On the extension of axially symmetric volume flow and mean curvature flow

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by  Sevvandi Kandanaarachchi

ON THE EXTENSION OF AXIALLY SYMMETRIC VOLUME FLOW AND MEAN
CURVATURE FLOW
SEVVANDIKANDANAARACHCHI
ABSTRACT. Weinvestigateconditionsofsingularityformationofmeancurvatureflowandvolume
3 preservingmeancurvatureflowinanaxiallysymmetricsetting. Weprovethatnosingularitiescan
1 developduringafinitetimeinterval,ifthemeancurvatureisbounded withinthattimeintervalon
0
theentire surface. Weprove this for volume preserving mean curvature flow as well asfor mean
2
curvatureflow.
n
a
J

7
1. INTRODUCTION

]
G Consider n-dimensional hypersurfaces M , defined by a one parameter family of smooth immer-
t
D sionsx :Mn Rn+1. Thehypersurfaces M aresaidtomovebymeancurvature, ifx = x(,t)
t t t
→ ·
.
h satisfies
t
a d
m (1.1) x(l,t) = H(l,t)ν(l,t), l Mn,t > 0.
dt − ∈
[
Byν(l,t)wedenoteasmoothchoiceofunitnormalonM atx(l,t)(outernormalincaseofcom-
t
1
pactsurfaces withoutboundary), andbyH(l,t)themeancurvature withrespecttothisnormal.
v
5
2
If the evolving compact surfaces M are assumed to enclose a prescribed volume V the evolution
1 t
1 equation changesasfollows:
.
1
d
0 (1.2) x(l,t) = (H(l,t) h(t))ν(l,t), l Mn,t > 0,
3 dt − − ∈
1
whereh(t)istheaverageofthemeancurvature,
:
v
Xi h(t) = MtHdgt,
R dg
ar Mt t
R
andg denotes themetriconM . Astheinitialsurface wechooseacompactn-dimensional hyper-
t t
surfaceM ,withboundary ∂M = . WeassumeM tobesmoothlyembeddedinthedomain
0 0 0
6 ∅
G = x Rn+1 : a < x < b , a,b > 0,
1
{ ∈ }
and ∂M ∂G. Here we have a free boundary. We consider an axially symmetric surface con-
0
⊂
tained in the region G between the two parallel planes x = a and x = b. Motivated by the fact
1 1
that the stationary solution to the associated Euler Lagrange equation satisfies a Neumann bound-
arycondition, wealsoassumethesurfacetomeettheplanesatrightanglesalongitsboundary. We
considerthegeneralquestionwhetherasingularitycandevelopifthemeancurvatureofasurfaceis
boundedforagiventimeinterval. LeandSesuminvestigatedrelatedproblemsin[12]. Inparticular
theyprovedthatformeancurvatureflow,ifallsingularities areoftypeI,themeancurvatureblows
2010MathematicsSubjectClassification. 53C44,35K93.
1
2 SEVVANDIKANDANAARACHCHI
upatthefirstsingular timeT .
Thispaperisorganisedasfollows. Insections3and4,westudyvolumepreservingmeancurvature
flow and investigate whether a singularity could develop if H is bounded. In sections 5 and 6 we
studythesamequestion formeancurvature flow. Weprovethefollowingtwotheorems.
Theorem 1.1. For x(l,t) M satisfying (1.2), if H(l,t) c for all l M2 and for all
t
∈ | | ≤ ∈
t [0,T)then,thereexistsaconstantc,suchthat A(l,t) c fort [0,T),i.e. theflowcanbe
′ ′
∈ | | ≤ ∈
extended pasttimeT.
Theorem 1.2. For x(l,t) M satisfying (1.1), if H(l,t) c for all l M2 and for all
t
∈ | | ≤ ∈
t [0,T)then,thereexistsaconstantc,suchthat A(l,t) c fort [0,T),i.e. theflowcanbe
′ ′
∈ | | ≤ ∈
extended pasttimeT.
Acknowledgements : This is the work of the author’s PhD dissertation. Theauthor would like to
thank her supervisor Maria Athanassenas for her guidance and support and Gerhard Huisken for
helpfuldiscussions andforassistingavisittotheMaxPlanckInstitute ofGravitational Physics. In
particular GerhardHuiskenpointedoutthetechnique ofrescalingbacktoobtainacontradiction as
doneinboththetheorems.
