Download 7. On the fundamental theorem of surface theory under weak

C. R. Acad. Sci. Paris, Ser. I 338 (2004) 71–76
Differential Geometry/Mathematical Problems in Mechanics
On the fundamental theorem of surface theory
under weak regularity assumptions
Sorin Mardare
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
Received 22 September 2003; accepted 4 October 2003
Presented by Philippe G. Ciarlet
Abstract
We consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that together
satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply connected open subset of R2 . If these fields are
of class C 2 and C 1 respectively, the fundamental theorem of surface theory asserts that there exists a surface immersed in the
three-dimensional Euclidean space with the given matrix fields as its first and second fundamental forms. The purpose of this
1,∞
Note is to prove that this theorem still holds true under the weaker regularity assumptions that these fields are of class Wloc
∞
and Lloc respectively, the Gauss and Codazzi–Mainardi equations being then understood in a distributional sense. To cite this
article: S. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).
 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Résumé
Sur le théorème fondamental de la théorie des surfaces sous des hypothèses faibles de régularité. On considère un
champ de matrices symétriques définies positives d’ordre deux et un champ de matrices symétriques d’ordre deux qui satisfont
ensemble les équations de Gauss et de Codazzi–Mainardi dans un ouvert connexe et simplement connexe de R2 . Si ces champs
sont respectivement de classe C 2 et C 1 , alors le théorème fondamental de la théorie des surfaces affirme qu’il existe une surface
plongée dans l’espace Euclidean tridimensionnel dont ces champs sont les première et deuxième formes fondamentales. L’objet
de cette Note est d’établir que ce théorème reste vrai sous les hypothèses de régularités affaiblies selon lesquelles ces champs
1,∞
et L∞
sont respectivement de classe Wloc
loc , les équations the Gauss et de Codazzi–Mainardi étant alors satisfaites aux sens
des distributions. Pour citer cet article : S. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).
 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Version française abrégée
On considère un champ (aαβ ) de matrices symétriques définies positives d’ordre deux et un champ (bαβ ) de
matrices symétriques d’ordre deux définis dans un ouvert connexe et simplement connexe ω de R2 . On suppose
que les fonctions aαβ et bαβ sont respectivement de classe C 2 et C 1 dans ω et qu’elles satisfont ensemble les
équations de Gauss et de Codazzi–Mainardi, à savoir
E-mail address: [email protected] (S. Mardare).
1631-073X/$ – see front matter  2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.
doi:10.1016/j.crma.2003.10.027
72
S. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 71–76
τ
τ
σ
σ
τ
∂γ Γαβ
− ∂β Γαγ
+ Γαβ
Γστγ − Γαγ
Γσβ
= bαβ bγτ − bαγ bβτ ,
σ
σ
bσ γ − Γαγ
bσβ = 0.
∂γ bαβ − ∂β bαγ + Γαβ
τ
associés à la métrique (aαβ ), ainsi que d’autre notations, sont définis dans la
Les symboles de Christoffel Γαβ
version anglaise.
Sous ces hypothèses, le théorème fondamental de la théorie des surfaces (voir, e.g., [1,3,4]) affirme qu’il existe
une application θ : ω → R3 de classe C 3 telle que les première et deuxième formes fondamentales de la surface
S = θ (ω) sont respectivement données par les champs des matrices (aαβ ) et (bαβ ), i.e.,
∂α θ · ∂β θ = aαβ
et ∂αβ θ ·
∂1 θ ∧ ∂2 θ
= bαβ
|∂1 θ ∧ ∂2 θ|
dans ω.
L’objet de cette Note est d’établir que le théorème fondamental de la théorie des surfaces reste vrai sous des
hypothèses de régularité plus faibles. Le résultat principal (voir Théorème 2.1 dans la version anglaise) s’énonce
1,∞
ainsi : si les champs de matrices (aαβ ) et (bαβ ) sont respectivement de classe Wloc
et L∞
loc dans ω et satisfont
ensemble les équations de Gauss et de Codazzi–Mainardi au sens des distributions, alors il existe une application
2,∞
θ : ω → R3 de classe Wloc
telle que les première et deuxième formes fondamentales de la surface S = θ (ω) sont
respectivement données par les champs de matrices (aαβ ) et (bαβ ). De plus, si le diamètre géodésique de l’ouvert
ω est fini, aαβ ∈ W 1,∞ (ω), bαβ ∈ L∞ (ω), et (aαβ )−1 ∈ L∞ (ω, M2 ), alors l’application θ appartient à l’espace
W 2,∞ (ω, R3 ).
