1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

INCOMPRESSIBILITY OF BOUNDARY TORI 241 LoopO(t) FIGURE 33.2. Disk^0 (t) and Loopo (t). The bound for d+d~' (to) involves the n ...

242 33. NONCOMPACT HYPERBOLIC LIMITS Jo (D^2 (1 + ry)). Since V (to) is normal to TA (to), we have that (33.49) JV^0 (to)I = I~~ ...

INCOMPRESSIBILITY OF BOUNDARY TORI 243 Let ""1 and ""2 denote the eigenvalues of II, i.e., the principal curvatures of V (to). ...

244 33. NONCOMPACT HYPERBOLIC LIMITS Since v^0 (to) -7 0 as t 0 -7 oo, there exists t"' (3) < oo such that for to 2 t"' (3), ...

INCOMPRESSIBILITY OF BOUNDARY TORI 245 so that h (8D) = a. By Sard's theorem, almost every r E (a, b) is a regular value of h. ...

246 33. NONCOMPACT HYPERBOLIC LIMITS Restricting the exponential map to the normal bundle we have the map exp" ~ exp^9 (t) INTA ...

INCOMPRESSIBILITY OF BOUNDARY TORI torus with area A FIGURE 33.3. The submanifolds CA,B, TA,p, SA,p, and SA,p· 247 Let rv deno ...

248 and (33.70) Clearly (33.71) 33. NONCOMPACT HYPERBOLIC LIMITS FIGURE 33 .4. The annulus AA, p and the loop SA, p (p). since S ...

INCOMPRESSIBILITY OF BOUNDARY TORI 249 Since the area of V(t) is less than or equal to t he area of D(t), (33.73) 1 15 L (p) d ...

250 33. NONCOMPACT HYPERBOLIC LIMITS Since ec;p 1P lim _ L(p)dp=L(O), p->0 1 - e-P O we deduce from Lemma 33.34 and (33.67) t ...

4. INCOMPRESSIBILITY OF BOUNDARY TORI 251 PROOF. All of the following discussion is at time to. There exists P• E [P#, p*] such ...

252 33. NONCOMPACT HYPERBOLIC LIMITS (3) Let D~,P• c D^2 be the disk bounded by the loop ft~^1 (SA,p.)· Define the immersed comp ...

INCOMPRESSIBILITY OF BOUNDARY TORI 253 PROOF. The geodesic loop [, : [O, l] ---+ T defines a translation T.c (deck transformat ...

254 33. NONCOMPACT HYPERBOLIC LIMITS It follows from Lemma 33.37 that L (0) s (1 + c)^2 (1+2c) (1 + e-P• L (P•)) Lo S ( 1 + c)^2 ...

NOTES AND COMMENTARY 255 §4. This section follows §11 and § 12 of Hamilton [143]. For background and more advanced topics on m ...

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CHAPTER 34 Constant Mean Curvature Surfaces and Harmonic Maps by IFT The mist across t h e window hides the lines. From "Steppi ...

258 34. CMC SURFACES AND HARMONIC MAPS BY IFT PROOF. STEP 1. Linearization of th e mean curvature of graphs in a cusp. Recall th ...

CONSTANT MEAN CURVATURE SURFACES 259 By (34.2), the linearization of 1> at (gcusp, r , 0, n - 1), with respect to the vari- ...

260 34. CMC SURFACES AND HARMONIC MAPS BY IFT 1.2. Existence of CMC spheres in necks. Analogous to Proposition 33.11 we have the ...