318 7. THE REDUCED DISTANCELEMMA 7.39 (£-Second variation under (7.66)). If f E (0, T), Yr E
T'Y(r)M and Y is a solution to (7.66) with Y (f) = Yr, then the second
variation of £-length is given by
( 5}£) (I) - 2Vf ("9y Y, X) (f) + 2Vf Rc (Y, Y) (f)(7.69) = - for y'TH (X, Y) dr + IY ~1
25.2. Hessian comparison for L. Corresponding to the £-second vari-
ation formula is an upper bound for the Hessian of the L-distance function,
which we derive in this subsection. Given any (q, f), let I : [O, f] ~ M
be a minimal £-geodesic from p to q so that L (q, f) = £(I). Fix a vector
Y E TqM - { 0} and define the vector field Y ( T) along I to be the solution
to (7.66) with Y (f) = Y. Let Is : [O, f] ~ M be a smooth family of curves
for s E (-c;, c;) withd[s I (r) = Y (r) and (V'yY) (f) = 0.
ds s=O.
Then there exists a small neighborhood U of q, 5 E (0, c;], and a smooth
family of curves /x, 7 : [O, r] ~ M for (x, r) EU x (f - 5, f + 5) satisfying/x,T(O)=p, f'Ys(r),r=/s, and /x,T(r)=x
for s E (-5, 5). We define L (x, r) ~ £ (/x, 7 ). Then L (is (f), f) =£(Is).
Since L (·, ·) is an upper barrier function for the L-distance function
L (·, ·) at (q, f), we have
(Hess(q,r) L) (Y, Y) ::::; ( Hess(q,r) L) (Y, Y)
when L (·, f) is C^2 at q. Since (\i'y Y) (f) = 0 and( Hess(q,r) L) (Y, Y) = :: 2 1s=O L (is (f), f) = ::21s=O £(is)'
combining this with Lemma 7.39, we get the Hessian Comparison The-
orem for L.
COROLLARY 7.40 (Inequality for Hessian of L). Given f E (0, T), q E
M, and Y E TqM, let/ : [O, f] ~ M be a minimal £-geodesic from p to q.
The Hessian of the L-distance function L (·, f) at q has the upper bound
(7.70) (Hess(q,r) L) (Y, Y)::::; - for y'TH (X, Y) ( 7 ) dr + IY ~1
2- 2VfRc (Y, Y) (f),
where Y (r) is a solution to (7.66) with Y (r) = Y and H is the matrix
Harnack expression defined in (7.63). Equality in (7.70) holds when L (·, f)
is C^2 at q and Y ( T) is the variation vector field of a family of minimal
£-geodesics.