2. NOTATION
We use the same notation as in [11], which was based upon Huisken’s [9] and Athanassenas’ [1]
notation in describing the 2-dimensional axially symmetric hypersurface. Let ρ : [a,b] R
0
→
be a smooth, positive function on the bounded interval [a,b] with ρ (a) = ρ (b) = 0. Consider
′0 ′0
the 2-dimensional hypersurface M inR3 generated byrotating the graph of ρ about the x -axis.
0 0 1
We evolve M along its mean curvature vector keeping its enclosed volume constant subject to
0
Neumannboundaryconditionsatx = aandx = b. Equivalently,wecouldconsidertheevolution
1 1
of a periodic surface defined along the whole x axis. By definition the evolution preserves axial
1
symmetry. Thepositionvectorxofthehypersurface satisfiestheevolution equation
d
x= (H h)ν = H+hν,
dt − −
where H is the mean curvature vector. Since ∆x = H, where ∆ denotes the Laplacian on the
surface, weobtain,
d
(2.1) x= ∆x+hν.
dt
Leti ,i ,i bethestandardbasisinR3,corresponding tox ,x ,x axesandτ (t),τ (t)bealocal
1 2 3 1 2 3 1 2
orthonormal frameonM suchthat
t
τ (t),i = 0, and τ (t),i > 0.
2 1 1 1
h i h i
Let ω = xˆ R3 denote the unit outward normal to the cylinder intersecting M at the point
xˆ ∈ t
x(l,t),wh|er|exˆ = x x,i i . Let
1 1
−h i
y = x,ω and v = ω,ν −1 .
h i h i
ONTHEEXTENSIONOFAXIALLYSYMMETRICVOLUMEFLOWANDMEANCURVATUREFLOW 3
Here y is the height function. We call v the gradient function. We note that v is a geometric
quantity, related to the inclination angle; in particular v corresponds to 1+ρ2 in the axially
′
symmetric setting. The quantity v has facilitated results such as gradienpt estimates in graphical
situations (seeforexample[5,6]).
Weintroduce thequantities (seealso[9])
(2.2) p = τ ,i y 1, q = ν,i y 1,
1 1 − 1 −
h i h i
sothat
(2.3) p2+q2 = y 2.
−
Thesecond fundamental form has one eigenvalue equal top = 1 and one eigenvalue equal
ρ√1+ρ′2
to
ρ
′′
k = ν,τ = − .
1 1
∇ (1+ρ2)3/2
(cid:10) (cid:11) ′
Wenote thatρ(x ,t)istheradius function such thatρ : [a,b] [0,T) R,whereasy(l,t)isthe
1
× →
heightfunction andy :M2 [0,T) R.
× →
3. EVOLUTION EQUATIONS AND PRELIMINARY RESULTS - VOLUME FLOW
Lemma3.1. Wehavethefollowingevolution equations:
(i) dy = ∆y 1 +hpy;
dt − y
(ii) dv = ∆v A2v+ v 2 v 2;
dt −| | y2 − v|∇ |
(iii) dk = ∆k+ A2k 2q2(k p) hk2;
dt | | − − −
(iv) dp = ∆p+ A2p+2q2(k p) hp2;
dt | | − −
(v) dH = ∆H +(H h)A2;
dt − | |
Proof. Evolution equations (i), (ii), (iii) and (iv) are proved in [3] Lemma5.1 and (v) is proved in
[8]. (cid:3)
Boundson h. Getting bounds for h(t) is very important as h(t) is a global term. This enables us
to use the maximum principles. Athanassenas ([2] Proposition 1.4) has obtained bounds for h(t)
for an axially symmetric hypersurface between two parallel planes having orthonormal Neumann
boundary data. The proof is of a geometric nature and works because of the axial symmetry. We
stateithere.
Proposition 3.2. (Athanassenas) Assume M to be a family of smooth, rotationally symmetric
t
{ }
surfaces, solving (1.2)fort [0,T).Thenthemeanvaluehofthemeancurvature satisfies
∈
0 < c h c ,
2 3
≤ ≤
withc andc constants depending ontheinitialhypersurface M .
2 3 0
Nextweproveaheightdependent gradient estimate.
Lemma3.3. Thereexistsaconstant c depending onlyontheinitialhypersurface, suchthatvy <
4
c ,independent oftime.
