TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 363, Number 4, April 2011, Pages 1789–1804
S 0002-9947(2010)04844-5
Article electronically published on November 17, 2010
HODGE-LAPLACE EIGENVALUES OF CONVEX BODIES
ALESSANDRO SAVO
Abstract. We give upper and lower bounds of the first eigenvalue of the
Hodge Laplacian acting on smooth p-forms on a convex Euclidean domain for
the absolute and relative boundary conditions. In particular, for the absolute
conditions we show that it behaves like the squared inverse of the p-th longest
principal axis of the ellipsoid of maximal volume included in the domain (the
John ellipsoid). Using John’s theorem, we then give a spectral geometric
interpretation of the bounds and relate the eigenvalues with the largest volume
of a p-dimensional section of the domain.
1. Introduction
1.1. Notation. Let M be a compact Riemannian manifold with smooth boundary,
and let
Δ = dδ + δd
be the Hodge Laplacian acting on smooth differential p-forms on M . Here d is the
exterior differential and δ is the co-differential (the formal adjoint of d with respect
to the L2 −inner product of forms). We consider the eigenvalue problem for the
absolute boundary conditions:
⎧
⎪
⎨ Δω = μω,
iN ω = 0 on ∂M,
(1.1)
⎪
⎩ i dω = 0 on ∂M,
N
where ω is a p-form, N is the unit vector, normal to the boundary and pointing
[p]
inward, and iN denotes interior multiplication. We denote by μ1 the first positive
eigenvalue of the absolute problem (1.1).
The dual boundary conditions are the relative ones for which J∂M ω = J∂M δω =
[p]
0, with J∂M denoting restriction of a form to ∂M . We denote by λ1 the first
positive eigenvalue of the relative problem on p-forms. The Hodge operator
intertwines the two boundary conditions, and in particular
[p]
[n−p]
λ 1 = μ1
.
Received by the editors June 10, 2008.
2010 Mathematics Subject Classification. Primary 58J50.
Key words and phrases. Laplacian on forms, eigenvalues, convex bodies, John ellipsoid.
This work was partially supported by the COFIN program of MIUR and by GNSAGA (Italy).
c
2010
American Mathematical Society
Reverts to public domain 28 years from publication
1789
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1790
ALESSANDRO SAVO
Hence giving bounds for the absolute conditions automatically implies bounds on
the relative one; moreover, one has that
[0]
μ1 = first positive eigenvalue on functions for the Neumann condition,
[0]
[n]
λ1 = μ1 = first (positive) eigenvalue on functions for the Dirichlet condition.
In fact the absolute (resp. relative) boundary conditions on p-forms generalize the
Neumann (resp. Dirichlet) conditions for functions. Moreover, the vector space of
harmonic p-forms satisfying the absolute (resp. relative) conditions is isomorphic
to the p-th de Rham absolute (resp. relative) cohomology space of M . General
facts about the Laplacian on forms can be found in [15] and [2].
The scope of this paper is to give upper and lower bounds of the first positive
[p]
eigenvalue μ1 when the manifold is a convex domain in the Euclidean space Rn ; see
[p]
Theorems 1.1, 3.1 and 3.2 below. A lower bound of μ1 for Riemannian manifolds
whose boundary has suitable degree of convexity is given in [7]. For this and other
results on the Hodge Laplacian on manifolds with non-empty boundary, see also
the survey paper [8]. Recent extrinsic estimates for submanifolds can be found in
[13].
1.2. The main estimate. Now let Ω be a convex Euclidean domain of dimension
n. It has been proven in Theorem 2.6 of [7] that the sequence of the first eigenvalues
[p]
μ1 is non-decreasing with respect to the degree p; that is,
[0]
[1]
[2]
[n]
μ1 = μ1 ≤ μ1 ≤ · · · ≤ μ 1 .
(1.2)
[0]
[1]
A brief comment on the equality μ1 = μ1 : by differentiating eigenfunctions, one
[0]
[1]
verifies the inequality μ1 ≥ μ1 (for an arbitrary manifold); however, equality does
hold under the convexity assumption. This fact somehow shows that for a convex
[1]
[2]
[n]
Euclidean domain the significant (first) eigenvalues are μ1 , μ1 , . . . , μ1 which, by
abuse of language, will also be called the “fundamental tones” of Ω.
Our estimates show the following geometric property of the eigenvalues: up to
constants (depending only on the degree and the dimension) the “p-th fundamental
wavelength” 1[p] is equivalent to the p-th longest principal axis of Ω, defined here
μ1
as that of the unique ellipsoid of maximal volume included in Ω.
Before giving the precise statement, let us also remark that the bounds in this
paper could be seen as a generalization to p-forms of the following classical bounds
for the Laplacian on functions, namely, the Payne-Weinberger inequality [12], valid
for convex domains,
[0]
μ1 (Ω) ≥
(1.3)
π2
diam(Ω)2
and the inequality due to Hersch [9] (later generalized by Li and Yau [11]), which
is valid more generally when the mean curvature of ∂Ω is everywhere non-negative:
(1.4)
[n]
[0]
μ1 (Ω) = λ1 (Ω) ≥
π2
,
4R(Ω)2
where R(Ω) is the inner radius of Ω (that is, the radius of a largest ball included
in Ω).
We remark that the above inequalities have been extended by Li and Yau to
Riemannian manifolds with non-negative Ricci curvature. From (1.2)-(1.4) and the
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HODGE-LAPLACE EIGENVALUES OF CONVEX BODIES
1791
[n]
monotonicity of the first Dirichlet eigenvalue (that is, μ1 ) with respect to inclusion
we get, for all degrees p,
[0]
λ1 (B n )
π2
[p]
≤
μ
(Ω)
≤
,
1
diam(Ω)2
R(Ω)2
(1.5)
[0]
where λ1 (B n ) is the first Dirichlet eigenvalue of the unit ball in Rn . In [5], Pierre
Guerini proves the lower bound:
(1.6)
[p]
μ1 (Ω) ≥
max{p(n − p), n − 1}
1
·
,
ne3
diam(Ω)2
which in many cases improves the one in (1.5).
Now, looking at the monotonicity property (1.2), one expects that it would be
possible to improve the lower bounds in (1.5) and (1.6) by showing that
an,p
[p]
(1.7)
μ1 (Ω) ≥
Dp (Ω)2
for a positive constant an,p depending only on the degree and the dimension and for
a non-increasing sequence of geometric invariants Dp (Ω), depending on the degree
p, reducing to (half) the diameter when p = 1 and to the inner radius when p = n.
Our main estimate, Theorem 1.1 below, states that this is in fact possible; moreover, one also has an upper bound of the same type.
Let us now define the invariants Dp (Ω). If Ω is a true ellipsoid, then Dp is just
the p-th longest principal axis of Ω. If Ω is an arbitrary convex body, then Dp will
be the p-th longest principal axis of the ellipsoid E of maximal volume included in
Ω. By a well-known theorem in convex geometry, due to Fritz John (see [10] or
also [1]), E is unique, and moreover,
E ⊆ Ω ⊆ nE,
(1.8)
where the homothety is taken with respect to the center of E. In other words, if
γ(Ω) = inf{t ≥ 1 : Ω ⊆ tE},
√
then γ(Ω) ≤ n (and actually γ(Ω) ≤ n, provided that Ω is centrally symmetric).
E is known in the literature as the John ellipsoid of Ω.
Let us now state the main bound in precise terms. We state it for the degree p
in the range 2, . . . , n − 1, but the type of the bound also holds for p = 1 and p = n:
these cases correspond in fact to the Laplacian on functions, the constants being
given by the inequalities (1.3), (1.4) just mentioned and by an upper estimate due
to Cheng (however, see Remark 1.2 below).
(1.9)
Theorem 1.1. Let Ω be a convex body in Rn , n ≥ 3, and E the (unique) ellipsoid of
maximal volume included in Ω, with principal axes: D1 (E) ≥ D2 (E) ≥ · · · ≥ Dn (E).
Then, for all p = 2, . . . , n − 1 one has
an,p
an,p
[p]
<
μ
(Ω)
<
,
1
Dp (E)2
Dp (E)2
where
an,p =
n2
·
4
n
p−1
,
an,p = 4p(n + 2)nn .
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1792
ALESSANDRO SAVO
We will actually prove the theorem with the following constants:
4
n , an,p = 4p(n + 2)γ(Ω)n ,
(1.10)
an,p =
γ(Ω)2 · p−1
which improve the previous ones when an estimate of γ(Ω), better than γ(Ω) ≤ n,
is available (that is, when Ω is close to being a true ellipsoid).
Remark 1.2. For completeness, we give the constants for p = 1 and p = n. First
observe that, as E ⊆ Ω ⊆ nE, one has
D1 (E) ≤ diam(Ω) ≤ nD1 (E) and
Dn (E) ≤ 2R(Ω) ≤ nDn (E).
1) The lower bound for p = 1 is just the Payne-Weinberger inequality; therefore
π2
an,1 = 2 .
n
2) The upper bound for p = 1 can be obtained by the following estimate of
Cheng [3], valid for Riemannian manifolds with non-negative Ricci curvature (it
also holds for convex Euclidean domains thanks to a doubling argument):
[0]
μ1 ≤
nπ 2
,
diam(Ω)2
hence
an,1 = nπ 2 .
[n]
3) From the Hersch inequality (1.4) and the domain monotonicity of μ1
finally has
π2
[0]
an,n = 2 and an,n = 4λ1 (B n ).
n
one
Remark 1.3. The constants an,p and an,p are not sharp, as the strict inequality suggests, and we believe that they can be (perhaps significantly) improved. However,
Theorem 1.1 is geometrically sharp in the sense that the invariant Dp (E) determines
[p]
the correct asymptotic behavior of the eigenvalue μ1 (Ω):
[p]
μ1 (Ω) → 0 ⇐⇒ Dp (E) → ∞ and
[p]
μ1 (Ω) → ∞ ⇐⇒ Dp (E) → 0.
For example, if Ω “collapses”, that is, it is contained in a smaller and smaller
tubular neighborhood of an m-dimensional subspace of Rn , then Dm+1 (E) → 0 (and
[p]
Dp (E) → 0 for all p ≥ m + 1). In that case one gets μ1 (Ω) → ∞ for all p ≥ m + 1.
Vice versa, one can detect collapsing (and the dimension of the subspace on which
it takes place) just by counting the number of fundamental tones which diverge to
infinity (this fact was conjectured by P. Guerini in his doctoral thesis). Note that
the collapsing is controlled by explicit constants.
Therefore, Theorem 1.1 improves (1.5) and (1.6) when D1 (E)/Dp (E) is suffi[p+1]
[p]
ciently large. It also improves (1.2), because the ratio μ1
/μ1 is bounded above
2
2
and below by a constant times the ratio Dp (E) /Dp+1 (E) , which can be arbitrarily
large.
1.3. Scheme of the proof. The proof of Theorem 1.1 will be given in Sections 3,
4 and 5. The upper (resp. lower) bound is given in terms of ellipsoids contained
in (resp. containing) Ω. John’s theorem is then used to relate the two bounds. It
[p]
is possible to estimate μ1 in terms of other invariants (equivalent to Dp (E) up to
constants); for example, the p-th longest side of a box containing Ω (see the remark
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HODGE-LAPLACE EIGENVALUES OF CONVEX BODIES
1793
in Section 3). In the next section we point out some easy consequences which,
perhaps, are worth mentioning.
2. Further remarks and consequences
From now on an,p , an,p will refer to the constants in Theorem 1.1.
2.1. Spectral geometry. The bounds of Theorem 1.1 are equivalent to the fact
that, if Ω is a convex body and Dp (E) is the p-th longest principal axis of the
ellipsoid of maximal volume included in Ω, then
an,p
an,p
<
D
.
(2.1)
(E)
<
p
[p]
[p]
μ1 (Ω)
μ1 (Ω)
Hence if one can hear the fundamental tones of Ω, then one can roughly guess the
shape of Ω, loosely interpreting the sequence of fundamental wavelengths; that is,
−
1 , . . . , 1
as the “principal axes of Ω”. In fact, let Espec
be the ellipsoid of
[1]
[n]
μ1
μ1
principal axes,
−
Dp (Espec
)
=
an,p
,
[p]
μ1 (Ω)
p = 1, . . . , n,
+
be the ellipsoid of principal axes,
and let Espec
an,p
+
, p = 1, . . . , n.
Dp (Espec ) =
[p]
μ1 (Ω)
−
+
⊂ E ⊂ Espec
. Recalling that E ⊆ Ω ⊆ nE, one finally
Then (2.1) says that Espec
obtains:
Corollary 2.1. For all convex bodies Ω in Rn :
−
+
Espec
⊂ Ω ⊂ nEspec
.
2.2. Products of eigenvalues and the volume of cross-sections. For p =
1, . . . , n denote by vol[p] (Ω) the largest volume of a p-dimensional section of Ω; that
is,
vol[p] (Ω) = sup{vol(Σ) : Σ = π ∩ Ω, π is a p-dimensional plane}.
Note that vol[1] is just the diameter and vol[n] is the usual volume. Then:
Corollary 2.2. Let Ω be a convex body and p = 1, . . . , n. Then:
cn,p
c
n,p
< vol[p] (Ω) < ,
[1]
[p]
[1]
[p]
μ1 · · · μ1
μ1 · · · μ1
where
k=p
an,k ,
cn,p = 2−p vol(B p ) Πk=1
cn,p =
n p
2
vol(B p )
and B p is the p-dimensional unit ball.
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k=p Πk=1
an,k
1794
ALESSANDRO SAVO
Proof. Since E ⊆ Ω ⊆ nE we obtain
vol[p] (E) ≤ vol[p] (Ω) ≤ np vol[p] (E).
We observe that
vol[p] (E) = 2−p vol(B p )D1 (E) · · · Dp (E),
since the p-dimensional section of E with largest volume is the one spanned by the
longest p principal axes. We now apply estimate (2.1).
Obviously the corollary is equivalent to an upper and lower estimate of the
[1]
[p]
[k]
product μ1 · · · μ1 by the invariant vol[p] (Ω)−2 ; in particular, recalling that μ1
increases with k, one gets the (weaker) estimate:
Corollary 2.3. For all convex bodies Ω and p = 1, . . . , n,
2/p
cn,p
[p]
.
μ1 (Ω) >
vol[p] (Ω)
2.3. Weak monotonicity. It is well known that the first Dirichlet eigenvalue is
monotonic with respect to inclusion: if Ω ⊆ Ω , then
μ1 (Ω) ≥ μ1 (Ω ).
[n]
[n]
This fact fails to hold in degrees different from n because, for each fixed degree
p < n, there exists a family of domains Ω , > 0, all contained in a ball of fixed
[p]
radius and such that lim→0 μ1 (Ω ) = 0. These examples are due to P. Guerini
and can be found in Theorem 2.1 of [6]. However, the domains Ω are not convex.
Using the lower bound of Theorem 3.2(a) we can prove the following weak monotonicity principle in the convex case.
Corollary 2.4. Let Ω and Ω be convex bodies in Rn , with Ω ⊆ Ω . Then for all
p < n,
[p]
[p]
μ1 (Ω) > bn,p μ1 (Ω ),
an,p
where bn,p = .
an,p
Proof. Let E be the ellipsoid of maximal volume included in Ω . By Theorem 1.1
an,p
[p]
.
μ1 (Ω ) <
Dp (E )2
Now the ellipsoid nE contains Ω ; hence it also contains Ω. By Theorem 3.2(a) one
has
an,p
[p]
.
μ1 (Ω) >
Dp (E )2
The two inequalities give the assertion.
3. The upper and lower bounds
The proof of Theorem 1.1 follows from two independent upper and lower estimates. We start from the upper bound, which is stated for any Euclidean domain
[p]
and is given in terms of any ellipsoid contained in the domain. In what follows, μ1
[p]
(resp. μm ) denotes the first (resp. m-th) eigenvalue of the Laplacian restricted to
the subspace of exact p-forms (see Section 4 for the relevant definitions; however,
[p]
[p]
one has μm ≤ μm ).
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HODGE-LAPLACE EIGENVALUES OF CONVEX BODIES
1795
Theorem 3.1. (a) Let Ω be an arbitrary domain of Rn and E− any ellipsoid
contained in Ω and having principal axes D1 (E− ) ≥ D2 (E− ) ≥ · · · ≥ Dn (E− ).
Then, for all p ≥ 1,
1
vol(Ω)
·
.
vol(E− ) Dp (E− )2
More generally, fix h = p, . . . , n and let m = hp . Then
[p]
μ1 (Ω) < 4p(n + 2) ·
μ[p]
m (Ω) < 4p(n + 2) ·
1
vol(Ω)
·
.
vol(E− ) Dh (E− )2
(b) If Ω is convex and E is the ellipsoid of maximal volume contained in Ω, then
[p]
μ1 (Ω) < 4p(n + 2)γ(Ω)n ·
1
,
Dp (E)2
where γ(Ω) = inf{t ≥ 1 : Ω ⊆ tE}. If h = p, . . . , n and m =
n
μ[p]
m (Ω) < 4p(n + 2)γ(Ω) ·
h
p , then
1
.
Dh (E)2
The proof of the upper bound is given in the next section.
The lower bound is given in terms of any ellipsoid containing Ω.
Theorem 3.2. (a) Let Ω be a convex body in Rn and E+ any ellipsoid containing
Ω with principal axes D1 (E+ ) ≥ D2 (E+ ) ≥ · · · ≥ Dn (E+ ). Then, for all p ≥ 2,
−1
n
1
[p]
μ1 (Ω) > 4
·
.
p−1
Dp (E+ )2
(b) If E is the ellipsoid of maximal volume contained in Ω, then
[p]
μ1 (Ω) >
4
1
n ·
.
Dp (E)2
γ(Ω)2 p−1
The proof of the lower bound will be given in Section 5.
It is clear that parts (b) of the above two theorems prove Theorem 1.1 with the
constants as in (1.10).
Remark 3.3. Instead of ellipsoids one can consider boxes. A box in Rn is a product
of intervals: B = [0, b1 ] × · · · × [0, bn ], where we assume b1 ≥ · · · ≥ bn > 0. Let us
set bp (B) = bp (which is the p-th longest side of B) and let
(3.1)
bp (Ω) = inf{bp (B) : B is a box containing Ω}.
We now verify that, for a convex body Ω,
(3.2)
n
4
n
p−1
·
n2 an,p
1
[p]
≤ μ1 (Ω) ≤
.
2
bp (Ω)
bp (Ω)2
In fact, let B be the box with bp (B) = Dp (E). Then B contains E and by John’s
1
theorem we see that Dp (E) ≥ bp (Ω). Hence Theorem 1.1 gives the upper bound
n
in (3.2). For the lower bound, let B be any box containing Ω and Eˆ the ellipsoid
of maximal volume included in B. Clearly Dp (Ê) = bp (B). By John’s theorem the
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1796
ALESSANDRO SAVO
√
ellipsoid nÊ contains B and hence also Ω, and we can apply Theorem 3.2(a) to
√
√
it. As Dp ( nÊ) = nbp (B) we get
1
4
[p]
,
μ1 (Ω) ≥ n n p−1 bp (B)2
which gives the lower bound because B was an arbitrary box containing Ω.
4. Proof of the upper bound
We start by recalling some facts concerning the Hodge Laplacian. A basic tool is
the Hodge (or Hodge-Morrey) orthogonal decomposition of the space Λp of p-forms
on a manifold with boundary (see for example [14], Theorem 2.4.2):
⊕ dΛp−1
⊕ Hp ,
Λp = δΛp+1
A
R
(4.1)
where
ΛpA = {ω ∈ Λp : iN ω = 0 on ∂Ω},
ΛpR = {ω ∈ Λp : J∂Ω ω = 0},
(4.2)
Hp = {ω ∈ Λp : dω = δω = 0}.
p
The space Hp is infinite dimensional, but its subspace HA
= Hp ∩ ΛpA is finite
dimensional and isomorphic to the p-th de Rham absolute cohomology space of Ω.
It consists of all harmonic forms satisfying the absolute boundary conditions.
We will use the following easy consequence of (4.1). We say that a form ω is
tangential if iN ω = 0 on ∂Ω.
Lemma 4.1. Let ω be a co-closed, tangential p-form on the domain Ω with p =
p
(Ω) = 0 (this is true if Ω is convex). Then ω ∈ δΛp+1
1, . . . , n−1. Assume that HA
A ;
that is, ω is the co-differential of a tangential (p + 1)-form.
Proof. By the Hodge-Morrey decomposition (4.1)
ω = δξ + dφ + h,
with iN ξ = 0 and dh = δh = 0. Now the scalar product of ω and dφ is zero by the
Stokes formula, so that dφ = 0. As ω is tangential one also has iN h = 0, so h is
p
(Ω), which is zero by assumption. Hence ω = δξ with ξ
a cohomology class in HA
tangential.
As Δ commutes with both d and δ, one sees that
[p]
[p]
[p]
μ1 = min{μ1 , μ1
(4.3)
[p]
[p]
},
where μ1 (resp. μ1 ) is the first positive eigenvalue of Δ when restricted to the
subspace of exact (resp. co-exact) forms. By differentiating eigenforms it follows
that
[p−1]
(4.4)
μ1
[p]
= μ1 .
Any exact p-form ω on Ω admits a canonical primitive. This is, by definition, the
unique primitive θ of ω which belongs to δΛpA . Hence θ satisfies the conditions
dθ = ω, θ = δξ
for some p-form ξ such that iN ξ = 0 on ∂Ω.
If ω is an exact 1-form its canonical primitive is the unique primitive f which
integrates to zero on the domain. The canonical primitive has the least L2 -norm
among all primitives of ω; it obviously depends on Ω and will be denoted by θ[ω,Ω] .
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HODGE-LAPLACE EIGENVALUES OF CONVEX BODIES
1797
The first positive co-exact eigenvalue satisfies the min-max principle:
dθ2
[p]
p+1
Ω
: θ ∈ δΛA − {0} .
(4.5)
μ1 = inf θ2
Ω
From this, we can finally state the variational principle which will be used for the
upper bound.
Lemma 4.2. Let Ω be an arbitrary domain and ω an exact p-form. Then
ω2
[p]
Ω
,
μ1 (Ω) ≤
θ
2
Ω [ω,Ω]
where θ[ω,Ω] is the canonical primitive of ω in Ω. In particular, if Ω is a subdomain
of Ω, then
ω2
[p]
,
μ1 (Ω) ≤ Ω
θ[ω,Ω ] 2
Ω
where θ[ω,Ω ] now denotes the canonical primitive of ω on Ω .
Proof. Given ω, we use its canonical primitive θ[ω,Ω] ∈ δΛpA as a test-form for the
[p−1]
[p]
eigenvalue μ1
= μ1 . The first assertion now follows immediately from the
min-max principle (4.5). The second statement follows from the first and from the
fact that θ[ω,Ω ] minimizes the L2 -norm among all primitives of ω on Ω :
2
2
2
θ[ω,Ω] ≥
θ[ω,Ω] ≥
θ[ω,Ω ] .
Ω
Ω
Ω
Proof of Theorem 3.1. Let Ω be an arbitrary domain of Rn and E− any ellipsoid
contained in Ω. We first show that, for all p ≥ 1,
[p]
μ1 (Ω) < 4p(n + 2) ·
(4.6)
1
vol(Ω)
·
.
vol(E− ) Dp (E− )2
We fix coordinates so that the included ellipsoid E− is expressed by the inequality
α1 x21 + · · · + αn x2n ≤ 1,
4
; by assumption, α1 ≤ · · · ≤ αn . We choose the exact form
Dk (E− )2
ω = dx1 ∧ · · · ∧ dxp as a test p-form. By Lemma 4.2
where αk =
vol(Ω)
[p]
μ1 (Ω) < 2,
θ
E−
(4.7)
where θ = θ[ω,E− ] is the canonical primitive of ω on the ellipsoid E− (the strict
inequality holds because ω is harmonic, hence certainly not an eigenform associated
[p]
to μ1 (Ω)).
It turns out that θ is explicitly computable. In fact, let ν be the vector
field ν = nk=1 αk xk ∂/∂k . Then
θ=
where α =
p
k=1 αk .
1
· iν (dx1 ∧ · · · ∧ dxp ),
α
Explicitly,
1
ˆ k ∧ · · · ∧ dxp .
(−1)k+1 αk xk dx1 ∧ · · · ∧ dx
α
p
(4.8)
θ=
k=1
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1798
ALESSANDRO SAVO
To verify this, first observe that θ is indeed a primitive of ω; moreover, it is co-closed
because δ anticommutes with interior multiplication and ω is parallel. As iν θ = 0,
we see that θ is tangential because ν is orthogonal to the boundary of E− at any
point of it. Then θ ∈ δΛp+1
by Lemma 4.1.
A
From (4.8) one gets
2
E−
θ =
p
1 2
α
x2k .
k
α2
E−
k=1
Direct computation gives
(4.9)
E−
hence
(4.10)
E−
2
θ = p
x2k =
1
k=1
αk
·
vol(E− )
,
(n + 2)αk
vol(E− )
vol(E− )
≥
.
n+2
p(n + 2)αp
Inserted in (4.7), this completes the proof of (4.6).
For the higher eigenvalues, we use the min-max principle,
2
ω
[p]
m
Ω
μm (Ω) = inf
sup − {0} ,
2 : ω ∈V
Vm
θ[ω,Ω] Ω
where V m ranges over all m-dimensional subspaces of dΛp−1 . Let h be an integer
between p and n. We take V m as the subspace generated by all parallel exact forms
dxi1 ∧ · · · ∧ dxip ,
with i1 < · · · < ip ≤ h. It has dimension m = hp . Note that each ω ∈ V m has
constant pointwise norm, which we may assume to be equal to 1. Hence
vol(Ω)
[p]
m
: ω ∈ V , ω = 1 .
μm (Ω) < sup θ[ω,E− ] 2
E−
Again, the form θ[ω,E− ] is explicit; proceeding as in (4.10) we see that for all such
ω one has
vol(E− )
Dh (E− )2
θ[ω,E− ] 2 ≥
4p(n + 2)
E−
(we omit the details because they are straightforward). Hence
(4.11)
μ[p]
m (Ω) < 4p(n + 2)
1
vol(Ω)
·
,
vol(E− ) Dh (E− )2
as asserted.
Now assume that Ω is convex. If we choose E− = E, the ellipsoid of maximal
vol(Ω)
≤ γ(Ω)n . Inserted in the
volume contained in Ω, then Ω ⊆ γ(Ω)E, and so
vol(E)
inequalities (4.6) and (4.11), this gives part (b) of Theorem 3.1.
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HODGE-LAPLACE EIGENVALUES OF CONVEX BODIES
1799
5. Proof of the lower bound (Theorem 3.2)
In what follows we assume that Ω is a convex body in Rn (unless otherwise
stated). Recall that a p-form ω is said to be “tangential” if iN ω = 0 on ∂Ω. Fix
an orthonormal family of parallel vector fields V1 , . . . , Vp . The proof of the lower
bound is based on two independent estimates of the L2 -norm of the smooth function
ω(V1 , . . . , Vp ), where ω is a co-closed, tangential p-form. This will lead to a lower
bound for the energy of co-closed tangential forms which is of independent interest
and is the main step in the proof (see Lemma 5.3). The final conclusion is then
obtained by applying the Bochner formula to a co-exact eigenform.
Denote by V1 ∧ · · · ∧ Vp the p-dimensional plane through the origin spanned by
V1 , . . . , Vp . The expression “Σ is a section orthogonal to V1 ∧ · · · ∧ Vp ” means that
Σ = π ∩ Ω, where π is a plane parallel to (V1 ∧ · · · ∧ Vp )⊥ .
Clearly Σ is a convex set in the plane π. For example, if a coordinate system
has been fixed and Vk is the k-th coordinate field ∂/∂xk , then such a section Σ is
described by the equations
Σ = {(x1 , . . . , xn ) ∈ Ω : x1 = ξ1 , . . . , xp = ξp }
for a suitable (ξ1 , . . . , ξp ) ∈ Rp . We set
(5.1) d(V1 ∧ · · · ∧ Vp ) = sup{diam(Σ) : Σ is a section orthogonal to V1 ∧ · · · ∧ Vp }.
Lemma 5.1. Let ω be a co-closed, tangential p-form on the convex body Ω and let
V1 , . . . , Vp be an orthonormal family of parallel vector fields and f = ω(V1 , . . . , Vp ).
Then:
(a) The function f has zero integral when restricted to any section Σ orthogonal
to V1 ∧ · · · ∧ Vp .
(b) One has
π2
df 2 ≥
f 2.
d(V1 ∧ · · · ∧ Vp )2 Ω
Ω
Proof of (a). By Lemma 4.1 we can write ω = δξ with ξ tangential. Consider the
1-form ξˆ on Σ defined by
ˆ
ξ(X)
= ξ(X, V1 , . . . , Vp ).
We show that ξˆ is tangential (on ∂Σ) and δ ξˆ = f on Σ; hence f integrates to zero
on Σ by the Stokes formula. Fix an orthonormal parallel frame (e1 , . . . , en−p ) on
Σ so that the vectors (e1 , . . . , en−p , V1 , . . . , Vp ) form an orthonormal parallel frame
in Rn . Then
n−p
ˆ k)
δ ξˆ = −
ek · ξ(e
=−
k=1
n−p
ek · ξ(ek , V1 , . . . , Vp )
k=1
= δξ(V1 , . . . , Vp )
= f.
ˆ Σ ) = 0, where NΣ is a unit vector, normal to ∂Σ at a
It remains to show that ξ(N
given point x ∈ ∂Σ. Denote by X̄ the orthogonal projection of the vector X ∈ Rn
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1800
ALESSANDRO SAVO
onto Tx ∂Ω. Since iN ξ = 0 we see that
ˆ Σ ) = ξ(NΣ , V1 , . . . , Vp )
ξ(N
= ξ(N̄Σ , V̄1 , . . . , V̄p ).
The p+2 vectors V1 , . . . , Vp , N, NΣ are linearly dependent because they all belong to
the orthogonal complement of the subspace Tx (∂Σ), which has dimension n − p − 1.
As N̄ = 0, the vectors V̄1 , . . . , V̄p , N̄Σ are dependent; hence ξ(N̄Σ , V̄1 , . . . , V̄p ) = 0
as asserted.
Proof of (b). By part (a) we can use f (or more precisely JΣ f ) as a test-function for
[0]
the eigenvalue μ1 (Σ); the Payne-Weinberger inequality (1.3) applied to Σ (which
is convex) gives
π2
2
JΣ df ≥
f 2.
2
diam(Σ)
Σ
Σ
As JΣ df 2 ≤ df 2 one then gets, recalling definition (5.1),
π2
2
df ≥
·
f 2.
d(V1 ∧ · · · ∧ Vp )2 Σ
Σ
The assertion now follows by integrating the above inequality over all sections Σ
orthogonal to V1 ∧ · · · ∧ Vp (applying Fubini’s theorem in the obvious way).
Lemma 5.2. Here Ω is an arbitrary domain of Rn . Let V1 , . . . , Vp be an orthonormal family of parallel vector fields, ω a co-closed, tangential p-form on Ω
and f = ω(V1 , . . . , Vp ). Then
2
2
2
f2
≤ max |ρp |2 ·
ω ·
df ,
Ω
Ω
Ω
Ω
where ρp is the distance (taken with sign) from the hyperplane through the origin
orthogonal to Vp (so that ∇ρp = Vp ).
Proof. Consider the 1-form η defined by
η(X) = ω(V1 , . . . , Vp−1 , X),
so that f = iVp η. As δ anticommutes with interior multiplication, one sees that η
is co-closed; moreover, iN η = 0 on ∂Ω. Given a smooth function g we observe the
identity:
δ(gη) = −i∇g η + gδη.
Taking g = ρp one has ∇g = Vp ; as δη = 0 we can express f as follows:
f = −δ(ρp η).
(5.2)
Any form α obeys Stokes’ formula:
2
(5.3)
δα = α, dδα +
Ω
Ω
iN α, δα.
∂Ω
If α = ρp η we then conclude by (5.2) that
2
f =−
ρp η, df .
Ω
Ω
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HODGE-LAPLACE EIGENVALUES OF CONVEX BODIES
By the Cauchy-Schwarz inequality, since η2 ≤ ω2 , we then get
2 2
2
2
f
≤ |ρp |2 ω ·
df Ω
Ω
Ω
2
2
2
≤ max |ρp | ·
ω ·
df .
Ω
Ω
1801
Ω
The following estimate on the energy is the final step in the proof of Theorem
3.2.
Lemma 5.3. Let Ω be a convex body in Rn and E+ any ellipsoid containing Ω.
Let ω be a co-closed, tangential p−form (p = 1, . . . , n − 1) on Ω. Then
−1
n
1
2
∇ω > 4
ω2 .
p
Dp+1 (E+ )2 Ω
Ω
Proof. We can assume that ω has unit L2 -norm. Fix a coordinate system (x1 , . . . ,
xn ) so that the ellipsoid E+ is expressed as
x2
x21
+ · · · + n2 ≤ 1
2
D1
Dn
1
with Dk = Dk (E+ ) being the k-th principal semi-axis of E+ . By assumption
2
D1 ≥ · · · ≥ Dn .
∂
, and we write for simplicity
From now on Vi will be the i-th coordinate field
∂xi
ωi21 ...ip ,
(5.4)
ωi1 ...ip = ω(Vi1 , . . . , Vip ), Pi1 ...ip =
Ω
so that
(5.5)
i1 <i2 <···<ip
ω2 = 1.
Pi1 ...ip =
Ω
If Σ = π ∩ Ω is any section by an (n − p)-dimensional plane π orthogonal to
V1 ∧ · · · ∧ Vp , then, on Σ, the coordinate functions x1 , . . . , xp are constant and Σ is
contained in the ellipsoid π ∩ E+ given by the inequality
x2p+1
x2n
+
·
·
·
+
≤ 1.
2
Dp+1
Dn2
This shows that diam(Σ) ≤ 2Dp+1 . If f = ω1...p , then by Lemma 5.1
π2
π2
2
(5.6)
dω1...p ≥
·
|ω1...p |2 =
P1...p .
2
2
4Dp+1 Ω
4Dp+1
Ω
On the other hand, given a multi-index i1 < i2 < · · · < ip and letting f = ωi1 i2 ...ip ,
Lemma 5.2 implies that
Pi2 ...i
2
(5.7)
dωi1 ...ip ≥ 1 2 p ,
Dip
Ω
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1802
ALESSANDRO SAVO
because max |ρip |2 ≤ Di2p . Hence
∇ω2 =
dω1...p 2 +
Ω
Ω
dωi1 ...ip 2
i1 <···<ip ,ip ≥p+1
Ω
Pi21 ...ip
π P1...p
· 2 +
4 Dp+1
Di2p
i1 <···<ip ,ip ≥p+1
⎡
⎤
1 ⎣ π2
P1...p +
≥ 2
Pi21 ...ip ⎦ .
Dp+1 4
≥
(5.8)
2
i1 <···<ip ,ip ≥p+1
Assume P1...p > 0. As
(5.9)
π2
2
> 1 and P1...p ≥ P1...p
we obtain
4
1
∇ω2 > 2 ·
P2
.
Dp+1 i <···<i i1 ...ip
Ω
1
p
By the Cauchy-Schwarz inequality
i1 <···<ip
Pi21 ...ip
⎧
−1 ⎨ n
≥
⎩
p
⎫2
⎬
i1 <···<ip
−1
n
Pi1 ...ip
=
,
⎭
p
which substituted in (5.9) gives
−1
−1
n
n
1
1
2
∇ω >
· 2 =4
·
,
(5.10)
2
p
p
D
D
p+1 (E)
Ω
p+1
−1
is replaced by the larger
as asserted. Note that if P1...p = 0 the constant np
n
−1
constant ( p − 1) . Hence the strict inequality in (5.10) would hold as well. We can now prove our main lower bound, Theorem 3.2. Let Ω be a convex body
in Rn and E+ an ellipsoid containing Ω, with principal axes D1 (E+ ) ≥ D2 (E+ ) ≥
· · · ≥ Dn (E+ ). We have to prove that, for all p ≥ 2,
−1
n
1
[p]
·
.
(5.11)
μ1 (Ω) > 4
p−1
Dp (E+ )2
Moreover, if E is the ellipsoid of maximal volume contained in Ω, then
4
1
[p]
n ·
(5.12)
μ1 (Ω) >
.
2
2
D
(E)
γ(Ω) p−1
p
Proof of (5.11). From the proof of Theorem 2.6 in [7] we know that, if Ω is convex,
[p]
[p]
then μ1 ≤ μ1 so that
[p]
[p]
μ1 = μ1
[p−1]
= μ1
.
For simplicity of notation we prove the assertion in degree p + 1: then let ω be
[p+1]
[p]
an eigen p-form associated to μ1
= μ1 . By definition, ω is co-exact and
tangential, and we can (and will) assume that it has a unit L2 -norm.
The Bochner formula applied to the eigenform ω gives
1
[p+1]
2
2
2
(5.13)
μ1
ω = ∇ω + Δ(ω ).
2
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HODGE-LAPLACE EIGENVALUES OF CONVEX BODIES
1803
As ω satisfies the absolute boundary conditions and Ω is convex (so that the principal curvatures of ∂Ω are all non-negative), one has, by Lemma 4.10(b) in [7],
2
∂ω
≥ 0.
Δ(ω2 ) =
(5.14)
Ω
∂Ω ∂N
In fact, on the boundary, the normal derivative of ω2 is given by 2S [p] ω, ω, where
S [p] denotes the shape operator acting on p-forms; by the convexity assumption it
is non-negative. Integrating (5.13) over Ω and taking into account (5.14) we finally
obtain, by Lemma 5.3,
[p+1]
2
≥
∇ω
μ1
Ω
−1
n
4
>
·
.
p
Dp+1 (E+ )2
Replacing p by p − 1 we obtain (5.11).
Proof of (5.12). If E is the ellipsoid of maximal volume included in Ω, then the
ellipsoid E+ = γ(Ω) · E contains Ω. Applying (5.11) to E+ we obtain (5.12) immediately.
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Dipartimento di Scienze di Base e Applicate per l’Ingegneria - Sezione di Matematica,
Sapienza Università di Roma, Via Antonio Scarpa 14, 00161 Roma, Italy
E-mail address: [email protected]
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