(20) < x,, N, >" (0) = -r/0o(e, , e2) + 271 leo(e, , e2)] - 4H42 2 + 27/422T+. +r. - 2Hq) + .... Penn, G., Simon, U.: Deformations of hypersurfaces in equiaffine differen-.
manuscrlpta math.
75, 3 5 -
41
manuscripta mathematica
(1992)
9 Sprmger-Verlag 1992
AFFINE AND
ISOPERIMETRIC SURFACES
AFFINE
WITH
MEAN
A. Martlnez
PROBLEMS CONSTANT
CURVATURE F. Milgn 1
The first affine isoperimetric property for a locally strongly convex surface with nonzero constant affine mean curvature H (briefly, CH-surface) in R a was obtained by Blaschke in 1916. He proved the following result:
Among all ovaloids with constant volume the ellipsoid, and only the ellipsoid, attains the greastest affine-area. In the present note we prove that the condition of nonzero affine mean curvature on a locally strongly convex surface is equivalent to the fact that the surface is a critical point of an "isoperimetric problem ". Actually, a locally strongly convex surface has stationary aNne-area under all deformations in the affine normal direction that leave constant the volume if and only if it is a CH-surface (see Lemma
3). We also prove, by using the same idea as in [3], the following result: M a i n T h e o r e m The second variation of the affine-area of a CH-surface with H < 0
under all deformations in the affine normal direction that leave constant the volume is negative definite.
1
A f f i n e variational problems for C H - s u r f a c e s
Let x : M ---* R 3 be a locally strongly convex affine surface in the affine real 3-space provided with a volume element given by the usual determinant function, Det, and the fiat connection, D. Then there exist a uniquc choice of transversal vector field such that (1) D x x . ( Y ) = x. ( V x Y ) + h(X,Y)~, (2)
Dx{ = - x . ( S X ) , 1Research partially supported by DGICYT Grant PB90-0014-Co3-02
35
MARTINEZ -MILAN
9(X,Y) = Det(x.(X),x.(Y),~) = Ch(X,X)h(Y,Y)
(3)
- h ( X , Y ) 2,
for any tangent vector fields X, Y. We call ~ the affine normal, V the induced affine connection, h the Blaschke metric, S the affine shape operator and 8 the affme volume element. The affine mean curvature and the affine Gauss-Kronecker curvature of the immersion are H = 89 and T = det S, respectively. As S is diagonalizahle, we can take {el, e2} a local orthonormal frame on an open dense subset of M such that
(4) h ( o , el) : h(e2, e2) : l, h(e b e2) = 0, S e a : (H + o~)el, Se2 : (U - a)e2. Moreover if we write
(5)
VOle ~ : pr
~Te2e2 = qel,
and (6)
/ / ( e l , el) ~---a e l -~ be2
where ~ is the Levi-Civita connection of h and K = ~ - ~ , then the fundamental equations of the immersion are given by, (see [2], [4]) H + 2(a 2 + b2)
(7)
=
el(q) + c2(p)
3(qb- pa) - ~ + 3(@ + q~) 2~(p - ~)
el(b) - e2(a)
=
~,(.) + ~(b)
=
e2(H + ~)
=
e,(a-H)
---- 2 a ( a + q ) .
--
p2
_
q2
We shall denote by {wl,w2,w} the dual local frame of {x.(el), x.(c2),(}, then (8)
w = < N,
>,
where N is the affine conormal vector field and < > is the canonical scalar product in R 3. And if we take 77 = < x, N > the affine distance function then we h ave (9) x=-VT/+~(, Ay=-2-2H~, where by A and ~7 we denote the Laplacian and the gradient with respect to the Blaschke metric h, respectively. Let Ft C M be a relatively compact domain of M. Then (10)
An(x) : f~ 8
is the affine-area of x(fl), that is, the area of x(~) in the metric h and we shall call (lI)
V~(z) =
~fa
O
the volume of x(fl). (When fl is small ]V~(x)] is the volume of the cone over x ( a ) with vertex at the origin of R3). D e f i n i t i o n A deformation xt : ~t ~ the a ~ n e normal direction if
R 3, t 6 ( - e , e), is called a deformation in
x~ = x + r
36
MARTINEZ-MILAN
where r = 0 and Ct, t G ( - e , e), are smooth functions with compact support in f~. If Va(xt) = Vn(x), 'v't E (-e,e), we call xt a volume-preserving deformation in the affine normal direction.
