1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

Since HEAT KERNEL ON NONCOMPACT MANIFOLDS 00 = L Mk+l (xo, r; z, cr) k=l = 2 L oo 1r1 av-aii (xo, r; w, p) Mk (w, p; z, cr) dμ ...

302 24. HEAT KERNEL FOR EVOLVING METRICS THEOREM 24.40 (Existence and uniqueness of heat kernel on noncom- pact manifolds - time ...

NOTES AND COMMENTARY 303 where 1/ 7 ,i is the outward unit normal to ani with respect to g ( T); here we used ~~ni r,i ::=; 0. ...

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Chapter 25. Estimates of the Heat Equation for Evolving Metrics You talk about things that nobody cares. From "Sweet Emotion" b ...

306 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS where the volume is measured with respect to the metric g + dt^2 • T ...

MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 307 be a positive subsolution to (25.6). If (xo, To) E n x (0, T] and r ...

308 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS on n x [O, T]. We shall integrate this inequality after localizing i ...

MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 309 LEMMA 25.3. (25.19) 1T 2 r IV' ('!j;vP)l~(T) dμg(T) dT :S L1T 2 r v^ ...

310 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS REMARK 25.5. If n = 2, then under the assumptions of the above lemma ...

MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 311 Let Pr~ P9 (xo, To, r, -r^2 ), where P9 is defined in (25.7). For th ...

312 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS Summarizing, we have proved for 0 < r' < r :'S 2ro that (25.28 ...

MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 313 Note that 87/Ji 2i 2i+l (25.32) 0 < - < - and l\77/Jilg-S -. - ...

314 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS N ow smce · Lii=l '\''00 (n+2)-i --;;;:- _ - n 2 , we h ave (25.36)^ ...

MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 315 STEP 4. Finishing the proof of the theorem by jumping from L^2 to L^ ...

316 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS since llvllL=(PPf) < llvllL=(P2ro) < 00 independent of e. Prov ...

LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 317 Applying the Sobolev inequality, we obtain for 0 < r' < r & ...

318 2S. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS ('potential function' for the heat-type equation below) be a C^00 fu ...

LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 31g for each TE [O, T]. If u: U Bg( 7 ) (p, 2R) x {r}-+ IR+ TE[O,Tj i ...

320 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS PROOF. Let 'Y: [Ti, T 2 ] --+ M be a smooth path joining x1 to x2. B ...