TRANSACTIONS
OF THE
AMERICAN MATHEMATICAL
Volume 309, Number 2, October
SOCIETY
1988
POSITIVE QUADRATIC DIFFERENTIAL FORMS
AND FOLIATIONS WITH SINGULARITIES ON SURFACES
VICTOR GUINEZ
ABSTRACT. To every positive C-quadratic
differential form defined on an
oriented two manifold is associated a pair of transversal one-dimensional Cfoliations with common singularities.
An open set of positive C-quadratic
differential forms with structural stable associated foliations is characterized
and it is proved that this set is dense in the space of positive C°°-quadratic
differential forms with C2-topology. Also a realization theorem is established.
1. Introduction.
A positive Cr-quadratic
differential form on an oriented
two-dimensional manifold (see §2 for definitions) has associated two transverse Cr
one-dimensional foliations with common singularities called the configuration of the
quadratic differential form. The basic problems to be considered here are the local
and global descriptions of these configurations, their (structural) stability under
small perturbations of the positive quadratic differential form, and the genericity
of the stability property.
Another question that we will consider is the realization by a positive Crquadratic differential form of an arbitrary configuration of two transverse Cr-onedimensional foliations with common singularities.
In 1952 Hartman and Wintner [3] studied the existence of spiral solutions in the
neighborhood of a singularity for continuous positive quadratic differential forms
and applied their results to lines of principal curvature and to asymptotic lines
around umbilical points.
Later in 1982, Sotomayor and Gutierrez [2, 12] considered immersions in R3 of
a compact oriented two manifold. They described a class of immersions with stable
configuration of lines of principal curvature and they proved that this class is dense
in the space of immersions with the C2-topology. We recall that the configuration of
lines of principal curvature is obtained as the configuration of a positive quadratic
differential form.
In our work we consider the space of positive C""-quadratic differential forms on
an oriented compact two manifold. We describe a class of such forms each of whose
elements has stable configuration and we prove that this class is dense in the space
of positive C°°-quadratic differential forms with the C2-topology.
Received by the editors November 10, 1986.
1980 Mathematics Subject Classification (1985 Revision). Primary
57R30; Secondary
58F10,
34C04.
The author acknowledges
the very kind hospitality
provided by IMPA/CNP
during the prepa-
ration of this paper. This work was partially supported by CNP, (Brasil) and PNUD-UNESCO
CHI-84-004.
©1988
American
0002-9947/88
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477
Mathematical
Society
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478
VICTOR GUINEZ
We also prove in this paper (Realization Theorem) that every configuration of
two transverse Cr-one-dimensional
foliations with common singularities is realized
as the configuration of a positive Cr-quadratic
differential form.
This paper is organized as follows. §2 contains the principal definitions and the
precise statement of the results, namely Theorem A and Theorem B. In §3 we define the set Sr(M) (Theorem B) whose elements are proved to be structural stable
positive Cr-quadratic differential forms. The proof of Theorem A (Realization Theorem) is presented in §4. The next sections are devoted to the proof of Theorem B.
In §5 we prove the openness of Gr(M), the set of positive Cr-quadratic differential
forms with simple singular points. In §6 we prove the openness of the set Sr(M)
and the structural stability of its elements. In §7 we prove the density of the set
Sr(M) in Gr(M) with the CMopology for any s < r. Finally in the last sections
we complete the proof of Theorem B proving the density of the set G^ (M) in the
set of positive C°°-quadratic
differential forms with with C2-topology.
This work corresponds to my doctoral thesis at IMPA. I wish to thank my adviser
C. Gutierrez who suggested this problem. Thanks are also due to J. Palis and
C. Gutierrez for their help and interest in this work.
2. Definitions
and statement
of results.
This section contains our principal
definitions and the precise statement of our results. By M we will denote a Cocompact, connected, oriented two-dimensional manifold.
2.1.
DEFINITION. A Cr-quadratic
differential form on M is an element of
the form w = Y17=i •AiV'iiwhere —{0}.
R
2.11.
COROLLARY. Given a Cr-one-dimensional foliation f on M —S, where
S is a closed subset of M, there exists w E ^(M)
such that S = Sing(w)
and
f = fi(w).
2.12. REMARK. The configuration of lines of principal curvature of an immersion of M in R3 [2, 12]; the configuration of a pseudo-Anosov diffeomorphism [10];
the measured foliations [4, 10]; the arational foliations [9], the line fields [1, 5],
and the flow of vector fields are all realized as configurations of positive quadratic
differential forms.