La démonstration complète de ce théorème, esquissée dans la version anglaise, se trouve dans [6].
1. Preliminaries
All functions and fields appearing in this paper are real valued and the summation convention with respect to
repeated indices and exponents is used.
For any integer d 2, the d-dimensional Euclidean space will be identified with Rd . Let u · v denote the
Euclidean inner product of u, v ∈ Rd and let |v| denote the Euclidean norm of v ∈ Rd .
Let Mq,l denote the set of all matrices with q rows and l columns and let Md := Md,d . The notations Sd and Sd>
respectively designate the set of all symmetric matrices, and of all positive definite symmetric matrices, of order d.
The notation (aij ) (or (aji )) designates the matrix whose entries are the elements aij (or aji ), where the first (or
upper) index is the row index and the second (or lower) index is the column index.
Let x = (x1 , x2 , . . . , xd ) denote a generic point in Rd and let ∂i := ∂/∂xi . The open ball centered at x ∈ Rd
of radius R > 0 is denoted B(x, R) and |B(x, R)| denotes the Rd -Lebesgue measure of B(x, R). The distance
between two subsets A and B of Rd is defined by
dist(A, B) =
inf
x∈A, y∈B
|x − y|.
The geodesic diameter of an open subset Ω of Rd is the number DΩ ∈ [0, ∞] defined by
DΩ := sup inf L(γ ),
x,y∈Ω γ
where γ ∈ C 0 ([0, 1]; Ω) is any path joining x to y and L(γ ) designates the length of the path γ .
Let D (Ω) denote the space of distributions defined over Ω, let W m,p (Ω; Mq,l ) denote the usual Sobolev space,
and let
m,p Wloc Ω; Mq,l := v ∈ D Ω; Mq,l ; v ∈ W m,p U ; Mq,l for all open set U Ω ,
where the notation U Ω means that the closure of U in Rd is a compact subset of Ω. For real-valued function
spaces we shall use the notation W m,p (Ω) instead of W m,p (Ω, R), D (Ω) instead of D (Ω, R), etc.
S. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 71–76
73
In what follows, we make the following convention for classes of functions with respect to the equality almost
˙
everywhere: if f˙ ∈ L∞
loc (Ω), we will always use the representative f of f given by
1
f˜(y) dy,
f (x) := lim inf
ε→0 |B(x, ε)|
B(x,ε)
where f˜ is any representative of the class f˙ ∈ L∞
loc (Ω) (this definition is clearly independent of the choice of the
representative f˜). This choice of the representative insures that f˙L∞ (U ) = supx∈U |f (x)| for all open subset U
1,∞
of Ω. Also, for any f˙ ∈ Wloc
(Ω), we will choose the continuous representative f of f˙. For simplicity, we will
use the same notation for a class of functions and its representative chosen as before, the distinction between them
being made according to the context.
The following result constitutes a key step towards establishing the main result of this Note, viz., Theorem 2.1.
Theorem 1.1. Let Ω be a connected and simply connected open subset of Rd and let a point x 0 ∈ Ω and a matrix
∞
l
q,l
Y 0 ∈ Mq,l be fixed. Let the matrix fields Aα ∈ L∞
loc (Ω; M ) and Bα ∈ Lloc (Ω; M ) be given such that
∂α Aβ + Aα Aβ = ∂β Aα + Aβ Aα
∂α Bβ + Bα Aβ = ∂β Bα + Bβ Aα
in D Ω; Ml for all α, β ∈ {1, 2, . . . , d},
in D Ω; Mq,l for all α, β ∈ {1, 2, . . . , d}.
1,∞
(Ω; Mq,l ) to the system
(i) Then there exists a unique solution Y ∈ Wloc
∂α Y = Y Aα + Bα
Y x0 = Y 0.
for all α ∈ {1, 2, . . ., d},
(1)
(2)
(ii) If in addition the geodesic diameter of the set Ω is finite and the matrix fields Aα and Bα respectively belong
to L∞ (Ω; Ml ) and L∞ (Ω; Mq,l ), then the solution Y to the above system belongs to W 1,∞ (Ω; Mq,l ).
Sketch of proof. The proof of part (i) is based on techniques similar to those used for establishing Theorem 2.1
of [5], but generalized to systems of the form (1) with coefficients Aα , Bα in L∞
loc over Ω (in [5], we considered
systems of the form (1) with coefficients Aα ∈ L∞ over Ω and Bα = 0, the elements of the matrix field Aα being
the Christoffel symbols associated with a Riemannian metric in Ω).