4
4 SEVVANDIKANDANAARACHCHI
Proof. Wecalculate fromLemma3.1
d 2
(yv c t) = ∆(yv) v, (yv) yv A2+h c .
3 3
dt − − v h∇ ∇ i− | | −
Ash c wegetbytheparabolic maximumprinciple
3
≤
yv c t maxyv,
3
− ≤ M0
yv maxyv+c T =:c .
3 4
≤ M0
(cid:3)
Proposition3.4. AssumeM tobeaxiallysymmetricsurfacesasdescribedinSection2thatevolve
t
by (1.2). Then there is a constant c depending only on the initial hypersurface, such that the
1
principal curvatures k andpsatisfy k < c ,independent oftime.
p 1
Proof. Similartoequation(19)of[9]wecalculate fromLemma3.1
d k k 2 k q2 hk
= ∆ + p, +2 (p k)(p+k)+ (p k) .
dt (cid:18)p(cid:19) (cid:18)p(cid:19) p (cid:28)∇ ∇(cid:18)p(cid:19)(cid:29) p2 − p −
If k 1then(p k) < 0. Bytheparabolic maximumprinciple weobtain
p ≥ −
k k
(3.1) max 1,max =: c .
1
p ≤ (cid:18) M0 p(cid:19)
(cid:3)
InthenextProposition wewillgetboundsfor |k| ifthemeancurvature isbounded forallt < T.
p
Proposition 3.5. If there exists a constant C such that H (l,t) C for all l M2 and for all
| | ≤ ∈
t < T, then there exists another constant c0 such that |kp|(l,t) ≤ c0 for all l ∈ M2 and for all
t < T.
Proof. As
C H = k+p C,
− ≤ ≤
and 1 = vy c fromLemma3.3
p ≤ 4
k
| | 1+Cc =:c .
4 0
p ≤
(cid:3)
We note that the above proposition holds for a hypersurface evolving by mean curvature or by
volumepreserving meancurvature.
4. RESCALED SURFACE - VOLUME FLOW
4.1. Rescaling procedure. Wefollow a rescaling technique similar tothat of Huisken and Sines-
trari used in [10]. Consider time intervals [0,T 1], for i 1,i N, and determine the point
− i ≥ ∈
l M2 andthe(latestinthatinterval)timet ,suchthat
i i
∈
(4.1) A(l ,t ) = max A(l,t);
i i
| | l M2 | |
∈ 1
t T
≤ −i
ONTHEEXTENSIONOFAXIALLYSYMMETRICVOLUMEFLOWANDMEANCURVATUREFLOW 5
asitisanaxially symmetrichypersurface wechoosel M2,suchthatx(l ,t )isonthe(x ,x )
i i i 1 3
∈
plane. Let α = A(l ,t ) and x = x(l ,t ). We consider the family of rescaled surfaces M
i i i i i i i,τ
| |
definedbythefollowingimmersions:
(4.2) x˜ ( ,τ) = α x( ,α 2τ +t ) x ,i i ,
i · i · −i i −h i 1i 1
(cid:0) (cid:1)
whereτ [ α2t ,α2(T t 1)].Heret = α 2τ +t wheret 0,T 1 .
∈ − i i i − i − i −i i ∈ − i
(cid:2) (cid:3)
therescaledsurface
b b
theoriginalsurface x
i
b b b
hxib,i1ii1 0 Theaxisofrotation
FIGURE 1. Therescaling
Notethat werescale from apoint onthe axis of rotation corresponding tothe maximum curvature
A2. ThustherescaledsurfacesM satisfyaxialsymmetryandwedenotebyρ˜ theirgenerating
i,τ i,τ
| |
curves - the cross section of M with the (x ,x ) plane . We look at a sequence of rescaled
i,τ 1 3
surfaces at different times t where t T. We denote by A˜ and H˜ the second fundamental
i i i i
→ | |
formandthemeancurvature associated withtheimmersionx˜ . Bythedefinitionofx˜ wehave
i i
H˜ ( ,τ) = α 1H( ,α 2τ +t ), and A˜ ( ,τ) = α 1 A( ,α 2τ +t ).
i · −i · −i i | i| · −i | | · −i i
Fort T 1 fromthedefinitionofα wehave
≤ − i i
(4.3) A( ,α 2τ +t ) A(l ,t ),thatis α 1 A( ,α 2τ +t ) 1.
