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.