Section S Mala Results

condition sup/? < + o o can be dropp ed. M oreover, from the p r o o f o f T h . 1, «||ff||* € L * (M ,g ) is clearly satisfied, i f ||lf|| is bounded. T h is guarantees the convergence o f the integral JM div*. (W )d V k» . Finally, one m ay su b stitu te this con­ dition by ||7 /1|* € L * (M ,p ) and work o u t a reasoning similar t o the on e presented, using now <R-Vi>(Z )dV t . The same remarks apply to the next results.

In what follow s, conditions (A ), (i), (ii), (iii) always refer to the o n e s given in T h . 1. A s expected , the case m = 2 is special.

P r o p o s it i o n 3 L et (M ,g ) be an oriented complete surface, U C ( N , h ) an open

te t supporting a strongly conform al (i.e . a = fi ) sector field X , and f : M - * U an immersion satisfying (A ) with u € L * (A f,if). Then, f is conform al, if f (i) holds with m = 2.

P roof. A simple m odification o f the previous p roof gives the sufficient part. In

fa ct, in this case we im m ediately have from Eq. (5) 2 « < ||d/||*, o b ta in in g Eq. (9) as well. A s 2fiu — = or(2u — Hd/’HJ) = 0, we conclude from E q . (6 ) that / is conformal w ith h * = ug.

N ow we prove necessity. Given a conform al im mersion / : (M , g ) —* (JV, h ), the following form ula is well-known (E o-O s/82 ]:

m B = i f , + log y ? ) , (11)

where m = d im M , h * = a g , and V , is the gradient w .r.t. g. So, i f M is a surface and / is conform al, then u — a and, from Eq. (1 1 ), Tj — 2 uH = 7 / — 7 / = 0, which proves necessity o f (i).

Theorem 1 and its proof, together with P rop. 3, give:

T h e o r e m 2 L et (A /, g) be an m -dim ensional, oriented, complete R iem a n n ia n man­

ifold, U C (N , h ) be an open set supporting a strongly conformal s e e to r field s X , and f : M -* U be an im m ersion satisfying (A ) ( with u€ L * (A /, g ) , i f m = 2 ).

Then, f is an isom etry, iff

(») <7) - m u H, X , ) k £ 0 and

Section ». Main Results 71

(»»*) u > 1 , that it, f it tolum e increating fo r tn = 1, (•"») * = 1 , that it , f it volume preterving fo r tn = 2.

P roof. N ecessity is obvious. A s fo r sufficiency, since Ar is conform al, 0 = 1 and,

for m > 3, form u la (10) gives h* = g, i.e. / is an isometry. The other cases are analogous.

R e m a r k T h eorem 2 was proved in Ref. [R i/87 ] under the assumptions ( N ,h ) = (IRn, < , > ) , X is the position v ecto r field, and M is com pact.

Consider now the case where a strongly alm ost conform al vector field X has the additional p r op erty in fo = v > 0. Set /t = sup/? and $ = £ > 1, which is a constant. R ep la cing 0 by 0, a by i/, and /? by p in T h . 1, thus obtaining the cor­ responding con d ition s (»'), (*'»'), ( m ) , we can formulate the following strengthened theorem :

T h e o r e m 3 L e t (M , g) be an tn-dim entional, with tn £ 2, oriented, complete

Riem annian manifold, U C ( N ,h ) be an open te t supporting a itronglg almost conformal v ec to r field X with the property in f a > 0, and f : M - * U be an im m er­

sion satisfying cond ition (A ). I f (*) and (»») or (H i) hold, then f it an isom etry

and X i t h om oth etic.

Proof. T h e p r o o f o f T h . 1 goes through till Eq. (1 0 ), which now becom es

A* - * r f e , , (12) whence / is a hom othety. So in this case u = i A . Com puting the tension field

T f, using E q . (1 1 ), we obtain

<r, - m . H , X , ) „ = *E?=r(l - #'-■) { m H ,X ,) k . (M ) Com bined w ith E q . (7) this gives

<T> - m . H , X , ) „ = # r t ( l _ # -* ){ d iv ,.(I V ) - • } , (14) where $ > m v > 0. Condition (i) and once more the Gaffney-Yau Stokes’ theorem yield

Section

S.

Raults

as 0 > 1. Consequently, 0 = 1, th a t is, X is hom othetic and, from E q. (12), / is an isometry.

N ext we give an application o f Theorem 2.

P r o p o s i t i o n 4 Let i : (N ', h ') —♦ ( N , h ) 6« an isom etric im m ersion o f an oriented

m anifold N 'f with dimN' = m and i( N ') C U an open te t in N tupporting a conform al vector field X and having the property a > 0 on U . L et ( M ,g ) be an m - dimensional, compaet, oriented Riem annian manifold and F : (Af , g ) —» (N \ h ' ) be an orientation-preserving harm onic diffeom orphitm with ratio o f the volume elem ents u. L et V d i be the secon d fundamental ten sor o f i : N ' —* N and H its m ean-curvature vector field. T hen, F is an isom etry, iff

I t ) ( l r a t,,V <IH dF ,dF ) - m , H , X , . , ) k > 0 and (2 ) F is volume decreasing for m > 3,

(S) F is volume preserving fo r rn = 2, (4 ) F is volume increasing for m = 1.

P roof. Let / = i o F . Since i is an isom etric im m ersion and dimJV' = dim A f, a

standard com position form ula o f Eells-Sampson [E e-Sa/64] gives

T, = TF + tra ce , V«h (dF , JF) and U , - H ,

where H j is the mean-curvature v e cto r w ith respect to / . M oreover, F is harmonic and the ratio « / o f volum e elem ents w .r.t. / satisfies « / = « , which yields

(T/ - m u H f , X f ) k = ^ tra ce,V d i(d F , d F ) - m u H , X ior ) k •

Since / is an isometry, iff F is so, th e result follows im m ediately from T h . 2. 9

R e m a r k P roposition 4 generalises the main result o f Hsiung and R hodes (Hs- R h /0 8 ] (and, earlier, o f Chern and Hsiung (C h-H s/63]), which in our form ulation can be stated in the form:

L et F : (M ,g ) -* (AT'.fi') be a harm onic, volume-preserving diffeomorphism. Let

x : (M , g ) —» ( N ,h ) and i : (A T',h') - * ( N ,h ) be isometric immersions o f com ­

pact submanifolds into the Riem annian manifold ( N , h ) which adm its a strongly conform al v ector field X . I f

(7>

i

Reference! P u l U 73

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Part III

A V A R IA T IO N A L P R O B L E M