Let M be a closed connected manifold. Let m(M) be the Morse number of M, that is, the minimal number of critical points of a Morse function on M. Let N be a finite cover of M of degree d. M.Gromov posed the following question: what are the asymptotic properties of m(N) as d goes to infinity? In this paper we study the case of high dimensional manifolds M with free abelian fundamental group. Let x be a non-zero element of H^1(M), let M(x) be the infinite cyclic cover corresponding to x, and t be a generator of the structure group of this cover. Set M(x,k)=M(x)/t^k. We prove that the sequence m(M(x,k))/k converges as k goes to infinity. For x outside of a finite union of hyperplanes in H^1(M) we obtain the asymptotics of m(M(x,k)) as k goes to infinity, in terms of homotopy invariants of M related to Novikov homology of M.
In this paper, we present a sufficient condition for finite Morse index of complete properly self-shrinkers. We prove that a complete properly embedded self-shrinker in $\mathbb{R}^{n+1}$ with finite asymptotically conical ends or asymptotically cylindrical ends must have finite Morse index. Moreover, as an application of this result, we show that a complete properly embedded self-shrinker in $\mathbb{R}^3$ with finite genus has finite Morse index.
Given a smooth bounded domain $Ø\subseteq \R^2$, we consider the equation $\D v = 2 v_x \wedge v_y$ in $Ø$, where $v: Ø\to \R^3$. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple-bubbling phenomena. This allows us to settle a particular case of a question raised by H. Brezis and J.M. Coron.
Let $X$ be a compact connected CR manifold of dimension $2n-1, n\geq 2$. We assume that there is a transversal CR locally free $S^1$ action on $X$. Let $L^k$ be the $k$-th power of a rigid CR line bundle $L$ over $X$. Without any assumption on the Levi-form of $X$, we obtain a scaling upper-bound for the partial Szegő kernel on $(0,q)$-forms with values in $L^k$. After integration, this gives the weak Morse inequalities. By a refined spectral analysis, we also obtain the strong Morse inequalities in CR setting. We apply the strong Morse inequalities to show that the Grauert-Riemenschneider cri…
We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds.
It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. I…
We develop the theory of twisted L^2-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on the first de Rham cohomology of M and they generalize the standard notions. A new feature of the twisted L^2-cohomology theory is that in addition to satisfying the standard L^2 Morse inequalities, they also satisfy certain asymptotic L^2 Morse inequalities. These reduce to the standard Morse inequalities in the finite dimensional case, and when the Morse 1-form is exact. We define the extended twisted L^2 de Rham cohomology and …
Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory.
A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same role as the Kahler class in Kahler geometry. If these classes for two LCK-structures coincide, the difference between these structures can be expressed by a smooth potential, similar to the Kahler case. We show that the Morse-Novikov class and the Bott-Chern class of a Vaisman ma…
Let $X$ be an abstract orientable not necessarily compact CR manifold of dimension $2n+1$, $n\geq1$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish asymptotics of the heat kernel of Kohn Laplacian with values in $L^k$. As an application, we give a heat kernel proof of Morse inequalities on compact CR manifolds. When $X$ admits a transversal CR $\mathbb R$-action, we also establish asymptotics of the $\mathbb R$-equivariant heat kernel of Kohn Laplacian with values in $L^k$. As an applicat…
Let $(X,ω)$ be a Hermitian manifold and let $(E,h^E)$, $(F,h^F)$ be two Hermitian holomorphic line bundle over $X$. Suppose that the maximal rank of the Chern curvature $c(E)$ of $E$ is $r$, and the kernel of $c(E)$ is foliated, i.e. there is a foliation $Y$ of $X$, of complex codimension $r$, such that the tangent space of the leaf at each point $x\in X$ is contained in the kernel of $c(E)$. In this paper, local versions of Demailly-Bouche's holomorphic Morse inequalities (which give asymptotic bounds for cohomology groups $H^{q}(X,E^k\otimes F^l)$ as $k,l,k/l\rightarrow \infty$) are presente…
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 Möbius 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 Möbius group, to small geodesic spheres with a small handle. By u…
Let F be a flat vector bundle over a compact Riemannian manifold M and let f be a Morse function. Let g be a smooth Euclidean metric on F, let g_t=e^{-2tf}g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g_t. Assuming that the vector field grad(f) satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for \log(ρ(t)) for t\to +\infty of the form a_0+a_1t+b\log\left(\frac tπ\right)+o(1), where the coefficient b is a half-integer depending only on the Betti numbers of F. In the case where all the critical values of f are rational, we…
Consider a flat vector bundle F over compact Riemannian manifold M and let f be a self-indexing Morse function on M. Let g be a smooth Euclidean metric on F. Set g_t=exp(-2tf)g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g_t. Assuming that the vector field $grad f$ satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log(ρ(t)) for t\to\infty of the form a_0+a_1t+b log(t)+o(1). We present explicit formulae for coefficients a_0,a_1 and b. In particular, we show that b is a half integer.
Firstly, we wish to motivate that Conley pairs, realized via Salamon's definition [17], are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces [21]. From this fell off quite naturally, firstly, an alternative proof [20] of the cell attachment theorem in Morse theory [13] and, secondly, some ideas [12] how to try to organize the closures of the unstable manifolds of a Morse-Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use t…
A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the $dd^c$-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds.…
Let $ω$ be a Morse form on a manifold $M$. Let $p:\hat M\to M$ be a regular covering with structure group $G$, such that $p^*([ω])=0$. Let $ξ:G\to\mathbf{R}$ be the corresponding period homomorphism. Denote by ${\hat Λ}_ξ$ the Novikov completion of the group ring $\mathbf{Z} G$. Choose a transverse $ω$-gradient $v$. Counting the flow lines of $v$ one defines the Novikov complex $\mathcal{N}_*$ freely generated over ${\hat Λ}_ξ$ by the set of zeroes of $ω$. In this paper we introduce a refinement of this construction. We define a subring $\hatΛ_Γ$ of ${\hat Λ}_ξ$ and show that the Novikov compl…
