2 GAUSSIAN CURVATURE
can writ e g = dx 2 + dy 2. I n thes e ne w coordinate s z(x, y) satisfie s (1.2). I t i s
likely that a good understanding of (1.1) will at the same time resolve the question
of whether one can always locally isometrically embed a 2-manifold i n R 3.
Once one understands th e local solvability of (1.1), it is natural t o seek a global
solution i n a bounded ope n set in R 2, possibly with boundary conditions. We will
discuss thi s late r i n Chapte r 4 unde r th e specia l assumptio n tha t K 0, s o on e
can us e technique s fo r ellipti c equations . Ther e seem s t o b e littl e informatio n i f
K 0 (see [Az]).
In th e earl y year s o f thi s century , Minkowsk i an d Wey l pose d th e followin g
problems:
Minkowski problem. Prov e th e existenc e o f a compac t conve x surfac e i n R
3
whose Gaussia n curvatur e i s prescribe d a s a give n positiv e functio n o f th e uni t
normal.
Weyl problem. Le t g be a given metric on th e sphere S 2 wit h positive Gaussia n
curvature K. Prov e th e existence of a compact conve x surfac e i n R 3 with g as it s
metric, i.e., isometrically embed (S
2
, g) i n
R3
assuming K 0.
Both o f thes e problems lea d t o nonlinea r ellipti c partial differentia l equations .
They wer e eventually solved by Nirenberg [N] and Pogorelo v [PI]. More recently,
a higher-dimensiona l versio n of th e Minkowski problem has also been solve d (se e
[P2]), bu t ther e i s no informatio n o n an y suc h versio n o f th e Weyl problem ; fo r
instance, if (S n, g) ha s positive sectional curvature, can it be embedded i n R^ fo r
some "good" N dependin g on nl
A relate d questio n wa s aske d b y Olike r [O] . Le t 2 " «- » R,l+1 b e a smoot h
compact grap h ove r th e standar d uni t spher e S" *- R" + 1, so i f x e S n the n th e
points y o n 2 " ca n b e writte n a s a radia l grap h y = u{x)x, wher e u i s som e
positive functio n define d o n S" (an d ca n b e extende d t o R" +1- {0 } t o b e
constant alon g radii) . Th e Gauss-Kronecke r curvatur e K o f 2 " i s give n b y a
Monge-Ampere equatio n involvin g th e determinan t o f th e hessia n o f w on S" .
Conversely, give n a positive function A^(|x| ) on R
w + 1
{0} , Oliker aske d if ther e
is a closed conve x surface 2
M
«-»
Rn+1
ove r S
n
whos e Gauss-Kronecker curvatur e
is K. Combinin g Oliker' s [O ] wor k wit h a significan t improvemen t b y Delano e
[Del], th e resul t i s tha t ther e i s a t leas t on e suc h hypersurfac e i f K satisfie s th e
additional assumptio n tha t ther e are numbers 0 R
x
R
2
suc h that fo r al l x o n
S"
(1.3) inf K{R xx) R{" an d supK(R 2x) ^ R^".
Moreover, th e proo f give s a hypersurfac e 2" i n th e shel l betwee n th e sphere s of
radii R
x
an d R
2
. I f assumptio n (1.3) i s satisfied fo r som e othe r number s
R2 R{ R
2
, the n o f cours e on e find s anothe r solution , s o th e solutio n i s no t
unique. Oliker gives some additional assumption guaranteeing the uniqueness—u p
to homothety—o f th e hypersurface. Oliker' s existenc e proof o f a solutio n o f th e
Monge-Ampere equatio n use s th e continuit y method , whil e Delanoe' s use s th e
Schauder fixed poin t theorem .
Previous Page Next Page