1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. MATCHED ASYMPTOTI C STUDIES OF DEGENERATE NECKPINCHES 321 equ ation 8T V = .CV can be written in series form Lk bke>.kT hk ...

322 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES 2.5. The tip region. The numerical simulations discussed in § 1 of this ...

MATCHED ASYMPTOTIC STUDIES OF DEGENERATE NECKPINCHES 323 As determined above, the evolving geometry at the tip is modeled by s ...

324 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES f(T) = e(l-f-)^7 • With these restrictions imposed, two free parameters ...

3. DEGENERATE NECKPINCH SINGULARITIES 325 with no solution inside 2{n , , k c·<} , ever contacting 33+{ n , k ,c,E. } and wit ...

326 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES The other necessary task is finding the set Bk and the corresponding se ...

Appendix K. Implicit Function Theorem I can endure no more, I demand you remember who you are. From "I Need a Doctor" by Dr. Dr ...

328 K. IMPLICIT FUNCTION THEOREM PROOF. The idea is to iterate the map f (starting at any point) and obtain a Cauchy sequence wh ...

THE IMPLICIT FUNCTION THEOREM 329 that (DJ (x 0 ))-^1 is a bounded linear operator. Without loss of generality we may assume, ...

330 K. IMPLICIT FUNCTION THEOREM Hence given y EB (0, 8/2) and h EX with 0 < [[h[[ ~ 8/2 - [[y[[, we have llr1 (y+ h) -r1 (y) ...

THE IMPLICIT FUNCTION THEOREM 331 We call each (Ui, <pi) a Banach space coordinate chart. A Banach manifold atlas defines a ...

332 K. IMPLICIT FUNCTION THEOREM Let B be the Banach space of Ck,a vector fields on M which when restricted to 8M are tangent to ...

HOLDER SPACES AND SOBOLEV SPACES ON MANIFOLDS 333 where the {¢%;ii···iJ are locally defined functions on U 13. For any a E (0 ...

334 K. IMPLICIT FUNCTION THEOREM Let (Mn,gM) and (Nm,gN) be Riemannian manifolds. Let (U, {xir= 1 ) be a coordinate neighborhood ...

HOLDER SPACES AND SOBOLEV SPACES ON MANIFOLDS 335 natural bundle isomorphism defined by k J: Jk (M,N)---+ EB (piSiM @p2TN)' i= ...

336 K. IMPLICIT FUNCTION THEOREM I and II denote the first and second fundamental forms, respectively. Let l (A, K,) > 0 be a ...

HARMONIC MAPS AND THEIR LINEARIZATION 337 where C ~ C o f , for C E C^00 (TN). In local coordinates, this is (K.17) ("Vrhcrr = ...

338 K. IMPLICIT FUNCTION THEOREM As a special case, if M = N and f is the identity map and if we choose the coordinates x and y ...

HARMONIC MAPS AND THEIR LINEARIZATION 339 coordinates as follows: :S ls=O (f;h)ij (p) = OS a I s=O ( (ho:(3 o fs) (p) of~ axi ...

340 K. IMPLICIT FUNCTION THEOREM To obtain the third equality above we used the divergence theorem and (K.27) div^9 ((V, df) 1 . ...