A classification theorem for Helfrich surfaces

0.22 MB·English

by  James McCoy

A CLASSIFICATION THEOREM FOR HELFRICH SURFACES
JAMES MCCOY1 AND GLEN WHEELER2
Abstract. In this paper we study the functional W , which is the the sum of
λ1,λ2
theWillmoreenergy,λ1-weightedsurfacearea,andλ2-weightedvolume,forsurfaces
2 immersedinR3. This coincideswiththe Helfrichfunctionalwithzero‘spontaneous
1
curvature’. Ourmainresultisacompleteclassificationofallsmoothimmersedcrit-
0
2 ical points of the functional with λ1 ≥0 and small L2 norm of tracefree curvature.
Inparticularweprovethenon-existenceofcriticalpointsofthefunctionalforwhich
n
the surface area and enclosed volume are positively weighted.
a
J

2
2

1. Introduction
]
G
Consider a surface Σ immersed via a smooth immersion f : Σ → R3. There are
D
several functionals of f relevant to our study:
.
h
t
a µ(Σ) = dµ, the surface area
m
ZΣ
[ Vol Σ = f∗(dH3), the signed enclosed volume

1 ZΣ
v 1
0 W(f) = |H|2dµ, the Willmore energy
4 4
ZΣ
5
1
4 Hc0 (f) = (H −c )2dµ+λ µ(Σ)+λ Vol Σ, the Helfrich energy.
. λ1,λ2 4 0 1 2
1 ZΣ
0 In the above we have used dµ to denote the area element induced by f on Σ, dH3 to
2
1 denote Hausdorff measure in R3, H to denote the mean curvature, and c ,λ ,λ are
0 1 2
: real numbers. When f : Σ → R3 is not an embedding, we still require the definition
v
i of the enclosed volume to make sense. For this reason we have used the pull-back of
X
the Euclidean volume element f∗(dH3) by the map f, as is standard (see [1, Lemma
r
a 2.1] for example). Our notation is further clarified in Section 2.
The Helfrich functional Hc0 is of great interest in applications. Although it
λ1,λ2
appears to have first been studied by Schadow [14], its modern form and subsequent
popularity is due to Helfrich [8] (just as the Willmore functional was considered
1James McCoy, Institute for Mathematics and Applied Statistics, University of
Wollongong, Northfields Ave, Wollongong, NSW 2500, Australia
2Glen Wheeler ( ), Institut fu¨r Analysis und Numerik, Otto-von-Guericke-
(cid:0)
Universita¨t, Postfach 4120, D-39016 Magdeburg, Germany
2000 Mathematics Subject Classification. 35J30 and 58J05 and 35J62 .
Key words and phrases. globaldifferentialgeometryandfourthorderandgeometricanalysisand
higher order elliptic partial differential equations.
Financial support for Glen Wheeler from the Alexander-von-Humboldt Stiftung is gratefully ac-
knowledged.
Email address: [email protected].
1
2 A CLASSIFICATION THEOREM FOR HELFRICH SURFACES
by Germain [2, 6], Poisson [13], Thomsen [15], and others, long before the time of
Willmore [17]), who famously proposed that the minimisers of the functional model
theshape ofanelastic lipidbilayer, such asa biomembrane. Since this timethe model
has enjoyed considerable popularity. Despite this, relatively few analytical results can
be found in the literature.
Critical points of the functional Hc0 are called Helfrich surfaces and solve the
λ1,λ2
Euler-Lagrange equation
1
Hc0 (f) := − ∆H +H|Ao|2 +c 2K − Hc −2λ H −2λ = 0,
λ1,λ2 0 2 0 1 2
(cid:18) (cid:19)
(cid:16) (cid:17)
where Ao and K denote the tracefree second fundamental form and Gauss curvature
of f respectively. For the reader’s convenience we briefly derive this equation in
Lemma 2.
