Related papers: Deformations of circle-valued Morse functions on s…

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…

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of…

We prove that if f is an orientation-preserving homeomorphism of a closed orientable surface M whose singular set is totally disconnected, then f is topologically conjugate to a conformal transformation.

Many of the existing results for closed Hamiltonian G-manifolds are based on the analysis of the corresponding Hamiltonian functions using Morse-Bott techniques. In general such methods fail for non-compact manifolds or for manifolds with…

On closed symplectically aspherical manifolds, Schwarz proved a classical result that the action function of a nontrivial Hamiltonian diffeomorphism is not constant by using Floer homology. In this article, we generalize Schwarz's theorem…

We show that a finite volume deformation retract $\mathcal{T}_{\varepsilon_t}^{-}(\mathcal{N}_g)/\mathrm{MCG}(\mathcal{N}_g)$ of the moduli space $\mathcal{M}(\mathcal{N}_g)$ of non-orientable surfaces $\mathcal{N}_g$ behaves like the…

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda}$-a-priori bound for surfaces for which this functional is finite. In fact, it turns out…

Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve…

In this article we construct a type of deformations of representations $\pi_1(M)\rightarrow G$ where $G$ is an arbitrary lie group and $M$ is a large class of manifolds including CAT(0) manifolds. The deformations are defined based on…

Let $G$ be a torus and $M$ a compact Hamiltonian $G$-manifold with finite fixed point set $M^G$. If $T$ is a circle subgroup of $G$ with $M^G=M^T$, the $T$-moment map is a Morse function. We will show that the associated Morse…

The paper suggests new topological lower bounds for the number of zeros of closed 1-forms within a given cohomology class. The main new technical tool is the deformation complex, which allows to pass to a singular limit and reduce the…

The main objective of this paper is to extend Morse-Forman theory to vector-valued functions. This is mostly motivated by the need to develop new tools and methods to compute multiparameter persistence. To generalize the theory, in addition…

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed…

Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of…

This paper investigates fractional torsional rigidity on compact, connected metric graphs, a novel extension of the classical concept to nonlocal operators. The fractional torsional rigidity is defined as the $L^1$-norm of the fractional…

Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies…

Let $M$ be an orientable connected closed surface and $f$ be an $R$-closed homeomorphism on $M$ which is isotopic to identity. Then the suspension of $f$ satisfies one of the following condition: 1) the closure of each element of it is…

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points…

In this paper, we study Homeo$^1(S)$, the group of homeomorphisms of a surface that preserve the set of one-dimensional $C^1$ submanifolds of that surface. The group Homeo$^1(S)$ belongs to a family of similarly defined groups Homeo$^k(S)$…

Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $\pi:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $f\o\pi$ on the geodesic balls of…