GAUSSIAN CURVATUR E 5
This last equation i s exactly
(1.11) A 0u = 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 (1.11ha ) 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 0w = c Ke
2u
mus t satisfy :
(1.12) f X(K)e
2udA0=(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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