Consider the functional W = H0 . The functional W differs from the
λ1,λ2 λ1,λ2 λ1,λ2
Willmore functional W = W by the surface area and volume terms only, which
0,0
serve to punish or reward additional surface area and volume depending on the sign
of λ and λ respectively. Critical points of W satisfy
1 2 λ1,λ2
(1) W (f) := ∆H +H|Ao|2 −2λ H −2λ = 0.
λ1,λ2 1 2
Clearly the effects of λ and λ on the solution f are local in nature. For this reason
1 2
we nameW thelocally constrained Willmore functional. Froma physical perspec-
λ1,λ2
tive, our simplification corresponds with anassumption that the fluidsurrounding the
membrane f(Σ) and the fluid contained inside the membrane f(Σ) induce zero spon-
taneous curvature in f(Σ). Thus solutions of (1) faithfully represent lipid bilayers in
certain settings.
Heuristically, oneexpects thesign ofλ ,λ to have a dramaticimpact onthe nature
1 2
of theminimisers ofW . If λ is negative, thefunctional isunbounded frombelow.
λ1,λ2 1
Furthermore, as we assume f is only immersed, one can not control the sign of the
volume term. This makes a direct minimisation procedure quite difficult to carry out.
Despite this, the presence of the Willmore term punishes self-intersections and other
complicated branching behaviour. One may thus hope to understand the structure of
the family of immersed critical points with Willmore energy slightly larger than that
of an embedded round sphere.
Our main result is a direct classification of all smooth properly immersed solutions
to the Euler-Lagrange equation (1) with Willmore energy close to that of the sphere.
This is more general than classifying the set of all immersed critical points of W
λ1,λ2
under the energy condition, as we do not assume f is closed. In this case the func-
tional W is not well-defined. Indeed any reasonable definition of the functional
λ1,λ2
for proper non-closed immersed surfaces f would set W (f) = ∞. We may never-
λ1,λ2
theless work directly with the differential equation W (f) = 0. In this manner one
λ1,λ2
may speak of Helfrich surfaces with unbounded area, not well-defined volume, and
so on. The main motivation for considering this class of surfaces is that one expects
proper non-closed immersed surfaces to arise in the analysis of singularities, through
a blowup procedure for example.
A CLASSIFICATION THEOREM FOR HELFRICH SURFACES 3
Theorem 1. Suppose f : Σ → R3 is a smooth properly immersed surface. If
1
(2) |Ao|2dµ < := ε
1
2c c
ZΣ 1 2
where c , c are the absolute constants defined in Corollary 5 and Proposition 6 re-
1 2
spectively, the following statements hold: (λ > 0)
1
(λ < 0) W (f) = 0 if and only if f(Σ) = S (x) for some x ∈ R3,
2 λ1,λ2 −2λ1
λ2
(λ = 0) W (f) = 0 if and only if f(Σ) is a plane,
2 λ1,λ2
(λ > 0) W (f) 6= 0.
2 λ1,λ2
If λ = 0 then
1
(λ = 0) W (f) = 0 if and only if f(Σ) is a plane or a sphere,
2 λ1,λ2
(λ 6= 0) W (f) 6= 0.
2 λ1,λ2
Here S (x) = ∂B (x) denotes the sphere of radius ρ centred at x ∈ R3.
ρ ρ
We note that the fourth statement above is an improvement of [9, Theorem 2.7]
for the case where n = 3.
Remark. The catenoid f : Σ → R3 is a properly immersed minimal surface with
c
|Ao|2dµ = 8π.
ZΣ
Furthermore W (f ) = 0 for all λ ∈ R and so Theorem 1 holds at best for ε ≤ 8π.
λ1,0 c 1 1
The methods we use in this paper are inspired by some recent progress on the
analysis of the Willmore functional [9, 10, 11] due to Kuwert and Sch¨atzle. There are
notable differences between the functional W and the Willmore functional W.
