274 34. CMC SURFACES AND HARMONIC MAPS BY IFT
Hence, if X and Y are vector fields on M, we have
lvxY-V7xYl
9= jxiyJ(f'7J - rt) a~k 1
9
(34.55) ::::: c IXl 9 IYl 9 11.9 -glb<M,g).Equivalently,
(34.56)LEMMA 34.19. Let Nn-l c Mn be an oriented C^00 hypersurface and let v and
i/ be C^00 unit normal vector fields to N with respect tog and g, respectively, where
v and i/ are on the same side of Nin M. We then have the following:
(1)
(34.57)(2)
Iv - vl 9 ::::; C 11 .9 - gllco(M,g) ,
in particular, lvl 9 ::::; C 11 .9 -gllco(M,g) + l.(34.58) 1Vvl 9 ::::; C, IV (i/ - v)l 9 ::::: C 11.9 -glb(M,g).
(3) The second fundamental form satisfies(34.59)
(4)
(34.60)Ill (X, Y) - II (X, Y) I::::; c IX lg IYlg 11.9 -glb(M,g).
IVll (X, Y) - V7 II (X, Y) I ::::; C IXl 9 IYl 9 11.9 - gllc2(M,g) ·
PROOF. (1) Choose local coordinates (U, {xi r=l) in a neighborhood of a point
in N so that N n U = {p EU: xn (p) = O} and 8 ~n is on the same side as v. On
N n U let N = NJ 8 ~j be the normal field to N satisfying Nn = l. Then for j < n
we have that g(N, 8 ~j) = 0 and
(34.61)
n-l
Nj = - L G ijgin,
i=l
where (GiJ)~7~ 1 denotes the inverse matrix to (%)~j~ 1. Using (34.61) we compute
that
n-l
INl^2 - ~ ki
9 - gnn - L..,, ginG gkn·
i ,k=l
Now we obtain
N ~ _ "'n-1 QiJ. ...2-.
(34.62) v - --- axn L...ti=l gin ()xj- INI 9 -. / °"n-l Qkt
y 9nn - L...ti,k=l gin 9kn
We also have the analogous formula for i/. From this we may derive (34.57).
(2) We obtain (34.58) from differentiating the formula (34.62) for both v and i/.(3) By the definition of the second fundamental form of N, with respect to g
and g,