Hamiltonian systems of negative curvature are hyperbolic

半离散线性双曲曲率流

Semi-Discrete Linear Hyperbolic Curvature flow

该研究针对带边界的分段线性曲线，探讨其在线性双曲曲率流下的演化，推导自相似解、有限傅里叶通解，还研究用非齐次线性双曲流实现曲线间演化的问题。

Average area ratio and normalized total scalar curvature of hyperbolic n-manifolds

On closed hyperbolic manifolds of dimension $n\geq 3$, we review the definition of the average area ratio of a metric $h$ with $R_h\geq -n(n-1)$ relative to the hyperbolic metric $h_0$, and we prove that it attains the local minimum of one at $h_0$, which solves a local version of Gromov's conjecture. Additionally, we discuss the relation between the average area ratio and normalized total scalar curvature for hyperbolic $n$-manifolds, as well as its relation to the minimal surface entropy if $n$ is odd.

Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is obtained in the case in which the perturbing function exhibits a time decay.

On homoclinic orbits to center manifolds of elliptic-hyperbolic equilibria in Hamiltonian systems

We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure. This leads to the study of a singularity which inherits certain structure from the Hamiltonian nature of the system. Under non-degeneracy assumptions, we classify the possible Morse indices of this singularity, permitting a local description of the set of homoclinic orbits. We also consider the case of time-reversible Hamiltonian systems.

Curvatures and hyperbolic flows for natural mechanical systems in Finsler geometry

We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{Jacobi curves}) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (involving the gradient of the potential) of the Finsler manifold and the Hessian of the potential w.r.t. a Riemannian metric induced from the Finslerian metric. As an application, we give sufficient conditions for the Hamiltonian flows of the least action to be hyperbolic and show…

On curvature and hyperbolicity of monotone Hamiltonian systems

Assume that a Hamiltonian system is monotone. In this paper, we give several characterizations on when such a system is Anosov. Assuming that a monotone Hamiltonian system has no conjugate point, we show that there are two distributions which are invariant under the Hamiltonian flow. We show that a monotone Hamiltonian flow without conjugate point is Anosov if and only if these distributions are transversal. We also show that if the reduced curvature of the Hamiltonian system is non-positive, then the flow is Anosov if and only if the reduced curvature is negative somewhere along each trajecto…

Equivariant and self-similar standing waves for a Hamiltonian hyperbolic-hyperbolic spin-field system

In this paper we study the existence of special symmetric solutions to a Hamiltonian hyperbolic-hyperbolic coupled spin-field system, where the spins are maps from $\mathbb R^{2+1}$ into the sphere $§^2$ or the pseudo-sphere $\H^2$. This model was introduced in \cite{Martina} and it is also known as the {\it hyperbolic-hyperbolic generalized Ishimori system}. Relying on the hyperbolic coordinates introduced in \cite{KNZ11}, we prove the existence of equivariant standing waves in both regular hyperbolic coordinates as well as similarity variables, and describe their asymptotic behaviour.

Persistence of Hyperbolic Tori in Generalized Hamiltonian Systems

In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub-manifolds.

2-Frame flow dynamics and hyperbolic rank rigidity in nonpositive curvature

This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-Λ^2 < K < -λ^2$ where $λ/Λ> >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank rigidity t…