λ1,λ2
The set of all critical points contain the set of all minimisers of the functional, and
the functional W is not bounded from below. Despite λ ≥ 0, one is not able
λ1,λ2 1
to control the sign of the volume term. It is conceivable that an exotic critical point
with enormous negative volume exists as the limit of a properly immersed minimising
sequence.
We prove here that the condition (2) prevents any critical points from appearing
which are not flat planes or round spheres. The operator W does not admit
λ1,λ2
a maximum principle, and thus we do not have access to the large assortment of
tools it brings. We instead rely throughout the paper on estimates for curvature
quantities on smooth immersed surfaces combined with the divergence theorem and
the Michael-Simon Sobolev inequality [12]. Using these we derive a series of local
integral estimates, which when globalised, yield Theorem 1. This is similar to the
idea used to prove [9, Theorem 2.7].
It is notable that the theorem does not require any smallness of the parameters λ
1
and λ , and does not require any smallness (or even boundedness) of kHk for any
2 p
p. Indeed, the surface f may a priori possess quite wild behaviour at infinity. This
implies in particular that these critical points of the functional may have arbitrarily
largeornoteven well-defined Helfrichenergy. Existence resultswithout anysmallness
condition at all are extremely difficult and only known in the class of rotationally
4 A CLASSIFICATION THEOREM FOR HELFRICH SURFACES
symmetric critical points for the Willmore functional [3, 4]. There one finds such a
plethora of solutions that the classification question is for the moment too difficult.
This paper is organised as follows. In Section 2 we set up our notation and briefly
derive the first variation of the functional Hc0 . In Section 3 we establish local
λ1,λ2
integral estimates for the Euler-Lagrange operator W in L2. In Section 4 we
λ1,λ2
prove Theorem 1. Finally, we have included several proofs and derivations of known
results in Appendix A for the convenience of the reader.
The authors would each like to thank their home institutions for their support and
their collaborator’s home institutions for their hospitality during respective visits.
Both authors would also like to thank Prof. Graham Williams for useful discussions
during the preparation of this work.
Acknowledgements
The research of the first author was supported under the Australian Research
Council’sDiscoveryProjectsscheme(projectnumberDP120100097). Thefirstauthor
is also grateful for the support of the University of Wollongong Faculty of Informatics
Research Development Scheme grant.
Part of this work was carried out while the second author was a research associate
supported by the Institute for Mathematics and Its Applications at the University
of Wollongong. Part of this work was also carried out while the second author was
a Humboldt research fellow at the Otto-von-Guericke Universit¨at Magdeburg. The
support of the Alexander von Humboldt Stiftung is gratefully acknowledged.
2. Preliminaries
We consider a surface Σ immersed in R3 via f : Σ → R3 and endow a Riemanain
metric on Σ defined componentwise by
(3) g = h∂ f,∂ fi,
ij i j
where∂ denotestheregularpartialderivativeandh·,·iisthestandardEuclideaninner
product. That is, we consider the Riemannian structure on Σ induced by f, where
in particular the metric g is given by the pull-back of the standard Euclidean metric
along f. Integration on Σ is performed with respect to the induced area element
(4) dµ = det g dH3,
where dH3 is the standard Hausdorff mpeasure on R3.
The metric induces an inner product structure on all tensor fields defined over Σ,
where corresponding pairs of indices are contracted. For example, if T and S are
(1,2) tensor fields,
hT,Si = g gjqgkrTi Sp , |T|2 = hT,Ti .
g ip jk qr g
In the above, and in what follows, we shall use the summation convention on repeated
indices unless otherwise explicitly stated.
