1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

258 34. CMC SURFACES AND HARMONIC MAPS BY IFT

PROOF. STEP 1. Linearization of th e mean curvature of graphs in a cusp.

Recall that the embedded slices {r} x V C (a, b) x V are CMC hypersurfaces with

respect to gcusp· Consider those hypersurfaces which are graphs over V. That is,

we associate to a function cp : V --+ (a, b) the graphical hypersurface

Vcp ~ {(cp(x) ,x): x EV}

in (a, b) x V. When cp = r is a constant function, we write Vr ~ {r} x V.

Let g be a C^2 metric on [a, b] x V and let cp E C^2 (V; (a, b)). Let H 9 (Vcp) denote

the mean curvature of the graph Vcp, with respect to the metric g, where H9 is

defined using the choice of unit normal N to Vcp satisfying (N, gr) > 0.

For any f E C^2 (V), corresponding to the normal deformation of the hypersur-

face Vcp with velocity vector field f N , the variation of the mean curvature of Vcp is

given by

(34.1) D (H9 (Vcp)) (! N) = b.9,cpf + (IIIl

2

+Re 9(N, N))f,

where b. 9 ,cp denotes the Laplacian on Vcp with respect to the induced metric, where
II denotes the second fundamental form of Vcp with respect tog. Equation (34.1)
is a straightforward generalization of equation (B .17) for the mean curvature fl.ow
in Appendix B of Part I.
Let r E (a, b). When g = gcusp and Vcp = Vri (34.1) yields

(34.2)

where b.r is with respect to the metric e-^2 r 9flat on Vr since in this case

(34.3)

~ 8

N = or, II= - 9cusplvr, and sect (9cusp) = -1.

STEP 2. Suitable map between Banach spaces to apply the IFT. The existence
of CMC hypersurfaces amounts to solving H 9 (Vcp) + c = 0 for some constant c. So
as to be able to invoke the IFT, we now address the issue of the kernel of b.r in
(34.2), which comprises the constant functions.
Let~ denote the Banach space of C^2 ,o. symmetric 2-tensors on (a, b) x V with
the usual C^2 ,o.-norm, with respect to 9cusp (see §2 of Appendix K for the definition
of Holder spaces). Let 9J1et be the open cone of all positive-definite tensors in ~.
Denote

C^2 ,o. (V; (a', b')) ~ { cp E C^2 ,o. (V; (a', b')) : iv cp (x) du 9 n a t (x) = 0}

for a', b' E JR. U { -oo, oo} with a' < b'.

Let Eo E (0, b;a). Define the map

: 9J1et x (a+ Eo, b - Eo) x 62 ,0. (V; (-Eo, Eo)) x JR.--+ ca (V; JR.)

(34.4)