- THE SECOND VARIATION FORMULA FOR .C AND THE HESSIAN OF L 313
L defined in subsection 4.2 of this chapt~r. At a point q where L (·, f) is
C^2 , the second variation formula gives an upper bound for the Hessian of
the £-distance function and tracing this estimate yields an estimate for
the Laplacian of L. The Hessian upper bound will be very important in
discussing the weak solution formulation later.
In this section (Mn,g(T)), TE [O,T], will denote a complete solu-
tion to the backward Ricci fl.ow satisfying the pointwise curvature bound
max{IRml, IRcj} ::::; Co < oo on M x [O, T], and p E M shall denote a
basepoint.
5.1. The second variation formula for £,, Let f E (0, T) and let'Y : [O, f] ---+ M be an £-geodesic from p to q. Let 'Ys : [O, f] ---+ M, s E
(-E, c), be a smooth family of paths with 'YO ( T) = 'Y ( T). Recall that our
convention about the smoothness of a variation 'Ys of "( is that f3s (O") ~
"Is ( u;) is required to be a smooth function of ( O", s). It is easy to see that
the nondifferentiability of "Is at T = 0 causes no trouble in the following
calculation. This would also be clear if we use (7.19) to do the calculation.
Define ~'.; (T) ~ X (T, s) and^88 ~ (T) ~ Y (T, s), so that [X, Y] = 0.^10 We
also write Y ( T) ~ Y ( T, 0). Note that Y ( u;, s) is a smooth function of
(O", s).
Recall the first variation formula (7.30)which holds for alls E (-E, c). Differentiating this again, we get
(^52 Y £) ( ) 'Y = · d2 [,("Is) ds 2 I
s=O= 1
7yfi (Y (Y (R)) + 2 (\7y\7yX,X) + 2 l\7yXl^2 ) dT.
Now since [X, Y] = 0,
(\ly \ly X, X) = (\ly \7 x Y, X) = (R (Y, X) Y, X) + (\7 x \ly Y, X).
Hence
(7.59) (^2) _ (7 c( Y(Y(R))+2(R(Y,X)Y,X))
Sy£("!)-Jo yT +2(\7x\7yY,X)+2l\7yXl^2 dT.10 Alternately, given any vector field Y along'"'(, there exists a family of paths 'Ys suchthat 'b..... 8 s • I s=O = Y. In this case, we extend Y by defining 'b..... (^8) s s = Y for s E ( -c:, c:) , so that
[X,Y] = 0. Technically, X and Y are sections of the bundle G*TM on [0,7] x (-c:,c:),
where G (T, s) ~ 'Ys (T).