Table Of ContentTHE GEOMETRIC TRIHARMONIC HEAT FLOW OF IMMERSED SURFACES NEAR
SPHERES
JAMESMCCOY,SCOTTPARKINS,ANDGLENWHEELER
Abstract. We consider closed immersed surfaces in R3 evolving by the geometric triharmonic heat flow.
5
Usinglocalenergyestimates,weproveinteriorestimatesandapositiveabsolutelowerboundonthelifespan
1
of solutions depending solely on the local concentration of curvature of the initial immersion in L2. We
0
furtheruseanε-regularitytyperesulttoproveagaplemmaforstationarysolutions. Usingamonotonicity
2
argument, we then prove that a blowup of the flow approaching a singular time is asymptotic to a non-
n umbilicembedded stationary surface. This allows us to conclude that any solution withinitial L2-norm of
a the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a
J roundspherewithradiusequalto p3 3V0/4π,whereV0denotesthesignedenclosedvolumeoftheinitialdata.
0
3
]
P 1. Introduction
A
Let f : Σ [0,T) R3, T > 0, be a one-parameter family of compact immersed surfaces f(,t) = f :
h. Σ f (Σ)=×Σ . We→say f is evolving under the geometric triharmonic heat flow if it satisfies the·followintg
t t
t →
a equation
m
∂
(GTHF) f = (∆2H)ν.
[ ∂t −
1 withgivensmoothinitialsurfacef(,0)=f . Above∆andH arerespectivelytheLaplace-Beltramioperator
0
v and mean curvature of the surface Σ· , and ν is the outer unit normal. Further notation and conventions are
t
1
set out in Section 2.
5
The mean curvature vector H~ satisfies ∆f = H~ = Hν. As H~ is a section of the normal bundle,
6
−
7 it is natural to apply to this the induced Laplacian in the normal bundle ∆⊥. In this case one recovers
0 (∆2H)ν = ∆⊥∆⊥∆f. This is natural from a geometric perspective as well as from the perspective of
1. curvature flow: only normal terms in the speed of the flow affect geometric quantities which are invariant
0 under the diffeomorphism group, and so any natural geometric operator should have image in the normal
5 bundle. One may however wish to interpret H~ as a vector in R3 and instead apply the rough Laplacian ∆
R
1 toit. Inthiscaseonerecoversalengthyexpressionfor∆ ∆ ∆f whichhasleadingorderterm(∆2H)ν with
: R R
v many other terms contributing in tangential and normal directions. We have termed (GTHF) the geometric
i triharmonic heat flow in order to distinguish it from the possible second interpretation.
X
As (GTHF) is a sixth-order quasilinear degenerate parabolic system of partial differential equations, the
r
a local existence of a solution is standard (see, eg. [14, 15]) for regular enough initial data.
Theorem 1. Suppose Σ is a complete, compact 2-manifold without boundary. Let f : Σ R3 be a C6,α
0
→
isometric immersion. There exists a maximal T > 0 and a corresponding unique one-parameter family of
smooth isometric immersions f :Σ (0,T) R3 satisfying (GTHF) and
× →
f(,t) f () as t 0,
0
· → · ց
locally smoothly in the C6,α topology.
In this paper we will always assume that the initial data f is smooth, and so then the flow generated by
0
f is f :Σ [0,T) R3, which is a smooth family on the half-open interval [0,T).
0
× →
Date:****Received 6December2013; receivedinfinalform*******.
1991 Mathematics Subject Classification. 53C44.
TheresearchofthesecondauthorwassupportedbyanAustralianPostgraduateAward. Theresearchofthefirstandthird
authors was supported by Discovery Project DP120100097 of the Australian Research Council. Results from this paper will
appearinthePhDthesisofthesecondauthor.
1
2 J.MCCOY,S.PARKINS,ANDG.WHEELER
Applications of equations involving a thrice-iterated Laplacian are growing, and include for example the
modelling of ulcers [25], computer graphics [1, 24], andinteractive design[8]. In eachof these applications it
is important that the Laplacian considered possesses some inherent geometric invariances, with the natural
candidate being the Laplace-Beltrami operator. We hope in future work to generalise equation (GTHF) to
include lower order terms; such sixth order equations appear as the phase field crystal equation proposed as
amodel for microstructureevolutionoftwo-phasesystems onatomic lengthanddiffusive time scales (see [4]
and the references contained therein) and in surface modelling [12, 23].
In Section6 we study properlyimmersedsurfaceswhich arenotnecessarilyclosed. Our mainresultis the
following tracefree curvature estimate, which holds quite generally. In particular, the immersion does not
need to satisfy ∆2H =0.
Theorem 2. Let x R3, ρ>0, and f :Σ R3 be a locally W5,2 immersion satisfying
∈ →
(1) Ao 2 dµ ε
0
Zf−1(B2ρ(x))| | ≤
for ε >0 sufficiently small. Then there is a universal constant c>0 such that
0
Ao 6 c Ao 4 ∆H 2 +ρ−6 .
k k∞,f−1(Bρ(x)) ≤ k k2,f−1(B2ρ(x)) k∇ k2,f−1(B2ρ(x))
(cid:16) (cid:17)
This has as an immediate corollary the following higher-order analogue of ε-regularity (see [9, 20, 26]
for the corresponding result for the Willmore operator, minimal surfaces, and the surface diffusion operator
respectively).
Corollary 3. Let f :Σ R3 be a locally W5,2 closed immersion satisfying ∆2H = 0 weakly. Suppose that
for x R3, ρ > 0, the lo→cal smallness condition (1) is satisfied for ε > 0 sufficiently small. Then there is
0
∈
an absolute constant c>0 such that
Ao 6 cε2ρ−6.
k k∞,f−1(Bρ(x)) ≤
This estimate is a key first step in a larger regularity program for weak solutions, improving the a-priori
regularityassumptiondramaticallyunderthenaturalassumptionthattheL2-normofthetracefreecurvature
tensor Ao is globally bounded.
