1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

HARMONIC MAPS NEAR THE IDENTITY OF sn 261 2.1. Linearization and its kernel of the map-Laplacian on sn. Let g and g be C^00 Ri ...

262 34. CMC SURFACES AND HARMONIC MAPS BY IFT so that the two lowest eigenvalues of 6..d acting on co-closed 1-forms are (34.16) ...

HARMONIC MAPS NEAR THE IDENTITY OF s n 263 First, we prove part (2). If n = 2, then equation (34.19) implies JM (Lg (W) ' W) ...

264 34. CMC SURFACES AND HARMONIC MAPS BY IFT Of course, KV (Sn) c C^00 (TSn) c Ck,a (TSn) for any k and a. Define a Banach subs ...

HARMONIC MAPS NEAR THE IDENTITY OF sn 265 Furthermore, if n = 2, then the above formula is true for any conformal Killing vect ...

266 34. CMC SURFACES AND HARMONIC MAPS BY IFT REMARK 34.11. By applying the above proof to the n = 2 case, one can show that any ...

HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Re < O 267 3.2. The linearization of <!? and the function spaces for t ...

268 34. CMC SURFACES AND HARMONIC MAPS BY IFT on 8 M. In particular, since forker (£ 9 ) = 0, we have that the formal cokernel i ...

HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Re < O 269 That is, U so lves the equation (34.43a) (34.43b) -6.U-Rc(U)=Q ...

270 34. CMC SURFACES AND HARMONIC MAPS BY IFT Let x E 8M and let ( E Tx8M - {O}. The characteristic equation det (oL (x) ((+TN)) ...

HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Re < 0 271 On the other hand, we have the following elementary result. Cl ...

272 34. CMC SURFACES AND HARMONIC MAPS BY IFT Now we can apply (34.52) and Lemma 34. 15 to obtain LEMMA 34.17. The linear operat ...

APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES 273 bundles E and F over M together with a linear elliptic^4 bou ...

274 34. CMC SURFACES AND HARMONIC MAPS BY IFT Hence, if X and Y are vector fields on M, we have lvxY-V7xYl 9 = jxiyJ(f'7J - rt) ...

4. APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES 275 for X, Y E TxN. We then have Ill (X, Y) - II (X, Y) I :::; ...

276 34. CMC SURFACES AND HARMONIC MAPS BY IFT then there exists an isometry I: (H, h) ~ (il, h) which is close to F in the sense ...

APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES 277 PROOF OF STEP 2. We first show the following. C la im. For e ...

278 34. CMC SURFACES AND HARMONI C MAPS BY IFT From this and the fact t hat areas of tori do not change much under the maps Fi , ...

CHAPTER 35 Stability of Ricci Flow Look out kid. It's something you did. God knows when, but you're <loin' it again. From "S ...

280 35. STABILITY OF RICCI FLOW one expects to encounter a center manifold on which the behavior of solutions is determined not ...