A proof of a general isoperimetric inequality for surfaces

Math. Z. 162, 245-261 (1978)

Mathematische

Zeitschrift

9 by Springer-Verlag 1978

A Proof of a General Isoperimetric Inequality

for Surfaces

Jogo Lucas Barbosa 1 and Manfredo do Carmo 2 1 Universidade Federal Ceara, Campus do Pici, Fortaleza-Ce, Brasil 2 Instituto de Matematica Pura e Applicada,

Rua Luiz de Camges 68, Rio de Janeiro- R J, Brasil

I. Introduction

(1.1) Let M be a two-dimensional C2-manifold endowed with a C2-Riemannian metric. We say that M is a generalized surface if the metric in M is allowed to degenerate at isolated points; such points are called singularities of the metric.

In this paper we use the method of Fiala-Bol (cf. [-,12, 91) to give a proof of the following general isoperimetric inequality.

(1.2) Theorem. Let M be a generalized surface. Let D be a simply connected domain in M with area A and bounded by a closed piecewise Cl-curve F with length L. Let K be the Gaussian curvature and K o be an arbitrary real number. Assume that in a neighborhood of a singular'point, K is bounded above. Then

( 1 -K~ ]

(1.3) LZ>4rcA 1 - ~ ~ ( K - K o ) + d M equality holds iff K - g o in D '

D

and D is a geodesic disk.

Below we present a brief sketch of the curious history of (1.3).

For K - K o =0, (1.3) is the classical isoperimetric inequality in the plane. For K = K o , (1.3) was proved by Bernstein [-8] (for K o > 0 ) and Schmidt [19"1 (for K o < 0 ). The case K__<0, K o = 0 was proved by Carleman [-11"1 (for minimal surfaces) and by Beckenbach and Rad6 [-7] (the general case).

For K > 0 and K o = 0 , (1.3) was proved by Fiala [12] in the analytic case. By using the same method, the case K < K 0 was proved by Bol [-9"1 in the C 2- setting. Apparently unaware of Bol's work, Alexandroff [1] gave a proof of (1.3) for K < K o under the assumption that any two points of D can be joined by a inique minimal geodesic. This assumption was later show to be superflous by Toponogov [20].

The result of Fiala was extended by Huber [14] who proved (1.3) for arbitrary K and Ko = 0.

At this point, we must mention the work of A.D. Alexandroff and his school. The basic notion is that of a "manifold (surface) of bounded curvature" (cf.

246 J i . Barbosa and M. do Carmo Alexandroff [23 and Alexandroff and Zalgaller [7]). By a theorem of Rest- cheniak [18] (see also [2] pp. 503-505 and Huber [15]) this is equivalent to an isothermal representation of the metric in the form

ds2=e(Z)ldzl 2,

z=u+iv,

where

u(z)

is given by

u(z)

= h ( z ) - l ~ l o g ( z - ~) co;

here

h(z)

is a harmonic function and co is a signed measure. (By using the Jordan decomposition c o = c o + - c o - one sees that

u(z)

is the difference of two sub- harmonic functions.) In this context, the curvature of a set A ~ D is given by co(A). Given a real number Ko, we set

+ = sup (co(A)-KoA

).

COKo

A ~ D

It is easily seen that if the metric is of class C 2 then

+

co~o= ~ ( K - K o ) +

din.

D

A special case of such metrics, called

polyhedral metrics,

occurs when

1

u ( z ) = - - ~ cojlog(z-zj),

7~'j= 1

where z 1 ... z m are m points (vertices) in D and the number coj, j = 1 ... m, is the curvature at the vertex z j; thus polyhedral metrics have their curvatures con- centrated at the vertices.

It can be proved (cf. [5]) that manifolds of bounded curvature can be approximated, in a certain sense, by polyhedral metrics, and this gives a powerful method for proving geometric inequalities. The case of equality usually needs a special treatment.

For manifolds of bounded curvature, (1.3) takes the following form:

(1.3)'

L z > 4 ~ A ( 1 - 1 ~ ( K - K o ) + d M - ~ A )

9

equality holds iff D is a circle

= \ 2~D . _ /

for which the specific curvature of any region not containing the center is K o and the curvature of the center is

~(K-Ko) + dM.

D

For K o = 0 , (1.3)' has been proved by Alexandroff and Streltsov ([3,4]). rc

Inequality for Surfaces 247 S (K - K0)+< 2 ~, equality holds for the situation described in (1.3)'. The restriction D

on the inner radius was later shown to be superfluous by Burago [10]; no mention of the equality case is made in [10]. Finally, for K o > 0 , (1.3)' was proved by Bandle [6]. In contrast to the previous papers on (1.3)', which use the method of approximating by polyhedral metrics, Bandle's paper uses the poten- tial-theoretic methods introduced in Huber [143 and Restcheniak [18].

