1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 421 a (0) = x, a (1) = y, and L (a)= de (x, y). Note that we may assu ...

422 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS STEP 3 (Regularity of the path a). This follows from showing that a satisfies L ...

CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 423 2.1.4. The infinitesimal structure of locally convex subsets. The ...

424 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS Hence, for any t E [O, 1], the unit vector l=!~:~~:J is an interior point of the ...

CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 425 2.2.1. Definition of convex functions. Let C C M be a connected l ...

426 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS (3) (B (p, 2r) , g) is lOO~n 2 -close to the Euclidean ball (B (0, 2r), 9Euc) in ...

CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 427 Geometrically it is clear that there exists r 0 E (0, r) such tha ...

428 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS DEFINITION H.22. Given a function f: S--+ IR and VE TpS, we define the direction ...

CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 429 (i) For p EC and 0 < s1 :::; s 2 , we have (H.11) for all V E ...

430 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS (ii) We compute for s > 0 and VE dom (Js) that when jsVI < c:, IJs (V)I = ...

CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 431 is closest to expP (s (fVi + (1 - t) V2)). From the pointed Cheeg ...

432 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS From Lemma H.25(i), we have for i 2: io (DvJ)(pi) ::::; f ( expPi (so Vi)) -_ f ...

GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 433 for all VE TpM. It follows that min{(Dvf) (p): VE s;-^1 }::; 0. 3. G ...

434 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS (iii) Let W be as in (H.16). If (Dminf) (p) :S 0, then (H.19) (D\lf(p)f) (p) = - ...

GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 435 DEFINITION H.32 (Convex functions (zero at the boundary) - <tXo ( ...

436 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS By the Toponogov comparison theorem (the sectional curvature is bounded from bel ...

GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 437 Note that \7 f (x, y) is not continuous in (x, y) (it takes eight di ...

438 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS Let U 1 be an open neighborhood of I' ( s1) such that C n U1 is a convex set. No ...

GRADIENTS OF CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 439 Then I! Vi +! Vz I < 1,! Vi +! Vz E T pC, and! Vi +! Vz -/= 0 by ...

440 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS for any A. E (0, oo ). By the definition of the generalized gradient, we have (D ...