4
GAUSSIAN CURVATURE
In practice , on e ofte n seek s th e unknow n metri c g b y pickin g som e metri c g
0
and the n someho w deformin g i t t o th e desire d metri c g . Th e simples t typ e o f
deformation i s th e pointwis e conforma l deformatio n (1.5). Fro m a geometri c
viewpoint, thi s doe s no t giv e muc h flexibilit y sinc e ther e i s onl y on e unknow n
function u whil e o n a n /i-dimensiona l manifol d th e metri c i s locall y a n n X n
symmetric matri x an d henc e ha s n(n + l)/ 2 components . Nonetheless , thes e
deformations ar e surprisingl y useful—an d thei r geometri c interpretation i s ofte n
significant fo r geometric and physical problems.
Throughout thes e lectures we shall see several other effective way s t o deform a
metric, but muc h work remains to be done on this issue.
(a) Compact surfaces. On e aspec t o f th e tw o abov e problems , whic h ma y b e
viewed a s provin g th e existenc e o f a solutio n o f a certai n partia l differentia l
equation suc h as (1.1) or (1.6), is first findin g th e obstructions there may be to the
existence of a solution. For instance , the Gauss-Bonnet theore m (1.4) gives some
topological obstruction s t o th e existence of certain metrics . We used thi s in (1.7).
For th e "invers e problem " o n th e toru s J
2
, sinc e x(T
2)
0 ther e i s no metri c
with K 0 (or K 0) everywhere, since (1.4) implies that eithe r K = 0 or else K
changes sign . Ther e i s a simila r obviou s sig n conditio n o n K fo r othe r Eule r
characteristics.
THEOREM
1.8
(KAZDAN-WARNER) .
On a compact M2 , a function K
G
C^iM 2)
is the Gaussian curvature of some metric g if and only if K satisfies the obvious
Gauss-Bonnet sign condition.
There ar e tw o proof s o f this . T o begi n with , fo r an y metri c g ther e i s a
complicated formul a fo r th e Gaussia n curvatur e K i n term s o f th e firs t tw o
derivatives of g. We write this briefly a s
(1.9) F(g) = K.
Thus, give n K
9
(1.9) i s a quasilinea r partia l differentia l equatio n fo r g . Th e
shortest proo f [KW5 ] begins with a known metri c g0 with constant curvatur e K
Qi
so F(g0) K
0
, an d then appeals to the inverse function theore m to solve (1.9) for
K nea r K
0
i n a n appropriat e topology . Nex t on e observe s tha t i f "K nea r K
Q
"
means i n th e L
p
norm , the n give n an y functio n K with K(x
0
) K
0
fo r som e x
0
,
there i s a diffeomorphismJ of M s o that \K ° j - K
0
\ e in L
p
(th e diffeomor -
phism just spread s a neighborhood o f x0 ove r most of M). Thu s one can solve
(1.10) F(
gl
) = Kop.
But no w th e metri c g = 4*~ l(gi) satisfie s (1.9) (i.e . fro m th e viewpoin t o f M ,
both K an d K ° j are the same functions). Th e only difficulty i n carrying out thi s
program i s that th e inverse function theore m does not immediately apply in som e
standard cases , namely on S
2
an d T
2\
bu t thi s can be circumvented by additiona l
technical devices .
The secon d proo f i s geometrically richer . To solv e (1.9) one picks som e metri c
g() o n M an d seek s th e desired metri c g pointwise conformal t o g 0, s o g = e
2ug0
for som e functio n u t o b e foun d i n orde r t o satisf y (1.9), that is , F(e
2ug0)
= K.
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