2.13.
THEOREM B.
For each 1 < r < oo, there exists a nonempty open subset
Sr(M) C Gr(M) of9rTl(M) such that:
(i) All elements in Sr(M)
are structurally
stable.
(ii) Sr(M) is dense in Gr(M)
with the Cs-topology for s 0
D3: 6i 1. Then {AJ~0 is a
=
locally finite open covering of M —S. Let ^°10 (R2,0) such that
(x,y)*(w) = (y + My(x,y))dy2 + (byx + b2y + M2(x,y))dxdy
+ (-y + M3(x,y))dx2
with Mi(x,y) = 0(x2 + y2), i = 1,2,3 and by ^ 0.
PROOF.
The equivalence between (1), (2), and (3) is a consequence of the
following facts:
(a) If {Uy,4>i} is an atlas over M, then {Q(Ui), (pi} is an atlas over Q(M), where
Q{Ui) = {(P,a) E Q(M)\p E U,} and cbt:Q(Ui) -+ R5 is defined by &(p,a) =
(x(p),y(p),a(p),b(p),c(p)),
where tpi = (x,y) and a = a(p)dy2 + b(p)dxdy+c(p)dx2.
(b) If (x,y): (U,p) —►
(R2,0) is a local chart and
(x,y)*(w) = (ayx + a2y + My(x,y))dy2 + (byx + b2y + M2(x,y))dxdy
+ (cyx + c2y + M3(x, y))dx2
with Mi(x,y) = 0(x2+y2),
then (u,v)*(Dwp) = (ayu+a2v)dv2 + (byu+b2v)dudv+
(cyu + c2v)du2, where (u, v) = D(x, y)p : TPM —>R2.
Therefore only (2) =^ (4) needs to be proved. For this, let (x, y): (U, p) —*(R2,0)
be a chart such that
(x,y)*(w) = (ayx + a2y + My(x,y))dy2 + (byx + b2y + M2(x,y))dxdy
+ (cyx + c2y + M3(x, y))dx2
with Mi(x, y) = 0(x2 + y2), i = 1,2,3. Suppose b2 - 4ayCy > 0, and
(bf - 4a,ci)(&2 - 4a2c2) - (byb2 - 2axc2 - 2a2cy)2 > 0.
Observe that if A: R2 —> R2 is a linear isomorphism (u,v) = A o (x,y),
A_1(u,v) = (au+/3v,^u+6v)
and (u,v)*(w) = a(u,v)dv2+b(u,v)dudv+c(u,v)du2,
then
S2
(a,b,c)(u,v)=
Pb
2^6 a6 + fa
72
07
p2 1 a(A~1(u,v))
2a(3
a2
b(A~1(u,v))
c(A_1(u,v))
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484
VICTOR GUINEZ
and
da
ay = ~-(0,0) = 62(aay + ^a2) + f36(aby + ib2) + (32(acy +ic2),
~a2 ^(0,0)
=
W
= 62(/3ay + 8a2) + p8([3by+ 6b2) + P2(fay + 8c2),
dc
cy = q-(0, 0) = l2(otay + ^a2) + a^(aby + 762) + a2(acy + 7C2),
dc
~c2 —(0,0) = 72(/?a, + Sa2) + ai(pby + 6b2) + a2(fay + 6c2).
=
From this we see that if ay — Cy = 0, then taking 7 = /? = 0 we leave this
condition invariant and equations (*) are a2 = 63a2, c2 = a26c2. Since a2 ■c2 < 0,
it is clear that there exist a, 6 such that a2 = —c2 = 1.
Now if Ci =0 taking 7 = 0 we leave ci = 0 invariant and it follows from equation
(*) that ai = a6(6ay + (Iby) with by 7^ 0. So if /? = 6ay/by we obtain ai = 0.
Finally if ci 7^ 0 to obtain c"i = 0 it is enough to make a = A7 with A a solution
of the equation ci A3+ (61+02) A2+ (01+62) A+ a2 = 0. Thus the proof is complete.
The openeness of the set GT(M) in SFrx(M) will be a consequence of the following
proposition:
5.2.
PROPOSITION. Let w E &r(M) andp E Sing(w). Ifp is a simple singular
point of w there exist neighborhoods JV(w) C ^}(M)
of w and V C M of p such
that every vf)E jV(w) has a unique singular point p(w) in V, which is simple.
