GAUSSIAN CURVATUR E 5 This last equation i s exactly (1.11) A 0 u = K 0 -Ke2\ where A 0 an d K 0 ar e th e Laplacia n an d Gaussia n curvatur e o f g 0 . I f on e wishes , one ca n ignor e th e geometric origins of thi s equation an d as k for whic h function s KQ an d K ca n on e solv e i t o n an y compac t manifol d M" o f an y dimension . I t turns ou t tha t th e theor y o f thi s equatio n i s surprisingl y intricate—wit h man y cases stil l no t understood . W e shal l onl y summariz e briefl y (se e [KWl ] fo r details). The simples t cas e i s i f bot h K 0 0 an d K 0 the n (1.11 ) ha s a uniqu e solution on M n fo r an y n. This is easily proved using sub and super solutions. If K Q = 0 , the n ther e ar e tw o necessar y condition s o n K ( ^ 0 ) fo r th e solvability: (i) K changes sign, (ii)/ KdA o 0. If di m M 2 , thes e condition s ar e als o sufficient . Th e existenc e proo f use s th e calculus o f variation s an d break s dow n i f di m M 3 , wher e n o sufficien t conditions ar e known , despit e th e fac t tha t th e equatio n A 0 w = Kelu look s s o simple. Finally, i f K Q 0, eve n les s i s known . Onl y th e cas e wher e di m M = 1 i s clearly understoo d (the n on e ca n solv e (1.11 ) i f an d onl y i f K i s positiv e somewhere). If dim M = 2 , then one can solve (1.10) if K0 i s sufficiently small—b y the calculu s o f variations . Onl y fo r th e specia l cas e o f M = S 2 o r P 2 wit h thei r canonical metric s i s a littl e mor e known . Sa y K 0 = c 0 i s a constant . The n Moser [M ] use d th e calculu s o f variation s t o prov e tha t o n S 2 (resp . P 2 ) i f 0 c 1 (resp . 0 c 2), the n ther e i s a solutio n i f an d onl y i f K i s positiv e somewhere. Sinc e K 0 = c = 1 o n S 2 an d P 2 wit h thei r standar d metrics , thi s proves Theore m 1. 8 o n P 2 , bu t jus t misse s o n S 2 . Indeed , i f c 1 , ther e i s a nonexistence resul t o n S 2 whic h i s implie d b y th e followin g identit y tha t ever y solution of A 0 w = c Ke 2u mus t satisfy : (1.12) f X(K)e 2u dA0=(c-l)f (div X)Ke2udA0, where X i s i n th e six-dimensiona l spac e o f conforma l vecto r field s (i.e . th e Li e algebra o f th e group of conformal maps ) on S 2 . Th e special case where X i s in th e three-dimensional spac e o f gradient s o f first-orde r spherica l harmonic s wa s proved b y Kazda n an d Warne r [KWl] . Later , Bourguigno n an d Ezi n [BE ] observed tha t on e shoul d ad d th e othe r conforma l vecto r fields , i.e. , th e three - dimensional orthogona l grou p 0(3) . I n particular , i f c = 1 an d K = \p i s a first-order spherica l harmoni c (plu s a constan t i f on e wishes) , the n (1.12 ) i s no t satisfied wit h X = v ^ becaus e the n X(K) = | V^|2 s o th e lef t sid e i s positiv e while the right side is zero. It i s not know n i f th e existence of a function u satisfying (1.12 ) is sufficient fo r the solvabilit y o f (1.11 ) on S 2 . I n fact , thi s is not eve n know n i n th e specia l cas e where K is rotationally symmetri c and one seeks a rotationally symmetric solutio n
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