1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Re < 0 271

On the other hand, we have the following elementary result.

Claim. For any c > 0 there exists Cc; < oo such that for any U E C^1 (T M),

(34.49)

Proof of the claim. Suppose that the claim is false for some E: > 0. Then there

is a sequence of vector fields Uk E C^1 (TM) such that

(34.50)

By scaling, we may assume that ll\7Ukllco = 1, so that llUkllco(TM) > €. Fix a

point p EM and let d ~ diam(M). Then

IUk(P)I - d 5 IUk(q)I 5 IUk(P)I + d for any q EM.

So, by (34.50), we have

IUk(P)I + d > c + kllUkllL2(TM) 2: c + k(IUk(P)I - d) Vol(M)^1 l^2.
Hence {IUk(P)I} is a bounded sequence. Now we can apply the Arzela-Ascoli the-
orem to Uk to conclude that Uk -t U 00 in C^0 (T M). Then (34.50) implies that

llUoollL2(TM) = O; i.e., U 00 = 0. The claim follows since this contradicts that

llUkllco(TM) > c for all k.

Now, by combining (34.47), (34.48), and (34.49), we obtain

llUllc2,a(TM) 5C11-D.U - Re (U)llc"(TM)

+ C 11(\7 NUh - II(U)llc1,a(T(8M))

• C (11U.Lllc2.a(T(8M)) + llUllL2(TM)) ·

In particular, for any U E C^2 •°' (TM) with U.L = 0 on aM, we have

(34.51) llUllc2." (TM) 5C11-D.U - Re (U)llc"(TM)

+ c (11(\7 NUh - II(U)lb.<>(T(DM)) + llUllL2(TM)).

By (34.41), (34.34), and the Cauchy- Schwarz inequality, we have

2 1

llUllw1.2(TM) 5 J 11-D.U - Re (U)llL2(TM) llUllL2(TM)

1

+ J ll(\7NUh - II(U)llL2(T(8M)) llUllL2(T(8M)).

Applying llUllL2(T(DM)) 5 C llUllw1.2(TM) from the trace theorem, there exists a

constant C < oo such that

llUllL2(TM) 5 llUllw1.2(TM)

5 C (11-D.U - Re (U)llL2(TM) + 11(\7 NUh - II(U)llL2(T(aM))) ·

In conclusion, we apply this estimate to (34.51) to obtain

LEMMA 34.16 (Schauder estimate). For any U E C^2 • °' (TM) with U .L = 0 on

aM,

(34.52) llUllc2." (TM) 5C11-D.U - Re (U)llc" (TM)

+ c ll(\7NUh - II(U)lb.<>(T(DM)).