124 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONSis a tensor with components R~, antisymmetric in (i, j) and also in (k, l). Thens = l:i,j Rg. For example, the scalar curvature of the round metric on 84 is 12.)
To prove this lemma we need the following background:
Exercise 3.5.- Prove the Bochner-Weitzenbock formula:
D~DA = \7~ \7 A+~+~ cl(FA)
as operators on C^00 ( S+ EBB-). (The curvature terms arise from commutators
of covariant derivatives as in Exercise 1.13.)- Show that if 'if; is a section of S+ then
(3.4) D~DA'ifJ = ( \7~ \7 A+~+~ cl+(F,t)) 'if;.Remark 3.6. It turns out that D~ =DA. So the Bochner-Weitzenbock formula
says that D~ equals the "covariant Laplacian" \7~ \7 A, modulo lower order terms.
This is basically the property that Dirac was looking for when he defined the Dirac
operator, and this requirement naturally leads to the notion of Clifford multiplica-
tion. (See [3].)
Proof of Lemma 3.4. Choose a point in X at which l'i/Jl^2 is maximized. Then
6l'i/Jl^2 ~ 0
at this point. (Here 6 = dd*.) Compatibility of A with the metric implies6l'i/Jl^2 = 2 Re(\7~ \7 A'i/J, 'if;) - 21\7 A'i/Jl^2
:::; 2Re(\7~ 'VA'ifJ,'if;).
Plugging both Seiberg-Witten equations into the Bochner-Weitzenbock formula
(3.4) gives
t s 10 = \7 A \7 A'i/J + 4'i/J + 2 cl+(q('if;) + iμ)'if;.
Putting this into the inequalities above and using the definition of q we get0:::; 2Re\ ( ~s - ~cl+(cli('i/; @'i/;t) +iμ)) 'i/;,'i/;).
Now cl+ : A~ ----; End(S+) is injective and multiplies lengths by 2, and its imageconsists of the traceless endomorphisms. It follows that
cl+(cli('i/;®'i/;t)) =2 ('i/;®'i/;t-~l'i/J12).
Putting this into the inequality above we get0 :::; ( ~s - l'i/Jl2
+ 21μ1) l'i/Jl2
at a point where l'i/Jl^2 is maximized. This implies the a priori estimate (3.3). D
Starting from this, the idea of the proof of compactness is to use "bootstrap-
ping" to get bounds on all the higher derivatives of A and 'if;. Schematically, the
Seiberg-Witten equations are of the form
(3.5) 1J ( ~) = (1~) +smooth