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Register a new userDoubling property for biLipschitz homogeneous geodesic surfaces
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Authors
Enrico Le Donne
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In this paper we discuss general properties of geodesic surfaces that are locally biLipschitz homogeneous. In particular, we prove that they are locally doubling and that there exists a special doubling measure analogous to the Haar measure for locally compact groups.
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This paper is connected with the problem of describing path metric spaces that are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. Let $X = G/H$ be a homogeneous manifold of a Lie group $G$ and let $d$ be a geodesic distance on $X$ inducing the same topology. Suppose there exists a subgroup $G_S$ of $G$ that acts transitively on $X$, such that each element $g in G_S$ induces a locally biLipschitz homeomorphism of the metric space $(X,d)$. Then the metric is locally biLipschitz equivalent to a sub-Riemannian metric. Any such metric is defined by a bracket generating $G_S$-invariant sub-bundle of the tangent bundle. The result is a consequence of a more general fact that requires a transitive family of uniformly biLipschitz diffeomorphisms with a control on their differentials. It will be relevant that the group acting transitively on the space is a Lie group and so it is locally compact, since, in general, the whole group of biLipschitz maps, unlikely the isometry group, is not locally compact. Our method also gives an elementary proof of the following fact. Given a Lipschitz sub-bundle of the tangent bundle of a Finsler manifold, then both the class of piecewise differentiable curves tangent to the sub-bundle and the class of Lipschitz curves almost everywhere tangent to the sub-bundle give rise to the same Finsler-Carnot-Caratheodory metric, under the condition that the topologies induced by these distances coincide with the manifold topology.
Suppose that $E$ and $E$ denote real Banach spaces with dimension at least 2 and that $Dvarsubsetneq E$ and $Dvarsubsetneq E$ are uniform domains with homogeneously dense boundaries. We consider the class of all $varphi$-FQC (freely $varphi$-quasiconformal) maps of $D$ onto $D$ with bilipschitz boundary values. We show that the maps of this class are $eta$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided Holder condition. Moreover, replacing the class $varphi$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if $f$ is a $varphi$-FQC map which maps $D$ onto itself with identity boundary values, then there is a constant $C,,$ depending only on the function $varphi,,$ such that for all $xin D$, the quasihyperbolic distance satisfies $k_D(x,f(x))leq C$.
Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces are geodesically complete, that some others admit an incomplete geodesic but model geodesically complete surfaces, and that there are also others which do not model any complete surface. Our main result provides a way of determining whether a given set of constant Christoffel symbols can model a complete surface.
We prove a sharp $L^2to H^{1/2}$ stability estimate for the geodesic X-ray transform of tensor fields of order $0$, $1$ and $2$ on a simple Riemannian manifold with a suitable chosen $H^{1/2}$ norm. We show that such an estimate holds for a family of such $H^{1/2}$ norms, not topologically equivalent, but equivalent on the range of the transform. The reason for this is that the geodesic X-ray transform has a microlocally structured range.
We classify the transitive, effective, holomorphic actions of connected complex Lie groups on complex surfaces.
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