(ii) Let c1 , c2 0 be two constants such that
Aα L∞ (Ω;Ml ) c1
and Bα L∞ (Ω;Mq,l ) c2 .
Let x ∈ Ω be a fixed, but otherwise arbitrary, point in Ω. Given ε > 0, the definition of the geodesic diameter DΩ
shows that there exists a path γ̃ joining x 0 to x such that L(γ̃ ) DΩ + ε.
Let r = 14 dist(γ̃ ([0, 1]), Ω c ) > 0, where Ω c = Rd \ Ω. Since γ̃ is uniformly continuous over [0, 1], there exists
numbers t0 , t1 , t2 , . . . , tN ∈ [0, 1] such that 0 = t0 < t1 < t2 < · · · < tN = 1 and
γ̃ (ti ) − γ̃ (ti−1 ) < r for all i ∈ {1, 2, . . . , N}.
ε
Let δ := min(r, 2N
). Then there exist points x 1 , x 2 , . . . , x N ∈ Ω ∩ B(γ̃ (ti ), δ) such that
Y γ (t) = Y γ (0) +
t
0
(Y Aα ) γ (s) + Bα γ (s) γα (s) ds
for all t ∈ [0, N],
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S. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 71–76
where γ = (γα ) : [0, N] → Ω maps the interval [i − 1, i] onto the segment [x i−1 , x i ] for all i ∈ {1, 2, . . . , N}.
Consequently,
√
√
Y γ (t) Y 0 + c2 dL(γ ) + c1 d
t
γ (s)Y γ (s) ds
for all t ∈ [0, N],
0
which next implies, by Gronwall inequality, that
√
N 0 √
Y x Y + c2 dL(γ ) ec1 dL(γ ) .
But the length of the path γ satisfies L(γ ) L(γ̃ ) + ε DΩ + 2ε thanks to the triangular inequality. Letting then
ε go to zero, we deduce from the above inequality (where the terms x N and L(γ ) depend on ε) that
√
√
Y (x) Y 0 + c2 dDΩ ec1 dDΩ ,
since x N goes to x when ε goes to zero. This implies that the matrix field Y belongs to the space L∞ (Ω, Mq,l )
and, since the derivatives of the field Y are given by ∂α Y = Y Aα + Bα , the field Y also belongs to the space
W 1,∞ (Ω, Mq,l ). ✷
2. The fundamental theorem of surface theory revisited
Throughout this section, Greek indices vary in the set {1, 2}, Latin indices vary in the set {1, 2, 3} and the
summation convention with respect to repeted indices is used in conjunction with these rules.
Let ω be a connected and simply-connected open subset of R2 and let y = (y1 , y2 ) denote a generic point
1,∞
2
in ω. Let there be given two matrix fields (aαβ ) ∈ Wloc
(ω; S2> ) and (bαβ ) ∈ L∞
loc (ω; S ) and define the Christoffel
symbols associated with (aαβ ) by letting
1
τ
Γαβ
= a τ σ (∂α aβσ + ∂β aσ α − ∂σ aαβ ),
2
τ
σ
where (a (y)) is the inverse of the matrix (aαβ (y)) and ∂α = ∂/∂yα . We recall that, according to the conventions
made in Section 1, the field (aαβ ) is the continuous representative of the class, still denoted by, (aαβ ). Therefore,
(aαβ (y)) ∈ S2> for all y ∈ ω. This implies that the inverse matrix (a τ σ ) is continuous over ω and that the Christoffel
τ belong to the space L∞ (ω).
symbols Γαβ
loc
Assume that the Gauss and Codazzi–Mainardi equations,
τ
τ
σ
σ
τ
− ∂β Γαγ
+ Γαβ
Γστγ − Γαγ
Γσβ
= bαβ bγτ − bαγ bβτ ,
∂γ Γαβ
σ
σ
bσ γ − Γαγ
bσβ = 0,
∂γ bαβ − ∂β bαγ + Γαβ
τ ,b
σ τ belong to L∞ (ω)).
are satisfied in D (ω) (these equations make sense in D (ω) since Γαβ
αβ and a
loc
Our aim is to prove the existence of a surface immersed in the three-dimensional Euclidean space whose first
and second fundamental forms are given by the matrix fields (aαβ ) and (bαβ ), respectively. Our main result is the
following:
Theorem 2.1. Assume that ω is a connected and simply-connected open subset of R2 and that the matrix fields
1,∞
2
(aαβ ) ∈ Wloc
(ω; S2> ) and (bαβ ) ∈ L∞
loc (ω; S ) satisfy the Gauss and Codazzi–Mainardi equations in D (ω). Then
2,∞
(ω, R3 ), unique up to proper isometries in R3 , such that
there exists a mapping θ ∈ Wloc
aαβ = ∂α θ · ∂β θ
and bαβ = ∂αβ θ ·
∂1 θ ∧ ∂2 θ
|∂1 θ ∧ ∂2 θ |
a.e. in ω.
