Related papers: Minimal diffeomorphisms with $L^1$ Hopf differenti…
We consider the set of points with high pointwise emergence for $C^{1+\alpha}$ diffeomorphisms preserving a hyperbolic measure. We find a lower bound on the Hausdorff dimension of this set in terms of unstable Hausdorff dimension of the…
We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel-Anosov-Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by…
We study the space of Hopf differentials of almost fuchsian minimal immersions of compact Riemann surfaces. We show that the extrinsic curvature of the immersion at any given point is a concave function of the Hopf differential. As a…
Given a quasisymmetric homeomorphism $\varphi$ of the circle, Bonsante and Schlenker proved the existence and uniqueness of the minimal Lagrangian extension $f_\varphi:\mathbb{H}^2\to\mathbb{H}^2$ to the hyperbolic plane. By previous work…
We study the class of norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such norm that is continuous with…
We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L^2 complex relative to a suitable metric on the bundle and a complete metric on the…
In this paper we further explore the L-shadowing property defined in [17] for dynamical systems on compact spaces. We prove that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere…
Lawson-Osserman constructed three types of non-parametric minimal cones of high codimensions based on Hopf maps between spheres, which correspond to Lipschitz but non-differentiable solutions to the minimal surface equations, thereby making…
In this paper we determine for relatively minimal elliptic surfaces with positive Euler number the image of the natural representation of the group of orientation preserving self-diffeomorphisms on $\Hbar$, the second homology group reduced…
We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at…
We show that the Holmes--Thompson area of every Finsler disk of radius $r$ whose interior geodesics are length-minimizing is at least $\frac{6}{\pi} r^2$. Furthermore, we construct examples showing that the inequality is sharp and observe…
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient…
We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular every smooth irrational…
Friedman and Morgan made the "speculation" that deformation equivalence and diffeomorphism should coincide for algebraic surfaces. Counterexamples, for the hitherto open case of surfaces of general type, have been given in the last years by…
We show that on a totally disconnected compact metric space every separating homeomorphisms is expansive except at periodic points. We conclude that minimal separating homeomorphisms are expansive and that every separating homeomorphism has…
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…
We extend the celebrated theorem of Kellogg for conformal mappings to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimiser of Dirichlet energy of Sobolev mappings between double connected domains $D$ and…
Hartman-Grobman theorem states that there is a homeomorphism H sending the solutions of the nonlinear system onto those of its linearization under suitable assumptions. Many mathematicians have made contributions to prove H\"older…
On any closed symplectic manifold we construct a path-connected neighborhood of the identity in the Hamiltonian diffeomorphism group with the property that each Hamiltonian diffeomorphism in this neighborhood admits a Hofer and spectral…
We develop further basic tools in the theory of continuous bounded cohomology of locally compact groups. We apply this tools to establish a Milnor-Wood type inequality in a very general context and to prove a global rigidity result which…