Remark. Another kind of deformations of a locally strongly convex surface called normal deformations have been investigated in [3] and [6]. Let xt : Ft ----. R a, xt = x + Ct{, be a deformation in the affine normal direction. We set An(t) = An(x~), Vn(t) = Vn(xt) and denote by {t, Nt and h, the affine normal, the affine conormal vector field and the Blaschke metric of the immersion xt, respectively, then, (12)
0t(el,e2)g t = xt.e 1 A xt.e2,
where 0t is the volume element of ht, and
h~(ei,ej) = < N,,D~,x,.ej > .
(13)
Moreover, the first variation of the affine-area (see [3], [7] ) is given by 2A~(0) + 3
(14) where r = ~r
Lemma 1
= r
fn Hr
= 0,
(From now on we shall denote derivative in t by prime).
V~(O) = fn ~bO.
Proof. Using (1), (2) and (3), Ot(ea, e2) = Det(el + e~(r thus (15)
- CtSe~, e2 + e2(r
O0(ea,e=) = - 2 H r
- CtSe2, ~,),
+ < N,{ o >,
and from (11) and (15) we have 3V~(0)=f~{r
0>-2HV+}0.
As < N,{~ > + < N~,{ > = 0 and ei(r expression and (9) give v~(0) =
1
=-
< N~,el >,
i = 1,2, then the last
/ n { r + h(~Tr/, V~b) - 2HTl~b}O.
The Lemma follows by using again (9) and Green's Theorem. We consider the functional Jn : ( - e , e) ~ (16)
R given by
J~(t) = 2An(t) + 3HVa(t) = / a { 2 + H < xt, Nt > }Ot~
where H = A--~ 1 f. HO. From Lemma 1, (14), (16) and using the same method as in [1], we can prove the following results:
37
MARTINEZ-MILAN
L e m m a 2 Let 42 : M --~ R be a smooth function with compact support in ~ such that fn 420 = O. Then there exists a volume-preserving deformation in the a ~ n e normal direction xt : ll ~ R 3, xt = x + r with r = 42.
L e m m a ~ The following statements are equivalent: 1. x is a CH-surface. 2. For each relatively compact domain f~ C M and each volume-preserving deformation in the affine normal direction, A'n(O ) = O. 3. For each relatively compact domain f~ C M and each deformation in the affine normal direction, J~(O) = O.
Next we shall prove that the second variation on Ja depends only of 42. P r o p o s i t i o n 1 If x is a CH-surface then (17)
d~(0)
-
-3
8 f• {(A42)2 + 20H2422 - 4Hh(V42, V42)-
4h(SV42, V42) -8T42 2} 0.
Proof. Using (16) we have
(18)
Jg(0)
=
~ {H < x,, Art >" (0) + 2H < x,, N, >' (0)0;(el, e2)+
+
(2 + Hr/) 0o(el, e2)} O.
Now by differentiating (12) and evaluating for t = 0, we have, (19)
< xt, Nt >' (0) = -rlao(el , e=) + 42 + h(Vr/, V42) - 2Hr/42,
II I (20) < x,, N, >" (0) = -r/0o(e, , e2) + 271 leo(e, , e2)] ] - 4H42 2 + 27/422T+ +r - 2Hq) + h(Vr/, Vr ) + 8o(el, e2)[4H~42 - 242 - 2h(Vr/, V42)] + +2Dct(Vr/, e1(42)~, 42Se2) q- 2Det(Vrh 42Sel, e2( 42)~ ),
and using Green's Theorem, (9), (18), (19) and (20) we get (21)
J~(0)
=
fn{3Hr
-
4H2422 + 2HDet(Vrl, 42Se,, e=(42)~) + 2O0'(el, e2) } O.
e,(42){,42Se2)-
Now, we can compute 8~(el,e=) and 8~(e~, e2) in two different ways. In fact, from the first equality of (3) we obtain (15) and (22)
ffo'(e,, e2) = -2HCg + 2422T + 2Dct(e,(42)r e 2 , ~ ) + +2Det(el, e 2(42)~,~o)-4H42 ' " ~ > - 2 < N~,~o ' ' >. < N , ~ > - < Na,
Moreover, from (13) we also have (23)
h{~(el,ej) = - (V~,ej)(42)+ h(ei, e~) < N~,~ > +ciej(42) - 42h(ei, Sej),
38
MART INEZ-MILAN
(24)
< N,~; > (V,,ei)(r
hg(ei, ei) = -(V,,ei)(r
(SV,,ei)(r
+ < N~',~ > -2eie,(r < N,~ o > +2(Se,)(r 2) + (V,,Se,) (r +2r Sei) < N, ~o > +eie,(r - r Sei).
+
Thus, a straightforward computation with the second equality of (3), (23) and (24) also give us (25) 0o(e,, e2) = *-Ar + < No, ~ > - H e , 1
(26)
0~(e,,e2)
=
-4
1 (Ar
2+