If the L2-normofAo is infactgloballysmall, thenwe mayapply againTheorem2to obtainthe following
much stronger result than Corollary 3 above. It is proven in Section 8.
Theorem 4 (Gap Lemma). Suppose f :Σ R3 is a proper locally C6 immersion with ∆2H 0. Then, if
→ ≡
f is closed and satisfies
(2) Ao 2dµ<ε <8π
0
| |
ZΣ
for ε >0 sufficiently small, there exist x R3 and ρ>0 such that
0
∈
f(Σ)=S2(x),
ρ
where S2(x) denotes a standard round sphere in R3 with radius ρ and centre x R3. If f is not closed, then
ρ ∈
under assumption (2) we instead conclude that
f(Σ)=P2,
where P2 denotes a standard flat plane in R3.
Gap Lemmata are known in a variety of contexts: for the Willmore operator [9], the surface diffusion
operator [26], and a family of fourth-order geometric operators [27]. We expect Theorem 2 to enjoy further
applications.
Using Vol(Σ ) and Σ to denote the signed enclosed volume and surface area respectively of Σ in R3,
t t t
| |
we compute that under the flow (GTHF)
d
(3) Vol(Σ )= ∆2Hdµ=0
dt t −
ZΣ
and
d
(4) Σ = H∆2Hdµ= ∆H 2dµ 0.
dt| t| − − | | ≤
ZΣ ZΣ
THE GEOMETRIC TRIHARMONIC HEAT FLOW OF IMMERSED SURFACES NEAR SPHERES 3
From (3), (4), the flow (GTHF) is isoperimetrically natural. The flow may develop singularities while
improving the isoperimetric ratio, manifested as curvature singularities. Our first main parabolic result is
the following characterisationof the singular time.
Theorem 5 (Lifespan Theorem). Suppose f :Σ [0,T) Rn satisfies (GTHF) with smooth initial data
× −→
f . Then there are constants ρ>0,ε >0 and C < such that if ρ is chosen with
0 0
∞
(5) A2dµ =ε(x) ε for any x R3,
0
Zf−1(Bρ(x))| | (cid:12)(cid:12)t=0 ≤ ∈
(cid:12)
then the maximal time T of smooth existence fo(cid:12)r the flow satisfies
(cid:12)
1
(6) T ρ6,
≥ C
and we have the estimate
1
(7) A2dµ Cε for 0 t ρ6.
Zf−1(Bρ(x))| | ≤ 0 ≤ ≤ C
By linearising (GTHF) around a round sphere, one finds that the spectrum is entirely real and non-
positive. Using standardcentre-manifoldmethods, onecanconcludethat anysolutionwith initialdataclose
toasphere S inC6,α existsforalltime andconvergesexponentially fastto asphereS inthe C∞ topology.
1 2
Note that S need not equal S in any sense: the perturbation could be a small translation, in which case
1 2
the centre changes, or it may change the enclosed volume of the initial data from that of S , in which case
1
the radius changes. Using completely different methods, we are able to improve this statement, weakening
the initial condition to geometric averagedcloseness in L2.
Let us briefly detail the argument. Note that if the initial immersion is locally smooth with finite total
curvature then for any ε >0 it is always possible to find a positive ρ=ρ(ε ,Σ ) such that assumption (5)
0 0 0
is satisfied. Theorem 5 is a characterisation of singularities in the following sense. It tells us that the only
way the flow can cease to exist and lose regularity in finite time T < is if we encounter a specific type of
∞
curvaturesingularity: ifρ(t)denotesthelargestradiussuchthat(5)holdsattimet,thenρ(t) 6 C(T t)
≤ −
so at least ε of the curvature concentrates in a ball f−1 B (x) . That is,
0 ρ(T) p
lim (cid:0)A2dµ ε(cid:1)
0
t→TZf−1(Bρ(t)(x))| | ≥
where x=x(t) is the centre of a ball where the integral is maximised at time t.
By considering a sequence of rescalings, we are able to extract a convergent subsequence asymptotic to a
stationary solution of (GTHF). This relies on the monotonicity of the L2-norm of the tracefree curvature
tensor. The stationary solution extracted is entire, with small total tracefree curvature in L2, and so by
applying the Gap Lemma we are able to show that this is a contradiction. Subconvergence of the flow to
a specific sphere results, which we then strengthen by a standard linearised stability argument. Our final
result is the following.
Theorem 6 (Long time existence and exponential convergence to round spheres). There exists an absolute
constant ε > 0 such that a geometric triharmonic heat flow f : Σ [0,T) R3 with smooth initial data
0
× →
satisfying
(8) Ao 2dµ ε <8π
0
ZΣ| | (cid:12)t=0 ≤
exists for all time, is a family of embeddings, and fo(cid:12)r some x R3 the family Σ converges to S2 (x)
(cid:12) ∈ t √3 3V0/4π
exponentially fast in the C∞ topology, where V >0 is the signed enclosed volume of the initial immersion.
0
The paper is organised as follows. After fixing notation in Section 2 and stating evolution equations
and interpolation inequalities in Section 3, we establish local control on the L2-norm of iterated covariant
derivatives of the curvature in Section 4. These provide us with all the requisite tools to prove the Theorem
5, whichwedo inSection5. Section6is devotedto elliptic analysis,andculminates inthe proofofTheorem
2. Section 7 establishes that Ao 2 is a Lyapunov functional for the flow if initially small enough. Section 8
k k2
contains the proof of the Gap Lemma. In Section 9 we use a compactness theorem and most of our earlier
worktoestablishtheexistenceandseveralkeypropertiesoftheblowupofasingulartimefortheflow. Thisis
4 J.MCCOY,S.PARKINS,ANDG.WHEELER
appliedinSection10toconcludeglobalexistenceandsubconvergencetoasphere,whichisthenstrengthened
by a standard linearised stability argument.