The second fundamental form A is a symmetric (0,2) tensor field over Σ with
components
(5) A = ∂2f,ν ,
ij ij
(cid:10) (cid:11)
A CLASSIFICATION THEOREM FOR HELFRICH SURFACES 5
where ν is an inward pointing unit vector field normal along f. With this choice one
finds that the second fundamental form of the standard round sphere embedded in
R3 is positive. There are two invariants of A relevant to our work here: the first is
the trace with respect to the metric
H = trace A = gijA
g ij
called the mean curvature, and thesecond thedeterminant with respect to the metric,
called the Gauss curvature,
K = det A = det gikA ,
g kj
where PikQ is used above to denote the ma(cid:0)trix wi(cid:1)th i,j-th component equal to
kj
PikQ .
kj
(cid:0) (cid:1)
The mean and Gauss curvatures are easily expressed in terms of the principal
curvatures: at a single point we may make a local choice of frame for the tangent
bundle TΣ under which the eigenvalues of A appear along its diagonal. These are
denoted by k , k and are called the principal curvatures. We then have
1 2
H = k +k , K = k k .
1 2 1 2
We shall often decompose the second fundamental form into its trace and its tracefree
parts,
1
A = Ao + gH,
2
where (0,2) tensor field Ao is called the tracefree second fundamental form. In a basis
which diagonalises A, a so-called principal curvature basis, its norm is given by
1
|Ao|2 = (k −k )2.
1 2
2
The Christoffel symbols of the induced connection are determined by the metric,
1
Γk = gkl(∂ g +∂ g −∂ g ),
ij 2 i jl j il l ij
so that then the covariant derivative on Σ of a vector X and of a covector Y is
∇ Xi = ∂ Xi +Γi Xk, and
j j jk
∇ Y = ∂ Y −ΓkY
j i j i ij k
respectively.
From (5) and the smoothness of f we see that the second fundamental form is
symmetric; less obvious but equally important is the symmetry of the first covariant
derivatives of A,
∇ A = ∇ A = ∇ A ,
i jk j ik k ij
commonly referred to as the Codazzi equations.
One basic consequence of the Codazzi equations which we shall make use of is that
the gradient of the mean curvature is completely controlled by a contraction of the
(0,3) tensor ∇Ao. To see this, first note that
1
∇ Ai = ∇ H = ∇ (Ao)i + giH ,
i j i i j 2 j
(cid:16) (cid:17)
6 A CLASSIFICATION THEOREM FOR HELFRICH SURFACES
then factorise to find
(6) ∇ H = 2∇ (Ao)i =: 2(∇∗Ao) .
j i j j
This in fact shows that all derivatives of A are controlled by derivatives of Ao. For
a (p,q) tensor field T, let us denote by ∇ T the tensor field with components
(n)
∇ Tk1...kp = ∇ ···∇ Tk1...kp. In our notation, the i -th covariant derivative
i1...in j1...jq i1 in j1...jq n
is applied first. Since
1
∇(k)A = ∇(k)Ao + g∇(k)H = ∇(k)Ao +g∇(k−1)∇∗Ao ,
2
(cid:16) (cid:17) (cid:16) (cid:17)
we have
(7) |∇ A|2 ≤ 3|∇ Ao|2.
(k) (k)
The fundamental relations between components of the Riemann curvature tensor
R , the Ricci tensor R and scalar curvature R are given by Gauss’ equation
ijkl ij
R = A A −A A ,
ijkl ik jl il jk
with contractions
gjlR = R = HA −AjAk, and
ijkl ik ik i j
gikR = R = |H|2 −|A|2.
ik
We will need to interchange covariant derivatives; for vectors X and covectors Y we
obtain
∇ Xh −∇ Xh = Rh Xk = (A A −A A )ghlXk,
ij ji ijk lj ik lk ij
(8) ∇ Y −∇ Y = R glmY = (A A −A A )glmY .
ij k ji k ijkl m lj ik il jk m
We also use for tensor fields T and S the notation T∗S (as in Hamilton [7]) to denote
a linear combination of new tensors, each formed by contracting pairs of indices from
T and S by the metric g with multiplication by a universal constant. The resultant
tensor will have the same type as the other quantities in the expression it appears. As
is common for the ∗-notation, we slightly abuse this constant when certain subterms
do not appear in our P-style terms. For example
|∇A|2 = h∇A,∇Ai = 1· ∇ A∗∇ A +0· A∗∇ A = P2(A).
