Positive Injectivity Radius Research Articles

Closed geodesics in the tangent sphere bundle of a hyperbolic three-manifold

Let $M$ be an oriented three-dimensional manifold of constant sectional curvature $-1$ and with positive injectivity radius, and $T^1M$ its tangent sphere bundle endowed with the canonical (Sasaki) metric. We describe explicitly the periodic geodesics of $T^1M$ in terms of the periodic geodesics of $M$: For a generic periodic geodesic $(h,v)$ in $T^1M$, $h$ is a periodic helix in $M$, whose axis is a periodic geodesic in $M$; the closing condition on $(h,v)$ is given in terms of the horospherical radius of $h$ and the complex length (length and holonomy) of its axis. As a corollary, we obtain that if two compact oriented hyperbolic three-manifolds have the same complex length spectrum (lengths and holonomies of periodic geodesics, with multiplicities), then their tangent sphere bundles are length isospectral, even if the manifolds are not isometric.

Optimal Sobolev inequalities on complete Riemannian manifolds with Ricci curvature bounded below and positive injectivity radius

The concept of best constants for Sobolev embeddings appeared to be crucial for solving limiting cases of some partial differential equations. A striking example where it has played a major role is given by the very famous Yamabe problem. While the situation is well understood for compact manifolds, things are less clear when dealing with complete manifolds. Aubin proved in '76 that optimal Sobolev inequalities are valid for complete manifolds with bounded sectional curvature and positive injectivity radius. We prove here that the result still holds if the bound on the sectional curvature is replaced by a lower bound on the Ricci curvature (a much weaker assumption). We also get estimates for the remaining constants.