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
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 gdAg=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
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
o n M an d see k g pointwise
conformal to g0, that is,
(1.5) g = e
for some function u to be determined. Using classical formulas one finds that
fl = KdA = (K
- A 0w) dA0 - Q
-(A0w) dA Q.
Thus one wants to solve the linear equation (A 0M) dA0 = Q
- 0 , which we can
(1.6) & 0u = *(0
where * is the Hodge operator in the g0 metric. Since
(1.7) / ( 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
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