Related papers: Boundary rigidity for Lagrangian submanifolds, non…
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current…
In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work…
The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles,…
Dissipative dynamical systems characterised by two basins of attraction are found in many physical systems, notably in hydrodynamics where laminar and turbulent regimes can coexist. The state space of such systems is structured around a…
In this paper, the relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds is studied. First, special lagrangian fibrations are constructed on some regions of bounded curvature and…
In the framework of the Lagrangian field theory, we derive a type of new non-classical natural boundary condition to be correlated with the mean curvature of boundary surface. Under the condition of homogeneity and isotropy, this type of…
We study the boundary behavior of the Kobayashi-Royden metric and the Kobayashi hyperbolicity of domains in Riemannian manifolds. As an application, we prove a Fatou type theorem on the existence, almost everywhere, of non tangential limits…
We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $\mathbb{C}^2$. We recall some classical obstructions and prove some deeper ones related to symplectic…
We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the…
The main theme of this paper is the introduction of a new type of polarizations, suited for some open symplectic manifolds, and their applications. These applications include symplectic embedding results that answer a question by…
We check that several properties of the Aubry set, first proven for finite-dimensional Lagrangians by Mather and Fathi, continue to hold in the case of the infinitely many interacting particles of the Vlasov equation on the circle.
We establish the extrinsic Bonnet-Myers Theorem for compact Riemannian manifolds with positive Ricci curvature. And we show the almost rigidity for compact hypersurfaces, which have positive sectional curvature and almost maximal extrinsic…
A synergetic model describing the state of an ultrathin lubricant layer squeezed between two atomically smooth solid surfaces operating in the boundary friction mode has been developed further. To explain the presence of different operation…
Non-spherical particles transported by an anisotropic turbulent flow preferentially align with the mean shear and intermittently tumble when the local strain fluctuates. Such an intricate behaviour is here studied for inertialess,…
Hamiltonian stationary Lagrangian submanifolds (HSLAG) are a natural generalization of special Lagrangian manifolds (SLAG). The latter only make sense on Calabi-Yau manifolds whereas the former are defined for any almost K\"ahler manifold.…
We introduce a curvature-dimension condition for autonomous Lagrangians on weighted manifolds, which depends on the Euler-Lagrange dynamics on a single energy level. By generalizing Klartag's needle decomposition technique to the Lagrangian…
This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability…
In this paper, we investigate some dynamical properties near a nonhyperbolic fixed point. Under some conditions on the higher nonlinear terms, we establish a stable manifold theorem and a degenerate Hartman theorem. Furthermore, the finite…
We survey the main extensions of the classical Hadamard, Liebmann and Cohn-Vossen rigidity theorems on convex surfaces of $3$-Euclidean space to the context of convex hypersurfaces of Riemannian manifolds. The results we present include the…
We introduce a version of Aubry-Mather theory for the length functional of causal curves in compact Lorentzian manifolds. Results include the existence of maximal invariant measures, calibrations and calibrated curves. We prove two versions…