1547671870-The_Ricci_Flow__Chow

RELATION TO THE HARMONIC MAP FLOW 89 REMARK 3.26. Ben Andrews [4] has observed that when the universal cover of the initial Ri ...

90 3. SHORT TIME EXISTENCE is a solution of the Ricci- DeTurck fl.ow (3.33). Indeed, using (3.35) we compute that :t9(t) = :t (( ...

5. THE RICCI FLOW AS A HEAT EQUATION 91 Existence of harmonic coordinates is a straightforward consequence of standard existence ...

92 3. SHORT TIME EXISTENCE LEMMA 3.32. In harmonic coordinates, the Ricci tensor is given by 2Rj = ~ (gij) + Qij (g-^1 , og), w ...

CHAPTER 4 Maximum principles Maximum principles are among the most important tools in the study of second-order parabolic differ ...

94 4. MAXIMUM PRINCIPLES THEOREM 4.2 (scalar maximum principle, second version: lower bounds are preserved for supersolutions). ...

l. SCALAR EQUATIONS 95 1.2. The heat equation with a linear reaction term. More gen- erally, one may add in a reaction term. We ...

96 4. MAXIMUM PRINCIPLES THEOREM 4.4 (scalar maximum principle, fourth version: ODE gives pointwise bounds for PDE). Let u : Mn ...

TENSOR EQUATIONS 97 Weak maximum principles for tensor equations The maximum principle is extremely robust: it applies to gene ...

98 4. MAXIMUM PRINCIPLES and then taking V to be independent of time. To see that the Laplacian of V vanishes at (xo, to), choos ...

TENSOR EQUATIONS fort E [to, to+ o] when we choose o > 0 sufficiently small, depending on Mi;i;[~,T] I :t^9 1. The evolutio ...

100 4. MAXIMUM PRINCIPLES Extend v to a vector field V defined in a space-time neighborhood of (xi, ti) by the method described ...

ADVANCED MAXIMUM PRINCIPLES 101 Let F : £ x [O, T] ---+ £ be a continuous map such that F (.,., t) : £ ---+ £ is fiber-preserv ...

102 4. MAXIMUM PRINCIPLES Then if every solution of the ODE d dt a = F (a) a (0) E JCx (0) defined in each fiber Ex remains in J ...

NOTES AND COMMENTARY 103 conclude that u is st rictly positive for all positive time. This is an example of a strong maximum pri ...

104 4. MAXIMUM PRINCIPLES A nice presentation of the scalar maximum principle for generalized heat equations may be found on pag ...

CHAPTER 5 The Ricci flow on surfaces One of the triumphs of nineteenth-century mathematics was the Uni- formization Theorem, whi ...

106 5. THE RICCI FLOW ON SURFACES attempt to present the most efficient proofs known. The organization of the chapter is as foll ...

1. A CONFORMAL CHANGE OF METRIC 107 FIGURE 1. A 2-sphere whose curvature is of mixed sign In the case of a surface M^2 , we have ...

108 5. THE RICCI FLOW ON SURFACES is an orthonormal frame field for g and wl ~ eurt1 w2 ~ eu172 is an orthonormal coframe field ...