2. Preliminaries
We consider a surface Σ immersed into R3 via a smooth immersion f :Σ R3. The induced metric g on
Σ is given by pulling back the standard Euclidean inner product on R3 alon→g f. It is defined pointwise by
(9) g = ∂ f,∂ f .
ij i j
h i
Here∂ denotesthe coordinatederivativesonΣand , the standardEuclideaninnerproduct. Thisinduces
h· ·i
aninner productonalltensorsalongf ofsimilartype,whicharedefined viatracesoverpairsofindices. For
example, the inner product on the (1,2)-tensors S,T is defined by
S,T =gjmgkng Si Tl .
h i il jk mn
Here we adopt the Einstein convention: repeated indices are summed from 1 to 2. The norm squared of a
tensorT istheinnerproductofT withitselfandisdenoted T 2. FortensorsS andT wewillalsofrequently
| |
utilisethenotationofHamilton[6],usingS⋆T todenoteanarbitrarylinearcombinationofnewtensorsthat
is formed by contracting pairs of indices of S and T by g. For example, S,T =S⋆T. A useful property of
h i
⋆ notation is that
−
S⋆T c S T
≤ | || |
for some constant c that depends only on the number of indices in S and T combined (recall the dimension
and codimension are fixed). For a tensor T, we define
Pi(T)= T ⋆ ⋆ T,
j ∇(r1) ··· ∇(rj)
r1+·X··+rj=i
a sum of terms, each of which contains j factors of T with total order of covariant derivatives equal to i.
This notation is particularly useful when only the number of factors of T and total number of derivatives is
important. As an example
T 2 = T ⋆ T =P4(T).
∇(2) ∇(2) ∇(2) 2
The mean curvature is defined to be
(cid:12) (cid:12)
(cid:12) (cid:12) H =gijA =Ai
ij i
where A are the components of the second fundamental form A:
ij
A = ∂ f,ν .
ij ij
−h i
The Gauss curvature K is the determinant of the Weingarten map Aj :=gikA ,
i jk
K =det Aj.
i
We also define the tracefree second fundamental form to be the symmetric tracefree part of A, denoted Ao,
with components
1
Ao =A g H.
ij ij − 2 ij
A short calculation shows that
1 1
(10) ( ∗Ao) := iAo =gip A g H = H;
∇ j ∇ ij ∇p ij − 2 ij 2∇j
(cid:18) (cid:19)
note that ∗ is the geometric divergence operator or formal adjoint of in L2. Here we have used the
−∇ ∇
total symmetry of the (0,3)-tensor A, known as the Codazzi equations:
∇
A = A = A .
i jk j ki k ij
∇ ∇ ∇
It follows that
H 2 =4 ∗Ao 2 4 Ao 2.
|∇ | |∇ | ≤ |∇ |
Moreover,for any k N we have
∈
(11) H 2 4 Ao 2
(k) (k)
∇ ≤ ∇
and hence (cid:12) (cid:12) (cid:12) (cid:12)
A 2 = (cid:12) Ao 2(cid:12)+ 1 (cid:12) H 2(cid:12) 3 Ao 2.
∇(k) ∇(k) 2 ∇(k) ≤ ∇(k)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
THE GEOMETRIC TRIHARMONIC HEAT FLOW OF IMMERSED SURFACES NEAR SPHERES 5
The Laplace-Beltrami operator acts on the components of an (m,n)-tensor S via
∆Si1...im =gpq Si1...im.
j1...jn ∇pq j1...jn
We define the integral of a compactly supported function h:Σ R as
→
hdµ,
ZΣ
where dµ is the induced measure on Σ:
dµ:= det(g )d 2.
ij
L
q
We denote the area by
µ(Σ)= Σ = dµ.
| |
ZΣ
We will frequently employ the Divergence Theorem to ‘integrate by parts’ over a non-compact immersion.
To do so we include a cut-off function γ : Σ R3 in the integrand. We keep this function arbitrary but
→
sufficientlysmoothwithcompactsupportsothatwemayeventuallyconcludeglobalresultsfromresultsthat
hold on the support of γ. In particular, we take γ =γ˜ f :Σ [0,1] for some C3 function γ˜ satisfying
◦ →
0 γ˜ 1 and γ˜ c < .
≤ ≤ k kC3(R3) ≤ γ˜ ∞
Usingthe chainrule andthe factthat ∂ f providesabasis forthe tangentbundle TΣ,there is a universal,
i
{ }
bounded constant c >0 such that γ satisfies
γ
(12) γ c , γ c (c + A), and γ c c2 +c A + A .
k∇ k∞ ≤ γ ∇(2) ∞ ≤ γ γ | | ∇(3) ∞ ≤ γ γ γ| | |∇ |
(cid:13) (cid:13) (cid:13) (cid:13) (cid:0) (cid:1)
3. Evol(cid:13)ution(cid:13)equations for elemen(cid:13)tary g(cid:13)eometric quantities
Under (GTHF), we have the following evolution equations for geometric quantities associatedto Σ . The
t
proof of Lemma 7 is standard and straightforward.
Lemma 7. Let f :Σ [0,T) R3 satisfy (GTHF). Then
× →
∂ ∂ ∂
g = 2 ∆2H A, dµ= H ∆2H dµ, ν = ∆2H,
∂t − ∂t − ∂t ∇
∂ A=∆3(cid:0)A+P(cid:1)4(A), and ∂ H =(cid:0)∆3H +(cid:1) (∆2H) A2.
