Appendix H. Convex Functions on Riemannian Manifolds
But not too many horns can make that sound.- From "Sultans of Swing" by Dire Straits
In this appendix we provide a detailed discussion of convex analysis on
Euclidean spaces and on locally· convex subsets in Riemannian manifolds,
which comprise a 'baby version' of the corresponding study of semi-concave
functions on Aleksandrov spaces (see for example Petrunin [156] and the
references therein). The discussion of convex functions leads to a proof
of Theorem I.24 in the next appendix regarding the existence of distance-
nonincreasing maps between the level sets of Busemann functions in com-
plete noncompact manifolds with nonnegative sectional curvature.
In §1 we review aspects of convex analysis on Euclidean space related to
the differentiability (e.g., Lipschitz) properties of convex functions.
In §2 we discuss the properties of convex sets and functions on Riemann-
ian manifolds, including generalizations of some of the results of the previous
section.
In §3 we discuss the generalized gradient of a convex function on a Rie-
mannian manifold.
In §4 we discuss integral curves of concave functions.
1. Elementary aspects of convex analysis on Euclidean space
Convex analysis enters the Ricci fl.ow from two perspectives. First of
all, it is useful in the study of the maximum principle for systems, where
invariant subsets of vector bundles for solutions of heat-type equations are
fiberwise convex (see Chapter 10 of Part II). Secondly, totally convex subsets
are fundamental in the study of Riemannian manifolds with nonnegative
sectional curvature (see §1 of the next appendix). In part as a warm-up,
in this section we give a brief review of some elementary aspects of convex
analysis on Euclidean space. In the next section we shall discuss convex
analysis on certain subsets of Riemannian manifolds.
1.1. Lipschitz functions.Let (X, dx) and (Y, dy) be metric spaces. A map f : X -+ Y is Lips-
chitz if there exists a constant C < oo such that
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