g (1) (1) (2) 2
(cid:0) (cid:1) (cid:0) (cid:1)
The Laplacian we will use is the Laplace-Beltrami operator on Σ, with the compo-
nents of ∆T given by
∆Tk1...kp = gpq∇ Tk1...kp = ∇p∇ Tk1...kp.
j1...jq pq j1...jq p j1...jq
A CLASSIFICATION THEOREM FOR HELFRICH SURFACES 7
Using the Codazzi equation withthe interchange of covariant derivative formula given
above, we obtain Simons’ identity:
gkl∇ A = gkl∇ A +gklgpqR A +gklgpqR A
ki lj ik lj kilp qj kijp lq
∆A = ∇ Ak +gklgpq(A A −A A )A
ij ik j pi kl kp il qj
+gklgpq(A A −A A )A
pi kj kp ij lq
= ∇ Ak +HgpqA A −gklgpqA A A
ik j pi qj kp il qj
+gklgpqA A A −A hA,Ai
pi kj lq ij g
(9) = ∇ H +HAkA −|A|2A ,
ij i kj ij
or in ∗-notation
∆A = ∇ H +A∗A∗A.
(2)
The interchange of covariant derivatives formula for mixed tensor fields T is simple
to state in ∗-notation:
(10) ∇ T = ∇ T +T ∗A∗A.
ij ji
We now briefly compute the first variation of the Helfrich functional.
Lemma 2. Suppose f : Σ → R3 is a closed immersed surface and φ : Σ → R3 is a
vector field normal along f. Then
d
Hc0 (f +tφ)
dt λ1,λ2
(cid:12)t=0
1 (cid:12)
= hφ,νi((cid:12)∆H +H|Ao|2 +2c K −(2λ +c2/2)H −2λ )dµ.
2 (cid:12) 0 1 0 2
ZΣ
In particular, if
Hc0 (f) := − ∆H +H|Ao|2 +2c K −(2λ +c2/2)H −2λ = 0
λ1,λ2 0 1 0 2
then
(cid:0) (cid:1)
d
Hc0 (f +tφ) = 0
dt λ1,λ2
(cid:12)t=0
and f is a critical point of Hc0 . (cid:12)
λ1,λ2 (cid:12)
(cid:12)
Proof. For the sake of brevity we shall suppress the dependence of various quantities
on (f + tφ). All derivatives with respect to t will be implicitly evaluated at t = 0.
Let us assume φ = ϕν for some function ϕ : Σ → R. From the formulae (3), (5) we
compute the variation of the mean curvature as
∂ ∂ ∂
H = gij A +A gij = ∆H −φ|A|2 +2φ|A|2 = ∆ϕ+ϕ|A|2,
ij ij
∂t ∂t ∂t
while for the induced area element differentiating the determinant in (4) gives
d
dµ = −ϕHdµ.
dt
Each of these formulae are proven in [5, Appendix A]. Using these and the divergence
theorem we compute
d
dµ = − ϕHdµ,
dt
ZΣ ZΣ
8 A CLASSIFICATION THEOREM FOR HELFRICH SURFACES
d
Hdµ = −2 ϕKdµ,
dt
ZΣ ZΣ
and
1 d
|H|2dµ = ϕ(∆H +H|Ao|2)dµ.
2dt
ZΣ ZΣ
The variation of the enclosed volume may be computed as in [1]. For the convenience
of the reader we outline the argument here. Let p ∈ Σ. Suppose {e ,e ,n} is a
1 2
positively oriented adapted orthonormal frame around f(p). Then
f∗(dH3) = a(t,p)dt∧dµ
where
∂ ∂f ∂f ∂f
a(t,p) = f∗(dH3) ,e ,e = dH3 , ,
1 2
∂t ∂t ∂e ∂e
1 2
(cid:16) (cid:17) (cid:16) (cid:17)
∂f ∂f ∂f
R3
= Vol , , = hφ,ni.
