Related papers: On Hamiltonian stable Lagrangian tori in complex h…

We present a novel geometric approach for determining the unique structure of a Hamiltonian and establishing an instability criterion for quantum quadratic systems. Our geometric criterion provides insights into the underlying geometric…

Motivated by the possibility of radiation driven instabilities in rotating magnetic stars, we study the stability properties of general linear perturbations of a stationary and axisymmetric, infinitely conducting perfect fluid configuration…

In this article, we consider the dynamics in a neighborhood of a quasi-periodic torus which is invariant by a Hamiltonian flow, we discuss several notions of stability and we prove several results of instability when the frequency of the…

Given a smooth Tonelli Hamiltonian on the torus $\mathbb{T}^{n}$ and a $C^{2}$ Lagrangian graph $W \subset T^{*}\mathbb{T}^{n}$ that is invariant under the Hamiltonian flow and contained within a Ma\~n\'e supercritical energy level, we…

In this paper we study the gradient steady K\"ahler-Ricci soliton metrics on non-compact toric manifolds. We show that the orbit space of the free locus of such a manifold carries a natural Hessian structure with a nonnegative Bakry-\'Emery…

We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are $C^0$-generic and have…

Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a…

This note proves the orbital stability in the energy space $H^{1/2}$ of the sum of widely-spaced 1-solitons for the Benjamin-Ono equation, with speeds arranged so as to avoid collisions.

In the moduli space of semistable $\text{SL}(r, \mathbb{C})$-Higgs bundles, we show that there exists a sublocus of the upward flow through a polystable $\mathbb{C}^{*}$-fixed point, which is Lagrangian on its intersection with the stable…

In 2004, F\'ejoz [D\'emonstration du 'th\'eor\'eme d'Arnold' sur la stabilit\'e du syst\`eme plan\'etaire (d'apr\`es M. Herman). Ergod. Th. & Dynam. Sys. 24(5) (2004), 1521-1582], completing investigations of Herman's [D\'emonstration d'un…

We consider various constructions of monotone Lagrangian submanifolds of $C P^n, S^2\times S^2$, and quadric hypersurfaces of $C P^n$. In $S^2\times S^2$ and $C P^2$ we show that several different known constructions of exotic monotone tori…

In this paper, we study the stability of catenoids and helicoids in the hyperbolic $3$-space $\mathbb{H}^3$. (1) For a family of spherical minimal catenoids $\{\mathcal{C}_a\}_{a>0}$ in $\mathbb{H}^3$, there exist two constants $0<a_c<a_l$…

We show that any weakly partially hyperbolic diffeomorphism on the 2-torus may be realized as the dynamics on a center-stable or center-unstable torus of a 3-dimensional strongly partially hyperbolic system. We also construct examples of…

We rigorously show that a local spin system giving rise to a slow Hamiltonian dynamics is stable against generic, even time-dependent, local perturbations. The sum of these perturbations can cover a significant amount of the system's size.…

We show that the quadratic interaction of fundamental and second harmonics in a bulk dispersive medium, combined with self-defocusing cubic nonlinearity, give rise to completely localized spatiotemporal solitons (vortex tori) with vorticity…

We prove the existence of normally hyperbolic invariant cylinders in nearly integrable hamiltonian systems.

We apply the concept of castling transform of prehomogeneous vector spaces to produce new examples of minimal homogeneous Lagrangian submanifolds in the complex projective space. Furthermore we verify the Hamiltonian stability of a low…

The persistence of invariant tori in multi-scale Hamiltonian systems is intrinsically linked to the stability of the N-body problem. However, the existing non-degeneracy conditions in disordered scenarios have been formulated too generally,…

We prove $L^2$ orbital stability of Dirac solitons in the massive Thirring model. Our analysis uses local well posedness of the massive Thirring model in $L^2$, conservation of the charge functional, and the auto--B\"{a}cklund…

We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.