Boundaries of Graphs of Harmonic Functions
Harmonic functions u: Rn → Rm are equivalent to integral manifolds of an exterior differential system with independence condition (M,I,ω). To this system one associates the space of conservation laws C. They provide necessary conditions for g: Sn–1 → M to be the boundary of an integral submanifold....
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2009 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2009
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149134 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Boundaries of Graphs of Harmonic Functions / D. Fox // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 8 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859807288543412224 |
|---|---|
| author | Fox, D. |
| author_facet | Fox, D. |
| citation_txt | Boundaries of Graphs of Harmonic Functions / D. Fox // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Harmonic functions u: Rn → Rm are equivalent to integral manifolds of an exterior differential system with independence condition (M,I,ω). To this system one associates the space of conservation laws C. They provide necessary conditions for g: Sn–1 → M to be the boundary of an integral submanifold. We show that in a local sense these conditions are also sufficient to guarantee the existence of an integral manifold with boundary g(Sn–1). The proof uses standard linear elliptic theory to produce an integral manifold G: Dn → M and the completeness of the space of conservation laws to show that this candidate has g(Sn–1) as its boundary. As a corollary we obtain a new elementary proof of the characterization of boundaries of holomorphic disks in Cm in the local case.
|
| first_indexed | 2025-12-07T15:17:19Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 068, 8 pages
Boundaries of Graphs of Harmonic Functions?
Daniel FOX
Mathematics Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK
E-mail: foxdanie@gmail.com
Received March 06, 2009, in final form June 16, 2009; Published online July 06, 2009
doi:10.3842/SIGMA.2009.068
Abstract. Harmonic functions u : Rn → Rm are equivalent to integral manifolds of an
exterior differential system with independence condition (M, I, ω). To this system one asso-
ciates the space of conservation laws C. They provide necessary conditions for g : Sn−1 →M
to be the boundary of an integral submanifold. We show that in a local sense these condi-
tions are also sufficient to guarantee the existence of an integral manifold with boundary
g(Sn−1). The proof uses standard linear elliptic theory to produce an integral manifold
G : Dn → M and the completeness of the space of conservation laws to show that this
candidate has g(Sn−1) as its boundary. As a corollary we obtain a new elementary proof of
the characterization of boundaries of holomorphic disks in Cm in the local case.
Key words: exterior differential systems; integrable systems; conservation laws; moment
conditions
2000 Mathematics Subject Classification: 35J05; 35J25; 53B25
1 Introduction
On Cm with complex coordinates z1, . . . , zm, let
Ω(p,q) =
{
fdza1 ∧ · · · ∧ dzap ∧ dzb1 ∧ · · · ∧ dzbq | ai, bj ∈ {1, . . . ,m}, f ∈ C∞(Cn,C)
}
.
A holomorphic curve in Cm is a holomorphic map φ : X → Cm, where X is a Riemann surface.
This is equivalent to a real 2-dimensional surface G : X → Cm for which
G∗
(
Ω(2,0)
)
= G∗
(
Ω(0,2)
)
= 0. (1.1)
This equivalence can be demonstrated using the following argument. Suppose that locally G(X)
can be written as a smooth graph za = Ga(z1, z̄1). Then G∗(dza∧dz1) = ∂za
∂z̄1 dz̄1∧dz1 and
thus (1.1) implies ∂za
∂z̄1 = 0.
The 1-forms ϕ satisfying dϕ ∈ Ω(2,0) ⊕ Ω(2,0) provide moment conditions for the boundaries
of holomorphic curves. That is, for any holomorphic curve G : X → Cm with boundary g :
∂X → Cm we find∫
∂X
g∗ϕ =
∫
X
G∗(dϕ) = 0. (1.2)
In particular, if ϕ ∈ Ω(1,0) is holomorphic then dϕ ∈ Ω(2,0) and so the integral of any holomorphic
(1, 0)-form around the boundary of a holomorphic curve is zero. For more details see Example 4
of Section 1.1 of [3].