PROOF. Let (x,y): (U,p) —>
(R2,0) be a chart such that (x,y)*(w) = a(x,y)dy2
+b(x, y)dxdy+c(x,y)dx2 with (a, b, c)(x, y) = (y, byx+b2y, -y)+(My, M2, M3)(x,y),
where by ^ 0 and Mt(x,y) =0(x2 + y2), i = 1,2,3.
It follows from the transversal intersection of the curves a = 0 and c = 0 with
6 = 0 at (0,0), that there exist neighborhoods V C (x,y)(U) of (0,0) and^(w) C
of w such that a_1(0) n 6_1(0) f~l
c-1(0) n V = {(0,0)} and for each w E
^(M)
yV(w), the
b = 0 at a
62 — 4ac >
Observe
corresponding curves a = 0 and c
single point V, say (x,y)(py(w))
0, we have pi(wi) = p2(w) = p(w)
that, reducing jV(w) if necessary,
62-4a,ct>0,
= 0 have transversal intersection with
and (x,y)(p2(w)) respectively. Since
and therefore Sing(w) nV = {p(w)}.
we have
i = l,2,
and
(62 - 4aici)(&2 - 4a2c2) - (6162 - 2fiic2 - 2a2ci)2 > 0,
where
(ai,6i,ci)
= (d/dx)(ct,b,c)((x,y)(p))
and
(a2,62,c2) = (d/dy)(a,b,c)((x,y)(p)).
Therefore the proof is complete.
5.3.
COROLLARY. The subset Gr(M)
of those w in ^%(M) whose singular
points are all simple is open in SFrx(M).
PROOF. Since M is compact, the proof follows from the previous proposition.
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POSITIVE
QUADRATIC
DIFFERENTIAL
FORMS
485
6. Structural
stability.
The openness of the set Sr(M) (defined in §3) in
fl^1 (M) will be a consequence of the local stability of the hyperbolic singular points
and the continuity on compact parts, under small C1 -perturbations
of singular
separatrices, together with the local stability of hyperbolic compact leaves.
6.1. DEFINITION. A form wq in J^(M) is said to be locally stable at a point p
in M if there exist neighborhoods y^"(u;o) of wo in ^fr1(M) and U(p) of p in M such
that for each w in jV(wq) there is a point q = q(w) in U(p) and a homeomorphism
h: V(q) —*V(p) between neighborhoods of q and p such that h(q) = p and such
that h maps the configuration of w/V(q) onto the configuration of wq/V(p).
6.2. PROPOSITION. Let p be a hyperbolic singular point of w inf^-(M).
w is locally stable at p.
Then
PROOF. Since this is similar to the case of an immersion of M in R3 that verifies
condition D at an umbilical point (Proposition 2.1 of [12, pp. 201-206]) only an
outline of the proof is given below.
Let p be a hyperbolic singular point of a w in !Fr(M) and (a;, y): (U,p) —►
(R2,0)
be a local chart such that (x,y)*(w) = a(x,y)dy2 + b(x,y)dxdy + c(x,y)dx2 with
(a, 6, c)(a;, y) = (y, byx+b2y, -y) + (My,M2, M3)(x,y), where M{(x,y) = 0(x2+y2),
i = 1,2,3 and by ^ 0.
The vector field Yi = Pd/dx + Qid/dy, where P(x,y) = 2a(x,y) and Qi(x,y) =
-b(x,y) + (-l)t+1 \J(b2 —Aac)(x,y) for i = 1 or i = 2, is tangent to one of the
foliations of w except possibly when P = 0.
Making
{x
= s,
y = ts + y(s),
where y = y(s) is the unique solution of P(s, y(s)) = 0 with y(0) = 0 (and therefore
y'(0) = 0) we obtain over R2 - R • (0,1) the vector fields
Zi = H,Yi = Sd/ds + Tid/dt.
Therefore
S(s, t) = 2ts[l + U(s, t)]
with U(0, t) = 0
and
Tl(s,t) = s-1
-(t + y'(s))S + Sy + (-1Y+Iy/S2
+ SS2 ,
where
Sy (S, t) = -3[by + b2t + Uy(s, t)]
with Uy(0, t) = 0
and
S2(s,t) = -2s[-t
+ U2(s,t)}
v/ithU2(0,t) = 0.
Let Zi be defined by
Zi(s,t) = ri(s,t)Zi = Si—+f—
ds
dt
where
ri(s,t)
= (ts)-1
-(t + y'(s))S + Sy + (-l)l^S2
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+ SS2 .
