1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

INTRODUCTION TO KAHLER MANIFOLDS 57 It is interesting that S^6 admits an almost complex structure. In fact S^2 and S^6 are the ...

58 2. KAHLER-RICCI FLOW Given an almost complex manifold (M, J), the complexified tangent bundle is TcM ~ TM ®JR C. For each p E ...

INTRODUCTION TO KAHLER MANIFOLDS 59 A covariant tensor of type (p, q) is a section of @p,q M ~ ( tg)P A l,O M) Q9 ( @q A O,l M ...

60 2. KAHLER-RICCI FLOW EXERCISE 2.3. Given a (p, q)-form n 'I = 'iil n. ·· ·tpJl. ~ ... ~ Jq di^1 /\ • • • /\ dziP /\ dz3^1 /\ ...

INTRODUCTION TO KAHLER MANIFOLDS 61 Given a ]-invariant symmetric 2-tensor b, we define a ]-invariant 2-form ,8 by (3(X,Y) ~b( ...

(^62) 2. KAHLER-RICCI FLOW An oriented Riemannian surface (M, 9) has a natural complex struc- ture and 9 is a Kahler metric with ...

CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION 63 SOLUTION. We compute r^1 &/3 --- ~ 29 18 (~ f)za9f38 - + f)z/39a8 ...

64 2. KAHLER-RICCI FLOW EXERCISE 2.11 ( J-invariance of curvature). Show that for a Kahler man- ifold Rm (X, Y) is J-invariant, ...

CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION 65 which is equivalent to the Ricci form being closed: dp = 0. We may exp ...

66 2. KAHLER-RICCI FLOW to be K (Z W)::::: Rm (X, JX, JU, U) = Rmic (Z, Z, W, W) ic ' · 1x1^2 1u1^2 1z1^2 1w1^2 Clearly we have ...

CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION 67 Let V '13 ~ V 81 ozf3. The Laplacian acting on tensors is given by 1 - ...

68 2. KAHLER-RICCI FLOW PROOF. Using the fact that the Christoffel symbols are zero unless all of the indices are unbarred or al ...

CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION 69 Hence \7 a \7 13a'Y1 ""/'pJ1 ... Jq - \7 f3 \7 aa'Y1 "''YpJ1 ... Jq = ...

70 2. KAHLER-RICCI FLOW Since the Kahler-Ricci flow is a heat-type equation for Kahler metrics, some evolution equations we shal ...

EXISTENCE OF KAHLER-EINSTEIN METRICS 71 where w = Hwa~dza /\ dz/3 and wa~ = Wf3a· PROOF. This is a standard result in the theo ...

72 2. KAHLER-RICCI FLOW Yau's proof of Theorem 2.28 involves solving the fully nonlinear equation above by using the continuity ...

EXISTENCE OF KAHLER-EINSTEIN METRICS 73 canonical case.) By scaling the metric, we may assume [w] = ci(M). By Lemma 2.26 there ...

74 2. KAHLER-RICCI FLOW Sturm [304]. The existence of Kahler-Einstein metrics on complete non- compact manifolds with c 1 < 0 ...

INTRODUCTION TO THE KAHLER-RICCI FLOW 75 r.p ( t) defined on all of M such that (2.27) By (2.6) we have _ _ _ _ det ( g~ 8 + t ...

76 2. KAHLER-RICCI FLOW 4.2. The normalized Kahler-Ricci fl.ow equation. Let (Mn, J, go) be a closed Kahler manifold. Now we mak ...