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.     

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     

Rigidity of hyperstable complexes

A module J over a ring is said to be hyperstable when . Over a module M for which Ext we show that the projective n-stems for which is hyperstable constitute a single homotopy type. Received: 17 November 2006     

Novikov algebras and Novikov structures on Lie algebras

We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.     

Numerical invariants of cochain complexes and the Morse numbers of manifolds

Homotopy invariants of free cochain complexes are studied. These invariants are applied to the calculation of exact values of the Morse numbers for smooth manifolds of large dimension. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 252, pp. 261–276.     

Engel Theorem for Novikov Algebras

We prove that, if A is left-nil Novikov algebra, then A ² is nilpotent.     

Uniform Morse lemma and isotopy criterion for Morse functions on surfaces

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let $ \mathcal{D}_0 $ \mathcal{D}_0 ⊂ Diff(M) be the group of diffeomorphisms homotopic to id _M . Two smooth functions f, g: M → ℝ are called isotopic if f = h ₂ ℴ g ℴ h ₁ for some diffeomorphisms h ₁ ∈ $ \mathcal{D}_0 $ \mathcal{D}_0 and h ₂ ∈ Diff⁺(ℝ). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from F to be isotopic is proved. For each Morse function f ∈ F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which continuously and Diff(M)-equivariantly depends on f in C ^∞-topology on F (“uniform Morse lemma”). Applications of these results to the problem of describing the homotopy type of the space F are formulated.     

The Novikov Conjecture for Linear Groups

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C^*-algebra theory.     

Generalized gluing for Einstein constraint equations

In this paper we construct a family of new (topologically distinct) solutions to the Einstein constraint equations by performing the generalized connected sum (or fiber sum) of two known compact m-dimensional constant mean curvature solutions (M ₁, g ₁, Π₁) and (M ₂, g ₂,Π₂) along a common isometrically embedded k-dimensional sub-manifold (K, g _K ). Away from the gluing locus the metric and the second fundamental form of the new solutions can be chosen as close as desired to the ones of the original solutions. The proof is essentially based on the conformal method and the geometric construction produces a polyneck between M ₁ and M ₂ whose metric is modeled fiber-wise (i. e. along the slices of the normal fiber bundle of K) around a Schwarzschild metric; for these reasons the codimension n : =  m − k of K in M ₁ and M ₂ is required to be  ≥  3. In this sense our result is a generalization of the Isenberg–Mazzeo–Pollack gluing, which works for connected sum at points and in dimension 3. The solutions we obtain for the Einstein constraint equations can be used to produce new short time vacuum solutions of the Einstein system on a Lorentzian (m + 1)-dimensional manifold, as guaranteed by a well known result of Choquet-Bruhat.     

Well-posedness and persistence properties for the Novikov equation

Recently, Novikov found a new integrable equation (we call it the Novikov equation in this paper), which has nonlinear terms that are cubic, rather than quadratic, and admits peaked soliton solutions (peakons). Firstly, we prove that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces (which generalize the Sobolev spaces Hs) with the critical index . Then, well-posedness in Hs with , is also established by applying Kato's semigroup theory. Finally, we present two results on the persistence properties of the strong solution for the Novikov equation.     

Classical r-Matrices and Novikov Algebras

We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of solvable Lie algebras.     

The Cauchy problem for the Novikov equation

In this paper we consider the Cauchy problem for the Novikov equation. We prove that the Cauchy problem for the Novikov equation is not locally well-posed in the Sobolev spaces ${H^s(\mathfrak{R})}$ with ${s < \frac{3}{2}}$ in the sense that its solutions do not depend uniformly continuously on the initial data. Since the Cauchy problem for the Novikov equation is locally well-posed in ${H^{s}(\mathfrak{R})}$ with s > 3/2 in the sense of Hadamard, our result implies that s =  3/2 is the critical Sobolev index for well-posedness. We also present two blow-up results of strong solution to the Cauchy problem for the Novikov equation in ${H^{s}(\mathfrak{R})}$ with s > 3/2.     

Refined gluing for vacuum Einstein constraint equations

We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in every case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus.     

Blow-up phenomenon for the integrable Novikov equation

In this paper we investigate a new integrable equation derived recently by V.S. Novikov [Generalizations of the Camassa–Holm equation, J. Phys. A 42 (34) (2009) 342002, 14 pp.]. Analogous to the Camassa–Holm equation and the Degasperis–Procesi equation, this new equation possesses the blow-up phenomenon. Under the special structure of this equation, we establish sufficient conditions on the initial data to guarantee the formulation of singularities in finite time. A global existence result is also found.     

Stability of peakons for the Novikov equation

In this paper we study the orbital stability of the peaked solitons to the Novikov equation, which is an integrable Camassa–Holm type equation with cubic nonlinearity. We show that the shapes of these peaked solitons are stable under small perturbations in the energy space.     

Soliton solutions to the Novikov equation and a negative flow of the Novikov hierarchy

The Novikov equation and a negative flow of the Novikov hierarchy are related to a negative flow of the Sawada–Kotera hierarchy and the Sawada–Kotera equation by reciprocal transformations, respectively. With the help of the Darboux transformations for the negative flow of the Sawada–Kotera hierarchy, the Sawada–Kotera equation and reciprocal transformations, we obtain a parametric representation for $N$-soliton solutions to the Novikov equation and the negative flow of the Novikov hierarchy.     

Morse inequalities for almost-periodic functions

Translated from Matematicheski Zametki, Vol. 50, No. 2, pp. 146–151, August, 1991.