If the geodesic diameter of the set ω is finite, (aαβ ) ∈ W 1,∞ (ω; S2> ), (bαβ ) ∈ L∞ (ω; S2 ), and (aαβ )−1 ∈
then the mapping θ belongs to W 2,∞ (ω, R3 ).
L∞ (ω, M2 ),
S. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 71–76
75
Sketch of proof. The proof is broken into six steps numbered (i) to (vi). Throughout the proof, we fix a point
y 0 ∈ ω, a vector θ 0 ∈ R3 , and two vectors a 0α ∈ R3 such that a 0α · a 0β = aαβ (y 0 ). We also define a unit normal vector
to a 0α by letting a 03 := (a 01 ∧ a 02 )/|a 01 ∧ a 02 |. Finally, we define the matrix fields Γα : ω → M3 by letting
 1

1
Γα1 Γα2
−bα1
2
2
Γα :=  Γα1
(3)
Γα2
−bα2  .
bα1 bα2
0
(i) The Gauss and Codazzi–Mainardi equations are satisfied if and only if the following matrix equation is
satisfied:
∂α Γβ + Γα Γβ = ∂β Γα + Γβ Γα in D ω; M3 , α, β ∈ {1, 2}.
3
3
Note that this equation make sense since Γα ∈ L∞
loc (ω; M ) ⊂ D (ω; M ). It suffices to prove that the Gauss and
Codazzi–Mainardi equations imply the relation
γ
τ
τ γ
∂α bβτ + Γαγ
bβ = ∂β bατ + Γβγ
bα .
γ
γ
From the definition of the Christoffel symbols, we deduce that ∂α aσ τ = Γασ aτ γ + Γατ aσ γ . Using this relation
and the identity bατ aτ γ = bαγ in the Codazzi–Mainardi equations, we obtain that
γ
γ
γ
γ
γ
γ
+ bβτ Γατ
aσ γ + Γβσ bαγ = aσ τ ∂β bατ + bαγ Γβσ + bατ Γβτ aσ γ + Γασ
bβγ .
aσ τ ∂α bβτ + bβγ Γασ
Multiplying the previous relation with a σ ψ gives the desired relation.
1,∞
(ii) There exists a solution F ∈ Wloc
(ω, M3 ) to the system
∂α F = F Γα in D ω; M3 and F y 0 = F 0 ,
(4)
where F 0 ∈ M3 is the matrix whose i-th column is a 0i ∈ R3 .
It suffices to apply Theorem 1.1 to the above system, the assumptions of this theorem being satisfied thanks to
the previous step.
2,∞
(iii) Let a i denote the i-th column of the matrix field F . Then there exists a solution θ ∈ Wloc
(ω; R3 ) to the
system
∂α θ = a α in ω and θ y 0 = θ 0 .
(5)
Since the field F satisfies the system (4), one can see that
σ
σ
∂α a β = Γαβ
a σ + bαβ a 3 = Γβα
a σ + bβα a 3 = ∂β a α
in D ω; R3 .
1,∞
Since a α ∈ Wloc
(ω; R3 ) and ω is simply connected, we can apply Theorem 1.1 to problem (5) to show that it
1,∞
2,∞
(ω; R3 ). Since ∂α θ = a α , the mapping θ belongs in fact to Wloc
(ω; R3 ).
possesses a solution θ ∈ Wloc
(iv) The mapping θ satisfies the relations
aαβ = ∂α θ · ∂β θ
and
bαβ = ∂αβ θ ·
∂1 θ ∧ ∂2 θ
|∂1 θ ∧ ∂2 θ|
a.e. in ω.