∂t ∂e ∂e
1 2
(cid:16) (cid:17)
We thus have
d d
Vol Σ = a(t,p)dt∧dµ
dt dt
(cid:12)t=0 ZΣ (cid:12)t=0
(cid:12) (cid:12)
(cid:12) (cid:12)
= a(0,p)dµ = − hφ,νi dµ = − ϕdµ.
(cid:12) (cid:12)
ZΣ ZΣ ZΣ
Putting these together we have
d 1
(H −c )2dµ+λ µ(Σ)+λ Vol Σ
0 1 2
dt 4
(cid:18) ZΣ (cid:19)(cid:12)t=0
1 c (cid:12)
= ϕ(∆H +H|Ao|2)dµ+ 0 2(cid:12)ϕKdµ
(cid:12)
2 2
ZΣ ZΣ
− λ +c2/4 ϕHdµ−λ ϕdµ
1 0 2
ZΣ ZΣ
1 (cid:0) (cid:1)
= ϕ ∆H +H|Ao|2 +2c K −(2λ +c2/2)H −2λ dµ,
2 0 1 0 2
ZΣ
(cid:0) (cid:1)
which is the first statement of the lemma. The remaining statement clearly follows
(cid:3)
from the first.
3. Control of geometry by the Euler-Lagrange operator
In this section we prove estimates which give us control of the geometry of a sur-
face in terms of the Euler-Lagrange operator W (f) and some error terms. The
λ1,λ2
geometric quantity which we aim to control is
|∇ A|2 +|∇A|2|A|2 +|A|4|Ao|2.
(2)
Our control is gained through the use of the divergence theorem and multiplica-
tive Sobolev inequalities, as in [9, 16]. The key difference here is that the Euler-
Lagrange operator is much more complicated and we must deal with several extra
terms. Through the usage of a novel test function we have managed to exploit this
extra complexity (in the form of a lack of scale invariance of the functional).
A CLASSIFICATION THEOREM FOR HELFRICH SURFACES 9
Definition. Set γ = γ˜ ◦f : Σ → [0,1], γ˜ ∈ C2(R3) satisfying
c
(γ) k∇γk∞ ≤ cγ, k∇(2)γk∞ ≤ cγ(cγ +|A|),
for some absolute constant c < ∞.
γ
Remark. There is a smooth cutoff function γ˜ on B (y) ⊂ R3, y ∈ R3, i.e.
2ρ
1 if |x−y| ≤ ρ
γ˜(x) = and 0 ≤ γ˜(x) ≤ 1,
0 if |x−y| ≥ 2ρ,
(
such that c ≤ c for an absolute constant c (see the appendix of [18]).
γ ρ
Lemma 3. Suppose f : Σ → R3 is a smooth immersed surface and γ is as in (γ).
Then for δ > 0,
(1−2δ) |∆H|2γ4dµ+2λ |∇H|2γ4dµ
1
ZΣ ZΣ
≤ W (f) 4h∇H,∇γi+γ∆H γ3dµ+δ |H|4|Ao|2γ4dµ
λ1,λ2
ZΣ ZΣ
(c )2 (cid:0) (4δ2 +1(cid:1))2
+8 γ |∇H|2γ2dµ+ |Ao|6γ4dµ.
δ (4δ)3
ZΣ ZΣ
Proof. Since
−2λ Hh∇H,∇γiγ3 −2λ h∇H,∇γiγ3
1 2
= W (f)h∇H,∇γiγ3 −∆Hh∇H,∇γiγ3 −H|Ao|2h∇H,∇γiγ3
λ1,λ2
we have for δ > 0
−2λ Hh∇H,∇γiγ3dµ−2λ h∇H,∇γiγ3dµ
1 2
ZΣ ZΣ
(11) ≤ W (f)h∇H,∇γiγ3dµ+δ |∆H|2γ4dµ
λ1,λ2
ZΣ ZΣ
(c )2
+δ |H|2|Ao|4γ4dµ+ γ |∇H|2γ2dµ.
