1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

ALEKSANDROV SPACES WITH· CURVATURE BOUNDED FROM BELOW 401 If we have a variation or = s, then (G.15) OL('Y) = !,"''"'') (r'+ ( ...

402 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY Ricci fl.ow, spaces of less regularity arise when we take limits of smooth so- luti ...

ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 403 (2) Hinge version. Let L be a hinge^16 in M with vertices (p, q, r), ...

404 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY there exists an open neighborhood U of x such that for any triangle 6.pqr contained ...

ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 405 We say that a path "( emanating from p has a direction at p if its an ...

406 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY DEFINITION G.41 (Strained points and strainers). Let X be an Alek- sandrov space wi ...

ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 407 A line in a length space Xis a (unit speed) path I: JR-+ X such that ...

408 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY 2.4.1. The tangent cone of Aleksandrov spaces with curvature bounded from below. Le ...

ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 409 by expP (v, t) ~ x. (iii) The cut locus C (p) of p is the set of x E ...

410 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY function is not necessarily smooth. In fact, in this case, d ( ·, p) is a Lipschitz ...

ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 411 is concave in some neighborhood of x. 2.4.3. Quasi-geodesics. Geodesi ...

412 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY THEOREM G.60 (Regularity of Aleksandrov spaces). Let X be an n- dimensional Aleksan ...

Appendix H. Convex Functions on Riemannian Manifolds But not too many horns can make that sound. From "Sultans of Swing" by Dir ...

414 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS for all x 1 , x 2 E X. The Lipschitz constant (or dilatation) off is the infimum ...

ELEMENTARY ASPECTS OF CONVEX ANALYSIS ON EUCLIDEAN SPACE 415 where f' is the derivative function defined a. e. on [a, b]. (See ...

416 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS DEFINITION H.7. Let U c ffi.n be an open set. A continuous function f : U -t ffi ...

CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 417 Hence, for any x, y E :!Rn, (H.3) i11x (x) - 1fK (y)l^2 ::::; (7r ...

418 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS 2.1.1. Interior tangent cone of a subset. Let S be a subset in a Riemannian mani ...

CONNECTED LOCALLY CONVEX SUBSETS IN RIEMANNIAN MANIFOLDS 419 Since S is convex, we have /(, ~ { expp (sW) : WE U, s E (0, ll} ...

420 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS LEMMA H.15 (Existence of minimal geodesics in C between points). If C is a conne ...