Appendix I. Asymptotic Cones and Sharafutdinov Retraction
I keep on goin' guess I'll never know why.- From "Life's Been Good" by Joe Walsh
This appendix mainly focuses on various results regarding the geometry
of complete noncompact Riemannian manifolds with nonnegative sectional
curvature. We shall find these results of interest for the Ricci fl.ow since, by
the Hamilton-Ivey curvature estimate, 3-dimensional finite time singularity
models have nonnegative sectional curvature. Some of these results should
also be useful in the study of 3-dimensional r;;-solutions.
In §1, as a continuation of our discussion of Sharafutdinov's distance
nonincreasing retraction in the previous appendix, we prove the Sharafutdi-
nov retraction Theorem I.25.
In §2 we prove the existence of asymptotic cones.
In §3 we discuss a monotonicity property of distance spheres in non-
negatively curved manifolds when their radii are less than the injectivity
radius.
In §4 we discuss critical point theory for the distance function. As an
application we prove that large radii distance spheres in a complete non-
compact manifold with nonnegative sectional curvature are Lipschitz hyper-
surfaces.
In §5 we discuss an almost distance-decreasing property of the nearest
point projection map in a small tubular neighborhood of a hypersurface in
any oriented Riemannian manifold.
In §6 we discuss the mollified distance function and an approach, pro-
posed by Gromov and considered by Kasue, toward constructing the space
of points at infinity.
1. Sharafutdinov retraction theorem
In this section we discuss the Sharafutdinov retraction theorem. We
shall use some elementary comparison geometry, including the Busemann
function and totally convex subsets, discussed in Appendix B of Volume
One.
In this section (Mn, g) denotes a (connected and oriented) complete
noncompact Riemannian rnanifold.
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