A number of generalizations of (1.3) have appeared in print. For such generalizations and further details on the history of isoperimetric inequalities we refer to the fairly complete account of Osserman [17].

We give a proof of (1.3) making no use of Alexandroffs methods. We rather use the method of interior parallel curves introduced in [12] and [9]. Further- more, the equality case of (1.3) is not a particular case of (1.3)'. The main ideas of our proof are as follows.

We first assume that everything involved is analytic. The proof depends on the properties of the "interior cut locus" of F, developed by Fiala in [12]. For completeness we describe such properties in the beginning of Section 2. The main fact is that the set of points F~ of/} that are at a constant distance r from F (for technical reasons, we take r to be negative) consists of a union of a finite number of closed piecewise analytic curves. The only delicate point in the proof of (1.3) is to choose carefully a function

f(r)

that depends on the geometry of F~ and is such that f ( 0 ) > 0 is equivalent to (1.3). The proof is completed by showing that f is piecewise differentiable, f ' ( r ) > 0 whenever it exists, and there exists a number t o < 0 such that lim

f(r)>O.

r ~ r o

To pass from the analytic case to the C2-case, it is fairly easy if we are only interested in proving the inequality. For the case of equality, we use an approximation theorem of Whitney (cf. Sect. 3) and then go back to the derivative of the function f to prove that equality in (1.3) implies that K = K 0. We then apply the classical isoperimetric inequality for surfaces of constant curvature ((1.3) for K-= Ko) to complete the proof.

We could free our proof from the dependence on the classical case of constant curvature by using a symmetrization process (since we already have the inequality). However, we will not consider this question.

2. Proof of Theorem (1.3); the Analytic Case

(2.1) Throughout this section M will be an analytic manifold of dimension two endowed with an analytic Riemannian metric and

D = M

will be a simply- connected domain with compact c l o s u r e / ) whose boundary is a closed analytic curve F with length L. We assume that F is parametrized by the arc length 0,

O<_O<L.

Through each point

p~F

there exists a unique unit speed geodesic 7p perpendicular to F. We denote by r the arc length of 7p and assume that r = 0 on F and r < 0 on the piece of yp that goes initially towards D.

248 J.L. Barbosa and M. do C a r m o

On each geodesic 7p ( r < 0 ) there exists a point q (to be called a cut point relative to F) where 7p stops minimizing the distance from yp(r) to F. By letting p vary in F, the cut point q describes a connected set C c D which we call the interior cut locus of F. It is not difficult to prove that each point q~C is either a focal point of F or there are at least two minimizing geodesics (perpendicular to F) that meet at q. If there are exactly two such geodesics and q is not a focal point of F, q is called a normal point of C.

There are only finitely m a n y points in C that fail to be normal ([12], 8.13 and 8.14). In a neighborhood of a normal point qEC, the set C is a regular analytic curve that bissects the angle made by the two minimizing geodesics that meet at q ([-12], 8.7).

We will denote by r=N(O) the arc length of the cut point q on the geodesic ~p, where 0 is the arc length of F at the point p. Thus N(O) is a piecewise analytic function of 0, 0 < 0 <_ L.

It follows that there exists a piecewise analytic m a p from the rectangle /~ = {(r, 0)eR2; N(O)<r<O, O<O<L}

into /} that takes the curve 0--*(0, 0) into F and the curve O~(N(O), O) into C. The restriction of such a m a p to

(2.2) {(r, 0)~R2; N(O)<r<__O, O<_O<_L}

is a regular, one-to-one, analytic m a p whose image is D - C . By using the coordinate functions (r, 0) in (2.2) we can write the metric in D - C as

ds 2 =dr z + f (r, O) dO 2,

where f is an analytic function such that f (0, 0) = 1, f~ (0, 0) = kg(0);

here kg denotes the geodesic curvature of F. The Gaussian curvature K in/} - C satisfies then the equation

f,(r, O)+ K (r, O)f(r, O)= O.

We define a parallel F~ to F to be the set of points i n / } at a fixed distance a from F. It can be shown ([12], 9) that F~ is a union of piecewise analytic arcs (or points) that are joined together to m a k e up a finite n u m b e r of connected d o s e d curves. We denote by L(a) the length of F~ and define r 0 to be the infimum of the set of all r for which L ( r ) > 0 ; notice that ro=infN(O), O<O<L.

