336 K. IMPLICIT FUNCTION THEOREM
I and II denote the first and second fundamental forms, respectively. Let l (A, K,) > 0
be as in Lemma 1.42 in Part III. Then for any smooth function VJ : X ---+ JR with
IVJI < l (A, K,), we have that the normal graph
Yc,o ~ {expx (cp (x ) Llx): x EX}
is a smooth hypersurface. For such a hypersurface, define
dck,a (Yc,o, X) = llVJJlck,a(x) ·
DEFINITION K.9. If d 0 k ,a (Yc,o, X) ~ c:, t hen we say that Yc,o is c:-close to X in
ck,C>_norm.
3. Harmonic maps and their linearization
In t his section we will derive several formulas related to h armonic maps and
the harmonic map heat flow. Some of these formulas are used in Chapters 33 and
- As throughout this book, we find it useful to compute and express formulas
using local coordinates. In addit ion, we often indicate invariant ways to calculate
the formulas that we present.
3.1. Harmonic maps.
Let (Mn, g) and (Nm, h) be Riemannian manifolds with or without boundary.Given a map f : M ---+ N, its derivative (i.e., tangent map) df : TM ---+TN
may b e considered as a section of the vector bundle £ ~ T M © f (TN), where
f* (TN) ---+ M is the pull-back vector bundle of TN by f. If (U , {xi} ~= 1 ) and
(V, {y"'}~ 1 ) are local coordinates on M and N, respectively, then
df = ar - i a
8. dx © -
8
,
x' y"'where f" ~ y^1 of and where the expression on the RHS is defined on Un f-^1 (V).
On £ we have the natural bundle metric g i:gJ h ~ g-^1 © f* h. Using this we define
the energy density
2 -'- i j a r a Jf3
ldfl 9 18lh...,... g (haf3 ° f) -
8. -
8
..
x' x J
Since (fh)ij = h(!r,£h) = (ha(3of)?/:ir-~, an invariant way to write the
energy density is
ldfl~l8lh = gi j (fh)i j =tr 9 (fh).
DEFINITION K.10 (Map energy). The map energy of the map f is defined by
(K.15) Eg,h (f) ~JM ldfl~ 181 h dμ 9 =JM tr 9 (f* h) dμ 9.
We sh all determine the critical points of E 9 ,h in the next subsection. First,
we recall some basic facts about the pull-backs of connections , t heir tensor prod-