1547671870-The_Ricci_Flow__Chow

l. VARIATION FORMULAS 69 As in Lemma 3.2, we use geodesic coordinates centered at p E Mn to cal- culate that gt Rf 7 k (p) = vi ...

70 3. SHORT TIME EXISTENCE The general formula for the evolution of volume is given by the following observation. LEMMA 3.9. The ...

THE LINEARIZATION OF RICCI 71 Since [U, V] = 0, we have d d rb 1/2 dt Lt( "It) = dt la g (U, U) du = ~lb (U U)-1/ 2 ag (U U) d ...

72 3. SHORT TIME EXISTENCE of degree at most k + £ in (. We now regard the Ricci tensor Re (g) as a nonlinear partial differenti ...

THE LINEARIZATION OF RICCI Differentiating at t = 0, we obtain [X, [Y, Z]] =! <p; [Y, Z] = [ :t ( <p;Y), <p; Z] + [&l ...

74 3. SHORT TIME EXISTENCE Combining this equation with (3.14) and again using the fact that X is arbitrary, we obtain the follo ...

THE LINEARIZATION OF RICCI 75 and compare terms, we get 0 = [D (Rmg) (.Cxg)J;jk - (.Cx Rm)fjk -\Ji [ ( Rjpk - Rkpj - Rjkp) xq] ...

76 3. SHORT TIME EXISTENCE Consider the composition D (Reg) o o; : C^00 (T Mn) -+ C^00 (S 2 T Mn). This is a priori a third-orde ...

THE LINEARIZATION OF RICCI defined by (3.24) The total symbol of B 9 in the direction ( is the bundle homomorphism S2T* Mn --+ ...

78 3. SHORT TIME EXISTENCE proves that K<;, = At, hence that dim K<;, = n (n - 1) /2. Now we further consider the kernel o ...

PARABOLICITY OF THE RICCI-DETURCK FLOW 79 REMARK 3.14. We shall see in Corollary 7.7 that the lifetime of a max- imal solution ...

80 3. SHORT TIME EXISTENCE Now let f' be a fixed torsion-free connection. (For instance, we could take f' to be the Levi-Civita ...

PARABOLICITY OF THE RICCI-DETURCK FLOW 81 parabolic system of partial differential equations. It is a standard result that for ...

82 3. SHORT TIME EXISTENCE LEMMA 3.15. If {Xt: 0 :St< T :S oo} is a continuous time-dependent family of vector fields on a co ...

PARABOLICITY OF THE RICCI-DETURCK FLOW 83 (To see this, note that if x > 0, then t < x -^1 implies that x < c^1 .) 3. ...

84 3. SHORT TIME EXISTENCE at p for any tensor Q, it follows from (3.34) that the identity Wj (t) = gjk (t) gPq (t) ( (r g(t)):j ...

4. RELATION TO THE HARMONIC MAP FLOW 85 Hence \7 (df) = L (\7df)0 dxi 0 dxj 0 0 ° a i,J,a .. y where The harmonic map Laplacian ...

86 3. SHORT TIME EXISTENCE be a diffeomorphism. Then if "' is a metric on Nn, we have Hence Since a./3 - ( * )ij ac.pa. ac.p/3 K ...

RELATION TO THE HARMONIC MAP FLOW 87 At the origin of a normal coordinate system for g, we have Aij k = "2 1 ( Re -1) kf. ( V' ...

88 3. SHORT TIME EXISTENCE EXERCISE 3.23. Verify that ( C -1)ij \JiCjk = 2 1 ( C -1)ij \JkCij· Use this to prove that if the sec ...