p
Define aα3 (y) = a3α (y) = 0 and a33 (y) = 1 for all y ∈ ω and let Γαi denote the coefficients of the matrix Γα
p
p
defined by relation (3). Then one can see that ∂α aij = Γαi apj + Γαj aip . This implies that the functions Xij : ω → R
defined by Xij (y) := a i (y) · a j (y) − aij (y) for all y ∈ ω satisfy the system
p
p
∂α Xij = Γαi Xpj + Γαj Xip
Xij y 0 = 0.
in L∞ (ω),
76
S. Mardare / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 71–76
Let A := {y ∈ ω; Xij (y) = 0 ∈ M3 }. This subset of ω is non-empty (since y 0 ∈ A) and closed in ω (since the
functions Xij : ω → M3 are continuous). But it is also open in ω since an inequality of Poincaré type shows that,
for all y ∈ ω, there exists an open ball B(x, r) ω such that Xij = 0 over this ball. Since the set A is non-empty,
closed, and open in ω, the conectedness of ω implies that A = ω. Hence a i · a j = aij in ω. These relations show
that the first relation announced in step (iv) holds. They also show that
a1 ∧ a2
a1 ∧ a2
either a 3 (y) =
(y) or a 3 (y) = −
(y), y ∈ ω, and F T F = (aij ) in ω.
|a 1 ∧ a 2 |
|a 1 ∧ a 2 |
Therefore, (det F (y))2 > 0, which implies in particular that det F (y) = 0 for all y ∈ ω. Since det F (y 0 ) > 0 and
the function det F is continuous over the connected set ω, we have det F > 0 over ω. Hence
a 1 (y) ∧ a 2 (y)
for all y ∈ ω.
a 3 (y) =
|a 1 (y) ∧ a 2 (y)|
σ a + b a . Hence
On the other hand, step (ii) implies that ∂α a β = Γαβ
σ
αβ 3
bαβ = ∂α a β · a 3 = ∂αβ θ ·
∂1 θ ∧ ∂2 θ
.
|∂1 θ ∧ ∂2 θ |
(v) The mapping θ is unique up to proper isometries of R3 .
Let another mapping φ satisfy the conditions of the theorem. Let F0 ∈ M3 be the matrix whose i-th column is
0
a i and let E0 ∈ M3 be the matrix whose first, second, and third, column are respectively ∂1 φ(y 0 ), ∂2 φ(y 0 ), and
(∂1 φ(y 0 ) ∧ ∂2 φ(y 0 ))/|∂1 φ(y 0 ) ∧ ∂2 φ(y 0 )|. Define the mapping θ̂ : ω → R3 by letting
for all y ∈ ω,
(6)
θ̂(y) = θ 0 + Q φ(y) − φ y 0
where Q := F0 E0−1 . Then one can see that the matrix Q is a proper ortogonal matrix and that the mapping θ̂
satisfies the conditions of the theorem. Therefore
= F
Γα in ω,
∂α F = F Γα and ∂α F
) is the matrix whose i-th column is a i (resp. â i ), with self-explanatory notations.
where F (resp. F
(y 0 ) = F0 . Then
On the other hand, relation (6) shows that θ̂ (y 0 ) = θ 0 and â i (y 0 ) = a 0i . Hence F (y 0 ) = F
0
Theorem 1.1 shows that F = F in Ω, which next implies that ∂α θ = ∂α θ̂ . Since θ(y ) = θ̂ (y 0 ) = θ 0 and ω is
connected, we finally obtain that θ = θ̂ , which means that the mapping θ satisfying the conditions of the theorem
is unique up to proper isometries of R3 .
(vi) The second part of the theorem is a consequence of part (ii) of Theorem 1.1. ✷
Remark 1. In [2], Hartman and Wintner established the fundamental theorem of surface theory under the
assumptions that aαβ ∈ C 1 (ω) and bαβ ∈ C 0 (ω) together satisfy the Gauss and Codazzi–Mainardi equations in
an integral form. Their assumptions are weaker than those used in the classical framework (where aαβ ∈ C 2 (ω)
and bαβ ∈ C 1 (ω)) but stronger than those of Theorem 2.1.
References
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167–185.
[2] P. Hartman, A. Wintner, On the fundamental equations of differential geometry, Amer. J. Math. 72 (1950) 757–774.
[3] H. Jacobowitz, The Gauss–Codazzi equations, Tensor (N.S.) 39 (1982) 15–22.
[4] W. Klingenberg, A Course in Differential Geometry, Springer-Verlag, Berlin, 1978.
[5] S. Mardare, On isometric immersions of a Riemannian space under weak regularity assumptions, C. R. Acad. Sci. Paris, Ser. I 337 (2003)
785–790.
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