∂t 3 ∂t | |
By combining the evolution equations in Lemma 7 with the interchange of covariant derivatives formula,
we obtain the following evolution equations.
Lemma 8. Let f :Σ [0,T) R3 satisfy (GTHF). Then for any k N :
0
× → ∈
∂
A=∆3 A+Pk+4(A).
∂t∇(k) ∇(k) 3
Corollary 9. Let f :Σ [0,T) R3 satisfy (GTHF). Then for any k N :
0
× → ∈
∂
A 2 =2 A, i3i2i1 A +Pk+4(A)⋆ A.
∂t ∇(k) ∇(k) ∇ ∇i1i2i3∇(k) 3 ∇(k)
Using Corollary 9, Le(cid:12)(cid:12)mma 7(cid:12)(cid:12), the p(cid:10)roduct rule, and integrating(cid:11)by parts thrice we obtain the following.
Corollary 10. Let f :Σ [0,T) R3 satisfy (GTHF). Then for any k,s N :
0
× → ∈
d
A 2γsdµ+2 A 2dµ= A 2∂ γsdµ
dt ∇(k) ∇(k+3) ∇(k) t
ZΣ ZΣ ZΣ
(cid:12) (cid:12) 3 (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) 3 (cid:12) (cid:12) (cid:12)
2 γs A , A dµ+ Pk+4(A)⋆ A γsdµ.
− j ∇(j) ∇(k+3−j) ∇(k+3) 3 ∇(k)
j=1(cid:18) (cid:19)ZΣ ZΣ
X (cid:10)(cid:0) (cid:1)(cid:0) (cid:1) (cid:11) (cid:0) (cid:1)
To deal with the extraneous intermediate terms above we will make use of (12) and an interpolation
inequality of Kuwert and Scha¨tzle [9, Corollary 5.3]. This gives us:
6 J.MCCOY,S.PARKINS,ANDG.WHEELER
Lemma 11. Let 2 p < ,k,m N and s kp. Then for δ > 0 there exists a constant c > 0 depending
≤ ∞ ∈ ≥
only on δ, s and p such that
1
A pγsdµ p1 δc−1 A pγs+pdµ p1 +c ck A2γs−kpdµ p .
(cid:18)ZΣ ∇(k) (cid:19) ≤ γ (cid:18)ZΣ ∇(k+1) (cid:19) δ γ Z[γ>0]| | !
(cid:12) (cid:12) (cid:12) (cid:12)
The proof o(cid:12)f Lemm(cid:12)a 11 is essentially the s(cid:12)ame as in(cid:12) [9]. However,we haveretained the derivative bounds
c to make more explicit the scale-invariance of each quantity. Using Lemma 11, the properties (12) and
γ
Corollary 10 we obtain the following estimate.
Proposition 12. Let f : Σ [0,T) R3 satisfy (GTHF). For any δ > 0, k N , and s 2(k+3) there
0
× → ∈ ≥
exists a constant c>0 depending only on δ, s and k such that
d
A 2γsdµ+(2 δ) A 2γsdµ Pk+4(A)⋆ A γsdµ+cc2(k+3) A 2 .
dt ∇(k) − ∇(k+3) ≤ 3 ∇(k) γ k k2,[γ>0]
ZΣ ZΣ ZΣ
(cid:12) (cid:12) (cid:12) (cid:12) (cid:0) (cid:1)
In ord(cid:12)er to co(cid:12)ntrol the localised n(cid:12)orms (cid:12) A 2 := A2γsdµ we will employ standard interpo-
k∇(k) k2,γs Σ|∇(k) |
lation and Sobolev inequalities, which we now state.
R
Theorem 13 (Michael-Simon Sobolev Inequality [19]). Let Σ be an immersed surface and u C∞(Σ).
∈ c
Then, for a universal, bounded constant cMSS >0,
2
u2dµ cMSS udµ+ u H dµ
≤ |∇ | | || |
ZΣ (cid:18)ZΣ ZΣ (cid:19)
Theorem 14 ([10, Theorem 5.6]). Let f :Σ R3 be a smooth immersion. For n<p , 0 β and
→ ≤∞ ≤ ≤∞
0<α 1, where 1 = 1 1 β+1, there is a constant c depending on p and β such that for all u C1(Σ),
≤ α 2 − p ∈ c
(cid:16) (cid:17) α
(13) u c u 1−α u + Hu .
k k∞ ≤ k kp k∇ kp k kp
Proposition 15 ([10, Corollary 5.5]). Let 0 i ,...(cid:16),i k, r i =(cid:17) 2k and s 2k. Then for any tensor
≤ 1 r ≤ j=1 j ≥
T defined over an immersed surface f :Σ R3 we have
→ P
T ⋆ ⋆ Tγsdµ c T r−2 T 2γsdµ+c2k T 2 .
∇i1 ··· ∇ir ≤ k k∞,[γ>0] ∇(k) γ k k2,[γ>0]
(cid:12)ZΣ (cid:12) (cid:18)ZΣ (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
Note that w(cid:12)e have included cγ explici(cid:12)tly on the right hand(cid:12)side ab(cid:12)ove. This will be important later when
(cid:12) (cid:12)
we select a particular cutoff function. The next estimate below is an adaptation of [10, Lemma 4.3] to our
situation. The proof is similar.
Proposition 16. Suppose T is a tensor field, s 6 and γ is as in (12). There is an ε > 0 such that
≥
A 2 ε implies that there exists a universal, bounded constant c >0 depending only on the order of
k k2,[γ>0] ≤ 0
T such that
kTk6∞,[γ=1] ≤ckTk42,[γ>0] ∇(3)T ·γ2s 22+ |∇T|2|A|4·γs 1
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) + ∇(cid:13)(2)T(cid:13)(cid:13)|A|·γ2s 22+ (cid:13)(cid:13)|T||∇A||A|·γ2s 22,+c6γ T ·γs−26 22,[γ>0]!.
