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 x x) R{" an d supK(R 2 x) ^ 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 .
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