Using complex function theory Wermer [8] showed that, in the analytic category, the moment
conditions provided by holomorphic (1, 0)-forms are sufficient to characterize the boundaries of
?This paper is a contribution to the Special Issue “Élie Cartan and Differential Geometry”. The full collection
is available at http://www.emis.de/journals/SIGMA/Cartan.html
mailto:foxdanie@gmail.com
http://dx.doi.org/10.3842/SIGMA.2009.068
http://www.emis.de/journals/SIGMA/Cartan.html
2 D. Fox
holomorphic disks. That is, if D ⊂ C is a domain with real analytic boundary ∂D, g : ∂D → Cm
is real analytic, and
∫
∂D
g∗ϕ = 0
for all holomorphic (1, 0)-forms on Cm, then g(∂D) is the boundary of a holomorphic disk
G : D → Cm. This was generalized to higher dimensional domains for graphs by Bochner [1]
and then a complete treatment was given by Harvey and Lawson [5, 6] using geometric measure
theory.
We generalize this in another direction. Using conservation laws (the analogue of the holomor-
phic (1, 0)-forms) we characterize the boundaries of the graphs of 1-jets of harmonic functions
∆u = 0 where u : D → Rm has domain D ⊂ Rn with C2 boundary (Theorem 3.1). From Theo-
rem 3.1 we extract Corollary 4.2 which gives a new proof of the characterization of boundaries
of embedded holomorphic disks.
We work using exterior differential systems (EDS) and their characteristic cohomology, the
relevant aspects of which we now review1. To every partial differential equation one can associate
an EDS with independence condition. (In Section 2 we do this for the Laplace equation.) An
EDS is a pair (M, I) where M is a manifold and I ⊂ Ω∗(M,R) is a homogeneous differential
ideal. An independence condition is a totally decomposable nowhere vanishing form ω whose
degree is the same as the dimension of the domain of the PDE. The integral submanifolds,
that is, those submanifolds for which the ideal pulls back to be zero but ω pulls back to be
nonzero, are equivalent to solutions of the PDE. In this sense one can associate to every PDE
a submanifold geometry defined by a non-degeneracy condition and the vanishing of differential
forms.
Each EDS (M, I) (not necessarily with an independence condition) defines cohomology groups
Hp(Ω∗/I,d) on M known as the characteristic cohomology [3]. A certain graded piece of the
characteristic cohomology constitutes the space of conservation laws. Suppose that one is interes-
ted in n dimensional integral submanifolds of (M, I). To each EDS one associates a positive
integer l, known as its characteristic number, that measures how overdetermined the EDS is
(see Section 4.2 of [3] for the definition). In [3] it is shown that over contractible open sets
Hp(Ω∗/I,d) = 0 for p < n− l when I is involutive2. The space of conservation laws C is defined
to be the first nontrivial cohomology group:
C = Hn−l(Ω∗/I,d).
For systems (M, I) that arise from Lagrangians l is equal to 1. In this case, a class in C is
represented by a form ϕ ∈ Ωn−1(M,R) that is not in the ideal but for which dϕ ∈ I. From Stokes’
theorem such forms lead to moment conditions on boundaries just as they did for holomorphic
curves in equation (1.2).
In Section 2 we introduce the well known EDS with independence condition associated to the
Laplace equation and a useful subspace of conservation laws. In Section 3 we use this set up to
prove Theorem 3.1, which characterizes the boundaries of graphs of harmonic functions using
the moment conditions arising from conservation laws. In Section 4 we deduce Corollary 4.2
which characterizes the boundaries of holomorphic disks that are graphs.
1For an introduction to EDS the reader might enjoy [7] or [2]. The fundamental paper on characteristic
cohomology is [3].
2See [7] or [2] for the definition of involutivity.
Boundaries of Graphs of Harmonic Functions 3
2 The EDS for graphs of harmonic functions
and its conservation laws
In this section we introduce the exterior differential system and the space of conservation laws
we will need. For the Laplace equation
∆u = 0 (2.1)
where u : Rn → Rm and ∆ = ( ∂
∂x1 )2 + · · ·+ ( ∂
∂xn )2, we define
M = J1(Rn,R) = Rn × Rm × Rm ⊗ (Rn)∗
to be the first jet space of maps from Rn to Rm. It has natural coordinates (xi, ua, pa
i ). The
relevant differential ideal is
I = 〈θ,dθ, ψ〉,
which is algebraically generated by the components of the vector valued differential forms
θ = du− pdx ∈ Ω1(M,Rm),
dθ = −dp ∧ dx ∈ Ω2(M,Rm),
ψ = dpi ∧ dx(i) ∈ Ωn(M,Rm),
where
dx(i) = ∗dxi = (−1)i−1dx1 ∧ · · · ∧ d̂xi ∧ · · · ∧ dxn.
