Finite time singularities for the locally constrained Willmore flow of surfaces

0.33 MB·English

by  James McCoy

FINITE TIME SINGULARITIES FOR THE LOCALLY
CONSTRAINED WILLMORE FLOW OF SURFACES
JAMES MCCOY1 AND GLEN WHEELER2
Abstract. In this paper we study the steepest descent L2-gradient flow of the
afurneac,tiaonndalλW2-wλ1e,λig2h,twedhiecnhcilsotshedetvhoelusmume,ofofrthsuerWfacilelmsiomremeenresergdyi,nλR1-3w.eTighhitsecdosinucrfidacees
2
with the Helfrich functional with zero ‘spontaneous curvature’. Our first results
1
0 are a concentration-compactness alternative and interior estimates for the flow.
2 For initial data with small energy, we prove preservation of embeddedness, and by
n directly estimating the Euler-Lagrange operator from below in L2 we obtain that
a the maximal time of existence is finite. Combining this result with the analysis of
J a suitable blowup allows us to show that for such initial data the flow contracts to

2 a round point in finite time.
2

]
G
D 1. Introduction
.
h Suppose we have a surface Σ immersed via a smooth immersion f : Σ R3 and
t →
a consider the functional
m
1
[ c0 (f) = (H c )2dµ+λ µ(Σ)+λ Vol Σ.
Hλ1,λ2 4 − 0 1 2
1 ZΣ
v In the above we have used dµ to denote the area element induced by f on Σ, d 3
1 to denote Hausdorff measure in R3, H to denote the mean curvature, µ(Σ) to denoHte
4
5 the surface area, Vol Σ to denote the signed enclosed volume, and c0,λ1,λ2 are real
4 numbers. Our notation is further clarified in Section 2.
.
1 Suppose f : Σ R3 is an embedded surface. The Helfrich flow is the steepest
0
0 descent L2-gradien→t flow for c0 (f), and is given by the one-parameter family of
2 Hλ1,λ2
1 immersions f : Σ [0,T) R3 satisfying
: × →
v
∂f 1
i = ∆H +H Ao 2 +c 2K Hc 2λ H 2λ ν,
X 0 0 1 2
∂t − | | − 2 − −
r (cid:18) (cid:16) (cid:17) (cid:19)
a f( ,0) = f ( ),
0
· ·
where ν is the inward pointing unit normal to f, Ao denotes the tracefree second
fundamental form and K = det A denotes the Gauss curvature. That this flow
1Institute for Mathematics and Applied Statistics, University of Wollongong,
Northfields Ave, Wollongong, NSW 2500, Australia
2Institut fu¨r Analysis und Numerik, Otto-von-Guericke-Universita¨t, Postfach
4120, D-39016 Magdeburg, Germany
Key words and phrases. global differential geometry, fourth order, geometric analysis, parabolic
partial differential equations.
Financial support for the secondauthor from the Alexander-von-HumboldtStiftung is gratefully
acknowledged.
E-mail address: [email protected].
1
2 FINITE TIME SINGULARITIES
represents the steepest descent L2-gradient flow for c0 (f) follows from its first
Hλ1,λ2
variation (see Lemma 2.1).
The Helfrich functional is of great interest in applications. The modern application
ofthefunctional tomodel theshapeofanelastic lipidbilayer, such asabiomembrane,
is due to Helfrich [10]. Despite the considerable popularity of the functional as a
model, there are relatively few analytical results to be found in the literature. With
c = 0, we have
0
∂f
= W (f)ν = (∆H +H Ao 2 2λ H 2λ )ν,
(CW) ∂t − λ1,λ2 − | | − 1 − 2
f( ,0) = f ( ),
0
· ·
which is the steepest descent gradient flow in L2 for the locally constrained Willmore
functional = 0 . We have used W to denote the Euler-Lagrange
Wλ1,λ2 Hλ1,λ2 λ1,λ2
operator of .
Wλ1,λ2
The above flow, from a physical perspective, corresponds with an assumption that
thefluidsurroundingthemembranef(Σ)andthefluidcontainedinsidethemembrane
f(Σ) induce zero spontaneous curvature in f(Σ). Thus the flow (CW) faithfully
representstheHelfrichflowincertainsettings. Fromamoremathematicalperspective
however, the flow (CW) is a locally constrained Willmore flow. (In contrast with
globally constrained flows, such as those considered in [17, 21].) The normal velocity
consists precisely of a linear combination of the normal velocity of Willmore flow,
mean curvature flow, and a constant scaling factor.