The lemma below is similar to [10,(cid:13)(cid:13)L(cid:12)(cid:12)emma(cid:12)(cid:12) 4.2] an(cid:13)(cid:13)d [9,(cid:13)(cid:13)Proposition 2.6](cid:13)(cid:13). We h(cid:13)(cid:13)(cid:13)ave incor(cid:13)(cid:13)(cid:13)porated the
smallness assumption and since the proof involves some new estimates, we present it.
Lemma 17. Let γ satisfy (12). Then for an immersion f :Σ R3 satisfying
→
(14) A2 dµ ε
0
| | ≤
Z[γ>0]
for ε >0 sufficiently small, we have
0
A 2 A2γsdµ + A2 A4γsdµ c A2 dµ A 2γsdµ + c6 A2 dµ .
∇(2) | | |∇ | | | ≤ | | ∇(3) γ | |
ZΣ(cid:12) (cid:12) ZΣ Z[γ>0] (cid:16)ZΣ(cid:12) (cid:12) Z[γ>0] (cid:17)
(cid:12) (cid:12) (cid:12) (cid:12)
THE GEOMETRIC TRIHARMONIC HEAT FLOW OF IMMERSED SURFACES NEAR SPHERES 7
Proof. The second term can be estimated via Theorem 13 with u= A A2γs2:
|∇ || |
A2 A4γsdµ
|∇ | | |
ZΣ
2
c (2)A A2γs2dµ+ A2 A γ2sdµ+cγ A A2γs2−1dµ+ A A3γ2sdµ
≤ ∇ | | |∇ | | | |∇ || | |∇ || |
(cid:16)ZΣ(cid:12) (cid:12) ZΣ ZΣ ZΣ (cid:17)
c A 2(cid:12) (cid:12) A 2γsdµ+ A 2 A2γsdµ+ A2 A4γsdµ
≤ k k2,[γ>0] ∇(3) ∇(2) | | |∇ | | |
(cid:16)ZΣ(cid:12) (cid:12) ZΣ(cid:12) (cid:12) ZΣ
(15) + A4γ6dµ(cid:12)+c6 A(cid:12) 2 . (cid:12) (cid:12)
|∇ | γk k2,[γ>0]
ZΣ (cid:17)
Here we have used integration by parts, the Cauchy-Schwarzinequality and Lemma 11. For the second last
term in (15) we use integration by parts and Lemma 11:
A4γsdµ 3 A A2 A γsdµ+sc A3 A γs−1dµ
(2) γ
|∇ | ≤ ∇ |∇ | | | |∇ | | |
ZΣ ZΣ ZΣ
1 A4γsdµ+c(cid:12)(cid:12) (cid:12)(cid:12) A 2 A2γsdµ+ A2 A4γsdµ +cc4 A2γs−4dµ,
≤ 2 |∇ | ∇(2) | | |∇ | | | γ |∇ |
ZΣ (cid:16)ZΣ(cid:12) (cid:12) ZΣ (cid:17) ZΣ
which implies that (cid:12) (cid:12)
A4γsdµ c A 2γsdµ + A 2 A2γsdµ + A2 A4γsdµ + c6 A 2 .
|∇ | ≤ ∇(3) ∇(2) | | |∇ | | | γk k2,[γ>0]
ZΣ (cid:16)ZΣ(cid:12) (cid:12) ZΣ(cid:12) (cid:12) ZΣ (cid:17)
For the second term above,(cid:12)we use(cid:12) Theorem 13(cid:12)with u(cid:12)= (2)A A γ2s:
∇ | |
(cid:12) (cid:12)
(16) A 2 A2γsdµ c A2 dµ A(cid:12)2γsdµ(cid:12)+c2 A 2γs−2dµ
∇(2) | | ≤ | | ∇(3) γ ∇(2)
ZΣ(cid:12) (cid:12) Z[γ>0] (cid:16)ZΣ(cid:12) (cid:12) ZΣ(cid:12) (cid:12)
(cid:12) (cid:12) + (cid:12) A 2(cid:12) A2γsdµ +c(cid:12) (cid:12) A 2γ4dµ A2γ2dµ.
(2) (2)
∇ | | ∇ |∇ |
ZΣ(cid:12) (cid:12) (cid:17) ZΣ(cid:12) (cid:12) ZΣ
The last term in (16) can be dealt with via inte(cid:12)gration(cid:12)by parts, the Cauc(cid:12)hy-Sch(cid:12)warz inequality and the
interpolation inequality from Lemma 11:
A 2γ4dµ A2γ2dµ
(2)
∇ |∇ |
ZΣ ZΣ
(cid:12) (cid:12)
c(cid:12) (cid:12) A A γ4dµ+c A A γ3dµ A A γ2dµ+c A A γdµ
(3) γ (2) (2) γ
≤ ∇ |∇ | ∇ |∇ | ∇ | | |∇ || |
(cid:18)ZΣ ZΣ (cid:19)(cid:18)ZΣ ZΣ (cid:19)
1 (cid:12)(cid:12) A 2(cid:12)(cid:12)γ4dµ A2γ2dµ(cid:12)(cid:12) +c A(cid:12)(cid:12) 2 (cid:12)(cid:12)A 2γ6d(cid:12)(cid:12)µ+c6 A 2 .