Above ∗ is the Hodge star on Rn with respect to the standard flat metric and volume form
ω = dx1∧ · · · ∧dxn. We will use a mixture of index and matrix notation. For example, pdx is
the Rm-valued 1-form with components
∑n
i=1 p
a
i dx
i and in dpi∧dx(i) the sum over i is implicit.
The exterior differential system (M, I, ω) for harmonic functions is involutive with charac-
teristic number l = 1. Solutions to the Laplace equation (2.1) are equivalent to n-dimensional
embedded submanifolds G : X → M such that G∗(I) = 0 and G∗ω 6= 0. A k-dimensional
submanifold F : U →M is defined to be isotropic if F ∗(θ) = 0.
The general theory of characteristic cohomology of an exterior differential system indi-
cates that for the EDS associated to Laplace’s equation the space of conservation laws is
C = Hn−1(Ω∗/I,d). The short exact sequence
0 → I → Ω → Ω/I → 0
induces a long exact sequence in cohomology which, due to the vanishing Hs
dR(M,R) = 0 for
s > 0, produces the isomorphism
ι : Hn−1(Ω∗/I,d) → Hn(I,d).
The map is given by exterior differentiation: a class in Hn−1(Ω∗/I,d) is represented by a diffe-
rential form ϕ ∈ Ωn−1(M) such that dϕ ∈ I. So if [ϕ] ∈ Hn−1(Ω∗/I,d) then [dϕ] ∈ Hn(I,d).
The class [dϕ] or its representative dϕ is referred to as the differentiated conservation law
and [ϕ] or its representative ϕ as the undifferentiated conservation law. We will need the
explicit form of differentiated conservation laws and so turn to them now.
An element of Hn(I,d) is represented by a closed n-form in I. Any element in I ∩ Ωn(M,R)
is determined by an Rm-valued (n− 1)-form ρ, an Rm-valued (n− 2)-form σ, and an Rm-valued
function H, by the formula
Φ = tρ ∧ θ + tσ ∧ dθ − tHψ.
4 D. Fox
We seek ρ, σ, H that make Φ closed, but first we make a standard simplification. Using the
relation
Φ =
(
tρ− (−1)n−2d tσ
)
∧ θ − tHψ + (−1)n−2d
(
tσ ∧ θ
)
,
and the fact that we are really only interested in the class [Φ] ∈ Hn(I,d), we see that, for any
class in Hn(I,d), we can always find a representative for which σ = 0.
The following special set of conservation laws will be sufficient for studying the boundaries
of integral manifolds that satisfy the independence condition. If H : Rn → Rm is a harmonic
function and ρ = (−1)n ∗ dH, then
Φ = tρ ∧ θ + tHψ (2.2)
is closed and represents a class in Hn(I,d).
Conservation laws are a natural source of moment conditions. Let ϕ be an undifferentiated
conservation law. By stokes theorem∫
∂D
g∗ϕ =
∫
D
G∗(dϕ)
for any G : Dn →M with g = G|∂D
. Thus if G(D) is integral∫
∂D
g∗ϕ = 0.
In the next section we show that in a local sense the moment conditions coming from conservation
laws are complete for the harmonic function system.
3 Boundaries of graphs of harmonic functions
Let π : M → Rn × Rm be the standard projection (x, u, p) 7→ (x, u).
Theorem 3.1. Let g : Sn−1 →M be a C2 isotropic submanifold such that x◦g : Sn−1 → Rn is an
embedding and x ◦ g(Sn−1) is the boundary of a domain D ⊂ Rn. Then there exists G : D →M
such that
G∗(I) = 0, G(∂D) = g
(
Sn−1
)
if and only if∫
g(Sn−1)
ϕ = 0 ∀ϕ ∈ C. (3.1)
The proof relies on the standard theory of linear elliptic PDE to produce an integral sub-
manifold and then uses the moment conditions arising from conservation laws to show that it
has the desired boundary. We use the following existence and uniqueness result [4].
