Related papers: Lens rigidity with trapped geodesics in two dimens…
We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its…
Gravitational lensing deflects light. A single lens deflector can only shear images, but cannot induce rotations. Multiple lens planes can induce rotations. Such rotations can be observed in quadruply imaged sources, and can be used to…
We investigate CAT(0) metric spaces whose associated Tits boundary is compact. Prominent examples of such spaces are of course the euclidean ones. However there exist non trivial geodesically complete CAT(0) spaces with compact Tits…
We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.
The paper is devoted to the development of the differential geometry of saccades and saccadic cycles. We recall an interpretation of Donder's and Listing's law in terms of the Hopf fibration of the $3$-sphere over the $2$-sphere. In…
In this paper, we prove a rigidity theorem for Poincar\'e-Einstein manifolds whose conformal infinity is a flat Euclidean space. The proof relies on analyzing the propagation of curvature tensors over the level sets of an adapted boundary…
Any two compact, complete, one-dimensional geodesic spaces with identical marked length spectrum have isometric $\pi_1$-hull. The present version contains errors, notably in Lemmas 2.2 and 2.3 (path cancellations can be more complicated),…
We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on…
We consider rigidity properties of compact symmetric spaces $X$ with metric $g_0$ of rank one. Suppose $g$ is another Riemannian metric on $X$ with sectional curvature $\kappa$ bounded by $0 \leq \kappa \leq 1$. If $g$ equals $g_0$ outside…
Tightly focused light beams can exhibit electric fields spinning around any axis including the one transverse to the beams' propagation direction. At certain focal positions, the corresponding local polarization ellipse can degenerate into…
We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral…
We study the fundamental algorithmic rigidity problems for generic frameworks periodic with respect to a fixed lattice or a finite-order rotation in the plane. For fixed-lattice frameworks we give an $O(n^2)$ algorithm for deciding generic…
The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds…
Dispersion, impedance matching and resolution characteristics of an isotropic three-dimensional flat lens ("superlens") are studied. The lens is based on cubic meshes of interconnected transmission lines and bulk loads. We study a practical…
For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving…
We obtain a complete classification of ruled zero mean curvature surfaces in the three-dimensional light cone. En route, we examine geodesics and screw motions in the space form, allowing us to discover helicoids. We also consider their…
We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll,…
We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility…
We characterise rigid graphs for cylindrical normed spaces $Z=X\oplus_\infty \mathbb{R}$ where $X$ is a finite dimensional real normed linear space and $Z$ is endowed with the product norm. In particular, we obtain purely combinatorial…
Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is…