1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

HA R M O N IC MAPS AND THEIR LINEARIZATION 341 3 .3. Second variation formula for the map energy. We adopt t he notation of t ...

342 K. IMPLICIT FUNCTION THEOREM Combining this with (K.32), we have :s ls=O (6.g,hfs)'y = (6.g,h V) 7 6.g (!^13 ) r (h);Q V 0 ...

HARMONIC MAPS AND THEIR LINEARIZATION 343 we may rewrite (K.34) as (K.35) :s/s=O (V,b.g,hfs)J;h = h<>1,V"' (b.g,hV),, + ...

344 K. IMPLICIT FUNCTION THEOREM 3.4. The harmonic map heat flow. In 1964, Eells and Sampson [100] proved the following existenc ...

3. HARMONIC MAPS AND THEIR LINEARIZATION In view of this, we compute that ("V/vjdf)~ - ("Vj"Vkdf)~ = (Vi"Vjdf)~ - ("Vj"Vidf)~ = ...

346 K. IMPLICIT FUNCTION THEOREM so that :t l6.g,hfl 2 = 2h-yc; (6.g,hft :t (6.g,hf)^7 + :t (h-yc; 0 f) (6g,hft (6g,hf)^7 = 2h 7 ...

SPECTRUM OF 6.d ON p-FORMS ON sn 347 Hence, in invariant form, the variation formula for the first component of <P(g, Fs) i ...

348 K. IMPLICIT FUNCTION THEOREM Let Pk denote the space of degree k homogeneous polynomials on JRn+l and let 1-lk denote the sp ...

SPECTRUM OF L:,.d ON p-FORMS ON s n 349 and nonpositive: (6.da, a) £2 = -(da, da) £2 - (Ja, fo) £2 ::::; 0. We also h ave that ...

350 K. IMPLICIT FUNCTION THEOREM second fundamental form of sn equals the first fundamental form of sn. Our sign conventions are ...

Hence SPECTRUM OF ~d ON p-FORMS ON sn n n = - L (V' e; 7/lsn) (ei, X1, · .. , Xp_i) + L (De,7/) (ei, X1, ... , Xp-1) i=l + (Dv ...

352 K. IMPLICIT FUNCTION THEOREM 4.4. Bochner-Weitzenbock formula for p-forms. Recall that the Lichnerowicz theorem says that if ...

Bibliography [l] Abraham, Ralph. Lectures of Smale on differential topology. Columbia University, New York, 1963. [2] Abresch, U ...

354 BIBLIOGRAPHY [22] Bamler, Richard H. Long-time behavior of 3 dimensional Ricci fiow -C: 3-manifold topology and combinatoric ...

BIBLIOGRAPHY 355 [49] Cao, Huai-Dong; Zhu, Xi-Ping. A complete proof of the Poincare and geometrization conjectures-application ...

356 BIBLIOGRAPHY [73] Chow, Bennett; Hamilton, Richard S. The cross curvature fiow of 3 -manifolds with negative sectional curva ...

BIBLIOGRAPHY 357 [99] Eells, J a mes; Lemaire, Luc. Two reports on harmonic maps. World Scientific Publishing Co., Inc., River E ...

358 BIBLIOGRAPHY [129] Guenther, Christine; Isenberg, James; Knopf, Dan. Linear stability of homogeneous Ricci solitons. Int. Ma ...

BIBLIOGRAPHY 359 [156] Izumiya, Shyuichi; Pei, Donghe; Takahashi, Masatomo. Singularities of evolutes of hyper- surfaces in hype ...

360 BIBLIOGRAPHY [180] Langer, Joel. A compactness theorem for surfaces with Lp-bounded second fundamental form. Math. Ann. 270 ...