2δ
ZΣ ZΣ
Multiplying (1) by ∆Hγ4 gives
|∆H|2γ4 = W (f)∆Hγ4 −H|Ao|2∆Hγ4 +2λ H∆Hγ4 +2λ ∆Hγ4,
λ1,λ2 1 2
which combined with the divergence theorem yields
|∆H|2γ4dµ = W (f)∆Hγ4dµ− H|Ao|2∆Hγ4dµ
λ1,λ2
ZΣ ZΣ ZΣ
+2λ H∆Hγ4dµ+2λ ∆Hγ4dµ
1 2
ZΣ ZΣ
= W (f)∆Hγ4dµ− H|Ao|2∆Hγ4dµ
λ1,λ2
ZΣ ZΣ
−2λ |∇H|2γ4dµ−8λ Hh∇H,∇γiγ3dµ
1 1
ZΣ ZΣ
−8λ h∇H,∇γiγ3dµ.
2
ZΣ
10 A CLASSIFICATION THEOREM FOR HELFRICH SURFACES
Using (11) to estimate the inner product terms,
(1−2δ) |∆H|2γ4dµ+2λ |∇H|2γ4dµ
1
ZΣ ZΣ
≤ W (f) 4h∇H,∇γi+γ∆H γ3dµ+δ |H|2|Ao|4γ4dµ
λ1,λ2
ZΣ ZΣ
(c )2 (cid:0) 1 (cid:1)
+8 γ |∇H|2γ2dµ+ |H|2|Ao|4γ4dµ
δ 4δ
ZΣ ZΣ
≤ W (f) 4h∇H,∇γi+γ∆H γ3dµ+δ |H|4|Ao|2γ4dµ
λ1,λ2
ZΣ ZΣ
(c )2 (cid:0) (4δ2 +1(cid:1))2
+8 γ |∇H|2γ2dµ+ |Ao|6γ4dµ.
δ (4δ)3
ZΣ ZΣ
(cid:3)
This proves the lemma.
Lemma 4. Suppose f : Σ → R3 is a smooth immersed surface and γ is as in (γ).
Then
|∇ H|2 +|H|2|∇H|2 +|H|4|Ao|2 γ4dµ+2λ |∇H|2γ4dµ
(2) 1
ZΣ ZΣ
(cid:0) (cid:1)
≤ c W (f) 4h∇H,∇γi+γ∆H γ3dµ+c(c )2 |∇H|2γ2dµ
λ1,λ2 γ
ZΣ ZΣ
(cid:0) (cid:1)
+c |Ao|6 +|Ao|2|∇Ao|2 γ4dµ+c(c )4 |Ao|2dµ,
γ
ZΣ Z[γ>0]
(cid:0) (cid:1)
where c is an absolute constant.
Proof. Applying the divergence theorem to the interchange of covariant derivatives
formula gives the following identity
1
(12) |∇ H|2γ4dµ+ |H|2|∇H|2γ4dµ = |∆H|2γ4dµ
(2)
4
ZΣ ZΣ ZΣ
+ A∗Ao ∗∇Ao ∗∇Hγ4dµ+ ∇H ∗∇ H ∗∇γ γ3dµ;
(2)
ZΣ ZΣ
which upon decomposing A = Ao + 1gH, using (6) and estimating yields
2
1 1
|∇ H|2γ4dµ+ |H|2|∇H|2γ4dµ ≤ |∆H|2γ4dµ
(2)
2 4
ZΣ ZΣ ZΣ
+δ |H|2|∇Ao|2γ4dµ+c |Ao|2|∇Ao|2γ4dµ+c(c )2 |∇H|2γ2dµ,
1 γ
ZΣ ZΣ ZΣ
where δ > 0 and c is a constant depending only on δ . For the convenience of the
1 1
reader we provide a proof of (12) in Appendix A. Combining this with Lemma 3 we