Related papers: Deformative Magnetic Marked Length Spectrum Rigidi…

Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and…

Given $k\in \mathbb{R},$ $v,$ $D>0,$ and $n\in \mathbb{N},$ let $\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }$ be a Gromov-Hausdorff convergent sequence of Riemannian $n$--manifolds with sectional curvature $\geq k,$ volume $>v,$ and…

Let $\rho_\Sigma=h(|z|^2)$ be a metric in a Riemann surface $\Sigma$, where $h$ is a positive real function. Let $\mathcal H_{r_1}=\{w=f(z)\}$ be the family of univalent $\rho_\Sigma$ harmonic mapping of the Euclidean annulus…

Let $(M,g)$ be an $n-$dimensional compact Riemannian manifold. Let $h$ be a smooth function on $M$ and assume that it has a critical point $\xi\in M$ such that $h(\xi)=0$ and which satisfies a suitable flatness assumption. We are interested…

A family of diffeomorphism-invariant Seiberg--Witten deformations of gravity is constructed. In a first step Seiberg--Witten maps for an SO(1,3) gauge symmetry are obtained for constant deformation parameters. This includes maps for the…

In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic…

In this paper we construct smooth Riemannian metrics on the sphere which admit smooth Zoll families of minimal hypersurfaces. This generalizes a theorem of Guillemin for the case of geodesics. The proof uses the Nash-Moser Inverse Function…

Using the notion of magnetic curvature recently introduced by the first author, we extend E. Hopf's theorem to the setting of magnetic systems. Namely, we prove that if the magnetic flow on the s-sphere bundle is without conjugate points,…

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $\Omega$ be a non-exact 2-form. Let $\lambda$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la \Omega$…

We obtain Kupka-Smale theorem and Franks lemma for magnetic flows on manifolds with any dimension. This improves Miranda result on surfaces. However our methods relies on geometric control theory, like in Rifford and Ruggiero articles.

Given a compact surface $M$, consider the natural right action of the group of diffeomorphisms $\mathcal{D}(M)$ of $M$ on $\mathcal{C}^{\infty}(M,\mathbb{R})$ defined by the rule: $(f,h)\mapsto f\circ h$ for $f\in…

To a Riemannian manifold $(M, g)$ endowed with a magnetic form ${\sigma}$ and its Lorentz operator ${\Omega}$ we associate an operator $M^{\Omega}$, called the magnetic curvature operator. Such an operator encloses the classical Riemannian…

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $\Omega$ be a 2-form. We show that the magnetic flow of the pair $(g,\Omega)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has…

On a closed Riemannian surface $(M,\bar g)$ with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume $A>0$ and the property that their Gauss curvatures $f_\lambda= f + \lambda$ are given…

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of KAM tori and trapping regions provided a natural…

We investigate length decreasing maps $f:M\to N$ between Riemannian manifolds $M$, $N$ of dimensions $m\ge 2$ and $n$, respectively. Assuming that $M$ is compact and $N$ is complete such that…

We consider a closed negatively curved surface $(M, g)$ with marked length spectrum sufficiently close (multiplicatively) to that of a hyperbolic metric $g_0$ on $M$. We show there is a smooth diffeomorphism $F:M \to M$ with derivative…

Let $(F_t)$ be a smooth flow on a smooth manifold $M$ and $h:M\to M$ be a smooth orbit preserving map. The following problem is studied: suppose that for every point $z$ of $M$ there exists a germ of a smooth function $f_z$ at $z$ such that…

Consider a $C^{\infty}$ closed connected Riemannian manifold $(M, g)$ with negative curvature. The unit tangent bundle $SM$ is foliated by the (weak) stable foliation $\mathcal{W}^s$ of the geodesic flow. Let $\Delta^s$ be the leafwise…

We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing…