The Relation of the Morse Index of Closed Geodesics with the Maslov-type Index of Symplectic Paths

In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov–type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincaré map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov–type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic. Project 10071040 supported by NNSF, 200014 supported by Excellent. Ph.D. Funds of ME of China, and PMC Key Lab. of ME of China     

Morse theory for periodic solutions of hamiltonian systems and the maslov index

In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over π₂ (M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every 1-periodic solution has at least one Floquet multiplier which is not equal to 1.     

Morse Novikov theory and cohomology with forward supports

We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson [HL]. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of the manifold. This complex is then used in Novikov theory, to obtain a geometric realization of the Novikov Complex as a complex of currents and a new characterization of Novikov Homology as cohomology with compact forward supports. Two natural ``backward-forward' dualities are also established: a Lambda duality over the Novikov Ring and a Topological Vector Space duality over the reals.     

A little microlocal Morse theory

If a complex analytic function, f, has a stratified isolated critical point, then it is known that the cohomology of the Milnor fibre of f has a direct sum decomposition in terms of the normal Morse data to the strata. We use microlocal Morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along f have isolated support. We also investigate an index-theoretic proof of this fact. Received: 15 September 2000 / Published online: 18 June 2001     

A relative morse index for the symplectic action

The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define for any pair of critical points of a a relative Morse index, which corresponds to the difference of the two Morse indices in finite dimensions. It is based on the spectral flow of the Hessian of a and can be identified with a topological invariant recently defined by Viterbo, and with the dimension of the space of trajectories between the two critical points.     

Rigidity and gluing for Morse and Novikov complexes

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold (M,) with c₁|_2(M)=[]|_2(M)=0. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently C⁰ close generic function/hamiltonian. The gluing result is a type of Mayer-Vietoris formula for the Morse complex. It is used to express algebraically the Novikov complex up to isomorphism in terms of the Morse complex of a fundamental domain. Morse cobordisms are used to compare various Morse-type complexes without the need of bifurcation theory.     

Maslov index and Morse theory for the relativistic Lorentz force equation

We study the Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold. The flow of the Jacobi equation along each solution preserves the so-called twisted symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We introduce the notion of F-conjugate plane for each conjugate instant; the restriction of the spacetime metric to the F-conjugate plane is used to compute the Maslov index, which is given by a sort of algebraic count of the conjugate instants. For a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field preserving the flow of the Killing vector field, we introduce a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. We prove that the value of this Morse index equals the Maslov index and we prove the Morse relations for the solutions of the Lorentz force equation in a static spacetime.Mathematics Subject Classification (2002): Primary: 58E10, 83C10; Secondary: 53D12     

New bounds for Morse clusters

This paper presents new, simple arguments improving the lower bounds for the total energy and the minimal inter-particle distance in minimal energy atom cluster problems with interactions given by a Morse potential, where the atom separation problem is difficult due to the finite energy at zero atom separation. Apart from being sharper than previously known bounds, they also apply for a wider range ρ ≥ 4.967 of the parameter in the Morse potential. Most results also hold for more general pair potentials.     

Finite Morse Index Implies Finite Ends

We prove that finite Morse index solutions to the Allen-Cahn equation in ℝ² have finitely many ends and linear energy growth. The main tool is a curvature decay estimate on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second-order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation. Using an indirect blowup technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the obstruction to the uniform second-order regularity of clustering interfaces in ℝⁿ is associated to the existence of nontrivial entire solutions to a (finite or infinite) Toda system in ℝ^n–1. For finite Morse index solutions in ℝ², we show that this obstruction does not exist by using information on stable solutions of the Toda system. © 2019 Wiley Periodicals, Inc.     

A Morse Index Theorem for Elliptic Operators on Bounded Domains

Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ? ? ⁿ is deformed. For a family of domains {Ω _t } _{t∈[a, b]} we prove that the Morse index of L on Ω _a differs from the Morse index of L on Ω _b by the Maslov index of a path of Lagrangian subspaces on the boundary of Ω. This is particularly useful when Ω _a is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the “original” problem (on Ω _b ) and the “simplified” problem (on Ω _a ). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.     

Improved bounds for interatomic distance in Morse clusters

We improve the best known lower bounds on the distance between two points of an optimal Morse cluster, with ρ[4.967,15]. We develop a generalization of a method previously applied to the Lennard-Jones potential, that also leads to improvements of lower bounds for the Morse potential.     

A Morse complex on manifolds with boundary

Given a compact smooth manifold M with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or relative to the boundary) homology of M with integer coefficients. Our approach simplifies other methods which have been discussed in more specific geometric settings.     

Computation of Morse decompositions for semilinear elliptic PDEs

A numerical method for computing all solutions of an elliptic boundary value problem Au + g[u, λ] = 0 and their Morse indices as steady‐states of the parabolic problem u_t + Au + g[u, λ] = 0 is presented. Morse decompositions are also determined. The method uses a finite element approach that is based on the method of alternative problems. Error estimates for the finite element approximations are verified and examples are given. © John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 290–312, 2001     

The scalar curvature equation on

We give existence results for solutions of the prescribed scalar curvature equation on S³, when the curvature function is a positive Morse function and satisfies an index-count condition.     

Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity

ABSTRACT

The existence of a nontrivial solution for quasi-linear elliptic equations involving the p-Laplace operator and a nonlinearity with p-linear growth at infinity is proved. Techniques of Morse theory are employed.     

The algebraic construction of the Novikov complex of a circle-valued Morse function

The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction of the real-valued Morse function to a fundamental domain of the pullback infinite cyclic cover of M. Received: 23 November 2000 / Revised version: 3 May 2001 / Published online: 28 February 2002     

Multiple nontrivial solutions for resonant Neumann problems

We consider semilinear second order elliptic Neumann problems, which are resonant both at infinity (with respect to an eigenvalue λ_k, k ≥ 1) and at zero (with respect to the principal eigenvalue λ₀ = 0). Using techniques from Morse theory, combined with variational methods, we are able to show that the problem has at least four nontrivial smooth solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topological invariants of real Milnor fibres

 Let and be analytic germs, let be a closed ball of small radius centred at the origin, and let . Let , where λ(p) is the Morse index at a critical point p of a Morse function which approximates g|W _z . For F and g satisfying some natural assumptions, we shall show that there is a finite family of analytic germs θ _i (z) such that for a generic regular value z. Received: 8 May 2001 / Revised version: 16 May 2002 Published online: 24 January 2003 Supported by grant BW 5100-5-0256-9 Mathematics Subject Classification (2000): 14P15, 32B10     

Antiperfect morse stratification

For an equivariant Morse stratification that contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincaré series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal G-bundle over a connected, closed, nonorientable surface Σ is locally equivariantly \mathbbQ{\mathbb{Q}}-antiperfect when G = U(2), SU(2), U(3), SU(3); we propose that the Morse stratification is actually equivariantly \mathbbQ{\mathbb{Q}}-antiperfect in these cases. Our proposal yields formulas of Poincaré series P_t^G(Hom(p₁(S),G);\mathbbQ){P_t^G({\rm Hom}(\pi_1(\Sigma),G);\mathbb{Q})} when G = U(2), SU(2), U(3), SU(3). Our U(2), SU(2) formulas agree with formulas proved by T. Baird, who also verified our conjectural U(3) formula.