Theorem 3.2. Let D ⊂ Rn be a bounded domain with C2 boundary ∂D and let v : ∂D → Rm
be continuous. Then there is a unique function V : D → Rm satisfying V|∂D
= v and ∆V = 0.
Proof of Theorem 3.1. Let v = u|∂D
. Then by Theorem 3.2 there exists a unique smooth
function V : D → Rm such that ∆V = 0 and V|∂D = v. Let J1(V ) : D → M be the 1-jet of V ,
J1(V )(x) = (x, V (x),∇V (x)). By construction π ◦ g(S1) = π ◦ J1(V )(∂D). We will now show
that
J1(V )(∂D) = g
(
Sn−1
)
.
Once this is accomplished, G = J1(V ) is the desired solution.
Boundaries of Graphs of Harmonic Functions 5
Let g̃ = J1(V )|∂D ◦ x ◦ g : Sn−1 →M . Then g̃, g : Sn−1 →M are isotropic submanifolds that
both satisfy the moment conditions (3.1) and π ◦ g = π ◦ g̃. Write
g(s) =
(
x(s), u(s), A(s)
)
, g̃(s) =
(
x(s), u(s), Ã(s)
)
,
where à = ∇V|∂D ◦ x ◦ g and s ∈ Sn−1. For i = 1, . . . , n− 1 let si be local coordinates on Sn−1.
The fact that g : Sn−1 →M is isotropic implies that
0 = g∗θa =
(
∂ua
∂si
−Aa
j
∂xj
∂si
)
dsi
so that
∂ua
∂si
= Aa
j
∂xj
∂si
.
Similarly we find that
∂ua
∂si
= Ãa
j
∂xj
∂si
.
Let N be the outward unit normal of x ◦ g(Sn−1) ⊂ Rn so that
0 = Ni
∂xi
∂sj
.
We can now decompose Ã(s) = A(s)+ζ(s) tN(s) ∈ Rm⊗(Rn)∗ for some function ζ : Sn−1 → Rm.
Let
χ : [0, 1]× Sn−1 →M
be the smooth map given in coordinates by
χ(r, s) = (x(s), u(s), p(r, s)),
where
p(r, s) = A(s) + (1− r)ζ(s) tN(s).
The cylinder χ(Sn−1 × I) has the following properties:
• χ(0, s) = g̃(s);
• χ(1, s) = g(s);
• χ∗θa = 0;
• χ(Sn−1 × I) ⊂ π−1
(
π ◦ g(Sn−1)
)
.
Therefore ∂(χ(Sn−1 × I)) = g(Sn−1) − g̃(Sn−1) and because both g and g̃ satisfy the moment
conditions induced by conservation laws Φ = dϕ,∫
χ(Sn−1×I)
Φ =
∫
g(Sn−1)
ϕ−
∫
g̃(Sn−1)
ϕ = 0. (3.2)
Now assume that Φ is of the type specified in (2.2). Because χ is contact,
χ∗(Φ) = χ∗( tH ψ)
6 D. Fox
and we calculate that
χ∗(ψa) = χ∗(dpa
i ∧ dx(i)) = −ζaNidr ∧χ∗(dx(i)) = −ζadr ∧χ∗(N ω) = −ζadr ∧ g∗(N ω).
The definition of χ also implies that χ∗(H) = g∗(H). By Theorem 3.2, the harmonic function
H : Bn → Rm, where Bn is the closed ball with boundary Sn−1, is uniquely determined by
choosing an arbitrary continuous function, h : Sn−1 → Rm, and specifying that g∗(H) = h.
We now calculate∫
χ(Sn−1×I)
Φ =
∫
I×Sn−1
χ∗( tHψ) = −
∫
I×Sn−1
thζdr ∧ g∗(N ω)
= −
∫
Sn−1
thζ ·
(∫ 1
0
dr
)
g∗(N ω) = −
∫
Sn−1
thζg∗(N ω).
