METRICS, CONNECTIONS AND GLUING THEOREMS 21 As remarke d above , thi s las t fac t i s th e fundamenta l caus e o f th e compactness result s tha t ar e describe d below . A mor e immediat e con - sequence is: Proposition 2.2 . No V with negative second Chern class admits a co- variant derivative with anti-self dual curvature. And, if V has zero second Chern class (and is therefore isomorphic to X x C 2 ), then V 's only anti-self dual covariant derivatives have vanishing curvature. To continue th e discussio n o f compactness , conside r no w th e secon d issue, that o f the ellipticit y o f the anti-sel f dualit y condition . T o begin , you should consider the very obvious fact that a connection with anti-self dual curvature satisfies a first orde r differential equation . Tha t is , if one restricts t o som e geodesi c bal l i n X an d ther e write s V = V o + a , wit h V0 som e fixed covarian t derivative , then th e anti-sel f dua l equations fo r the su(2)-valued 1-for m a o n this bal l rea d (2.20) (Linea r combinatio n o f 1 st derivatives o f a) + a2 = smooth . Here is an obviou s ramification o f (2.20) : I f a were known to b e contin - uous, the n thi s equatio n woul d asser t tha t som e linea r combinatio n o f a's firs t derivative s i s also continuous . Supposin g tha t sai d linea r com - bination's continuity implied continuity for all first derivatives , then on e could differentiat e (2.19 ) an d rea d tha t som e linea r combinatio n o f a' s second derivative s wa s continuous . And , the n on e coul d conclud e tha t all o f a' s secon d derivative s wer e continuous , an d differentiat e agai n to ge t a' s thir d derivative s .. . etc . Thi s bootstrappin g woul d lea d t o bounds o n a' s derivative s t o an y orde r i n term s o f a boun d o n a. Now , a linear differential operato r i s called elliptic precisely when a variant of the preceeedin g bootstra p works . (I n reality , on e run o f the bootstrap - ping argumen t give s slightl y les s tha n a boun d o n a' s firs t derivatives , only a Holde r estimate . Bu t thi s i s enoug h t o ru n th e bootstra p a s outlined above. ) In th e gaug e theory case , the firs t orde r derivativ e ter m fo r th e anti - self dual equation s ca n be assume d ellipti c as long as a is not to o large . (If [V ] is close to [V 0 ], then on e ca n augmen t (2.15 ) wit h th e conditio n that d Vo * a 0, whic h i s t o sa y tha t [V ] i s i n th e imag e o f som e exponential map from TB°\{^ 0 ] int o B°. Th e pair (2.15 ) plus d Vo * a = 0 form a n ellipti c system. ) So, w e lear n tha t a boun d o n a E V - V o give s estimate s o n th e size o f a' s derivative s (t o an y order ) i n term s o f th e siz e o f a . Thus , if we have a sequenc e {V a }a=ioo o f covarian t derivative s wit h anti - sel f dual curvature , al l wit h a unifor m boun d o n th e siz e o f {a a = V a Vo} i n som e geodesi c bal l i n X, the n w e ge t unifor m bound s o n thei r
Previous Page Next Page