Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature
نویسنده
چکیده
We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates. In contrast with earlier results based on a global bound for derivatives of the curvature, our method requires only a sup-norm bound on the curvature near the given observer.
منابع مشابه
Local canonical foliations of Lorentzian manifolds with bounded curvature
We consider pointed Lorentzian manifolds and construct “canonical” foliations by constant mean curvature (CMC) hypersurfaces. Our result assumes a uniform bound on the local sup-norm of the curvature of the manifold and on its local injectivity radius, only. The prescribed curvature problem under consideration is a nonlinear elliptic equation whose coefficients have limited regularity. The CMC ...
متن کاملHyperbolic conservation laws and spacetimes with limited regularity
Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows andgeneral relativity. Recentworkby the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existence and qualitative behavior of solutions. The metric on the manifold may either ...
متن کاملInjectivity Radius of Lorentzian Manifolds
Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzianmanifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically asso...
متن کاملSomeApplications of Collapsing with Bounded Curvature
In my talk I will discuss the following results which were obtained in joint work with Wilderich Tuschmann. 1. For any given numbers m, C and D, the class of m-dimensional simply connected closed smooth manifolds with finite second homotopy groups which admit a Riemannian metric with sectional curvature |K| ≤ C and diameter ≤ D contains only finitely many diffeomorphism types. 2. Given any m an...
متن کاملOn Lorentzian two-Symmetric Manifolds of Dimension-four
‎We study curvature properties of four-dimensional Lorentzian manifolds with two-symmetry property‎. ‎We then consider Einstein-like metrics‎, ‎Ricci solitons and homogeneity over these spaces‎‎.
متن کامل