≤ 2 ∇(2) |∇ | k k2,[γ>0] ∇(3) γk k2,[γ>0]
ZΣ ZΣ (cid:18)ZΣ (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
Hence (cid:12) (cid:12) (cid:12) (cid:12)
A 2γ4dµ A2γ2dµ c A 2 A 2γ6dµ+c6 A 2 .
∇(2) |∇ | ≤ k k2,[γ>0] ∇(3) γk k2,[γ>0]
ZΣ ZΣ (cid:18)ZΣ (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
Substituting thi(cid:12)s result(cid:12) into (16) and combining with (15) then(cid:12)yields(cid:12)
A 2 A2γ6dµ+ A2 A4γ6dµ
(2)
∇ | | |∇ | | |
ZΣ ZΣ
(cid:12) (cid:12)
c(cid:12) A 2 (cid:12) A 2γ6dµ+ A 2 A2γ6dµ+ A2 A4γ6dµ+c6 A 2 ,
≤ k k2,[γ>0] ∇(3) ∇(2) | | |∇ | | | γk k2,[γ>0]
so that absorbing and(cid:16)mZΣul(cid:12)(cid:12)tiplyin(cid:12)(cid:12)g out yieldZsΣt(cid:12)(cid:12)he stat(cid:12)(cid:12)ement of the lZeΣmma. (cid:17) (cid:3)
4. A-priori estimates for A 2 along the flow
k∇(k) k2,γs
The structure of this argument is similar to that used for [10, Proposition 4.6], although the details are
necessarily different. We begin by applying the evolution equations and interpolation inequalities above to
obtain local control on the concentration of curvature A 2 .
k k2,γs
8 J.MCCOY,S.PARKINS,ANDG.WHEELER
Proposition 18. Let f : Σ [0,T∗] R3 satisfy (GTHF), and let γ be a cutoff function satisfying (12).
× →
Then there is an absolute constant ε >0 such that if
0
sup A2 dµ ε
0
[0,T∗]Z[γ>0]| | ≤
then for any t [0,T∗] we have
∈
t
A2 dµ+ A 2+ A 2 A2+ A2 A4 dµdτ A2 dµ +cc6ε t.
Z[γ=1]| | Z0 Z[γ=1](cid:16)(cid:12)∇(3) (cid:12) (cid:12)∇(2) (cid:12) | | |∇ | | | (cid:17) ≤Z[γ>0]| | (cid:12)(cid:12)t=0 γ 0
Proof. Utilising Proposition1(cid:12)2 with(cid:12)s=(cid:12)6,k=0(cid:12), yields (cid:12)
d
(17) A2γ6dµ+(2 δ) A 2γ6dµ P4(A)⋆A γ6dµ+cc6 A 2 .
dt | | − ∇(3) ≤ 3 γk k2,[γ>0]
ZΣ ZΣ ZΣ
(cid:12) (cid:12) (cid:2) (cid:3)
We estimate the P-style terms via integr(cid:12)ation b(cid:12)y parts and the Cauchy-Schwarz inequality, whilst utilising
our inequalities from Lemma 17:
P4(A)⋆A γ6dµ= A⋆A⋆A⋆A γ6dµ
3 ∇(4)
ZΣ ZΣ
(cid:2) (cid:3) (cid:0) (cid:1)
+ A⋆ A⋆A+ A⋆ A⋆A+ A⋆ A⋆ A ⋆A γ6dµ
(3) (2) (2) (2)
∇ ∇ ∇ ∇ ∇ ∇ ∇
ZΣ
(cid:2)(cid:0) (cid:1) (cid:3)
A⋆ A⋆A⋆A γ6dµ+ A⋆A⋆A⋆A⋆ γ γ5dµ
(3) (3)
≤ ∇ ∇ ∇ ∇
ZΣ ZΣ
(cid:0) (cid:1) (cid:0) (cid:1)
+δ A 2γ6dµ+c A 2 A2+ A2 A4+ A4 γ6dµ
(3) δ (2)
∇ ∇ | | |∇ | | | |∇ |
ZΣ(cid:12) (cid:12) ZΣ(cid:16)(cid:12) (cid:12) (cid:17)
δ (cid:12)A 2γ6(cid:12)dµ+c (cid:12)A 2 A(cid:12)2+ A2 A4+ A4 γ6dµ
(3) δ (2)
≤ ∇ ∇ | | |∇ | | | |∇ |
ZΣ(cid:12) (cid:12) ZΣ(cid:16)(cid:12) (cid:12) (cid:17)
(18) +c(cid:12)c2 (cid:12)A6γ4dµ. (cid:12) (cid:12)
γ | |
ZΣ
The last term in (18) can be estimated via Theorem 13, the Cauchy-Schwarzinequality and Lemma 17:
2
c2 A6γ4dµ cc2 A A γ2dµ+c A3γdµ+ A4γ2dµ
γ | | ≤ γ |∇ || | γ | | | |
ZΣ (cid:18)ZΣ ZΣ ZΣ (cid:19)
c A 2 A 2γ6dµ+ A2 A4γ6dµ+c2 A6γ4dµ+c6 A 2 .
≤ k k2,[γ>0] ∇(3) |∇ | | | γ | | γk k2,[γ>0]
(cid:16)ZΣ(cid:12) (cid:12) ZΣ ZΣ (cid:17)
Hence for ε sufficiently small, Lemm(cid:12) a 17 t(cid:12)ells us that
0
c2 A6γ4dµ c A2 dµ A 2γ6dµ+c6 A2 dµ .
γZΣ| | ≤ Z[γ>0]| | ZΣ ∇(3) γZ[γ>0]| | !
