hypersurfaces o f the M obiu s space, w hich generalises the special case o f im m ersed surfaces in S* treated in Ref. [R i/86 ].
Let F : M * —» HI* b e an oriented W illm ore surface immersed into the Euclidean 3-space, i.e. F satisfies Eq. (2.6). L et v F : M - » IR* be the spherical Gauss m ap given b y uF ( x ) = i / „ V* e M , w here */* is the positive unit normal to F . Let
o F : h i —* 2R* be the map defined by
aF {x ) = i/F ( x ) + H F ( x ) .
Then, the follow ing theorem can be form ulated [R i/86]:
T h e o r e m ( R J g o li) L et F : M * —» JR* be a com plete, oriented im m erted W illm ore
eurfacc. I f there cxieU an a e IR* with v = < aF ,a > « • # 0 on M , then F ( M ) it either a tphere o r a plane.
This theorem is the analogue o f the weak form o f the parametric Bernstein theo rem, which states th a t a com plete, oriented, minimal immersed surface F : M * —»
IR* with spherical G auss map i/F lyin g in a hemisphere o f S * is a plane. Further
more, it was reform ulated in the conform al geom etry o f surfaces o f S* by the same author:
Consider the im m ersion / = * o F : M * —» S* in to the M obius space, where » : IR* —» 5 *\ {X oo} is the diffeom orphism as defined in diagram (1.91). Let
E — [E o ,E i,E t , E t , E+\ : A / —* G* be a D arbou x frame along F o f the type de
scribed in Remark 1.4. Then, using the identification (1.93), we can consider E 0 and E t as vectors o f IR*, being E 0 = F and E t the positive unit normal to F . Then, in the latter fram e, we can w rite <rF( x ) — (E t + H E 0) ( x ) . Let e : M —* G b e the second-order fram e constructed from E as described in Sec. 1.3. Thus,
¿t = E t + H Eq. T h a t is, o F corresponds to the hyperbolic Gauss m ap 7 / o f / .
T h e following th eorem is the conform al version o f the previous one:
T h e o r e m ( R i g o i l ) L et f : M —* S* be a compact, connected, oriented W illmore
tu r f ace with hyperbolic conformal G a tin map 7 /. I f there ex iett an a G JR*, inch that < 7 /, a > j t 0 o n M , then / ( M ) i t a 2 ■ tphere.
Chapter I, Section 4. A Conformal Bernstein-type Theorem 162
N ow we derive a generalisation o f th is theorem . Let / : A / " -1 —» S n be an im m er sion o f a hypersurface into the M öb iu s space, and let -y : M —* Q be the hyperbolic con form a l Gauss m ap o f / defined in E q . (2.65). O bserve that, if M is the M öbius
/ is a trivial W illm ore hypersurface and 7 / = q*. In particular, < 'jj, if* 0 on all M . The following theorem show s that this prop erty (with an additional con d ition ) characterises the hyperspheres o f S n.
T h e o r e m 2 .1 Suppose n 4 and » / 6. Let f : M n~ l —» S n be a compact,
orien ted , connected Willmore hypersurface im m erted in to S n with hyperbolic co n fo r m a l Gauss map 7 /. I f there exists an a £ JRn+i, such that < T/,a 0 on all
M , an d i f f satisfies the condition (£ .7 2 ), then f ( M ) is an (n - 1 )• sphere. P ro o f. Set m = n — 1. O bviously, w ithout loss o f generality, we m ay assume
(7 /, a ) > 0 on all M . Let e : M —* G b e a second-order G -fram e along / and let ||h|| b e as in E q. (2 .46 ), relative t o th is frame. Consider the local (m — l)-fo r m on M given by
“ = ( - 1 ) ' ‘ ll * ir ~ ’ ( ( m - l ) p ? < e ,,a > - * , " » < c , a >
O ne can straightforwardly verify, u sing the transform ation laws for second-order fram es given in Sec. 1.2.C, that u. is a well-defined glob al (m — l)-fo r m on A f. Using Eqs. ( l .« 0 ) , (1 .62 ), (1.67), (2 .4 7 ), and (2.48), we have
du. = ( - l ) ' - ‘ (m - 1 ) , ; < e « ,a > d(||A||— *) A 4 , J ~ m + ( ~ 1)' '( m — l)||fh||"*“ * < eo,a > dp" A d 1" 1-"*
+ ( - l ) i-'(m - 1)|*N— V d (< «o,« >) Ad1“
+ ( - > ) - ' ( • » - 1)11*11"-’ < « . , . > pTddt J - " + ( - l ) ' A ’, < « » , « > d ( | * r ' , ) A d , - ; - “ + ( - 1 ) , U * | - ’ < « » , . > d * r . A d 1- ' ” + ( - l ) ' l l * l l ”- ’ *.,i ^ ( < . . , « > ) A d 1 " - ' "
+ ( - l ) ' I M - ’ <«»,•> *radd‘- ;-”
sp ace S m 1 and / is the inclusion m a p given by / , then
Ckmpter