Assume now r4: r o. It can then be shown that F~ can be divided into a finite n u m b e r of analytic arcs, say F~ i, i = 1 .... , n(r) = n, so that the interior of each Fk i is contained in/} - C; furthermore, the singular points of F~ occur on the cut locus C and they are, except perhaps for finitely m a n y r, normal points in C. Let us denote by Odr)< 0 < O~(r) the interval corresponding to the arc F, i. Then

n 0~(r)

C(r)= Z ~ f (r, O) dO.

I n e q u a l i t y f o r S u r f a c e s 2 4 9

It can be proved that L(r) is a continuous piecewise analytic function of r ([12], 9.5). At the points where the derivative of L exists, it is given by

, ( ~ i ~ r ) _ _ )

(2.3) E ( r ) = ~ f

f~(r,O)dr +(f(r, Oi(r))O'~(r)-f(r, Oi(r))O'i(r)) ~.

i k 0 d r ) )

The terms of the right hand side of (2.3) can be given a geometric in- terpretation. In fact, by computing the geodesic curvature of F~ we find

kg(r, O) =s

O)/f (r, 0).

Thus the total geodesic curvature

k(r)

of F~ is

0i (r)

k(r)= E

f (r,O)dO,

i Oi(r)

which is precisely the first term of the right hand side of (2.3).

For the other two terms, we proceed as follows (cf. [12], p. 239). Let F S and F / m e e t at the singular point q. Since q is a normal point of the cut locus C, it is represented by a pair of coordinates (r, Oj(r)), (r, Oi(r)). Denote by

~i(r) and

~i(r)

the acute angles formed by C and the minimizing geodesics that join q to

(0, Oi(r))~F and (0, Oj(r))eF,

respectively. It can be proved that the angle of F~ and F/is given by

~i(r)+~(r)

and that

tan cq (r) = - f (r, 0 i(r)) O' i (r), tan ~j(r)= f (r, Oj(r)) O)(r).

Thus (2.3) can be written in a more geometric form as (2.4)

E(r)=k(r)+ ~ tan~i(r)-~ tan~i(r ).

i i

This completes all the information we need from [12].

(2.5) We now start the preparation for the proof of Theorem (1.3) in the analytic case. To simplify computations, let us agree on the following notation:

D(r,s)={(t,O)cR2; ro<r<_t<_s<_O , O<_O<_L},

s

(2.6)

A(r,s)= ~ dM= ~ f(t,O)dtdO=~L(t)dt,

D (r, s) D (r, s) r

(2.7)

C(r)= ~ KdM,

D ( r , O)

s

(2.8)

~o(r,s)=~C(t)dt-C(s)(s-r)=~( ~ KdM)dt,

r r D ( t , ~ )

s

(2.9) ~ (r, s) = ~ {~ (tan c~ i(t) - ct i (t) + tan ~i(t) - $i(t)} dt,

r i

m(r) = n u m b e r of components of F~

s

(2.10)

M(r,s)=2Tc~(m(t)-l)dt.

250 J.L. Barbosa and M. do Carmo N o t e that the functions (2.6) to (2.10) except p e r h a p s c~, can be extended continuously to r - - r 0. W e will show that the s a m e is true with ~..

In fact, by applying the G a u s s Bonnet's t h e o r e m to D(t, 0) we obtain C(t) + k(O) - k(t) - ~ (c~i(t) + ~i(t)) = (1 - re(t)) 2~,

i = 1

which together with (2.4) gives (2.11) E ( t ) = k ( O ) + C ( t ) - 2 n ( 1 - m ( t ) )

+ ~ (tan cq(t) - ai(t) + tan ~i(t) - ~i(t)).

i = 1

I n t e g r a t i o n of (2.11) f r o m r to s gives

(2.12) L(s) - L(r) = k(O) (s - r) + (p (r, s) + C (s) (s - r) + M (r, s) + ~ (r, s). F r o m (2.12) it follows that a can be extended continuously to r = r o.

F r o m n o w on, we let r vary in the interval ro<r<O. By setting s = 0 in (2.12), we obtain that the area A of D=D(ro, O ) is given by

0

(2.13) A = A ( r o , O)= S L ( r ) d r

ro

2 0

= - L(0) r o - k(0) ~ - - S (~o (r, 0) + M(r, O) + ~ (r, 0)) dr.

rO

If we multiply (2.13) by - 2 k ( 0 ) and add L2(0) to b o t h sides, we obtain

0

(2.14) L2 (O)- 2k(O) A =(L(O) + k(O) ro)a + 2k(O) S (q)(r, O) + M (r, O) + ~(r, O)) dr.

ro

Since D is simply-connected we have that (2.15) k ( O ) = 2 n - [ K d M ,

D

and, therefore, the left h a n d side of (2.14) is

If one observes that M(r,O) and ~(r,0) are nonnegative, and ~o(r, 0) is non- negative whenever K is nonnegative, then one concludes f r o m the a b o v e Fiala's inequality referred to in the Introduction. In w h a t follows we will obtain a m o r e general inequality that is essentially T h e o r e m (1.3) in the analytic case.