Using (3.2) this implies that
0 =
∫
Sn−1
thζg∗(N ω)
for all continuous functions h : Sn−1 → Rm. The (n− 1)-form g∗(N ω) is the induced volume
form on x ◦ g : Sn−1 → Rn. Therefore, using the assumption that x ◦ g is an embedding, we can
conclude that ζ = 0. This implies that A = Ã and thus g̃ = g. �
4 Boundaries of holomorphic disks
On Cm let Ĵ = 〈Ω2,0 ⊕ Ω0,2〉. Then a holomorphic curve is a real surface that is an integral
manifold for Ĵ. Using distinct approaches, Wermer [8] and Harvey and Lawson [5] prove
Theorem 4.1. Let Y ⊂ Cm be a compact, connected, oriented submanifold of dimension one
and of class C2. Suppose that∫
Y
ϕ = 0
for all holomorphic 1-forms ϕ ∈ Ω(1,0). Then there exists an irreducible holomorphic curve
X ∈ Cm \ Y such that ∂X = Y .
Wermer provided the first such result using complex function theory. Harvey and Lawson
actually prove a much stronger result that characterizes the boundaries of complex submanifolds
in which the boundary may have multiple connected components. We can deduce a local version
of this from Theorem 3.1.
Corollary 4.2. Let g : S1 → Cm be a C2 embedded curve for which there exists a projection to
a complex line ζ : Cm → C such that ζ ◦ g(S1) is the boundary of a domain D ⊂ C. Then there
exists a holomorphic map G : D → Cm such that
G(∂D) = g
(
S1
)
if and only if∫
g(S1)
ϕ = 0 (4.1)
for all holomorphic 1-forms ϕ on Cm.
Boundaries of Graphs of Harmonic Functions 7
Proof. We make an integrable extension and then rely on Theorem 3.1. Let (M, I, ω) be the
system for harmonic functions u : R2 → Rm−1, so that
M = R2 × Rm−1 × Rm−1 ⊗
(
R2
)∗
.
Let
ζ : M → R2 × Rm−1 ⊗
(
R2
)∗
be the standard projection and identify the image with Cm by defining the holomorphic coordi-
nates za = pa
1 +
√
−1pa
2 for a = 1, . . . ,m− 1 and zm = x1 −
√
−1x2. We define the differential
ideal J = 〈dza∧dzm〉 on Cm and let Ω = −
√
−1
2 dzm∧dz̄m define an independence condition. The
integral manifolds of (Cm, J,Ω) are holomorphic disks that can be graphed as functions of zm. It
is readily checked that ζ∗(dza∧dzm) = dθa −
√
−1ψa and that ζ∗(Ω) = ω. Therefore the projec-
tion ζ : M → Cm makes (M, I, ω) an integrable extension of (Cm, J,Ω): that is, I is algebraically
generated by ζ∗J and the 1-forms θa, and the independence conditions are compatible.
First we must show that when the conservation laws of (Cm, J,Ω) are pulled back using ζ,
they surject onto the special class of conservation laws used in the proof of Theorem 3.1. Then
we must show that we can lift the supposed boundary g : S1 → Cm to f : S1 →M so that f(S1)
satisfies all of the moment conditions for (M, I, ω). For the first part we must show that for any
harmonic function H : D → Rm−1 and Φ defined from H as in (2.2), there is a conservation
law for J that pulls back under ζ to give the same class as [Φ] ∈ H2(I,d). To see this, let H
be the desired harmonic function and let K : D → Rm−1 be its harmonic conjugate, so that
Ka +
√
−1Ha is a holomorphic function of zm. Let
Υ =
(
Ka +
√
−1Ha
)
dza ∧ dzm ∈ J.
Then dΥ = 0 and Υ is a differentiated conservation law for J. When pulled up to M we find
ζ∗(Re(Υ)) = Kadθa +Haψa.
We can rewrite this as
ζ∗(Re(Υ)) = tρ ∧ θ + tHψ + d
(
tHθ
)
,
where ρ = ∗dH, which is consistent with (2.2). Since dΥ = 0, the form tρ∧θ + tHψ is also
closed and therefore a differentiated conservation law of the form (2.2) with the desired harmonic
functionH. Therefore if g : S1 → Cm satisfies all of the moment conditions from the conservation
laws of J, then an appropriate lift to M will satisfy all of the moment conditions for I that are
needed to apply Theorem 3.1.
