344 K. IMPLICIT FUNCTION THEOREM3.4. The harmonic map heat flow.
In 1964, Eells and Sampson [100] proved the following existence theorem for
harmonic maps. Essentially, this result began the study and application of non-linear parabolic equations in differential geometry. It was a main motivation for
Hamilton's discovery of the Ricci flow.
THEOREM K.17 (Existence of harmonic maps into nonpositively curved tar-gets). Let (Mn,g) and (Nm,h) be closed Riemannian manifolds where (N, h) has
nonpositive sectional curvature. If Jo : M --t N is a smooth map, then there exists
a harmonic map f 00 : M --t N which is homo topic to f o.The original proof^3 of Eells and Sampson uses the harmonic map heat flow
of
(K.37) Ft = 6.g,hf.
They proved that if (N, h) has nonpositive sectional curvature, then for any C^00
map f o : M --t N, there exists a C^00 solution f ( t) : M --t N to the harmonic map
heat flow with f (0) =Jo and defined for all time t E [O, oo). Moreover, as t --too,
the map f ( t) converges in C^00 to a C^00 harmonic map f 00. When the target has
negative sectional curvature, Hartman [145] proved a uniqueness result.
In this subsection we discuss a couple of energy-type estimates used in the proof
of Theorem K.17; these estimates give indications that solutions to the harmonic
map heat flow behave well. The interested reader may see [100] for a complete
proof.
First, we compute the evolution of the energy density under the harmonic mapheat flow. By taking V = 6. 9 ,hf in (K.28) we have that
:t ldf l~1811i = 2 \ vr h (6.g,hf), df)
9181
h.To rewrite this as a heat-type equation we shall deduce a Bochner- Weitzenbock-
type formula. We compute6. 9 ldfl^2 9181 1i = 2 (6. 9 ,1i (df), di)+ 2 I V^9 18lh (di)^12 gl8lh,
which implies(K.38) :t ldfl~1811i = 6.g ldfl~181h - 2 IV^9181 h (df)l:
181 1i
+ 2 \ I v t*h (6.g,hf) - 6.g,h (elf)' df ) g181h.We claim that as sections of £ = T M ® J (TN) we have
(K.39) 9f*h (6. 9 ,hf) - 6. 9 ,h (di)= - (RcM) (df) + tr^2 •^3 9 ((RN) (df, df) df),
where the trace tr^2 •^3 9 is with respect to the second and third components of RN.
Recall that for a 1-form won M, we have
VNjwk -Vj\liwk = -(RM):jkwe,
whereas for a vector field W on N we have
V °' V 13 W^6 - V 13 V °' W^6 = (RN) ~/3-y W-Y.
(^3) Schoen [348] later gave a variational proof of this theorem using a convexity property of
the map energy functional which holds when (N, h) has nonpositive sectional curvature.