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
Previous Page Next Page