1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

so CENTER OF MASS AND NONLINEAR AVERAGES We compute using the Jacobi equation ~ Jll = ~ (g (\le;erl, 1)) dr^2 dr Ill g(\le;erl ...

178 4. PROOF OF THE COMPACTNESS THEOREM and that¢ (r) is nonnegative for r E [O, 1] when K::::; 0 or when K > 0 and VK l'YI : ...

CENTER OF MASS AND NONLINEAR AVERAGES 179 which implies that the maximum of f ( "( ( t)) occurs at the endpoints. Hence d(O,"( ...

180 4. PROOF OF THE COMPACTNESS THEOREM Then (fk(l,y,z)) theorem to (xk) = expy z. We will apply the implicit function F (x, y, ...

CENTER OF MASS AND NONLINEAR AVERAGES 181 Using lzl::; 2c/JCQ, we can choose co~ sn 4 c2 (1). We have proved ( 4.22) Now we ca ...

182 4. PROOF OF THE COMPACTNESS THEOREM 5.2. Nonlinear averages. Let (Mn,g) be a complete Riemannian manifold with sectional cur ...

CENTER OF MASS AND NONLINEAR AVERAGES 183 PROPOSITION 4.51 (Dependence of cm on weights and points). Suppose (Mn, g) is a Riem ...

is4 4. PROOF OF THE COMPACTNESS THEOREM By the previous lemma, cm(μi, ... ,μk) {qi, ... , qk} is the unique solution of the equa ...

CENTER OF MASS AND NONLINEAR AVERAGES This proves the first two convergences. Next we estimate {) \l qi oμi cm(μ1, .. .,μk) {q ...

186 4. PROOF OF THE COMPACTNESS THEOREM Furthermore F ( x) is smooth and its derivatives \le F ( x) are bounded by constants dep ...

NOTES AND COMMENTARY 187 4.49(ii) that by choosing k large and x to be in a very small neighborhood of xo, we can make cm^9 (^ ...

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CHAPTER 5 Energy, Monotonicity, and Breathers Truth is ever to be found in the simplicity, and not in the multiplicity and confu ...

190 5. ENERGY, MONOTONICITY, AND BREATHERS expanding and steady cases in Proposition 1.13. To prove the nonexistence of nontrivi ...

ENERGY, ITS FIRST VARIATION, AND THE GRADIENT FLOW 191 1.1. The energy functional F. Let C^00 (M) denote the set of all smooth ...

192 5. ENERGY, MONOTONICITY, AND BREATHERS define V ~ gij Vij. Routine calculations give (5.6) ovrij k (g) = 2,g^1 kl ('\liVjl + ...

ENERGY, ITS FIRST·VARIATION, AND THE GRADIENT FLOW 193 So using (5.8), 6 [(R+L1f)e-fdμ] = gij5 [(Rij + "Vi"Vjf) e-f dμJ - 6gij ...

194 5. ENERGY, MONOTONICITY, AND BREATHERS The first quantity vanishes on steady gradient solitons fl.owing along \7 f, whereas ...

ENERGY, ITS FIRST VARIATION, AND THE GRADIENT FLOW 195 where we think of Re ( e -n!q f (g + hq)) and R ( e -n!q f (g + hq)) as ...

196 5. ENERGY, MONOTONICITY, AND BREATHERS LEMMA 5.8 (Modified contracted second Bianchi identity). (5.21) 1.4. The functional ; ...