Morse theory on Banach manifolds

LetM be aC ²-Finsler manifold modeled on a Banach space, and letf be aC ²-real-valued function defined onM. Using theA-gradient vector field which was introduced in [31] we give a suitable definition for nondegenegacy of critical points off, then generalize the Morse handle-body decomposition theorem and the Morse inequalities to a kind of Banach manifolds. A generalization in the reflexive case has been done in [31].     

Some aspects of analysis on almost complex manifolds with boundary

We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds, and the elliptic regularity of some diffeomorphisms of almost complex manifolds with boundary. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 47, Complex Analysis, 2007.     

Minimal Morse flows on compact manifolds

In this paper we prove, using the Poincaré-Hopf inequalities, that a minimal number of non-degenerate singularities can be computed in terms only of abstract homological boundary information. Furthermore, this minimal number can be realized on some manifold with non-empty boundary satisfying the abstract homological boundary information. In fact, we present all possible indices and types (connecting or disconnecting) of singularities realizing this minimal number. The Euler characteristics of all manifolds realizing this minimal number are obtained and the associated Lyapunov graphs of Morse type are described and shown to have the lowest topological complexity.     

On the geometry of model almost complex manifolds with boundary

We study some special almost complex structures on strictly pseudoconvex domains in ℝ^{2 n} . They appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. We determine explicitely some geometric invariants of these model structures and derive necessary and sufficient conditions for their integrability. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.     

Morse theory and Lyapunov stability on manifolds

The aim of this article is to recall the main theorems of Morse theory and to infer some corollaries for the problem of Lyapunov stability on manifolds. It makes a link between Morse theory and the general theory of the Lyapunov stability for dynamical systems.     

Linear independence of Morse functions on manifolds

It is proved that on a smooth n-dimensional manifold there exist n linearly independent Morse functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 835–838, June, 1990.     

A Morse index for geodesics in static Lorentz manifolds

Sponsored by MURST, research founds, 40%, 60%     

Collapsing manifolds with boundary

This paper studies manifolds-with-boundary collapsing in the Gromov– Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The first group of results provide a fiber bundle structure to the manifolds-with-boundary. One of the main theorems establishes a disc bundle structure for any manifold-with-boundary having two-sided bounds on sectional curvature and second fundamental form, and a lower bound on intrinsic injectivity radius, which is sufficiently close in the Gromov–Hausdorff topology to a closed manifold. Another result is a rough version of Toponogov’s Splitting Theorem. The second group of results identify Gromov–Hausdorff limits of certain sequences of manifolds with non-convex boundaries as Alexandrov spaces of curvature bounded below.     

Minimal Morse functions on a pair of manifolds

The existence theorem for a minimal Morse function on a pair of manifolds (M ⁿ,N ^k), wheren-k 3,k 6, is proved.Translated from Ukrainskii Matematiccheskii Zhurnal, Vol. 45, No. 1, pp. 143–144, January, 1993.     

A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M ⁿ , g) be a compact Riemannian manifold with boundary ?M. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have ?M as a constant mean curvature hypersurface. We prove that this set is compact for dimensions n ?? 7 under the generic condition that the trace-free 2nd fundamental form of ?M is nonzero everywhere.     

Morse theory for minimal surfaces in manifolds

A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \({\mathcal E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \({\mathcal E}\). We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \({\mathcal E}\) is non-degenerated.     

Nonlinear Hodge theory on manifolds with boundary

Summary The intent of this paper is first to provide a comprehensive and unifying development of Sobolev spaces of differential forms on Riemannian manifolds with boundary. Second, is the study of a particular class of nonlinear, first order, ellipticPDEs, called Hodge systems. The Hodge systems are far reaching extensions of the Cauchy-Riemann system and solutions are referred to as Hodge conjugate fields. We formulate and solve the Dirichlet and Neumann boundary value problems for the Hodge systems and establish the ℒ^p for such solutions. Among the many desirable properties of Hodge conjugate fields, we prove, in analogy with the case of holomorphic functions on the plane, the compactness principle and a strong theorem on the removability of singularities. Finally, some relevant examples and applications are indicated. Entrata in Redazione il 4 dicembre 1997. The first two authors were partially supported by NSF grants DMS-9401104 and DMS-9706611. Bianca Stroffolini was supported by CNR. This work started in 1993 when all authors were in Syracuse.