The principal object of study for this paper is the flow (CW). Local existence for
(CW) is explicitly established in [11] using results of Amann [1, 2, 3, 4]. We quote
the result in the following (weaker) form.
Theorem 1.1 (Kohsaka-Nagasawa). Suppose f : Σ R3 is a closed immersed
0
→
surface. There exists a maximal T, T (0, ], and a corresponding unique one-
parameter family of smooth immersions∈f : Σ∞ [0,T) R3 satisfying (CW) and
× →
f( ,0) = f ( ).
0
· ·
Remark. The evolution equation (CW) is invariant under tangential diffeomor-
phisms, and depending on the choice of λ and λ may be also invariant under
1 2
subgroups of the full M¨obius group of R3. (If λ = λ = 0 then the equation is
1 2
invariant with respect to the entire M¨obius group.) The uniqueness in the local
existence theorem above is understood to be modulo these invariances.
It is an easy exercise to see that for λ > 0 and λ 0 spheres S shrink self-
1 2 ρ
≥
similarly along the flow. It is thus natural to wonder if this property of the flow is
robust in the sense that solutions nearby spheres also shrink in finite time to round
points. It could a priori be the case that there exist local minimisers of the functional
intheneighbourhoodofspheres, whichprevent thefamilyofspheres S : ρ (0, )
ρ
{ ∈ ∞ }
from being local attractors for the flow. The following classification theorem assures
us that this is not the case.
Theorem 1.2 ([16, Theorem 1]). Suppose f : Σ R3 is a smooth properly immersed
→
surface. There exists an absolute constant ε > 0 such that if
1
(1) Ao 2dµ < ε
1
| |
ZΣ
FINITE TIME SINGULARITIES 3
then 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) = 0.
2 λ1,λ2 6
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 = 0) W (f) = 0.
2 6 λ1,λ2 6
Here S (x) = ∂B (x) denotes the sphere of radius ρ centred at x R3.
ρ ρ
∈
Clearly this implies the following partial result.
Corollary 1.3. Suppose f : Σ [0,T) R3 is a one-parameter family of closed
× →
immersions evolving by (CW) with λ > 0, and λ 0. Suppose assumption (1) is
1 2
≥
satisfied for each t [0,T). Then f( ,t) is never stationary; that is, for all p Σ
∈ · ∈
and t [0,T) we have
∈
∂f
(p,t) = W f( ,t) ν(p,t) = 0.
∂t − λ1,λ2 · 6
This partial result indicates that a cond(cid:0)ition s(cid:1)uch as (1) on the L2-norm of the
tracefree second fundamental form is appropriate to use as a ‘distance’ from the
family of round spheres. It is not obvious however that if (1) is initially satisfied, it
remains satisfied for all time. Most importantly, the statement that the flow never
reaches a critical point is not anywhere near as strong as stating that the flow is
asymptotic to a shrinking sphere. It does not even imply that the maximal time of
existence is finite.
In this paper we offer the following more comprehensive answer as our main result.
Theorem 1.4. Suppose f : Σ [0,T) R3 is a one-parameter family of closed
× →
immersions evolving by (CW) with λ > 0, λ 0, and Vol Σ > 0. There exists an
1 2 0
≥
ε > 0 depending only on λ and λ such that if
2 1 2
(2) (f ) < 4π +ε
Wλ1,λ2 0 2
then
1
T < (f )+1,
4λ2πWλ1,λ2 0
1
and f(Σ,t) shrinks to a round point as t T.
→
We note that the smallness of ε required may be computed explicitly; it is not the
2
result of a contradiction argument.
The methods we use in this paper are inspired by recent progress on the analysis
of the Willmore functional [12, 13, 14] due to Kuwert & Sch¨atzle. There are some
notable differences between the functional and the Willmore functional .
Wλ1,λ2 W0,0
The extra terms in break theconformal invariance ofthe functional andadd to
Wλ1,λ2
the complexity of the Euler-Lagrange operator W . Furthermore, for the steepest
λ1,λ2
descent gradient flow of , one loses the a priori global monotonicity of the
Wλ1,λ2
Willmore energy. Indeed, one loses not only the a priori monotonicity of the Willmore
4 FINITE TIME SINGULARITIES
energy but also the a priori uniform bounds on the Willmore energy. Since the
flow (CW) is fourth order and highly non-linear, one should expect that the flow
could drive initially embedded data to a self-intersection. It is then conceivable that
Vol Σ < 0 for some t > 0 and therefore one loses all control on the Willmore energy
t
(and the surface area). The situation could continue to worsen, with the Willmore
energy growing without bound while the energy continues to satisfy (2).