(2.16) Proposition. For any real number Ko, we have

(

KoCh;

L2>_4nA 1 - I ~ ( K - K o ) + d M

- z ~ D 4 ~ ]

Inequality for Surfaces 251 Proof. F r o m (2.15) it follows that to prove (2.16) is equivalent to prove that (2.17) L 2 ( 0 ) - 2 k ( 0 ) A - 2 A ~ K d M + 2 A ~ ( K - K o ) + d M + K o A2 >=0.

D D

Define the following functions: (2.18)

(2.19)

~(s) = k(o) + C(s),

f (s) = (L(s) - k(s) (s - ro)) 2

+ 2 k(s) i (~o (r, s) + M(r, s) + ~ (r, s)) ds ro

- 2 A ( r o , S) ~ K d M + 2 A ( r o , S) ~ ( K - K o ) + d M

D ( r o , s) D(ro, s)

+ KoA2(ro, s).

By using (2.14), one observes that the conditions f ( 0 ) > 0 is equivalent to (2.17). Furthermore, f(s) is clearly a continuous, piecewise differentiable function. Thus to prove (2.17) it suffices to show that:

(2.20) lim f'(s)=>0, and f'(s)>=O whenever it exists.

8 ~ r o

The first condition follows from the definition of f. T o prove the second one, we need some lemmas.

( 2 . 2 1 ) L e m m a .

a) i (r M(r,s)+~(r,s))dr

r o

= L(s) (s - ro) - A (ro, s) - 89 ~(s) (s - ro) 2.

b) ~s ( r , s ) = ( r - s ) C'(s).

Proof. a) follows from (2.12) in the same way as we proved (2.13), except that now we do not set s = 0 in (2.12).

b) follows immediately from the definition of ~o. q.e.d. N o w define r to be the function

s

(s) = (L(s) - k(s) (s - to)) 2 + 2/~(s) ~ (~o (r, s) + M(r, s) + ~ (r, s)) dr. ro

(2.22) L e m m a .

4' (s) = - 2 C' (s) A (to, s) + 4 ~ L (s) (m (s) - 1)

n (st

+ 2 L(s) ~ (tan ~,(s) -- c~ i(s) + tan c~ i(s) -- c7 i (s)).

2 5 2 J i . B a r b o s a a n d M. d o C a r m o

Proof.

By using (2.21) b), the definitions of

M(r,s)

and

c~(r,s),

and the fact that

go (s, s) = M(s,

s) = c~(s, s) = 0, we obtain

~' (s) = 2 (L(s) - k(s) (s - ro) ) (E (s) - C' (s) (s - ro) - f~(s))

+ 2 C' (s) i (go (r, s) + M(r, s) + c~(r, s)) dr

ro

+ 2/~(s) { - 89 - ro) 2

C'(s) + 2n(s - ro) (re(s) -

1) + (s o - ro) ~ (tan c~ i(s) - el(S) + tan 5i(s) - c~ i(s))}.

i = 1

By using now (2.21) a) and (2.11), one simplifies the above to (2.22). q.e.d. We now return to the p r o o f of Proposition (2.16). Since

re(s)>

1 and tan c~ >

(a < n/2),

we have

, >

(2.23) ~

(s)=2C'(s)A(ro, S ).

If we differentiate (2.18) and apply (2.23), we obtain

s) d .

f'(s)> -C'(s)A(ro, s ) - Z L ( S ) D ( ~ , s ) K d M - ZA(ro, ~s DOo,s)KdM

s) dsd

+ 2L(S)D(!,s)(K-Ko)+_

dM +

2A(ro,

D()~,s)

dM

+ 2KoA(ro,S) L(s).

By using that

K d M =

~ K d M - C ( s )

D(ro, s) D(ro, O)

and that (K - Ko) + - (K - Ko) = (K - Ko)-, one obtains (2.24)

f'(s)>__2L(s)

~ ( K - K o )

dM

D (to, s) d

+2A(ro, S)~s

~ ( K - K o ) + d M .