Now we turn to the lift. Any lift f : S1 →M of g is defined by choosing a map u : S1 → Rm−1.
For the lift to be contact we must have
0 = f∗(θa) =
∂ua
∂s
−
(
pa
1
∂x1
∂s
+ pa
2
∂x2
∂s
)
,
so define
ua(s) =
∫ s
0
(
pa
1(t)
∂x1
∂t
+ pa
2(t)
∂x2
∂t
)
dt.
This is a periodic function since g satisfies the moment condition∫
g(S1)
(
pa
1dx
1 + pa
2dx
2
)
= 0.
Now by Theorem 3.1 there exists F : D2 → M such that F|∂D = f and F ∗(I) = 0. Then
G = ζ ◦ F : D2 → Cm is the desired holomorphic disk. �
8 D. Fox
Acknowledgements
I would like to thank Dominic Joyce and Yinan Song for useful conversations. This work was
carried out with the support of the National Science Foundation grant OISE-0502241.
References
[1] Bochner S., Analytic and meromorphic continuation by means of Green’s formula, Ann. of Math. (2) 44
(1943), 652–673.
[2] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems,
Springer-Verlag, New York, 1991.
[3] Bryant R.L., Griffiths P.A., Characteristic cohomology of differential systems. I. General theory, J. Amer.
Math. Soc. 8 (1995), 507–596.
[4] Gilbarg D., Trudinger N.S., Elliptic partial differential equations of second order, Springer-Verlag, Berlin,
2001.
[5] Harvey F.R., Lawson H.B. Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975),
223–290.
[6] Harvey F.R., Lawson H.B. Jr., On boundaries of complex analytic varieties. II, Ann. Math. (2) 106 (1977),
213–238.
[7] Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior
differential systems, Graduate Studies in Mathematics, Vol. 61, American Mathematical Society, Providence,
RI, 2003.
[8] Wermer J., The hull of a curve in Cn, Ann. of Math. (2) 68 (1958), 550–561.
1 Introduction
2 The EDS for graphs of harmonic functions and its conservation laws
3 Boundaries of graphs of harmonic functions
4 Boundaries of holomorphic disks
References
|
| id | nasplib_isofts_kiev_ua-123456789-149134 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:17:19Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fox, D. 2019-02-19T17:36:22Z 2019-02-19T17:36:22Z 2009 Boundaries of Graphs of Harmonic Functions / D. Fox // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 8 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 35J05; 35J25; 53B25 https://nasplib.isofts.kiev.ua/handle/123456789/149134 Harmonic functions u: Rn → Rm are equivalent to integral manifolds of an exterior differential system with independence condition (M,I,ω). To this system one associates the space of conservation laws C. They provide necessary conditions for g: Sn–1 → M to be the boundary of an integral submanifold. We show that in a local sense these conditions are also sufficient to guarantee the existence of an integral manifold with boundary g(Sn–1). The proof uses standard linear elliptic theory to produce an integral manifold G: Dn → M and the completeness of the space of conservation laws to show that this candidate has g(Sn–1) as its boundary. As a corollary we obtain a new elementary proof of the characterization of boundaries of holomorphic disks in Cm in the local case. This paper is a contribution to the Special Issue “Elie Cartan and Differential Geometry”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Boundaries of Graphs of Harmonic Functions Article published earlier |
| spellingShingle | Boundaries of Graphs of Harmonic Functions Fox, D. |
| title | Boundaries of Graphs of Harmonic Functions |
| title_full | Boundaries of Graphs of Harmonic Functions |
| title_fullStr | Boundaries of Graphs of Harmonic Functions |
| title_full_unstemmed | Boundaries of Graphs of Harmonic Functions |
| title_short | Boundaries of Graphs of Harmonic Functions |
| title_sort | boundaries of graphs of harmonic functions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149134 |
| work_keys_str_mv | AT foxd boundariesofgraphsofharmonicfunctions |