1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

270 34. CMC SURFACES AND HARMONIC MAPS BY IFT

Let x E 8M and let ( E Tx8M - {O}. The characteristic equation

det (oL (x) ((+TN))= 0

has unique conjugate roots T+ ( () and T-( () of multiplicity n with positive and

negative imaginary parts, respectively. This is because the characteristic equation

is equivalent to the quadratic equation

0 =A ((+TN)= L ak£ (x) (k(£ + 2T L akn (x) (k + ann (x) T^2.

k,£'.'O,n-1

Define M± : T(aM) x JR -7 JR by

M± ((, T) = (T - T±(()r,

so that M+ ((, T) M-((, T) = det (uL (x) ((+TN)).

The principal symbol uB (x) (~) : TxM -7 TxM of the boundary operator B

in (34.44b)- (34.44c) is given by

uB (x) (~) (V) = ~ (N) VT+ Vi ..

We compute that

(34.46) [uB (x) ((+TN) o adj (uL (x) ((+TN))] (V)

=I(+ TNl^2 (n-l) (TVT + V.L)

= (T-T+(or-l (T -T+(()r-l (TVT + v.L).

Therefore the row vectors of the matrix corresponding to the transformation on the

LHS of (34.46) are linearly independent modulo M+ ((, T). This verifies the Com-

plementing Boundary Condition on pp. 42-43 of [4]. Regarding this condition,

in the introduction to [4], the authors write:.

This Complementing Condition is necessary and sufficient in order

that it be possible to estimate all the derivatives occurring in the

system, without loss of order, i.e., that inequalities of "coercive"

type be valid.

More precisely, by Theorem 9.3 in [4] we conclude that if Q E c= (TM) and

fE c= (T(aM)), then the solution U to (34.43) satisfies U E c= (TM).

3.4. Schauder estimates for the linearization.

Now we consider the related Schauder theory. Note that the boundary condi-

tions U .L = 0 and (V NUh -II(U) = 0 have different orders of derivatives. In view

of this, we invoke Theorem 9.3 of [4] (or Theorem 5 on p.406 of Simon [381]) to

obtain the following estimate. For any U E C^2 '°'(T M),

(34.47)

[V

2

UJa :SC (11-b.U - Re (U)llc<>(TM) + ll(V NUh - II(U)ll 0 1,a(T(oM)))

+ C (llU.Lllc2.a(T(8M)) + llUllco(TM)) ·

By an interpolation estimate (see §6.8 of [122]), one can show that for any

E: > 0 there exists CE < 00 such that for any u E C^2 (T M),

(34.48) llUllc2(TM) :S c['Y^2 U]a + CcllUllco(TM)·