S D(ro, s)

The first term of (2.24) is clearly nonnegative. T o show that the same holds for the second term, observe that

( K - K o ) + d M = ~

-Ko)+ dO dr.

O(ro, s) ro O~

Thus

n(r) O~(r)

d ~ ( K _ K o ) + = Z ~

(K_Ko)+dO>=O"

(2.25) d s D ( , s ) i=1 o~(,)

Inequality for Surfaces 253 Assume now that equality holds in (2.16). Then

(2.26) l i m f ( s ) - - 0 , and f'(s)=-O.

s ~ r o

F r o m (2.22) and (2.24), one sees that the second condition above implies K = K o , m ( s ) = l , and cq(s)-c~i(s)=0.

This implies that all parallel are simply-connected closed analytic curves. F r o m the first condition of (2.26) and (2.19), one concludes that lira L(r)=0. This

r ~ r o

implies that the cut locus C of F is a point. Thus all geodesics perpendicular to C meet at one point, hence have the same length. Therefore, F is a geodesic circle, and this completes the proof of Proposition (2.16). q.e.d.

Thus we have proved Theorem (1.3) in the analytic case. In the rest of this Section we make a remark on Theorem (1.3) and establish a Proposition that will be needed to extend Theorem (1.3) to the C2-case.

The inequality just proved depends on a real number K o. It is a natural question to ask for what value of K o we get the best inequality, that is, for what value of K o is the difference between the two sides of inequality (2.16) a minimum. To answer this question, we define the function

(K-O aM-=-].

F(t)=L2 4~zA(1 1 + At

\

ZTr, D q-Tr, /

A simple computation yields

F ' ( t ) = A 2 - 2 A area { p e b ; K>t}.

Therefore the only critical point of F is the value of t that satisfies area {peD; K > t} =A/2.

Since F'(t) is clearly an increasing function, such a critical point is a minimum for F. This proves the following Proposition.

(2.27) Proposition. The unique value of K o for which the function

attains its minimum is given by area {p~/5; K > Ko} =A/2.

2 5 4 J . L . B a r b o s a a n d M . d o C a r m o

(2.28)

Proposition.

(a) I f ( K - K o ) + ~ 0 and ( K - K o ) - ~0, then 2F(Ko) > [(y (K - Ko)- dM)2/sup (K - Ko)-]

D D

+ [(5 (K - Ko) + dM)2/sup (K - Ko) + 3,

D D

(b) I f ( K - K o ) + ~-0 and ( K - K o ) - ~0, then 2F(Ko) > (5 (K - Ko) + dM2/sup (K - Ko) +.

D D

(c) I f ( K - K o ) + - 0 and ( K - K o ) ~-0, then 2F(Ko) >= (5 (K - Ko)- dM)2/sup (K - Ko)-.

D D

Proof. By integrating (2.24) from r o to 0 and by using that lira f(r)>O, one

obtains r~ ~o

0

f(O)=>25 (L(s) ~ ( K - K o ) - d M ) d s ro D (ro, s)

o d

+25(A(ro, S ) ~ s ~ ( K - K o ) + d M ) d s .

ro D(ro, s)

Thus, for any real numbers r 1 and r 2 such that r o < q , r z <0, we have 0

f(O)>=25(L(s) 5 ( K - K o ) - d M ) d s ro D(ro, s)

o d

+25(A(r0, s ) ~ s 5 ( K - K o ) + d M ) ds"

ro D (to, s)

Since 5 ( K - K o ) dM and A(ro,S ) are increasing functions of s, we obtain D (to, s)

(2.29) _{f ( 0 ) > 2 5} ( K - K o ) - d M ~ L ( s ) d s 0 D ( r o , r O rl

+2A(ro, r2) i d ( 5 ( K - K o ) + d M ) ds r2 ds D(ro,rl)

>2A(rl, 0) 5 ( K - K o ) - d M O(ro, r O

Inequality for Surfaces 255 N o w , assume the hypothesis in (a) and choose rl and r 2 such that:

(2.30) ~ ( K - K o ) d M = 8 9

D(rl, O) D

( K - K o ) + dM= 89 ~ ( K - K o ) + dM.

D(r2, O) D

W i t h such a choice, we have

(2.31) ~ ( K - K o ) - d M = ~ ( K - K o ) - d M < = A ( r l , 0 ) s u p ( K - K o )

D(ro, rl) D(rl, O) D

( K - K o ) + dM = ~ ( K - K o ) + dM<=A(ro,r2)sup(K-Ko) +.

