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boundary of tubular neighborhood of convex subset with piecewise smooth boundary

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The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is known to be smooth, e.g., by page 73 of Hyperbolic Manifolds and Kleinian Groups. I wonder whether this is true for negatively curved manifold with pinched sectional curvature $kappa in [-b, -1]$, where $b$ is a positive constant greater than 1. The classical paper Geometrical finiteness with variable negative curvature by Bowditch does not seem to discuss the regularity issuse of the boundary of the r-neighborhood of the convex core. Any reference for this issue, preferably with a proof would help. In general the $r$-neighborhood of a convex set with piecewise smooth boundary is not smooth. A counterexample would be the convex hull of two intersecting geodesic segment in hyperbolic 2-space, one going from $(0,1)$ to $infty$, one is a quarter circle from $(0,1)$. Its r-neighborhood has 3 parts. By observing the curvature of each part we can see that its boundary is not smooth.

hyperbolic-geometry

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asked yesterday

HanHan

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0

$begingroup$

The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is known to be smooth, e.g., by page 73 of Hyperbolic Manifolds and Kleinian Groups. I wonder whether this is true for negatively curved manifold with pinched sectional curvature $kappa in [-b, -1]$, where $b$ is a positive constant greater than 1. The classical paper Geometrical finiteness with variable negative curvature by Bowditch does not seem to discuss the regularity issuse of the boundary of the r-neighborhood of the convex core. Any reference for this issue, preferably with a proof would help. In general the $r$-neighborhood of a convex set with piecewise smooth boundary is not smooth. A counterexample would be the convex hull of two intersecting geodesic segment in hyperbolic 2-space, one going from $(0,1)$ to $infty$, one is a quarter circle from $(0,1)$. Its r-neighborhood has 3 parts. By observing the curvature of each part we can see that its boundary is not smooth.

hyperbolic-geometry

share|cite|improve this question

asked yesterday

HanHan

161

$endgroup$

add a comment | 

$begingroup$

The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is known to be smooth, e.g., by page 73 of Hyperbolic Manifolds and Kleinian Groups. I wonder whether this is true for negatively curved manifold with pinched sectional curvature $kappa in [-b, -1]$, where $b$ is a positive constant greater than 1. The classical paper Geometrical finiteness with variable negative curvature by Bowditch does not seem to discuss the regularity issuse of the boundary of the r-neighborhood of the convex core. Any reference for this issue, preferably with a proof would help. In general the $r$-neighborhood of a convex set with piecewise smooth boundary is not smooth. A counterexample would be the convex hull of two intersecting geodesic segment in hyperbolic 2-space, one going from $(0,1)$ to $infty$, one is a quarter circle from $(0,1)$. Its r-neighborhood has 3 parts. By observing the curvature of each part we can see that its boundary is not smooth.

hyperbolic-geometry

share|cite|improve this question

asked yesterday

HanHan

161

$endgroup$

share|cite|improve this question

asked yesterday

HanHan

161

share|cite|improve this question

asked yesterday

HanHan

161

asked yesterday

HanHan

161

asked yesterday

HanHan

161