundary conditions, or first order periodicity conditions. We also prove a Morse index theorem for elastica and use the Morse inequalities to give lower bounds for the number of elastica of each index in terms of the Betti numbers of the path space.

Morse inequalities for Fourier components of Kohn–Rossi cohomology of CR manifolds with $$S^1$$ S 1 -action

Let X be a compact connected CR manifold of dimension \(2n-1, n\ge 2\) with a transversal CR \(S^1\)-action on X. We study the Fourier components of the Kohn–Rossi cohomology with respect to the \(S^1\)-action. By studying the Szego kernel of the Fourier components we establish the Morse inequalities on X. Using the Morse inequalities we have established on X we prove that there are abundant CR functions on X when X is weakly pseudoconvex and strongly pseudoconvex at a point.

G-invariant holomorphic Morse inequalities

Consider an action of a connected compact Lie group on a compact complex manifold M, and two equivariant vector bundles L and E on M, with L of rank 1. In this note, we give holomorphic Morse inequalities in the spirit of Demailly for the invariant part of the Dolbeault cohomology of high tensor powers of L twisted by E, via the induced geometric data on the reduced space.

Optimal control in prescribing Webster scalar curvatures on 3-dimensional pseudo Hermitian manifolds

In this work, we give new existence and multiplicity results for the solutions of the prescription problem for the Webster scalar curvature on a 3-dimensional Pseudo Hermitian Manifold. The critical points of prescribed functions verify mixed conditions. We establish some Morse Inequalities at Infinity and a Poincaré–Hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the Morse index of such solutions.

The cohomological span of [formula omitted]-Conley index

In this paper we introduce a new homotopy invariant – the cohomological span of LS-Conley index. We prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form Lx+K(x), where L is bounded linear and K is completely continuous. We give examples of Hamiltonian systems for which our methods give better results than the Morse inequalities. We also give a formula for the LS-index of an isolated critical point, which is an extension of the classical Dancer theorem for the case of LS-index.

On the Logarithmic Green–Griffiths Conjecture: Table 1.

We establish that for complements of general hypersurfaces |$H\subset \mathbb {P}^{n}$| of degree |$d\geqslant (5n)^{2}\,n^{n}$|⁠, entire holomorphic curves |$f\colon \mathbb {C}\to \mathbb {P}^{n}\setminus H$| are algebraically degenerate. The proof follows in outline the recent implementation by Diverio–Merker–Rousseau of the variational method of Voisin–Siu combined with Demailly's algebraic Morse inequalities, into which it incorporates two new key ingredients, namely an iterated residue formula following Bérczi and low pole order logarithmic slanted vector fields, thereby reducing the reputedly challenging main result to a careful use of majorant series.

The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds

In a recent work (Bagchi and Datta, 2014) with Datta, we introduced the mu-vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu-vectors. With the new definition, most results of Bagchi and Datta (2014) become correct without the hypothesis of 2-neighbourliness. In particular, the combinatorial Morse inequalities of Bagchi and Datta (2014) are now true of all simplicial complexes.As an application, we prove the following generalized lower bound theorem (GLBT) for connected locally tame combinatorial manifolds. If M is such a manifold of dimension d, then for 1≤ℓ≤d−12 and any field F,gℓ+1(M)≥(d+2ℓ+1)∑i=1ℓ(−1)ℓ−iβi(M;F). Equality holds here if and only if M is ℓ-stacked.We conjecture that, more generally, this theorem is true of all triangulated connected and closed homology manifolds. A conjecture on the sigma-vectors of triangulated homology spheres is proposed, whose validity will imply this GLB Conjecture for homology manifolds. We also prove the GLB Conjecture for all connected and closed combinatorial 3-manifolds. Thus, any connected closed combinatorial manifold M of dimension three satisfies g2(M)≥10β1(M;F), with equality iff M is 1-stacked. This result settles a question of Novik and Swartz (2009) in the affirmative.

Sufficient bigness criterion for differences of two nef classes

We prove the qualitative part of Demailly's conjecture on transcendental Morse inequalities for differences of two nef classes satisfying a numerical relative positivity condition on an arbitrary compact K\"ahler (and even more general) manifold. The result improves on an earlier one by J. Xiao whose constant $4n$ featuring in the hypothesis is now replaced by the optimal and natural $n$. Our method follows arguments by Chiose as subsequently used by Xiao up to the point where we introduce a new way of handling the estimates in a certain Monge-Amp\`ere equation. This result is needed to extend to the K\"ahler case and to transcendental classes the Boucksom-Demailly-Paun-Peternell cone duality theorem if one is to follow these authors' method and was conjectured by them.

Generalized Poincaré–Hopf theorem and application to nonlinear elliptic problem

In this paper we get an extended version of Poincaré–Hopf theorem. Without the assumption that critical point set between two level sets of energy functional is finite, this result actually generalizes Morse inequality. And the isomorphism between Cq(J,∞) and Cq(J(a,b),0) is yielded as (a,b)∉Σ, where Cq(J,∞) denotes the critical groups of energetic functional J at infinity and Cq(J(a,b),0) stands for the critical groups of functional J(a,b) at zero, and Σ is the set of points (a,b)∈R2 for which the problem{−Δu+αu=au−+bu+,x∈Ω,∂u∂ν=0,x∈∂Ω, has a nontrivial solution, u+=max⁡{u,0}, u−=min⁡{u,0}. (Concerning the definitions of J and J(a,b), see Section 4.) As to application aspects, we are mainly concerned with nonlinear elliptic problem with Neumann boundary condition provided that the origin is a non-isolated critical point and obtain the existence of multiple solutions.

Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere𝕊+3

In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.

Linking and the Morse complex

Pour une fonction de Morse f sur une variété compacte orientée M, nous montrons que f a un nombre de points critiques supérieur au nombre requis par les inégalités de Morse si, et seulement si, il existe une certaine classe d’entrelacs dans M, dont les composantes ont un nombre d’enlacement non trivial, telle que la valeur minimale de f sur l’une des composantes est supérieure à sa valeur maximale sur l’autre composante. Nous définissons le nombre exact de points critiques de f en fonction des nombres de Betti de M et du comportement de f par rapport aux entrelacs. Ce résultat peut être vu comme un raffinement, dans le cas des variétés compactes, du théorème du point selle de Rabinowitz. Notre approche, partiellement inspirée des techniques de théorie symplectique de Floer au niveau des chaînes, est basée sur l’association d’opérations algébriques sur le complexe de Morse de f à certaines collections de chaînes de M, ce qui induit des relations entre les nombres d’enlacement des (pseudo-)cycles homologiquement triviaux de M d’une part, et un accouplement d’enlacement algébrique sur le complexe de Morse d’autre part.

AN ANALYTIC APPROACH TO THE STRATIFIED MORSE INEQUALITIES FOR COMPLEX CONES

In a previous paper the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. It is well-known from stratified Morse theory that the singular points of the complex cone contribute to the stratified Morse inequalities in middle degree only. In this paper, an analytic proof of this fact is given.

On a geometric equation involving the Sobolev trace critical exponent

Abstract In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of R n , n ≥ 4 . Under the assumption that the prescribed function is flat near its critical point, we give precise estimates on the losses of the compactness, and we provide a new existence result of Bahri-Coron type. Moreover, we establish, under generic boundary condition, a Morse inequality at infinity, which gives a lower bound on the number of solutions to the above problem. MSC:58E05, 35J65, 53C21, 35B40.