D(r2, O) D(ro, r2) D

Substitution of (2.30) and (2.31) into (2.29) yields p a r t (a) of the Proposition. The proofs of (b) and (c) are n o w immediate, and this proves P r o p o s i t i o n (2.28). q.e.d.

3. Proof of Theorem (1.3); the General Case

We start by generalizing P r o p o s i t i o n (2.16) for the case where F is a piecewise C 1 curve.

(3.1) Proposition. Let M be an analytic Riemannian manifold of dimension two and D ~ M be a simply connected domain of area A whose boundary F is a closed piecewise C 1 curve of length L. Let K be the Gaussian curvature of M and K o be any real number. Then

L2>4~A 1--~S~D(K--Ko)+dM-- .

Equality holds if and only if M has constante curvature K = K o in D, and D is a geodesic disk in M.

Proof D e n o t e by C~(S 1, M), j = l , 2, ..., 0% co (co=analytic), the space of all j-differentiable m a p s f r o m S ~ into M with the C J-topology, and by C~(S ~, M), k = 1, 2, ..., the subspace of CJ(S 1, M) consisting of those m a p s that are regular except at k points at most. A t a point where a m a p a: S 1 ~ M is not regular its image m a y have a corner. By r o u n d i n g off corners, one shows that C~o(S 1, M) is dense in C~(S ~, M) for any k = 1, 2, .... Since M is analytic, G r o m o v ' s version of N a s h t h e o r e m (cf. e.g., [13], p. 13) tell us that we m a y assume M to be isometrically and analytically e m b e d d e d in R" for s o m e n. N o w , f r o m T h e o r e m 2 of W h i t n e y [21], it follows that C~ ~, M) is dense in CJ(S 1, M). If one observers that C~o(Sa,M) is an o p e n set in CJ(S~,M), then one concludes that C~d(S~,M) is dense in CJo(S1,M) and so in Ck(S , M). J 1 In particular we have:

(3.2) C~ ~, M) is dense in C k (S , M) for any k. 1

256 J.L. B a r b o s a a n d M . d o C a r m o

Suppose

aoeC~(S1,M)

parametrizes the boundary of a simply connected domain. Let B~ (ao) denote the ball of radius e around a 0 in

CI(S 1, M).

Of course if e is sufficiently small, B~ (ao) contains only curves that are homotopic to a o and, furthermore, as ao, each one bounds a simply connected domain. Therefore, in B~(o'o) it makes sense to define the following functions"

S 1

(3.3) d ( a ) = f ~

dM

D~

fd( ) = SS f dM

D~

where D~ is the domain bounded by o-eB~(ao) and

f:M---rR

is any continuous function. Our argument will depend on the following lemma whose proof is postponed.

(3.4) Lemma. 5 ~ d

and f d are continuous functions in B~(ao).

Now the proof of (3.1) proceeds as follows. Define @ = 5 0 2 _ 4 7 z d (1 1 + K 0 d ~

(3.5)

_{- ~ ( K - K o ) d -}

_{47z ]"}

We have, from the above lemma, that ~- is continuous in B~(o-o), and from (2.16) we know that J~(a)=>0 for all

aeC~(S1,M)c~B~(ao).

Now, by using (3.2) and the continuity of @, we obtain that ~(o-)>=0 for all

a~B~(r

and, therefore, for a o. This proves the inequality of Proposition (3.1).

For the case of equality, one starts by observing that, from the classical result (1.3) for

K=-Ko,

it suffices to prove that: ~(O-o)=0

implies K=Ko .

To show this, we define in B~(o-o) the functions:

_ ~ = 0 , if

(K-Ko)-=O

in D ,

(r [ = ( K - K o ) - / s u p ( K - K o ) ,

if

( K - K o ) - +0

in D, qo

~ = 0 , if

(K-Ko) +

= 0 i n D, (3.6)

(P+ (a) [=((K-Ko)+d)2/sup(K-Ko)+,

if

( K - Ko) + +0

in D, fq(a) = ((p + (a) + (p- (a))/2.

From (3.4) one sees that q0 +, q~- and f# are continuous, and from Proposition (2.28) we have

(3.7) ~ ( a ) > f#(a),

for all

a~C'~(S1,M)~B~(ao).

Since this set is dense in B~(ao) and ~ and fr are continuous, it follows that (3.7) is true for any

r

in particular for a 0. From (3.7) and the definition of f#, one obtains that if ~ ( a o ) = 0 then

K=-Ko

I n e q u a l i t y for Surfaces 257

Proof of Lemma (3.4). Take S ~ parametrized by the arc length s. Then for o':, a2~B~(~o) we have

2re 2r~

(3.8) Is

< j"

rl~q(s)l- iG~(s)rl ds < j" I~;(s)-,z~(s)r

ds

0 0

< 2?: sup [a i (s) - cr~ (s)[ < 2~ d 1 (al, er2),

where d j ( , ) stands for the C J-metric. Therefore 2" is Lipschitz and thus continuous in B~(ao).

