GAUSSIAN CURVATURE 3 Treibergs and Wei [TW, T] discuss similar problems for the mean curvature, so they are led to a quasi-linear elliptic equation. 2. Prescribin g th e curvatur e for m o n a surface . I f (M 2 , g) i s a compac t two-dimensional Riemannia n manifol d withou t boundary , the n th e Gauss - Bonnet theorem asserts that (1.4) / K g dAg=2wX(M). If one thinks of 0 = K dA as the curvature two-form, then one type of converse to Gauss-Bonne t is , given an y two-for m 0 satisfyin g j M 0 = 2w%(M% fin d a metric g so that 0 = K dA. This was solved by Wallach and Warner [WW], who gave th e followin g simpl e proof . Fi x a metri c g 0 o n M an d see k g pointwise conformal to g0, that is, (1.5) g = e 2 »g0 for some function u to be determined. Using classical formulas one finds that fl = KdA = (K 0 - A 0 w) dA0 - Q 0 -(A0w) dA Q . Thus one wants to solve the linear equation (A 0 M) dA0 = Q 0 - 0 , which we can rewrite as (1.6) & 0 u = *(0 0 -°) » where * is the Hodge operator in the g0 metric. Since (1.7) / ( 0 0 - 0 ) - 0 , by linear elliptic theory, equation (1.6) has a solution u which is unique up to an additive constant. This solves the problem. We d o no t believ e anyon e ha s solve d th e analogou s proble m o n compac t surfaces wit h boundary , o r th e mor e difficul t questio n i n highe r dimension s (except in the case of Kahler manifolds, in which case this is the Calabi problem to be discussed in Chapter 3). 3. Prescribing the Gaussian curvature on a surface. For a surface M 2 ther e are two types of questions: 1. Fin d a metri c g wit h constan t Gaussia n curvature . Here—an d i n simila r higher-dimensional problems—the point is to obtain metrics with special proper- ties which often help clarify the geometry of M. 2 (Invers e problem) . Give n a functio n K, fin d a metri c g whos e Gaussia n curvature is K. If th e manifold M is not compact, one may wish to require that the metric be complete, whil e i f M i s compac t wit h boundary , the n w e ma y impos e som e boundary conditions.
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