486
VICTOR GUINEZ
H6I1C6
-
j(S' '
(2s[l + U(s,t)}fi(s,t)
iis>0,
1 2s[l + U(s,t)]f3-i(s,t)
ils 0 is small enough we have \\w\ - wa\\a < e/2
and then \\w\ — w\\a < s. Thus the proof of Proposition 8.1 is complete.
The proofs of Propositions 8.2 and 8.3 will be given in §§10 and 11 respectively,
using some preliminary lemmas that we introduce in the next section.
9. Technical
lemmas.
In this section we state some technical lemmas that
are used in the proofs of Propositions 8.2 and 8.3 (§§10 and 11).
the following sets.
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494
VICTOR GUINEZ
If U is an open subset of R2, let Ar(U) be the set of triples (a, 6,c) of realvalued functions defined in U such that the quadratic differential form a(a;, y)dy2 +
b(x,y)dxy + c(x,y)dx2 is in &r(U). That is, the functions a, b,c are of class Cr,
(b2 - Aac) >0inU,
and (62 - 4ac)"1(0) = a'^O) n &-x(0) n c~l(0).
If Uy,U2 are open subsets of R2 with Uy C U2 and U2 a compact set, let
b(Uy,U2) be the set of smooth functions 0, and an open
neighborhood V of (0,0) contained in U, there exist (a, 6, c) 6 Ar(U) such that:
(a) (b2 - 4ac)-1(0) = (62 - 40c)-1 (0).
(b) (a, 6, c) = (a, 6, c) outside V.
(c)
(d) max{||a
- a||s, ||6 - 6||s, ||c - d||s} < e.
PROOF. Let sSN,
s < r, e > 0 and V be an open neighborhood of (0,0)
contained in U. Consider 6 E b(Uy,U2) with (0,0) 6 Uy and U2 C V. Then if
(dk/dxldyk-i)(a,b,c)(0,0)
= (a0,b0,c0), for 8 > 0 we define
' (a, 6, c) + 8Q(-b, 2(a - c), b) if [60^ 0 or 60 = 0 and
a0 c0 7^0 and a0 ^ c0],
(a, 6, c) = < (a, 6, c) + 8@(0, 2a, 6) if 6n = 0 and ao = Co,
(a, 6, c) + 8Q(0,2a, b + 8Qa)
if b0 = c0 = 0,
. (a, b, c) + 8e(b + 8Qc, 2c, 0) if 60 = a0 = 0.
Therefore (a, 6, c) E Ar(U) and verifies conditions
small enough also verifies conditions (c) and (d).
(a) and (b) and if 8 > 0 is
9.2.
LEMMA. Let (a,b,c) be a triple in Ar(U). Then given s E N, s < r,
e > 0, and any open set V o/R2 with V a compact set contained in U, there exists
a triple (a,b,c) in Ar(U) such that:
(a) (a, 6, c) = (a,b,c) outside V.
(b) (b2 - 4acJ)-1(0) = (62 - 4ac)"1(0).
(c) If(x,y) isinV, (dk/dyk)(a,b,c)(x,y)
? (0,0,0) and
((0) -40-0)(x0'
- a||s, ||6 - 6||s, ||c - c||s} < £.
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.
n
POSITIVE QUADRATIC DIFFERENTIAL
FORMS
495
PROOF. Let SEN, s < r, e > 0 and V be an open set such that V is a compact
set contained in U. Let 0 such that 8 ■ max{|| ■ (a — c)\\a,
\\2tp ■ b\\a} < e. Then the triple (a, 6, c) = (a, 6, c) + 8(p(a — c, 26, c — a) belongs to
Ar(U) and satisfies (a), (b), (c), and (d).
9.3.
LEMMA.
Given r > 0, for all k E N there exists a smooth function
ipk: R —►
[0,1] such that:
Vfc-1(l) = [-r,r],
|Vfei)(z)l 0. It is sufficient to prove that for all fc E N, there exists a
smooth function 9fc: R -> [0,1] such that: Qkl(0) =] - oo,0], 9^(1) = [r,+oo[,
and \eki](x)\ < 2'(i+1)/2 • r~* for all x G R and t = 1,..., fc.
To obtain Ofc we see by induction that for all fc E N there exists a smooth
function/fc: R - [0,1] such that: /fc_1(0)=]-oo,0]; 4_1(1) = [2fc,+oo]; \f^(x)\
2t(t+i-2fc)/2 for gji j- _ 1?