Despite these considerations, we show here that the condition (2) is quite suitable
for the study of . The operator W does not admit a maximum principle,
Wλ1,λ2 λ1,λ2
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 in-
equality [18].
Our proofofTheorem 1.4relies uponaconcentration-compactness alternative (also
called a lifespan theorem), which classifies finite singular times as being local concen-
trations of the curvature in L2, for a class of flows larger than those generated by only
considering the L2 gradient flow of c0 . The methods used here are classical inter-
Hλ1,λ2
polation inequalities and energy estimates, such as was used for a large class of higher
order equations in [6, 7, 8] and successfully applied to the study of the Willmore flow
in [13].
The global analysis of (CW) requires that one first obtain good control on the
Willmore energy and the surface area along the flow. As mentioned above, due to the
possibility of self-intersections occuring along the flow, we must be quite careful in
using the monotonicity of the energy . We first prove (cf. [22]) in Proposition
Wλ1,λ2
4.1 that under (2) a conservation law holds for the Willmore energy, and is itself
monotonically decreasing along the flow. This implies by a well-known result of Li
and Yau [15] that the evolving surface remains embedded for all time. Using this, we
prove L1 estimates for Ao 4 (Proposition 4.2), which we then apply to estimate the
L2 norm of W fromk bekl∞ow. This immediately gives a quantifiable finite estimate
λ1,λ2
of the extinction time for the flow (Proposition 4.3). Employing a blowup analysis,
we find that the blowup along any blowup sequence is a round sphere. This implies
that the area of the evolving surface vanishes as t T while the surfaces themselves
→
become asymptotically round.
This paper is organisedas follows. In Section 2we set up our notationandstate the
first variation of the functional c0 . In Section 3 we establish parabolic regularity
Hλ1,λ2
theory for a general class of flows. The main results are the lifespan theorem and the
interior estimates, Theorem 3.1 and Theorem 3.11 respectively. Section 4 contains
the demonstration of a finite time singularity, including the proof that the maximal
existence time is finite and the blowup classification. Finally, we 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.
FINITE TIME SINGULARITIES 5
Acknowledgements
TheresearchofthefirstauthorwassupportedundertheAustralianResearchCoun-
cil’sDiscoveryProjectsscheme(projectnumbersDP0556211andDP120100097). The
first author 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 = ∂ f,∂ f ,
ij i j
h i
where∂ denotestheregularpartialderivativeand , isthestandardEuclideaninner
h· ·i
product. That is, we consider the Riemannian structure on Σ induced by f, where
in particular the metric g is given by the pullback of the standard Euclidean metric
along f. Integration on Σ is performed with respect to the induced area element
(4) dµ = det g d 3,
H
where d 3 is the standard Hausdorff mpeasure on R3.
H
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,
T,S = g gjqgkrTi Sp , T 2 = T,T .
h ig ip jk qr | | h ig
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
where ν is an inward pointing unit vector(cid:10)field n(cid:11)ormal 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, andthesecond the determinant with respect to themetric,
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)
6 FINITE TIME SINGULARITIES
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)
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
A = Ao + g H = Ao +g Ao ,
(k) (k) (k) (k) (k 1) ∗
∇ ∇ 2 ∇ ∇ ∇ − ∇
(cid:16) (cid:17) (cid:16) (cid:17)
we have
(7) A 2 3 Ao 2.
(k) (k)
|∇ | ≤ |∇ |
FINITE TIME SINGULARITIES 7
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 [9]) 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.
We denote polynomials in the iterated covariant derivatives of T by
Pi(T) = c T T,
j ij∇(k1) ∗···∗∇(kj)
k1+X...+kj=i
where the constants c R are absolute. We use P0(T) to denote a constant. As
ij ∈ 0
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 = A, A = 1 A A +0 A A = P2(A).
|∇ | h∇ ∇ ig · ∇(1) ∗∇(1) · ∗∇(2) 2
The Laplacian we will use is the(cid:0)Laplace-Beltra(cid:1)mi op(cid:0)erator on Σ(cid:1), with the compo-
nents of ∆T given by
∆Tk1...kp = gpq Tk1...kp = p Tk1...kp.
j1...jq ∇pq j1...jq ∇ ∇p j1...jq
Using the Codazzi equation with theinterchange 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 +gklgpq(A A A A )A
ij ∇ik j pi kl − kp il qj pi kj − kp ij lq
= Ak +HgpqA A gklgpqA A A +gklgpqA A A A A,A
∇ik j pi qj − kp il qj pi kj lq − ijh ig
(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
∇ ∇ ∗ ∗
WenowstatethefirstvariationoftheHelfrichfunctionalforeaseoffuturereference.