To study the continuity of d and f a g at a point %GB*(a0), one proceeds as follows. Given : > 0 , we build a neighborhood U a of al(S 1) whose area is less than :. This can be done as follows. We cover a:(S 1) by a family of geodesic balls of radius 6 centered in the points of O-l(S: ) and choose the value of a in such way that area(Ua)<g. Now it is easily seen that if dl(~rl,ae)<a then

d o ( O - l , o - e ) < a , and thus

[ff(G1)--ff(O'2)]<g

and [ f d ( ~ : ) - f a g ( ~ 2 ) l < M g , where

M is the maximum of If[ on D~: • ~ . Therefore d and f a g are continuous in B~(a0). q.e.d.

Next, we extend the result of (3.1) for the case where M is analytic with a C g- metric. Our main tool will still be Theorem 2 of Whitney [213 that will allow us to approximate a C2-metric by an analytic one.

(3.9)

Proposition.

Let M be an analytic manifold of dimension two endowed with a C2-metric and D c M be a simply connected domain of area A whose boundary is a closed piecewise C: curve of length L. Let K be the Gaussian curvature of M and K o be any real number. Then

(

L2>4rcA 1 - ( K - K o ) + d m - ~ 7 - ] .

Equality holds if and only if M has constant curvature K = K 0 along D, and D is a geodesic disk in M.

Proof. Let B(R2,R) be the space of symmetric bilinear forms in R 2 and let Ca(M,B(R 2, R)) be the space of all C ~ maps from M into B(R 2, R) endowed with the C j metric. In particular, if M is a noncompact simply connected surface, hence contractible, CY(M, B(R 2, R)) contains all j-differentiable metrics on M. In fact we may assume throughout this proof this to be the case; otherwise we would restrict ourselves to an open simply connected neighborhood of D that exists because D is simply connected. Since M is analytic, we may assume that M is analytically embedded in R" for some n. Since B(R2,R) is also analytic, it follows from T h e o r e m 2 of Whitney [21] that:

(3.10) CO(M,B(R2,R) is dense in CJ(M,B(R2,R)).

258 J.L. B a r b o s a a n d M. d o C a r m o

Let/5 be the unit closed disk in R 2. Let X:/5 ~ M be a continuous map such that:

a) X restricted to the open disk is a C j parametrization of D, b)

a=XlobeC~(S1,M)

for some k.

This defines local coordinates x~, x 2 on D. As usual set

gii = g ((3x/Sxi, 8x/Oxj),

i, j = 1, 2,

det(g) = gl 1 g22-g12 g21.

Let

ds

be the element of arc of S 1. With these notations, we define on B{(g-) the functions:

(3.11) ~(g)= ~

~ d s ,

geB~(g-),

S 1

(3.12) ~(g) =S~l/det(g)

dx 1

A

dx2,

g~B{(~),

b

and, for any map f:B~(~)--*

C~

(3.13)

f~(g)=S~f(g)(xl, x2) dl/de~dxlmdX2,

gEB~(~).

b

Our reasoning will depend on the following lemma whose proof is postponed to (3.19).

j ~

(3.14) Lemma. The

functions ~ , sd and f s~ are continuous on B~(g).

Now, the proof of (3.9) proceeds as follows. Define in

BJ(~,), j>2,

the function

(3.15) ~ = ~ 2 - 4 ~ 7 ( 1 - 2 ~ ( / ( - K o ) + f f - ~ ) ,

where

I(:B{(~,)---,C~

is the map that associates to each metric g~B{(g') over D its Gaussian curvature. (Notice that for

j>2,

/((g) is a C o function). It follows from the previous lemma that ~ is continuous. From (3.1) we have that

~ ( g ) > 0 ,

for g in

B~(~,)c~C'~

Since this set is dense in B~(g ~) and g is continuous it follows that ~ ( g ) > 0 , for all

geB~(g~),

in particular for ~. This proves the inequality of (3.9). For the case of equality, one starts by observing that, from the classical result (1.3) for

K = K o

it suffices to prove that: ~-(g~)=0

implies K = K o.

