Related papers: Effective bounds in E.Hopf rigidity for billiards …
In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round…
In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that…
We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric…
We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed…
We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets…
We compute a lower bound for the scalar curvature of a gradient Einstein soliton under a certain assumption on its potential function. We establish an asymptotic behavior of the potential function on a noncompact gradient shrinking Einstein…
By analytical mapping of the eigenvalue problem in rough billiards on to a band random matrix model a new regime of Wigner ergodicity is found. There the eigenstates are extended over the whole energy surface but have a strongly peaked…
We obtain a global weighted $L^p$ estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one…
Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius.…
We prove that there exists a residual set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument,…
A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces in the elliptic region to the equations for harmonic 1-forms on the projective disc is studied in detail. A…
In this note we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. $\operatorname{Ric}_g=\lambda g$ for some real number $\lambda$.…
An approach due to Wojtkovski [9], based on the Jacobi fields, is applied to study sets of 3-period orbits in billiards on hyperbolic plane and on two-dimensional sphere. It is found that the set of 3-period orbits in billiards on…
We introduce the concepts of rotation numbers and rotation vectors for billiard maps. Our approach is based on the birkhoff ergodic theorem. We anticipate that it will be useful, in particular, for the purpose of establishing the…
The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper,…
We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with…
We obtain a bound for the area of a capillary $H-$surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side…
We show that two-dimensional billiard systems are Turing complete, in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a specified open set. Billiards…
We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to…
We prove generalized Fefferman-Stein type theorems on sharp functions with $A_p$ weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted…