Table Of ContentDeforming metrics of foliations
Vladimir Rovenski and Robert Wolak
∗ †
Abstract
We study geometry of a manifold endowed with two complementary orthogonal distributi-
2 ons (plane fields) and a time-dependent Riemannian metric. The work begins with formulae
1 concerningdeformationsofgeometricquantitiesasthemetricvariesconformallyalongoneofthe
0
distributions. Then we introduce the geometric flow depending on the mean curvature vector
2
of the distribution, and show existence/uniquenes and convergence of a solution as t → ∞,
n when the complementary distribution is integrable with compact leaves. We apply the method
a
to the problem of prescribing mean curvature vector field of a foliation, and give examples for
J
harmonic and umbilical foliations and for the double-twisted product metrics, including the
5
codimension-one case.
] Keywords: Riemannian metric; foliation; distribution; geometric flow; second fundamental
G
tensor; mean curvature; harmonic; umbilical; heat equation; laplacian; double-twisted product
D
Mathematics Subject Classifications (2010) Primary 53C12;Secondary 53C44
.
h
t
a 1 Introduction
m
[ Geometric Flows (GFs) are important in many fields of mathematics and physics. A GF is an
3 evolutionofageometricstructureunderadifferentialequationrelatedtoafunctionalonamanifold,
v usually associated with some curvature. The most popular GFs in mathematics are the heat flow
8
([5], [7] etc), the Ricci flow ([2], [19] etc) and the mean curvature flow. They all correspond to
6
8 dynamical systems in the infinite-dimensional space of all appropriate geometric structures on a
1
given manifold. GF equations are quite difficult to solve in all generality, because of nonlinearity.
.
9 Although the short time existence of solutions is guaranteed by the parabolic or hyperbolic nature
0
of the equations, their (long time) convergence to canonical geometric structures is analyzed under
1
1 various conditions.
: Extrinsic geometry of foliated Riemannian manifolds describes properties which can be ex-
v
i pressed in terms of the second fundamental form of the leaves and its invariants (mean curvature
X
vector, higher mean curvatures and so on). One of the principal problems of extrinsic geometry
r
a of foliations reads as follows: Given a foliation F on a manifold M and an extrinsic geometric
property (P), does there exist a Riemannian metric g on M such that F enjoys (P) w.r.t. g?
Such problems (first posed by H.Gluck in 1979 for geodesic foliations) were studied already
in the 1970’s when D.Sullivan provided a topological condition (called topological tautness) for a
foliation, equivalent totheexistence ofaRiemannianmetricmakingalltheleaves minimal(see[3]).
In recent decades, several tools providing results of this sort have been developed. Among them,
one may find foliated cycles [18] and new integral formulae, [14, Part 1], [15], [16], [20], the very
first of which is Reeb’s vanishing of the integral of the mean curvature. Few works consider GFs
on foliated manifolds, see [1], [6], [10]). A GF on a foliated manifold is extrinsic, if the evolution
dependson thesecond fundamentaltensor of the(leaves of the) foliation. Recently, thefirstauthor
and P.Walczak introduced and studied extrinsic GFs on codimension-one foliations, see [14, Parts
2 and 3]. The paper extends this field of research for foliations of arbitrary codimension.
∗Mathematical Department,University of Haifa, E-mail: [email protected]
†FacultyofMathematicsandComputerScience,TheInstituteofMathematicsofJagiellonianUniversity,Krakow,
E-mail: [email protected]
1
Let(M,g)beaconnectedclosedRiemannianmanifoldwithapairofcomplementaryorthogonal
distributionsDandD (planefields)ofdimensionsnandp,respectively. IfD(orD )isintegrable,
⊥ ⊥
itistangenttoafoliation F (respectively, F ). Denotebybandb thesecondfundamentaltensors
⊥ ⊥
of D and D (w.r.t. to g), H = Tr b ∈ Γ(D ) and H = Tr (b ) ∈ Γ(D) their mean curvature
⊥ g ⊥ ⊥ g ⊥
vectors. We call D (or D ) umbilical, harmonic or totally geodesic if
⊥
nb = H ·g , H = 0 or b = 0 (respectively, pb = H ·g , H = 0 or b = 0).
D ⊥ ⊥ D⊥ ⊥ ⊥
| |
By means of the natural representation of the structure group O(p)×O(n) on TM, Naveira [11]
obtained thirty-six distinguished classes of Riemannian almost-product manifolds. Following this
line of research, several geometers, [4], [8] and [9], completed the geometric interpretation, gave
nontrivial examples for each class, and studied the behavior of the different conditions (and hence
of the different classes of almost-product structures) under a conformal change of the metric.
The notion of the D-truncated (r,k)-tensor field Sˆ (where r = 0,1, and denotes the D-
component) will be helpful: Sˆ(X ,...,X ) = S(Xˆ ,...,Xˆ ) (X ∈ TM). We introduce the Ex-
1 k 1 k i
b
trinsic Geometric Flow (EGF) as a family g of Riemannian metrics on M satisfying the PDE
t
2
∂ g = − (div H )gˆ, (1)
t t ⊥ t t
n
where g = g is given. Here gˆ is the D-truncated metric tensor g on M, i.e., gˆ(X,Y) = g(X,Y)
0
and gˆ(ξ,·) = 0 for all X,Y ∈ D and ξ ∈ D . Some operations w.r.t. EGF metrics g restricted
⊥ t
on D are t-independent (e.g., ∇ and div ). Given a Riemannian metric g, the mean curvature
⊥ ⊥ ⊥
vector H can be expressed in terms of the first partial derivatives of g. Therefore g 7→ div H is a
⊥
second-order partial differential operator.
We show that EGFs serve as a tool for studying the following:
Question: Under what conditions on (M,D) the EGF metrics g converge to one for which D
t
enjoys a given extrinsic geometric property (P), e.g., is harmonic, umbilical, or totally geodesic?
The structure of the paper is as follows. Section 2 collects main results. Section 3 develops
formulae for the deformation of extrinsic geometric quantities of a foliation as the Riemannian
metric varies conformally along D. Section 4 introduces EGF and shows the existence/uniquenes
and converging of a global solution g (t ≥ 0). In particular, is is verified that
t
(i) the 1-form θ (dual to H) satisfies the heat equation along D .
H ⊥
(ii) the metrics g preserve the ”umbilical” (”totally geodesic”, etc.) property of D.
t
(iii) for appropriate g the metrics g converge to a metric g with “harmonic distribution D”.
0 t
∞
In Theorem 1, the orthogonal distribution D is integrable and the method of proof is based on
⊥
solvingtheheatequationfor1-formsontheleavesofD . InTheorem2themethodisappliedtothe
⊥
problem of prescribing mean curvature vector field of D. Section 5 contains results and examples
for codimension-one foliations (Proposition 6), and double-twisted product metrics. Appendix
(Section 6) collects the necessary facts about the heat equation and the heat flow for 1-forms.
2 Main Results
We define the connection ∇ induced on D by ∇ ξ = (∇ ξ) (i.e., ∇ ξ is projected onto D ),
⊥ ⊥ ⊥X X ⊥ X ⊥
where ξ ∈ D and X ∈ TM. In this section we suppose that D is integrable with all leaves
⊥ ⊥
compact and orientable. (If both D and D are integrable and the foliation tangent to D is
⊥ ⊥
totally geodesic then the foliation tangent to D is Riemannian).
Denote θ the 1-form on D dual to the vector field ξ ∈ Γ(D ). The 1-form θ is D -harmonic
ξ ⊥ ⊥ ξ ⊥
(see Section 6.2) if and only if δ θ = 0 (i.e., div ξ =0) and d θ = 0 (i.e., ∇ ξ is symmetric).
⊥ ξ ⊥ ⊥ ξ ⊥
i π
Theorem 1. Let the leaves of D compose a fibration L ֒→ M → B. If dimD = p > 1,
⊥ ⊥ ⊥
suppose that d θ = 0. Then (1) admits a unique smooth solution g for all t ≥ 0 that converges
⊥ H t
in C -topology as t → ∞ to a Riemannian metric g for which D is harmonic.
∞
∞
2
Example 1. Let the leaves of integrable distribution D be flat tori Tp. Any differential 1-form
⊥
on Tp can be written as ω = ω dxi. The form ω is harmonic if and only if the functions ω are
i i i
harmonic, and therefore constPant. In this case, the vector field dual to ω is constant. Indeed, the
space of constant vector fields on a flat torus Tp is isomorphic to H1(Tp,R) ≃ Rp.
A foliation, whose normal plane field is umbilical, is locally conformally equivalent to a Rie-
mannian foliation, see [9]. The next corollary completes this fact.
Corollary 1. Under the assumptions of Theorem 1, let D be g -umbilical. Then D is g -totally
0
∞
geodesic (g is the limit of metrics (1) as t → ∞).
∞
A foliation with vanishing mean curvature is called harmonic. Every leaf of such foliation is a
minimal submanifold of M. A foliation F is taut if there is at least one metric on M for which
F is harmonic. In particular, if there is an immersed closed transversal manifold that intersects
each leaf, then F is taut (see [13] and survey in [3]). For example, a Reeb foliation on S3 is
not taut. The known proofs of existence of “taut” metrics use the Hahn–Banach Theorem and
are not constructive. Corollary 2 (of Theorem 1) shows how to produce in some cases a family
of metrics converging to the metric for which F is harmonic (i.e., H = 0). Certain results for
∞
codimension-one foliations are given in Section 5.
Corollary 2. Let F be a foliation on (M,g) tangent to D of codimension p > 1. Suppose that the
i π
leaves of D compose a fibration L ֒→ M → B, and the equality d θ = 0 is satisfied. Then (1)
⊥ ⊥ ⊥ H0
admits a unique solution g (t ≥ 0), converging in C -topology as t → ∞ to a Riemannian metric
t ∞
g , for which F is harmonic.
∞
Let F be a foliation of any codimension of a closed manifold M and X be a vector field on M.
Recently, P. Schweitzer and P. Walczak [17] provided some necessary and sufficient conditions for
X to become the mean curvature vector of F with respect to some Riemannian metric on M.
Extending the definition of EGF and method of Theorem 1, we show how to produce in some
cases (e.g., Theorem 2) a one-parameter family of metrics converging to the metric with prescribed
mean curvature vector field of F.
i π
Theorem 2. Let p > 1 and the leaves of D compose a fibration L ֒→ M → B. Then for any
⊥ ⊥
smooth vector field X on M orthogonal to F and satisfying d θ = 0, the PDE
⊥ H X
−
2
∂ g = − div (H −X)gˆ, g = g (2)
t t ⊥ t t 0
n
admits a unique solution g (t ≥ 0), converging in C -topology as t → ∞ to a Riemannian metric
t ∞
g with D -harmonic 1-form θ . If H1(L ,R) = 0 for the leaves of D , then H = X.
⊥ H∞ X ⊥ ⊥
∞ − ∞
Example 2. (a) By Bochner theorem (see [5]), if the Ricci curvature of any leaf L of D is
⊥ ⊥
non-negative everywhere and positive for some point then H1(L ,R) = 0 (see Theorem 2).
⊥
(b)Thefollowingexample(communicatedtoauthorsbyP.Walczak)showsusthatthecondition
d θ = 0 (see Theorem 2) and the assumption d θ = 0 (see Theorem 1 and Corollary 2) are
⊥ H X ⊥ H
neede−d. Let X be a divergence free (e.g., a Killing) vector field on the leaves Sp of the product
M = M ×Sp of a unit p-sphere and a Riemannian manifold (M ,g ). Let the distribution D on
1 1 1
M corresponds to TM . Then the product metric g has the mean curvature H = 0, and g is a
1
fixed point of the dynamical system (2). Consequently, H = 0 for all t ≥ 0 and H = 0 6= X.
t
∞
3 D-conformal variations of geometric quantities
In this section we develop formulae for deformations of geometric quantities as the Riemannian
metric varies conformally along one of the distributions.
3
3.1 Preliminaries
Denote by M the space of smooth Riemannian metrics of finite volume on M such that D is
⊥
orthogonal to D. Elements of M are called (D,D )-adapted metrics. Let M ⊂ M be the
⊥ 1
subspace of (D,D )-adapted metrics of unit volume, and π : M → M , where π(g) = g¯ =
⊥ 1
vol(M,g) 2/ngˆ ⊕g be the D-conformal projection. Let g ∈ M (with 0 ≤ t < ε) be a family of
− ⊥ t
m(cid:0) etrics with g o(cid:1)f unit volume. Consider the D-truncated tensor field
0
S = ∂ g .
t t t
Recall that the musical isomorphism ♯ : T M → TM sends a covector ω = ω dxi to ω♯ = ωi∂ =
∗ i i
gijω ∂ , and ♭ : TM → T M sends a vector X = Xi∂ to X♭ = X dxi = g Xjdxi. We denote by
j i ∗ i i ij
S♯ the (1,1)-tensor field on M which is g-dual to a symmetric (0,2)-tensor S,
S(X,Y) = g(S♯(X),Y) for all vectors X,Y.
Thevolume form vol of g evolves as d vol = 1 (Tr S♯)vol , see [19]. If S = s gˆ with s :M → R
t t dt t 2 t t t t t
(i.e., g are conformally equivalent along D and gˆ is the D-truncated metric g ) then
t t t
d n
vol = s vol . (3)
t t t
dt 2
Hence, the metrics g˜ = (φ gˆ)⊕g with dilating factors φ = vol(M,g ) 2/n, belong to M .
t t t t⊥ t t − 1
Recall that the Levi-Civita connection ∇t of a metric g on M is given by
t
2g (∇t Y,Z) = X(g (Y,Z))+Y(g (X,Z))−Z(g (X,Y))
t X t t t
+ g ([X,Y],Z)−g ([X,Z],Y)−g ([Y,Z],X) (4)
t t t
for all vector fields X,Y and Z on M. Since the difference of two connections is always a tensor,
Π := ∂ ∇t is a(1,2)-tensor fieldon(M,g ). Differentiation (4)w.r.t. tyieldstheformula, see[19],
t t t
1
g (Π (X,Y),Z) = (∇t S )(Y,Z)+(∇t S )(X,Z)−(∇t S )(X,Y) (5)
t t 2 X t Y t Z t
(cid:2) (cid:3)
for all X,Y,Z ∈ TM. Indeed, if the vector fields X = X(t), Y = Y(t) are t-dependent, then
∂ ∇t Y = Π (X,Y)+∇ (∂ Y)+∇ Y. (6)
t X t X t ∂tX
Notice the symmetry Π (X,Y)= Π (Y,X) of the tensor Π .
t t t
We will use the following condition for convergence of evolving metrics (see [2, Appendix A]).
Proposition 1. Let ∂ g = s gˆ (t ≥ 0) be a one-parameter family of Riemannian metrics on
t t t t
a closed manifold M with complementary distributions D and D . Define functions u (t) =
⊥ m
supM |(∇t)mst|g(t) and assume that 0∞um(t)dt < ∞ for all m ≥ 0. Then, as t → ∞, the metrics
g converge in C -topology to a smoRoth Riemannian metric g .
t ∞
∞
Proof. Our assumptions ensure that g converge in C -topology to a symmetric (0,2)-tensor g .
t ∞
The metrics are uniformly equivalent: c 1gˆ ≤ gˆ ≤cgˆ for some c> 0 and all t ≥ 0. Hence, g ∞is
− 0 t 0
∞
positive definite.
Let ∇ φ be the D -component of the gradient of a function φ ∈ C1(M). The second funda-
⊥ ⊥
mental tensor of D (similarly of D ) is defined by
⊥
1
b(X,Y) = (∇ Y +∇ X) , X,Y ∈ D. (7)
X Y ⊥
2
The second fundamental forms of D with respect to metrics g and g˜= (e2φgˆ)⊕g are related
⊥
by the following lemma.
4
Lemma 1 (see[14]forcodimension-onefoliations). Let (M, g = gˆ⊕g ) be a Riemannian manifold
⊥
with complementary orthogonal distributions D and D . Given φ ∈ C1(M), define a metric g˜ =
⊥
(e2φgˆ)⊕g . Then the second fundamental forms and the mean curvature vectors of D w.r.t. g˜
⊥
and g are related by
˜b = e2φ b−(∇ φ)gˆ , H˜ = H −(dimD)∇ φ. (8)
⊥ ⊥
(cid:0) (cid:1)
So, if φ is constant then, by (8), we have: ˜b = e2φb and H˜ = H.
Proof. By (4), for any X,Y ∈ D and ξ ∈ D we have
⊥
1
g(∇˜ Y,ξ)= e2φg(∇ Y,ξ)−e2φg(X,Y)ξ(φ)− (e2φ −1)g([X,Y],ξ).
X X
2
From this and definition (7), formula (8) follows. Since H = Tr b, we have (8) .
1 g 2
3.2 The integral formula
The divergence of a vector field X on (M,g) is given by divX = g(∇ X,e ), where (e ) is a
s es s s
local orthonormal frame on (M,g). Recall the identity for a smootPh function f : M → R,
div(f ·X) = f ·divX +X(f).
For a compact manifold M with boundary and inner normal n, the Divergence Theorem reads as
divXdvol = g(X,n)dω. (9)
Z Z
M ∂M
For a closed manifold M, we have divXdvol =0. The D -divergence, div ξ, of a vector field
M ⊥ ⊥
ξ ∈ Γ(D ) is defined similarly to dRivξ, using a local orthonormal frame (ε ) of D .
⊥ α ⊥
Lemma 2. For a vector field ξ ∈ Γ(D ) on a closed manifold M, we have the identity
⊥
(div ξ)dvol = g(H,ξ)dvol. (10)
⊥
Z Z
M M
So, (div ξ)dvol = 0 for any vector field ξ ∈ Γ(D ) if and only if H = 0.
M ⊥ ⊥
R
Proof. Using the definition H = b(e ,e ), we have
i p i i
P≤
divξ−div ξ = g(∇ ξ,e )= − g(b(e ,e ),ξ) = −g(H,ξ).
⊥ Xi p ei i Xi p i i
≤ ≤
By the Divergence Theorem, div ξdvol = 0, we obtain (10).
M
R
Remark 1. (i) By Lemma 2 with ξ = H, we have
(div H)dvol = g(H,H)dvol ≥ 0. (11)
⊥
Z Z
M M
(ii) By Lemma 2 with ξ = ∇ f, for a function f ∈ C2(M), we have
⊥
(∆ f)dvol = g(∇f,H)dvol.
⊥
Z Z
M M
Here ∆ f = div (∇ f) is the D -Laplacian of f.
⊥ ⊥ ⊥ ⊥
(iii) In analogy with the fact that on a closed connected Riemannian manifold, every harmonic
function (i.e., ∆f = 0) is constant, we claim: If ∆ f = g(∇f,H) then ∇ f = 0. Indeed,
⊥ ⊥
div(f∇ f)+f(H(f)−∆ f)= g(∇ f,∇ f).
⊥ ⊥ ⊥ ⊥
Using the Divergence Theorem, we obtain g(∇ f,∇ f)dvol = 0, and then ∇ f = 0.
M ⊥ ⊥ ⊥
R
5
3.3 D-related geometric quantities
Let {e , ε } (i ≤ n, α≤ p) be a local g -orthonormal frame on TM adapted to D and D .
i α 0 ⊥
Lemma 3. Let {e } be a local g -orthonormal frame of D (on a set U ⊂ M), and ∂ g = sgˆ.
i 0 q t t t
Suppose that {e (t)} evolves according to
i
1
∂ e (t) = − se (t). (12)
t i i
2
Then {e (t)} is a g -orthonormal frame of D on U for all t.
i t q
Proof. We have
∂ (g (e ,e )) = g (∂ e (t),e (t))+g (e (t),∂ e (t))+(∂ g )(e (t),e (t))
t t i j t t i j t i t j t t i j
1 1
=sgˆ(e (t),e (t))− g (se (t),e (t))− g (e (t),se (t)) = 0.
t i j t i j t i j
2 2
The following lemma is compatible with Lemma 1.
Lemma 4. Let g ∈ M and ∂ g = s gˆ for some s ∈ C1(M). Then the second fundamental
t t t t t t
tensor b, its mean curvature vector H and the D -divergence div H are evolved by
⊥ ⊥
1 1
∂ b(X,Y) = sb(X,Y)− gˆ(X,Y)∇ s+ (∇ ∂ Y +∇ Y +∇ ∂ X +∇ X) , (13)
t 2 ⊥ 2 X t ∂tX Y t ∂tY ⊥
n n n
∂ H = − ∇ s, ∂ (div H) = − ∆ s, ∂ θ = − d s. (14)
t ⊥ t ⊥ ⊥ t H ⊥
2 2 2
Proof. Let S = ∂ g be D-truncated. By (5), (7), symmetry of S and S(·,D )= 0, we have
t t ⊥
1
g (∂ b(X,Y),ξ) = g ∂ (∇t Y)+∂ (∇t X), ξ
t t 2 t t X t Y
(cid:0) (cid:1)
1
= (∇t S)(Y,ξ)+(∇t S)(X,ξ)−(∇tS)(X,Y) +Q
2 X Y ξ
(cid:2) (cid:3)
for all ξ ∈ D and t-dependent X,Y ∈ D. Here, Q := 1 g (∇t ∂ Y +∇t Y +∇t ∂ X+∇t X, ξ)
⊥ 2 t X t ∂tX Y t ∂tY
due to (6). Substituting S = sgˆ, we obtain the required (13):
1
g(∂ b(X,Y), ξ) = − sgˆ(Y,∇t ξ)+sgˆ(X,∇t ξ)+ξ(s)gˆ(X,Y) +Q
t 2 X Y
(cid:2) (cid:3)
1
= sg(b(X,Y), ξ)− gˆ(X,Y)ξ(s)+Q.
2
Let {e (t)} be a local g -orthonormal frame of D on U for all t, hence (12) holds, see Lemma 3.
i t q
By the above we obtain (14) (see also alternative proof in Remark 2):
1
∂ H = ∂ b(e (t),e (t))
t t i i
Xi
1 n
= sb(e (t),e (t))− g (e (t),e (t))∇ s − sb(e (t),e (t)) = − ∇ s.
i i t i i ⊥ i i ⊥
Xi(cid:2) 2 (cid:3) Xi 2
To show (14) , let S = ∂ g be D-truncated (i.e., S(D ,·) = 0). Using (5), we have
2 t t ⊥
∂ (div H) = ∂ (g(∇ H,ε )) = (∂ g)(∇ H,ε )+g (∂ (∇ H),ε )
t ⊥ Xα≤p t εα α Xα≤p(cid:2) t εα α t t εα α (cid:3)
1
= S(∇ H,ε )+ (∇ S)(ε ,ε ) +div (∂ H) = div (∂ H).
Xα≤p(cid:2) εα α 2 H α α (cid:3) ⊥ t ⊥ t
Here, weusedS(ε ,·) = 0and(∇ S)(ε ,ε ) = H(S(ε ,ε ))−2S(∇ ε ,ε )= 0. Now, assuming
α H α α α α H α α
S = sgˆ, and using (14) , we obtain (14) : ∂ (div H)= div (∂ H)= −n div (∇ s)= −n ∆ s.
1 2 t ⊥ ⊥ t 2 ⊥ ⊥ 2 ⊥
To show (14) , we use (14) to calculate for any X ∈ D :
3 1 ⊥
n
∂ θ (X) = ∂ (g (H ,X)) = sgˆ(H ,X)+g (∂ H ,X)+g (H ,∂ X) = − g (∇ s,X)+g (H ,∂ X).
t H t t t t t t t t t t t ⊥ t t t
2
Hence, (∂ θ )(X) = −n (∇ s)♭(X)+g (H ,∂ X)−θ (∂ X) = −nd s(X).
t H 2 ⊥ t t t H t 2 ⊥
6
Remark 2. The alternative proof of (14) is based on the identity (see [14, Lemma 2.4] for k = 0)
1
∂ (Tr B)= Tr (∂ B)−hB,Si ,
t gt gt t gt
where S = ∂ g, B – a t-dependent symmetric (k,2)-tensor on (M,g), and hB,Si= BijS .
t ij
In our case, k = 1, B = b, S = sgˆ and Tr B = H . Thus, using (13), we have
t gt t
n
∂ (Tr B) = ∂ H, Tr (∂ B)= Tr (∂ b) = sH − ∇ s,
t gt t gt t gt t 2 ⊥
hB,Si = hb,sgˆi = shb ,gˆi = sTr b = sH .
gt t gt t t gt gt t t
Next we show that D-conformal variations of metrics preserve the umbilicity of D.
Proposition 2. Let ∂ g = s gˆ (s : M → R), be a D-conformal family of Riemannian metrics
t t t t t
on a manifold (M,D,D ). If D is umbilical for g , then D is umbilical for any g .
⊥ 0 t
Proof. Since D is g -umbilical, we have b = 1 Hgˆ at t = 0, where H is the mean curvature
0 n
vector field of D. Applying to (13) the theorem on existence/uniqueness of a solution of ODEs, we
conclude that b = 1 H˜ gˆ for all t, for some H˜ ∈ Γ(D ). Tracing this, we see that H˜ is the mean
t n t t t ⊥ t
curvature vector of b , hence D is umbilical for any g .
t t
3.4 D -related geometric quantities
⊥
Lemma 5. Let g ∈ M and ∂ g = s gˆ for some s ∈ C1(M). Then the second fundamental
t t t t t t
tensor b and its mean curvature vector H are evolved as
⊥ ⊥
∂ b = −sb , ∂ H =−sH . (15)
t ⊥ ⊥ t ⊥ ⊥
Proof. We shall show for the more general setting S = ∂ g that
t t
∂ b = −S♯◦b , ∂ H = −S♯(H ). (16)
t ⊥ ⊥ t ⊥ ⊥
Using (5), we compute for any X ∈ D and ε ,ε ∈ D ,
α β ⊥
1
g (∂ b (ε ,ε ),X) = g (∂ (∇t ε )+∂ (∇t ε ), X)
t t ⊥ α β 2 t t εα β t εβ α
1
= (∇t S)(X,ε )+(∇t S)(X,ε )−(∇t S)(ε ,ε )
2 εα β εβ α X α β
(cid:2) (cid:3)
1
= − S(∇t ε ,X)+S(∇t ε ,X) = −S(b (ε ,ε ),X).
2 εα β εβ α ⊥ α β
(cid:2) (cid:3)
From this (16) follows when S = sgˆ. Next, for any X ∈ Γ(D), we have
1
1
g (∂ H ,X) = g (∂ (∇ ε ),X) = 2(∇ S)(ε ,X)−(∇ S)(ε ,ε )
t t ⊥ Xα t t εα α 2(cid:2) εα α X α α (cid:3)
= − S(∇ ε ,X) = −S(H ,X),
Xα εα α ⊥
which confirms (16) . From (16) for S = sgˆ we obtain (15).
2
Next we show that D-conformalvariations of metrics preserve“umbilical” (i.e., b = H ·g )
⊥ ⊥ D⊥
|
and “harmonic” (i.e., H = 0) properties of D .
⊥ ⊥
Proposition 3. Let ∂ g = s gˆ (s :M → R), be a D-conformal family of Riemannian metrics on
t t t t t
a manifold (M,D,D ). If D is either umbilical (e.g., totally geodesic) or harmonic for g , then
⊥ ⊥ 0
D is the same for any g .
⊥ t
Proof. If D is g -umbilical, then b = 1 H g at t = 0, where H is the mean curvature vector
⊥ 0 ⊥ p ⊥ ⊥ ⊥
field of D . Applying to (15) the theorem on existence and uniqueness of a solution of ODEs, we
⊥ 1
conclude that b = 1H˜ g for all t, where H˜ ∈ Γ(D). Tracing this, we show that H˜ is the mean
⊥t p t t⊥ t t
curvature vector of b , hence D is umbilical for any g . Indeed, H˜ ≡ 0 when D is g -totally
⊥t ⊥ t t ⊥ 0
geodesic. The remaining property, i.e., D is harmonic, can be proved similarly.
⊥
7
4 Proofs of main results
4.1 Introducing the Extrinsic Geometric Flow
Definition 1. Given g = g, a family of (D,D )-adapted Riemannian metrics g , t ∈ [0,ε), on M
0 ⊥ t
will be called (a) an Extrinsic Geometric Flow (EGF) if (1) holds; (b) a normalized EGF if
2 2
∂ g = − div H +r(t) gˆ, where r(t) = − (div H )dvol /vol(M,g ). (17)
t t n ⊥ t t nZ ⊥ t t t
(cid:0) (cid:1) M
TheEGF(1)andits normalized companion (17)providesomemethodsof evolving Riemannian
metrics on foliated manifolds. Obviously, both EGFs preserve harmonic (and totally geodesic)
foliations. If g ∈ M , then all metrics g (t ≥ 0) of (17) belong to M , because, see (3),
0 1 t 1
d n
vol(M,g )= − div H + r(t) dvol = r(t)vol(M,g )− r(t)dvol = 0.
dt t ZM (cid:16) ⊥ t 2 (cid:17) t t ZM t
Substituting (11) in the definition (17) of r(t), we have
2
r(t)= − g(H ,H )dvol /vol(M,g )≤ 0.
n Z t t t t
M
ByProposition2, EGFspreservethefollowing propertiesofD : (i)umbilical(b = H ·g ),
⊥ ⊥ ⊥ D⊥
|
(ii) totally geodesic (b = 0), (iii) harmonic (H = 0); which in case of integrable D mean that
⊥ ⊥
the foliation tangent to D is (i) conformal, (ii) Riemannian, (iii) transversally harmonic.
Let g be a family of Riemannian metrics of finite volume on (M,D,D ). Metrics g˜ = (φ gˆ)⊕
t ⊥ t t t
g with φ = vol(M,g ) 2/n have unit volume: dvol = 1. The next proposition shows that
t⊥ t t − M t
unnormalized and normalized EGFs differ only byRrescaling along the distribution D.
f
Proposition 4. Let g be a solution (of finite volume) to (1) on (M,D,D ). Then the metrics
t ⊥
g˜ = (φ gˆ)⊕g , where φ = vol(M,g ) 2/n,
t t t ⊥ t t −
evolve according to the normalized EGF
2 2
∂ g˜ = − div H˜ +ρ gˆ˜, where ρ = − (div H)dvol /vol(M,g ). (18)
t t n ⊥ t t t t n Z ⊥ t t
(cid:0) (cid:1) M
Proof. Since φ depends only on t, by Lemma 1, H˜ = H for metrics g˜ and g . Hence, div H˜ =
t t t t ⊥ t
div H . From (3) with s = −2(div H ) we get the derivative of the volume function
⊥ t n ⊥ t
d d
vol(M,g ) = dvol = − (div H )dvol .
dt t dt Z t Z ⊥ t t
M M
Thus φ = vol(M,g ) 2/n is a smooth function of variable t. By Lemma 1, we have ˜b = φ ·b .
t t − t t t
Therefore
2 φ
∂ g˜ = φ ∂ g +φ gˆ = − div H − ′t gˆ˜.
t t t t t ′t t n ⊥ t φ t
(cid:0) t(cid:1)
e
n/2
Notice that dvol = φ dvol . Using this and (3), we obtain
t t t
fddt volt = ddt (φtn2 volt) = (cid:0)n2 φtn2−1φ′t+ 21φtn2ns(cid:1)volt = n2(cid:16)φφt′t +s(cid:17)volt.
f f
Let ρ be the average of s, see (18). From the above we get
t
d φ φ
0 = 2 dvol = n ′t +s dvol = n ′t +ρ .
dt ZM t ZM (cid:16)φt (cid:17) t (cid:0)φt t(cid:1)
f f
This shows that ρ = −φ /φ . Hence, g˜ evolves according to (18).
t ′t t
8
4.2 Proof of Theorems 1–2 and Corollaries 1–2
Proposition 5. Let D be integrable. The mean curvature vector H of D and its D -divergence
⊥ t ⊥
with respect to g of EGF (1) or (17) satisfy the following PDEs on any leaf of D :
t ⊥
∂ H = ∇ (div H), ∂ (div H) = ∆ (div H). (19)
t ⊥ ⊥ t ⊥ ⊥ ⊥
Proof. By Lemma 4 with s = −2 (div H) or s = −2 div H −r(t), we obtain (19) . Similarly,
n ⊥ n ⊥ 1
from (14) we deduce (19) .
2 2
The eigenvalue problem, −∆ u = λu, on a leaf L (of D when it is integrable) has solution
⊥ ⊥ ⊥
with a sequence of eigenvalues with repetition (each one as many times as the dimension of its
finite-dimensional eigenspace) 0 = λ < λ ≤ λ ≤ ··· ↑ ∞. Let φ be an eigenfunction with
0 1 2 j
eigenvalue λj satisfying L⊥φ2j(x)dx = 1. Then G⊥(t,x,y) = jeλjtφj(x)φj(y) is a fundamental
solution of the heat equaRtion on L , see details in Section 6.1. APsolution satisfying u(x,0) =u (x)
⊥ 0
is given by u(x,t) = G (t,x,y)u (y)dy. Moreover, if u ∈ L2(L ) then the solution converges
L⊥ ⊥ 0 0 ⊥
uniformly, as t → ∞,Rto a D -harmonic function (constant λ when L is closed).
⊥ 1 ⊥
Proof of Theorem 1. This is based on the heat flow for 1-forms, see Section 6.2. Let θt be the
H
dual 1-form on D to the mean curvature vector field H (with respect to g for t ≥ 0). Using
⊥ t t
div H = −δ θt and applying (14) with s = −2 div H, we show similarly to (19) that
⊥ t ⊥ H 3 n ⊥ 1
∂ θt = −d δ θt , (20)
t H ⊥ ⊥ H
where θ0 = θ is known. We obtain (for L -product along the leaves of D )
H H 2 ⊥
1
∂ (kθt k2)= ∂ hθt , θt i = −hd δ θt , θt i= −kδ θt k2 ≤ 0.
2 t H t H H ⊥ ⊥ H H ⊥ H
By the above, we have uniqueness of a solution of the linear PDE (20).
By Theorem B in Section 6.2, the heat equation (considered on the leaves of D )
⊥
∂ ω = ∆ ω, where ω = θ , (21)
t ⊥d 0 H
admits a unique solution ω (t ≥ 0). As t → ∞, the 1-form ω converges exponentially to a
t t
D -harmonic 1-form ω , i.e., kω − ω k ≤ ce λt for some constants c,λ > 0. Since ω = θ
⊥ t − 0 H
∞ ∞
is D -closed, again by Theorem B, we have d ω = 0 for all t ≥ 0. (Notice that for p = 1 the
⊥ ⊥ t
1-form θ is always D -closed). Comparing (20) with (21), we conclude that θt = ω is a unique
H ⊥ H t
solution of (20). By the above, div H = 0 (here H ∈ Γ(D ) is dual to θ = ω ). Applying
⊥ ⊥ H∞
the integral formula (11), we conclude t∞hat lim kH k2∞=0, hence H = 0. ∞
t
t ∞
With known H , the PDE (1) also has a →un∞ique smooth global solution, and
t
2 t
gˆ = gˆ exp − (div H )ds .
t 0 n Z ⊥ s
(cid:0) 0 (cid:1)
By Lemma 5, we also have b = b exp(2 t(div H )ds) for t ≥ 0. Since the leaves of D are
⊥t ⊥0 n 0 ⊥ s ⊥
compact, and div H satisfies (19) (on anyR leaf of D ), we have |div H |≤ e λ1t|div H | and
⊥ t 2 ⊥ ⊥ t − ⊥ 0
t t t 1−e λ1t c˜
(div H )ds ≤ div H ds < c˜ e λ1tds = c˜ − <
(cid:12)Z0 ⊥ s (cid:12) Z0 (cid:12) ⊥ s(cid:12) Z0 − λ1 λ1
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
for some c˜>0 and all t ≥ 0. Now, for some c≥ 1 and all t ≥ 0, we have the uniform bounds
c 1gˆ ≤ gˆ ≤ cgˆ , c 1b ≤ b ≤ cb .
− 0 t 0 − ⊥0 ⊥t ⊥0
From the above and Proposition 1, g → g and b → b as t → ∞ in C -topology.
t ⊥t ⊥ ∞
∞ ∞
9
Proof of Corollary 1. By Theorem 1, the EGF metrics g converge to a smooth metric g with
t
∞
H = 0. By Proposition 2, EGFs preserve the umbilicity of D, hence D is g -umbilical. Note
∞ ∞
that an umbilical foliation with vanishing mean curvature is totally geodesic.
Proof of Corollary 2. The leaves of orthogonal foliation are immersed compact cross-sections,
hence F is taut. On the other hand, by Theorem 1, the EGF metrics g converge as t → ∞ to a
t
smooth Riemannian metric g with H = 0.
∞ ∞
Proof of Theorem 2. The vector field H = H −X satisfies PDEs of Proposition 5,
t t
e
∂ H = ∇ div H , ∂ (div H )= ∆ (div H ). (22)
t t ⊥ ⊥ t t ⊥ t ⊥ ⊥ t
e e e e
Since ∂ X = 0, the 1-form θt := θ −θ satisfies the PDE, see (20),
t He Ht X
∂tθHe = −d⊥δ⊥θHe (23)
withtheinitial valueθ0 = θ −θ . As in theproofof Theorem1, wehave uniquenessof a solution
e H X
H
of (23) for t ≥ 0. By Theorem B in Section 6.2, the heat equation (considered on the leaves of D )
⊥
∂ ω = ∆ ω (24)
t ⊥d
admits a unique smooth solution ω (t ≥ 0) with the initial value ω = θ −θ . As t → ∞, the
t 0 H X
1-form ω converges exponentially to a D -harmonic 1-form ω , and kω −ω k ≤ ce λt for some
t ⊥ t −
∞ ∞
constants c,λ > 0. Since ω is D -closed, again by Theorem B, we have d ω = 0 for all t ≥ 0.
0 ⊥ ⊥ t
Comparing (23) with (24), as in the proof of Theorem 1, we conclude that θt = ω is a unique
e t
H
solution of (23). By the above, div (H −X) = 0 (here H −X is dual to ω ). With known H ,
⊥ t
the PDE (2) also has a unique smooth∞global solution, and∞gˆ = gˆ exp − 2 ∞tdiv (H −X)ds .
t 0 n 0 ⊥ s
ByLemma5,wealsohaveb = b exp(2 tdiv (H −X)ds)forallt ≥(cid:0)0. UsRing|div (H −X)| ≤(cid:1)
⊥t ⊥0 n 0 ⊥ s ⊥ t
e λ1t|div (H −X)|, we obtain R
− ⊥ 0
t t t 1−e λ1t c˜
(div (H −X))ds ≤ div (H −X) ds < c˜ e λ1tds = c˜ − <
(cid:12)Z0 ⊥ s (cid:12) Z0 (cid:12) ⊥ s (cid:12) Z0 − λ1 λ1
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
for some c˜ > 0 and all t ≥ 0. As in the proof of Theorem 1, we conclude that g converges in
t
C -topology as t → ∞ to a Riemannian metric g with D -harmonic 1-form θ .
∞ ⊥ H∞ X
Certainly, if H1(L ,R) = 0 for the leaves of D∞, then H = X. −
⊥ ⊥
∞
5 More examples
5.1 The codimension-one case
Let(M,g)beaclosedRiemannianmanifoldwithacodimension-onedistributionD(i.e.,p = 1). Let
N be the unit vector field (orthogonal to D), and b the scalar second fundamental form of D with
respecttoN. Indeed,2b(X,Y)= g(∇ Y +∇ X, N). Hence, H = τ N andτ = g(N,H) = TrA,
X Y 1 1
where A= b♯ is the shape operator. By Lemma 4, we find the variations (see also [14])
1 n
∂ A= − N(s)iˆd, ∂ τ = − N(s), (25)
t t 1
2 2
where ∂ g = s gˆ. For a codimension-one case, the EGF definitions (1) and (17) read as:
t t t t
2
∂ g = − N(τ )gˆ, (26)
t t 1 t
n
2 2
∂ g = − N(τ )+r(t) gˆ, r(t) = − N(τ )dvol /vol(M,g ), (27)
t t n 1 t nZ 1 t t
(cid:0) (cid:1) M
10
