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 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

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 0w) dA0 - Q

0

-(A0w) dA Q.

Thus one wants to solve the linear equation (A 0M) dA0 = Q

0

- 0 , which we can

rewrite as

(1.6) & 0u = *(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.