To show this, we define in B~(g ~) the functions

_f=0,

if ( / ( - K o ) - - = 0 in D,

~o ~=((l~_Ko)_S#)2/sup(K_Ko) -,

if

( K - K o ) - ~O in D,

(3.16)

~~ § { = 0 , if

( K - K o ) + =-0

in

D,

Inequality for Surfaces 259

and set

(3.17) ~=(~o + +q~-)/2.

From Lemma (3.14) one sees that ~o +, ~o- and f~ are continuous on B~(g), and from (2.28) we then have

(3.18) ~ ( g ) > ~(g),

for all g E B~ (g')c~

C~

B(R 2, R)). Since this set is dense in B 1 (g), and ~ and f~ are continuous, it follows that (3.18) is true for any geB~(ff), in particular for g. From (3.18) and the definition of ff it follows that ~'(ff)=0 implies

K=-Ko

along D. Therefore Proposition (3.12) is proved.

1 2 @ B l e ~

(3.19)

Proof of Lemma

(3.14). For g , g ~ tg) we set

=sup ]gi(a', o-')- g2(o -', o-')1.

S t

Then we have

I5s163 < 5

I]/TiTa( ~r') -

g]/7~, ~r')l ds<= 5 ]/~l ds= 2 ~z ]~.

S x S 1

If we write

cr' = ~ a i OX/Ox i

then we have that

i

l ~ < d l ( g l g2) sup (~ a2).

S 1 i

Therefore 2 is Lipschitz and thus continuous. For the case of s~7, we set

D

and observe that

gl~ <g2~ +(,

g~2<g~2+~,

g~2>g~2-~.

Hence

] / ~ (gl) - - < ~ + ] f ~ ] ~ a a +g222 +2922,

and thus

1 ~ - / 2 ..~ 2

+2g~2

d x 1 d x 2

[ ~ ( g l ) - - ~ ( g 2 ) [ ~ f l l / g 1 / g l l g 2 2

< ~z ]/~ sup ]/g21 + gz2z + 2 g22 .

D

260 J.L. Barbosa and M. do Carmo

= ( a - b) (c + d) + (a + b) ( c - d). We omit the details. This completes the proof of Lemma (3.19). q.e.d.

(3.20) We now complete the proof of Theorem 1.3 for non-degenerate metrics. Let 9,1 be a C 2 atlas that defines the C 2 differentiable structure on M. Since M is two dimensional, the existence of isothermal parameters in M implies that 9,1 admits a subatlas ~1 that defines on M an analytic structure. Therefore we can apply Proposition (3.12) to obtain Theorem (1.3) for non-degenerate metrics, q.e.d. (3.21) It remains to consider the case where the metric is degenerate. Let

H(D)c C2(M, B(R 2, R))

be the space of all symmetric bilinear forms that are positive on the closure 15 olD. It makes sense to define on

H(D)

the functions 2,~ and considered in (3.11) and (3.12). Furthermore the proof of Lemma 3.14 shows that they are continuous. Suppose that geH(D) has only isolated singularities. Then there are finitely many singularities i n / 5 and we easily can find a sequence of positive definite metrics {g,} ~

H(D)

that converges to g in the C2-sense. By using the notation established in (3.15) and the C2-case of Theorem (1.3), we find that ~ ( g , ) > 0, for all n. Thus, by letting n--+ oo, we obtain

/Co~(g)~

~2(g)-4rcag(g) ( 1 - 1 lira ( K ( g , ) - K o ) + ag(g,) >0. (3.22)

2~ ~oo ~-~ -] -

Notice that K(g) may not be defined at the singular points ofg. However since K(g) is bounded above and gn--+ g, the sequence {(K(gn)- Ko)+} is bounded above by a certain function. Thus, by the dominated convergence theorem, we obtain (3.23) lim ( K ( g , ) - K o ) + ~ ( g n ) = lim 5 ( K ( g , ) - K o ) +

dM,

=5 (K(g)-Ko) + dM,.

1)

This together with (3.21) shows the inequality in (1.3) holds.

For the case of equality, we proceed as follows. By using the notation established in (3.15), (3.16) and (3.17) and the fact that ~ >= (~ for positive definite metrics, we have that ~ ( g , ) > (~(g,), for all n. By the argument which led to (3.22), we obtain

~(g) = lim ~ ( g . ) > lim ~(g.)= ~(g).

n ~ o o n ~ c o

Now, if ~,~(g)=0 then ~(g)=0, and thus

K = K o

in /5. Therefore g has no singularities in D. The result that D is a geodesic circle now follows from the classcal case of (1.3) for K - K 0. The converse is obvious and this completes the proof of Theorem (1.3). q.e.d.

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Inequality for Surfaces 261

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