<
j fc; yfc^j _ ! _ yfc(2*- x) and therefore /„2*fk(d)dx =
2k-i
If we define Qk(x) = fk(2k • r_1x) the lemma is proved.
9.4. COROLLARY. For r > 0 let us consider the rectangles Ry = {(x,y)\\x\ <
r> \y\ < r} and R2 = {(x,y) \ \x\ < 2r, \y\ < 2r}. Then given s E N there exists
4>sE b(intRy,intR2) such that 1(3*0,/^9y 0, Mt(0,0) ^ 0 for z = 1,2,3 and ^n)(0) = fc,(n)(0)=
g\n) (0) = 0 for all n E N and i = 1,..., fc.
Let e > 0. Let V2 be an open neighborhood of (0,0) such that V2 is a compact
set contained in Vi; Mt(x,y) / 0 for i = 1,2,3 and (M% - 4MyM3)(x,y) > 0 for
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POSITIVE
QUADRATIC
DIFFERENTIAL
FORMS
497
all (x,y) E V2, and
(*)
max{\\(H + G- 2K)My\\a, \\(H - G)My\\a, \\(K - G)My\\a,
\\(K-G)M3\\a} 0 small
enough to have R3 cV2.
Let (py E 6(int/?2,intr?3)
(**)
dx^dyi-i&y)
and (p2 E 6(inti?i,inti22)
^2'(8+1)/2 • (r/3)"
such that
for a11(*•»)e r2>
fc= 1,2, i = 1,2,..., s, and j = 0,1,..., i (Corollary 9.4).
It is sufficient to prove that there exists (5,6, c) E Aoo(V2) such that:
(1) (5,6, c) = (a, 6, c) outside inti23,
(2) max{||5
- a||s, ||6 - 6||s, ||c - c||s} < e,
(3) (d,b,c)(x,y) = (yk - K(x,y)) ■(My,M2,M3)(x,y) for all (x,y) E Ry,
(4) (62 - 45c)-1 (0) n (V2 - Ry) = (b2 - Aac)-1 (0) n (V2 - Ry).
Let us consider the cases Mi (0,0) • M3(0,0) > 0 and Mi (0,0) • M3(0,0) < 0
separately.
Case A: Mi(0,0) • Af3(0,0) > 0.
Let
(a,b,c)(x,y) = (a,b,c)(x,y)+ 0 for all (x,y) E Rp.
b\ - 4axcx > 0 in Ipx ([-28,-8] U [8,28]) for 0 < 8 < 1 where (aA,6A,cA) =
yk(My,M2,M3)(x,y)
Since S00(I,w)
+ X(H,K,G)(x,y).
is a compact set, there exist pi,...
,pk E T,00(I,w) with x(pi) <
■• • < x(pfc) such that E00(7, w) C U,=i int^J
and RPi n Rpi+1 is empty. For i =
1,... ,fc consider two points: q\,q2 of E°(7, w) fl RPl such that (x,y)(Y,°(I,w)) C\
Rp,C[x(ql),x(q2)]x{0}.
Let J = \Jl=y[x(ql),x(q2)] and H,K,G
be the smooth functions defined by
(HKGMx
) = { (0'°,0)
ifx€j'
' ' >[X,y) \ (H,K,G)(x,y)
iixtl
Let (p: R —►
[0,1] be a smooth function such that 0_1(1) = [—£,£], 0_1(O) =
K-]-28,28[,
and \(p{l](y)\ 0 small we obtain Hi(0,0) ■Gi(0,0) 7^0.)
So £>Ai(0,0) 7^ 0 and DA2(0,0) 7^ 0 and the curves Ax = 0 and A2 = 0
determine at a small neighborhood U2 of (0,0) two sectors where the discriminant
A is not negative. Since d2h/dy2(0,0)
7^ 0, the function h is always positive or
negative where A is positive and we choose 0 for all
(x,y)G(72nA-1(]0,+oo[).
Observe that if (a,b,c) is in Aoo((x,y)(U)), taking the neighborhood Uy of (0,0)
with Uy C U2 and 8 small enough we have solved our problem. If (a, 6, c) is not
Aoo((x, y)(U)) we define (5,6, c) = (a, (1 + eO)6, c) with e > 0 small enough.
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Departamento
Chile,
Casilla
de Matematicas,
653, Santiago,
Facultad
de Ciencias,
Chile
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Universidad
de