8 FINITE TIME SINGULARITIES
Lemma 2.1. Suppose f : Σ R3 is a closed immersed surface and φ : Σ R3 is a
→ →
vector field normal along f. Then
d 1
c0 (f +tφ) = φ,ν (∆H +H Ao 2+2c K (2λ +c2/2)H 2λ )dµ.
dtHλ1,λ2 (cid:12)t=0 2 ZΣh i | | 0 − 1 0 − 2
In particular, if (cid:12)
(cid:12)
∂f
φ = = ∆H +H Ao 2 +2c K (2λ +c2/2)H 2λ ν
∂t − | | 0 − 1 0 − 2
then (cid:0) (cid:1)
d 1
c0 (f) = ∆H +H Ao 2 +2c K (2λ +c2/2)H 2λ 2dµ
dtHλ1,λ2 −2 | | 0 − 1 0 − 2
ZΣ
(cid:12) (cid:12)
and the one-parameter famil(cid:12)y f : Σ [0,T) R3 is the steepest descent L(cid:12)2-gradient
× →
flow of c0 .
Hλ1,λ2
Proof. For the proof of the first statement see [16, Lemma 2.1]. The remaining state-
ments follow from the definition of the L2-gradient. (cid:3)
3. Parabolic regularity
In this section we first prove that, analagous to the cases of Willmore flow, sur-
face diffusion flow and the constrained variants thereof [13, 17, 21, 22], so long as
the concentration of curvature remains well-controlled the flow continues to exist
smoothly. This statement not only holds for c = 0, but also more generally for flows
0
f : Σ [0,T) R3 of the form 6
× →
3
∂f
(11) = ∆H + P0(A) ν.
∂t − α
!
α=0
X
The speed F is a second order elliptic differential operator on the Weingarten map
Dν, that is, a fourth order differential operator on f.
Our main result in this section is the following concentration-compactness alterna-
tive for the class of flows (11).
Theorem 3.1. Let f : Σ R3 be a smooth immersion. There are absolute constants
→
ε > 0 and c < such that if ρ > 0 is chosen with
0 0
∞
(12) A 2dµ ε < ε
0
| | ≤
Zf−1(Bρ(x)) (cid:12)(cid:12)t=0
(cid:12)
for any x R3, then the maximal time T of (cid:12)existence of the flow (11) satisfies
∈ (cid:12)
1
T ρ4,
≥ c
0
and for 0 t 1ρ4,
≤ ≤ c0
A 2dµ c ε .
0 0
| | ≤
Zf−1(Bρ(x))
FINITE TIME SINGULARITIES 9
Remark. It is possible to weaken the regularity requirement on the initial data
by exploiting the instantaneous smoothing property [5] of the flow. In the proof of
Theorem 3.1, smoothness of the initial data is only needed to bound the derivatives of
curvature at final time. However for this argument we may use in place of the initial
data the immersion at any earlier time: in particular f = f ,δ , δ (0,T),
δ
· · ∈
which is smooth (Theorem 3.2).
(cid:0) (cid:1) (cid:0) (cid:1)
In the proof of Theorem 3.1 we shall use local coordinate notation as well as the
- and P-style notation introduced in Section 2, which is most convenient for our
∗
computations, as for example in [9]. We briefly note that the more general flow (11)
also enjoys local existence. The proof is an essentially identical (to that found in [11])
verification that the general existence theory of Amann [1, 2, 3, 4] applies. Note again
that the uniquness statement below is understood modulo the natural invariances of
(11), which includes at least the family of diffeomorphisms tangential along f.
We note that the initial regularity required by Theorem 3.2 below is not optimal.
Theorem 3.2. For any C4,α initial immersion f : Σ R3, there exists a unique
0
solution f : Σ [0,T) R3 to the flow (11) on a maxim→al time interval [0,T) with
× →
initial value f and for which f ( ) := f( ,t) is smooth for every t (0,T).
0 t
· · ∈
The following evolution equations follow from straightforward computations. Their
derivations in a slightly more general setting can be found in Lemma A.1 and Lemma
A.2.
