2 HENRY C. WENTE

where (assuming that c o £ 0) a(u), /?(u) will be solutions to a

v

second order system of differential equations from which the

solution co(u,v) is to be recovered. The system is

a" = aot - 2a 2 /? - 2A/?

( 1 . 4 )

2

/?" = a/? - 2a/? - 2Ba a = c o n s t a n t .

Furthermore one finds

, . . , , 2

, .

A

2 , 2 CO . . „ , 2 .

-200

_ . 00 ^ . -00

( 1 . 5 ) 4oo = - (4A + a )e - (4B + /? )e - 4 a ' e + 4/?'e + 6 ^

wher e 6y = 6a/? - 4 a .

If one solves the system (1.4) then (1.3) and (1.5) may be

used to recover oo(u,v). This development is carried out in

Section II. The system (1.4) is an algebraic completely

integrable Hamiltonian system. To solve it we follow the method

described in Darboux [4] which is to solve the relevant

Hamilton-Jacobi equation by the method of separation of variables.

A general feature of any solution oo(u,v) obtained in this manner

is that it will be periodic in the v-direction (with infinite

period allowed) and quasi-periodic in the u-direction.

In section III we study the case with mean curvature H = 1/2

3

in R . Here the Gauss Equation (1.2) may be written in the form

(1.6) Aoo + sinhoo coshoo =0, A = B = 1/4.

This is the P.D.E. which initially attracted my attention when

3

constructing immersed cmc tori in R [13]. One finds here that if

an immersion of Enneper type is defined locally then it extends to

3

a global mapping of the plane into R . In particular the

immersion will not develop any umbilic points. One can explicitly

compute the centers and radii of the spheres determined by the

lines of curvature from the solutions to the system (1.4). It

turns out that any immersion has an axis I on which the centers

of all spheres lie. This fact enables one to give fairly explicit

formulae for the immersion itself.

All solutions to the elliptic sinh-Gordon equation (1.6)

which are doubly periodic have been classified recently in a