(cid:12) (cid:12)
(cid:12) (cid:12)
Combining this with (18) and substituting into (17) then yields
d A2γ6dµ+ 3 A 2γ6dµ c A 2 A2+ A2 A4 γ6dµ+cc6 A2 dµ.
dt | | 2 ∇(3) ≤ ∇(2) | | |∇ | | | γ | |
ZΣ ZΣ(cid:12) (cid:12) ZΣ(cid:16)(cid:12) (cid:12) (cid:17) Z[γ>0]
Using Lemma 17 and allowi(cid:12)ng ε t(cid:12)o be sufficiently(cid:12)small(cid:12)we then obtain
0
d
A2γ6dµ+ A 2+ A 2 A2+ A2 A4 γ6dµ cc6 A2 dµ.
dt | | ∇(3) ∇(2) | | |∇ | | | ≤ γ | |
ZΣ ZΣ(cid:16)(cid:12) (cid:12) (cid:12) (cid:12) (cid:17) Z[γ>0]
Integrating over [0,t] and using [(cid:12)γ =1](cid:12) [γ(cid:12)>0], 0(cid:12) γ 1 completes the proof. (cid:3)
⊂ ≤ ≤
We now control the evolution of A 2 for k 1. Our goal is to eventually apply the interpolation
k∇(k) k2,γs ≥
inequalities to the resultantestimate in orderto provethat A 2 satisfies a differentialinequality and
k∇(k) k2,γs
is therefore bounded via Gronwall’s inequality.
THE GEOMETRIC TRIHARMONIC HEAT FLOW OF IMMERSED SURFACES NEAR SPHERES 9
Proposition 19. Let f : Σ [0,T) R3 satisfy (GTHF) and let γ be a cutoff function satisfying (12).
× →
Then for s 2k+6 the following estimate holds:
≥
d
A 2γsdµ+ A 2γsdµ
dt ∇(k) ∇(k+3)
ZΣ ZΣ
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) c A(cid:12) 6 A 2γsdµ+cc2k A2 dµ c6 + A 6 .
≤ k k∞,[γ>0] ∇(k) γ | | γ k k∞,[γ>0]
ZΣ(cid:12) (cid:12) Z[γ>0] (cid:16) (cid:17)
Proof. We use integration by parts, Proposition 1(cid:12)5 and(cid:12)the Cauchy-Schwarz inequality to deal with the
P-style terms that appear in Proposition 12:
Pk+4(A)⋆ A γsdµ
3 ∇(k)
ZΣ
(cid:0) (cid:1)
= A⋆A⋆A+ A⋆ A⋆A ⋆ A γsdµ+ P2k+4,k+2(A) γsdµ
∇(k+4) ∇(k+3) ∇ ∇(k) 4 ·
ZΣ ZΣ
(cid:2)(cid:0) (cid:1) (cid:3)
A⋆ A⋆ A⋆A+ A⋆A⋆A γs dµ
(k+3) (k) (k+1)
≤ ∇ ∇ ∇ ∇
ZΣ
(cid:2) (cid:0) (cid:1) (cid:3)
+cc A A A2γs−1dµ+ P2k+4,k+2(A)γsdµ
γ ∇(k+3) ∇(k) | | 4
ZΣ ZΣ
(cid:12) (cid:12)(cid:12) (cid:12)
δ (cid:12)A 2γsdµ(cid:12)+(cid:12) P(cid:12)2k+4,k+2(A)+P2k+2,k+1(A) γsdµ+c2 P2k,k(A)γs−2dµ
≤ ∇(3) 4 6 γ 6
ZΣ(cid:12) (cid:12) ZΣ(cid:16) (cid:17) ZΣ
δ (cid:12) A(cid:12)2γsdµ+c A 2 A 2γsdµ+c2(k+2) A 2
≤ ∇(3) k k∞,[γ>0] ∇(k+2) γ k k2,[γ>0]
ZΣ(cid:12) (cid:12) (cid:16)ZΣ(cid:12) (cid:12) (cid:17)
+c (cid:12)A 4 (cid:12) A 2γsdµ+(cid:12)c2(k+1) A(cid:12) 2
k k∞,[γ>0] ∇(k+1) γ k k2,[γ>0]
(cid:18)ZΣ (cid:19)
(cid:12) (cid:12)
(19) +cc2 A 4 (cid:12) A(cid:12)2γs−2dµ+c2k A 2 .
γk k∞,[γ>0] ∇(k) γ k k2,[γ>0]
(cid:18)ZΣ (cid:19)
(cid:12) (cid:12)
Integrating by parts, applying the(cid:12) Cauch(cid:12)y-Schwarz inequality and absorbing and multiplying out similarly
as in the proof of Lemma 11 we obtain the following interpolation inequalities:
(20) A 2 A 2γsdµ δ A 2γsdµ
k k∞,[γ>0] ∇(k+2) ≤ ∇(k+3)
ZΣ ZΣ
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) +c A 4 (cid:12) (cid:12) A 2γsdµ+cc2k A 2 c6 + A 6 ,
k k∞,[γ>0] ∇(k+1) γ k k2,[γ>0] γ k k∞,[γ>0]
ZΣ(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
(21) A 4 A 2γsdµ δ A 2 A 2γsdµ
k k∞,[γ>0] ∇(k+1) ≤ k k∞,[γ>0] ∇(k+2)
ZΣ ZΣ
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) +c A 6 (cid:12) A 2γsd(cid:12)µ+cc2k A 2 c6 + A 6 ,
k k∞,[γ>0] ∇(k) γ k k2,[γ>0] γ k k∞,[γ>0]
ZΣ(cid:12) (cid:12) (cid:16) (cid:17)
and (cid:12) (cid:12)
(22) c2 A 4 δ A 2γsdµ+c A 6 A 2γsdµ+cc2(k+3) A 2 .