Lemma 3.3. Under the flow (11) we have the following evolution equations for var-
ious geometric quantities associated with f:
∂ ∂ ∂
g = 2FA gij = 2FAij ν = F
ij ij
∂t ∂t − ∂t −∇
∂ d
Γi = F A+FP1(A) dµ = HFdµ
∂t jk ∇ ∗ 1 dt −
3 5
∂
A = ∆2A + P2(A)+ P0(A)
∂t ij − ij α α
α=1 α=2
X X
3 5
∂
A = ∆2 A + Pm+2(A)+ Pm(A).
∂t∇(m) ij − ∇(m) ij α α
α=1 α=2
X X
Corollary 3.4. Under the flow (11),
∂
A 2 = 2 β α A i1 imAkl
∂t ∇(m) − ∇ ∇ ∇α∇β∇i1 ···∇im kl∇ ···∇
(cid:12) (cid:12) 3 5
(cid:12) (cid:12) + Pm+2(A)+ Pm(A) A.
α α ∗∇(m)
" #
α=1 α=2
X X
Proof. This follows by direct computation using Lemma 3.3 and (10) as in the proof
(cid:3)
of Lemma A.2.
We now establish energy estimates for the flow.
10 FINITE TIME SINGULARITIES
Lemma 3.5. Let η : Σ [0,T] R be a C2 function. While a solution to the flow
× →
(11) exists,
d ∂η
2 2 2
η A dµ = ηH∆H A dµ+ A dµ
(m) (m) (m)
dt ∇ − ∇ ∂t ∇
ZΣ ZΣ ZΣ
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) 2 (cid:12) (cid:12) A α β (cid:12)η i1 (cid:12) imAkl dµ
− ∇α∇β∇i1 ···∇im kl∇ ∇ ∇ ···∇
ZΣ
3 5 (cid:0) (cid:1)
+ η Pm+2(A)+ Pm(A) Adµ.
α α ∗∇(m)
ZΣ "α=1 α=2 #
X X
Proof. The Lemma follows by differentiating using Lemma 3.3 and Corollary 3.4 and
(cid:3)
applying the divergence theorem.
Weshall further specialise by setting η to beasmoothcutoff functionontheinverse
image under f of balls from R3.
Definition. Set γ = γ˜ f : Σ [0,1], γ˜ C2(R3) satisfying
◦ → ∈ c
(γ) γ c , γ c (c + A ),
γ (2) γ γ
k∇ k∞ ≤ k∇ k∞ ≤ | |
for some absolute constant c < .
γ
∞
Lemma 3.6. Suppose η = γs where γ is as in (γ), s 4 and θ > 0. While a solution
≥
to the flow (11) exists,
d
A 2γsdµ+(2 θ) A 2γsdµ
(m) (m+2)
dt ∇ − ∇
ZΣ ZΣ
(cid:12) (cid:12) ∂γ (cid:12) (cid:12)
s(cid:12) (cid:12) A 2γs 1 dµ+C(cid:12) (cid:12)A 2γs 4 γ 4 +γ2 γ 2 dµ
(m) − (m) − (2)
≤ ∇ ∂t ∇ |∇ | ∇
ZΣ ZΣ (cid:16) (cid:17)
(cid:12) 3 (cid:12) 5 (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
+ γs Pm+2(A)+ Pm(A) Adµ,
α α ∗ ∇(m)
ZΣ "α=1 α=2 #
X X
where C is a constant depending only on θ and s.
Proof. This follows from Lemma 3.5 using the divergence theorem and Cauchy’s in-
equality ab δa2 + 1 b2. (cid:3)
≤ 4δ
Lemma 3.7. Suppose γ is as in (γ), s 2m+4 and θ > 0. While a solution to the
≥
flow (11) exists, we have
d
A 2γsdµ+(2 θ) A 2γsdµ
(m) (m+2)
dt ∇ − ∇
ZΣ ZΣ
(cid:12) 3(cid:12) 5 (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
C Pm+2(A)+ Pm(A) A γsdµ+C A 2γs 4 2mdµ,
≤ α α ∗∇(m) | | − −
ZΣ"α=1 α=2 # Z[γ>0]
X X
where C is a constant depending only on θ, s, m and c .
γ
Proof. Estimate the time derivative of γ by
∂γ
c ∆H +P0(A) c P2(A)+c P0(A),
∂t ≤ γ| 3 | ≤ γ 1 γ 3