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 2u g0 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 2u g0) = K.

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