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Register a new userInjectivity radius of manifolds with a Lie structure at infinity
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Quang-Tu Bui
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Using Lie groupoids, we prove that the injectivity radius of a manifold with a Lie structure at infinity is positive.
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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.
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $operatorname{conv}(p)ge min{frac{1}{2}operatorname{inj}(p),operatorname{foc}(B_{operatorname{inj}(p)}(p))}$, where $operatorname{inj}(p)$ is the injectivity radius of $p$ and $operatorname{foc}(B_r(p))$ is the focal radius of open ball centered at $p$ with radius $r$; 2) for any two points $p,q$ in $M$, $operatorname{inj}(q)ge min{operatorname{inj}(p), operatorname{conj}(q)}-d(p,q),$ where $operatorname{conj}(q)$ is the conjugate radius of $q$; 3) for any $0<r<min{operatorname{inj}(p),frac{1}{2}operatorname{conj}(B_{operatorname{inj}(p)}(p))}$, any (not necessarily minimizing) geodesic in $B_r(p)$ has length $le 2r$. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.
We propose a new notion called emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $pto infty $. Infinity harmoncity appears in many familiar contexts. For example, metric projection onto the orbit of an isometric group action from a tubular neighborhood is infinity harmonic. Unfortunately, infinity-harmonicity is not preserved under composition. Those infinity harmonic maps that always preserve infinity harmonicity under pull back are called infinity harmonic morphisms. We show that infinity harmonic morphisms are precisely horizontally homothetic mas. Many example of infinity-harmonic maps are given, including some very important and well-known classes of maps between Riemannian manifolds.
It is given a topological pinching for the injectivity radius of a compact embedded surface either in the sphere or in the hyperbolic space
Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $bar M$, some of them may appear on $partial M$. We describe the asymptotic behavior of the blowup solutions around each blowup point and derive an energy estimate as a consequence.
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