γk k∞,[γ>0] ≤ ∇(k+3) k k∞,[γ>0] ∇(k) γ k k2,[γ>0]
ZΣ ZΣ
(cid:12) (cid:12) (cid:12) (cid:12)
Combining(20),(21)and(22)a(cid:12)ndsubst(cid:12)itutinginto(19)estimates(cid:12)ofallt(cid:12)heP-styleterms. Substitutingthis
result into Proposition 19 then yields the desired result. (cid:3)
We are now in a position to prove an analogue of [10, Proposition 4.6].
Proposition 20. Let f : Σ [0,T∗] R3 satisfy (GTHF) and let γ be a cutoff function satisfying (12).
× →
Then there is a ε >0 such that if
0
(23) sup A2 dµ ε ,
0
[0,T∗]Z[γ>0]| | ≤
10 J.MCCOY,S.PARKINS,ANDG.WHEELER
we can conclude that for all k N there are constants c˜ , depending only on T∗, c , ε and
k γ 0
∈
k+3
2
(24) α (k+3):= A
0 ∇(j) 2,[γ>0] t=0
Xj=0(cid:13) (cid:13) (cid:12)(cid:12)
such that (cid:13) (cid:13) (cid:12)
2
(25) A c˜ .
∇(k) ∞,[γ=1] ≤ k
Proof. Weintroduceasetofcutofffunctions(cid:13)γ fo(cid:13)r0 σ <τ 1satisfyingγ =1forγ τ andγ =0
(cid:13) σ,τ (cid:13) ≤ ≤ σ,τ ≥ σ,τ
for γ σ. Applying Proposition 18 with γ =γ yields
≤ 0,12
T∗
A 2+ A 2 A2+ A2 A4 dµdτ cε 1+c6T∗ .
Z0 Z[γ≥21] ∇(3) ∇(2) | | |∇ | | | ≤ 0 γ
(cid:12) (cid:12) (cid:12) (cid:12) (cid:0) (cid:1)
Next, we apply Proposition 1(cid:12)6 with(cid:12)T =(cid:12)A and(cid:12)γ . This gives
1,3
2 4
T∗
A 6 dτ cε3 1+c6T∗ .
Z0 k k∞,[γ≥43] ≤ 0 γ
(cid:0) (cid:1)
Here we have used γ 3 γ =1 , γ γ 1 and (26). Next we look at Proposition 19 with
≥ 4 ⊆ 21,34 21,34 ⊆ ≥ 2
γ43,87 and integrate o(cid:2)ver [0,(cid:3)T∗]h. This givieshus foriany(cid:2)t∈[0,(cid:3)T∗]:
t
A 2γs dµ A 2 +cc2kε 1+c6T∗ +c A 6 A 2γs dµdτ.
ApZpΣl(cid:12)(cid:12)y∇in(gk)Gr(cid:12)(cid:12)om43w,a78ll’s≤Ine(cid:13)(cid:13)q∇u(akl)ity(cid:13)(cid:13)w2,[hγi>ls0t](cid:12)(cid:12)(cid:12)tta=k0ing inγto a0c(cid:0)countγiden(cid:1)tity (Z206)kgikv∞es,[γ≥43]ZΣ(cid:12)(cid:12)∇(k) (cid:12)(cid:12) 43,78
Z[γ≥87](cid:12)(cid:12)∇(k)A(cid:12)(cid:12)2 dµ≤cck(cid:16)((cid:13)(cid:13)α∇0((kk))A,(cid:13)(cid:13)c22γ,,[γε>00,]T(cid:12)(cid:12)(cid:12)t∗=)0.+cc2γkε0(cid:0)1+c6γT∗(cid:1)(cid:17)ecε30(1+c6γT∗)
≤
Combining this with Proposition 16 with γ ,
7,15
8 16
A 6 cc2 c + P2(k+2),k+2(A)+P2(k+1),k+1(A) γs dµ+c2(k+3)ε
(cid:13)∇(k) (cid:13)∞,[γ≥1156] ≤c(ck(cid:16),ck+,3c Z,Σch 4,ε ,T∗,c ), 6 i 78,1165 γ 0(cid:17)
(cid:13) (cid:13) 3 k k+2 k+3 0 γ
≤
which proves (25). Here we have used the inequality
A 6 cε2 c +c6ε
k k∞,[γ=1] ≤ 0 3 γ 0
which can be obtained by combining Proposition 16 and L(cid:0)emma 17(cid:1). (cid:3)
We now have sufficient tools to prove Theorem 5. However, we will first sharpen our derivative bounds
from Proposition 20 for later application in combination with Theorem 32 where we construct a blowup.
Theorem 20 will also be used to obtain our preliminary compactness result (Lemma 38) for the flow once
global existence has been obtained.
Theorem 21 (Interior Estimates). Suppose f :Σ (0,T∗] R3 satisfies (GTHF) and
× →
(26) sup A2 dµ ε for T∗ cρ6.
0
0<t≤T∗Zf−1(Bρ(x))| | ≤ ≤
Then for any m N we have at time t (0,T∗] we have the estimates
0
∈ ∈
∇(m)A 2,f−1(Bρ(x)) ≤cm√ε0t−m6 and ∇(m)A ∞,f−1(Bρ(x)) ≤ck√ε0t−m6+1,
2 2
(cid:13) (cid:13) (cid:13) (cid:13)
where the const(cid:13)ants c (cid:13)depend only upon m, ρ, T∗ and(cid:13) A(cid:13) .
m ∇(m) 2,f−1(Bρ(x0)) t=0
The idea of the proof is to consider a family of cuto(cid:12)(cid:12)ff funct(cid:12)(cid:12)ions in time (cid:12)(cid:12)(cid:12)and integrate over (0,T∗]. An
inductive argument then gives the desired the result. Since this is similar to [9, Theorem 3.5], we have
omitted it.
