Morse theory for analytic functions on surfaces

In this paper we deal with analytic functions defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values −1, 0 or 1, and prove that Q is made up of finitely many (critical) points with non zero index and embedded circles. Further, we generalize the famous Morse result by showing that the sum of the indexes of the critical points of f equals χ (S), the Euler characteristic of S. As an intermediate result we locally describe the level set of f near a point p ∈Q. We show that the level set f ⁻¹(f (p)) is either a) the set {p}, or b) the graph of a smooth curve passing through p, or c) the graphs of two smooth curves tangent at p or d) the graphs of two smooth curves building at p a cusp shape.     

Special framed Morse functions on surfaces

Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M and $\mathbb{F}^1$ be the space of framed Morse functions both endowed with the C ^??-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping is a homotopy equivalence. In the case when at least x(M) + 1 critical points of each function of F are marked, the homotopy equivalences and are proved, where is the complex of framed Morse functions, is the universal moduli space of framed Morse functions, is the group of self-diffeomorphisms of M homotopic to the identity.     

Deformations of circle-valued Morse functions on surfaces

Let M be a smooth connected orientable compact surface and let F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) be a space of all Morse functions f : M → S ¹ without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S ¹ is either a constant map or a covering map. The space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) is endowed with the C ^∞-topology. We present the classification of connected components of the space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) . This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M.     

Morse lemma for functionals of variational calculus

Institute for Control Problems, Academy of the Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 1–11, July–September, 1991.     

Uniform approximation by meromorphic functions on Riemann surfaces

A closed subsetE of a Riemann surfaceS is called a set of uniform meromorphic approximation if every functionf continuous onE and holomorphic onE ⁰ can be approximated uniformly onE by meromorphic functions onS. We show that ifE is a set of uniform meromorphic approximation, then so is for every compact parametric diskD. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A. G. Vitushkin. Partially supported by a grant from NSERC of Canada.     

The number of critical elements of discrete Morse functions on non-compact surfaces

This paper is focused on looking for links between the topology of a connected and non-compact surface with finitely many ends and any proper discrete Morse function which can be defined on it. More precisely, we study the non-compact surfaces which admit a proper discrete Morse function with a given number of critical elements. In particular, given any of these surfaces, we obtain an optimal discrete Morse function on it, that is, with the minimum possible number of critical elements.     

Morse functions on cobordisms

We study the homotopy invariants of crossed and Hilbert complexes. These invariants are applied to the calculation of the exact values of Morse numbers of smooth cobordisms.     

Morse inequalities for almost-periodic functions

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

Minimal surfaces and Schwarz lemma

We prove a sharp Schwarz lemma type inequality for the Weierstrass–Enneper parameterization of minimal disks. It states the following. If $F:\mathbf{D}\to \Sigma$ is a conformal harmonic parameterization of a minimal disk $\Sigma \subset {\mathbf{R}}^3$, where $\mathbf{D}$ is the unit disk and $\left|\Sigma \right|=\pi R^2$, then $\left|F_x\left(z\right)\right|(1-{\left|z\right|}^2)\le R$. If for some $z$ the previous inequality is equality, then the surface is an affine image of a disk, and $F$ is linear up to a Möbius transformation of the unit disk.     

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.     

The Schwarz lemma for nonpositively curved Riemmanian surfaces

In this paper, we prove that if f is a conformal map between two Riemannian surfaces, and if the curvature of the target is nonpositive and less than or equal to the curvature of the source, then the map is contracting.     

On a Morse conjecture for analytic flows on compact surfaces

The aim of this paper is to prove a Morse conjecture; in particular it is shown that a topologically transitive analytic flow on a compact surface is metrically transitive. We also build smooth topologically transitive flows on surfaces which are not metrically transitive.     

Optimal discrete Morse functions for 2-manifolds

Morse theory is a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a Morse function has been defined on a manifold, then information about its topology can be deduced from its critical elements. The main objective of this paper is to introduce a linear algorithm to define optimal discrete Morse functions on discrete 2-manifolds, where optimality entails having the least number of critical elements. The algorithm presented is also extended to general finite cell complexes of dimension at most 2, with no guarantee of optimality.     

On Drewnowski lemma for non-additive functions and its consequences

In this paper we state an extension of a Drewnowski lemma to non-additive functions which are defined on an orthomodular structure and attain values into a uniform space, where no algebraic structure is required and the uniformity is induced by a complete metric. As consequences we prove Brooks–Jewett as well as Cafiero theorems for such class of functions.