I. Gaussian Curvature
1. Surfaces i n R
3.
As sometimes occurs, one of th e most interestin g unresolve d
problems involvin g Gaussia n curvatur e i s als o on e o f th e easies t t o understand .
Let z = u(x, y) b e a surface i n R 3. Its Gaussian curvature K(x, y) i s given by th e
formula
0.1)
U "U»-U»2-K(x,y).
( 1 + | V « |
2
)
Turn thi s around an d le t K be a given C
00
functio n o f x an d y. I s there a surfac e
z = u(x, y) wit h curvatur e Kl Thu s on e want s t o solv e (1.1 ) fo r w . The mos t
modest questio n i s to at leas t fin d thi s surface locally , say in som e neighborhoo d
of th e origin . I f A ' is a rea l analyti c function , the n a loca l solutio n exist s b y th e
Cauchy-Kovalevsky theorem . I f AT(0,0 ) 0 , the n (1.1 ) i s ellipti c a t th e origin ,
while if K(0,0) 0 , then it is hyperbolic there. In either case standard machiner y
allows on e t o conclud e tha t ther e i s a solutio n i n som e neighborhoo d o f th e
origin. Th e cas e AT(0,0 ) = 0 i s no t yet full y understood . I f K(x, y) ^ 0 o r
VAT(0,0) # 0 , then Li n [Li] has recently shown tha t (1.1) is locally solvable, using
the Nash-Mose r implici t functio n theorem . Nothin g mor e i s known . I t i s eve n
quite possibl e tha t ther e ar e som e function s K e C
00
fo r whic h (1.1 ) ha s n o
solution i n an y neighborhoo d o f th e origin—jus t a s fo r th e classica l Lew y
example. Equation s suc h a s (1.1) that involv e the determinant o f th e hessian o f a
function ar e usually called Monge-Ampere equations .
The proble m o f solvin g (1.1) i s closel y relate d t o th e proble m o f realizin g a n
abstract Riemannia n metri c ds
2
= Ed£
2
+ IFdi-dr} + G
di)2
o n R
2
locall y a s
the graph o f a smooth surface z = Z(JC , y) i n
R3,
so then
(1.2) ds 2 = dx 2 + dy 2 + dz 2.
Rewrite thi s a s ds 2 - dz 2 = dx 2 + dy 2. Sinc e dx 2 + dy 2 i s th e standar d fla t
metric o n R 2, on e seek s z(£ , TJ) with z(0,0 ) = 0 , Vz(0,0 ) = 0 s o tha t th e metri c
g = ds 2 dz 2 i s flat , i.e. , ha s zer o Gaussia n curvatur e i n som e neighborhood o f
the origin . Thi s lead s t o a Monge-Amper e equatio n simila r t o (1.1), with simila r
difficulty i f th e Gaussia n curvatur e o f ds
2
i s zero . I f on e ca n fin d z(£ , TJ), the n
since g is flat a n eas y geometric argumen t show s that i n geodesi c coordinates we
l
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