1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

442 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

4. Integral curves to gradients of concave functions

In this section we shall discuss integral curves for the generalized gra-

dients of concave functions and their properties. The main result is Shara-

futdinov's Theorem H.59. The proof of this theorem meshes better with

concave functions as opposed to convex functions. Our prior discussion of

convex functions applies to concave functions by changing the sign.

Throughout this section (Mn, g) shall denote a connected complete Rie-

mannian manifold. Let C c M be a closed connected locally convex set

with nonempty interior and boundary. We denote by

(H.30) ~Wo (C)

the collection of locally Lipschitz concave functions f : C -+ IR satisfying

flac = O; i.e., f E ~S)Jo (C) if and only if -f E ~Xo (C), where ~Xo (C) is

given by Definition H.32. For f E ~S)Jo (C) we define the generalized gradient

by

'7 f =';: \J (-f)

(by our convention, without the 'extra' minus sign on the RHS).

By Lemma H.36, we have for p E C,

\Jf(p) = l(Duf)(P)IU

for some u E TpC n s;;-^1 with

(H.31) (Du J) (p) =max { (Dv f) (p) : v E TpC n s;-^1 } =';: (Dmaxf) (p).

From Lemma H.30(i), if f E ~Wo (C) is differentiable at p, then \JJ (p) is

equal to the (standard) gradient off.

For a concave function f E ~S)Jo ( C), we say that a point p E C is a

critical point if (Dma;xf) (p) :::; 0.

4.1. Right tangent vectors.
Extending the definition of tangent vector for a 01 path, we have the
following.

DEFINITION H.40 (Right tangent vector). Let 'Y : [a, b] -+ (Mn, g) be a
continuous path and let s E [a, b). A vector

i'+ (s) E T'Y(s)M

is said to be the right tangent vector to 'Y at s if for every 000 function
h : M -+ IR the right derivative

__!},_ (h 0 'Y) =';: lim h ('Y (s +er)) - h ('Y (s))

ds+ CT-tO+ er

exists and

(H.32) ds+ d (ho 'Y) = 'Y+. (s) (h).