Universal radius of injectivity for locally quasiconformal mappings

Ifn>2 and iff is a locally quasiconformal mapping from the ballB ⁿ= {x∈R ⁿ:⋎x⋎<1} intoR ⁿ ∪ {∞} thenf is injective inB ⁿ (r)={x∈R ⁿ:⋎x⋎ <r} wherer>0 depends only onn and the maximal dilatation off. Supported in part by the Samuel Neaman Fund, Special Year in Complex Analysis, Technion, I.I.T., Haifa, Israel, 1975/76.     

Volume, surface area and inward injectivity radius

We show some relations among the volumes of domains in Euclidean spaces, their surface areas and the inward injectivity radii from their boundaries. In particular, we give an estimate for the upper bound of the ratios of their surface areas and volumes by means of inward injectivity radii. The upper bound seems to depend on their topological structures.

     

Induced Riemannian structures on null hypersurfaces

Given a null hypersurface L of a Lorentzian manifold, we construct a Riemannian metric on it from a fixed transverse vector field ζ. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold and the vector field ζ. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.     

Three-dimensional manifolds of nonpositive curvature with small injectivity radius

It is proved that if at every point of a closed, three-dimensional, Riemannian manifold with bounded sectional curvature the injectivity radius does not exceed a specific absolute constant, then the manifold is a special graph and its metric splits locally.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 10–19, 1991.     

Curvature and injectivity radius estimates for Einstein 4-manifolds

Let denote an Einstein -manifold with Einstein constant, , normalized to satisfy . For , a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on , under the assumption that the -norm of the curvature on is less than a small positive constant, which is independent of , and which in particular, does not depend on a lower bound on the volume of . In case , we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, . These estimates provide key tools in the study of singularity formation for -dimensional Einstein metrics. As one application among others, we give a natural compactification of the moduli space of Einstein metrics with negative Einstein constant on a given .

     

On the injectivity of the Poisson transform

The injectivity of the Poisson transform associated to the Cartan motion groups is proved with the aid of a Fatou-type theorem.     

Extrinsic radius pinching for hypersurfaces of space forms

We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M and the L^∞-norm of the second fundamental form B such that the pinching condition (where H is the mean curvature) implies that M is diffeomorphic to an n-dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the Lp-norm of H, p?2, instead of the L^∞-norm.     

On Ricci coefficients of null hypersurfaces with time foliation in Einstein vacuum space-time: part I

The main objective of this paper is to control the geometry of null cones with time foliation in Einstein vacuum spacetime under the assumptions of small curvature flux and a weak condition on the deformation tensor for the future directed unit normal T to each leaf. We establish a series of estimates on Ricci coefficients, which play a crucial role to prove the improved breakdown criterion in Wang (Commun. Pure Appl. Math. 65(1):0021–0076, 2012).     

On the curvature of algebraic hypersurfaces

Using algebraic residue theory, we try to generalize a theorem of Chasles about osculating circles of plane algebraic curves to algebraic hypersurfaces over algebraically closed fields of characteristic zero.     

On the hyperbolicity of general hypersurfaces

We introduce and study stochastic \(N\)-particle ensembles which are discretizations for general-\(\beta \) log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, \((z,w)\)-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as \(N\to \infty \). The covariance is universal and coincides with its counterpart in random matrix theory.Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work of Nekrasov and his collaborators.     

On injectivity of the combinatorial Radon transform of order five

In the present work, we give a proof of the injectivity of the combinatorial Radon transform of order five.