| | · −i i ≤ | | i i −i | | · −i i ≤
We observe that M flow by mean curvature, preserving the enclosed volume of the rescaled
i,τ
surface: Toseethis,let
H˜ ( ,τ)dg˜
M i τ
h˜ = i,τ · ,
i R
dg˜
M τ
i,τ
R
andnotethatthemetricsatisfies
detg˜ (τ) = αn detg (t),
ij i ij
q q
where (with a slight abuse of notation) we denoted by g˜ the metric on M , and by g˜ (τ) the
τ i,τ ij
components ofg˜ . Therefore
τ
α 1H( ,α 2τ +t )αndg
h˜ (τ) = Mi,τ −i · −i i i t = α 1h(α 2τ +t ) = α 1h(t),
i R αndg −i −i i −i
Mi,τ i t
R
andfinally
d dx dt
(4.4) x˜ = α = α 1(H h)ν = (H˜ h˜ )ν.
dτ i i dt dτ − −i − − i− i
6 SEVVANDIKANDANAARACHCHI
τ
b
i
i=1
τ=−α21t1 τ=α21(T−t1−1)
b b
i=2
τ=−bα22t2 bτ=α22(T−t2−1/2)
i=j
τ=−α2jtj τ=α2j(T−tj−1/j)
b b
b
b
b
FIGURE 2. Schematic representation of the flow of the rescaled surfaces for dif-
ferenti
NowweproveTheorem1.1
ProofofTheorem1.1. Assume a singularity develops at t = T, i.e. A2 as t T. We
| | → ∞ →
rescale the surface as discussed above in (4.2). As the rescaled flow satisfies (4.4), the curvature
bound A˜ 2 1forany iimplies analogous bounds onall itscovariant derivatives as inTheorem
i
| | ≤
4.1 in [8]. On the rescaled surfaces we have the maximum curvature achieved at (l ,0) where
i
l Mn, which means A˜ (l ,0) = 1 for all i . As H (l,t) is bounded for all t < T, we have
i i i
∈ | |
|kp| ≤ c0 onMt byLemma3.5. Inregardstothecurvature wehave;
A = k2+p2 c p c y 1,
5 5 −
| | ≤ ≤
p
(4.5) A˜ = α 1 A α 1c y 1 = c (α y) 1 = c y˜ 1.
| i| −i | | ≤ −i 5 − 5 i − 5 −
We will show that for a fixed τ , a subsequence of the rescaled surfaces converges as i goes to
0
infinity. Wecallthelimitinghypersurface M .Wewillexplainthisprocessindetail.
τ0
Converging sequence ofpoints. Withoutlossofgenerality letusassumethatthesingularity devel-
ops at the origin. We translate each generating curve to have x(l ,t ),i = 0. Werename them
i i 1
h i
againwiththesameρandworkwiththemfortherestofthesection. Wechoose ¯l M2 suchthat
∈
x(¯l(t),t)isonthe(x ,x )plane,and
1 3
¯l(t)= l M2: x(l(t),t),i = 0 ,
1
∈ h i
(cid:8) (cid:9)
We note that ¯l(t ) = l . For a fixed τ as i goes to infinity, t goes to T making the max A on
i i 0 Mt| |
thecorresponding originalhypersurfaces gotoinfinity. Thereforeontherescaledhypersurfaces for
afixedτ wecanfindN Nsuchthatfori > N
0 0 0
∈
1
A˜ (¯l(α 2τ +t ),τ ) .
| i| −i 0 i 0 ≥ 2
ONTHEEXTENSIONOFAXIALLYSYMMETRICVOLUMEFLOWANDMEANCURVATUREFLOW 7
As
A˜ ( ,τ )y˜( ,τ ) c , from(4.5),
i 0 0 5
| | · · ≤
wehave
y˜(¯l(α 2τ +t ),τ ) 2c for i > N .
−i 0 i 0 ≤ 5 0
Forτ = τ andi > N welookatthepoints ontherescaled surface generating curves ρ˜ which
0 0 i,τ0
areonthex axis. As
3
0 ρ˜( ,τ ) 2c ,
≤ i · 0 x1=0 ≤ 5
(cid:12)
(cid:12)
xn+1
b 2c5
b
b
b
b
b
b
b
b
x1
0
FIGURE 3. Theconvergence ofpoints.
wefindasubsequence ofpoints thatconverge. Wecallthesubsequence ofcorresponding rescaled
generating curvesρ˜1 andthelimitingpointonthex axisc := lim ρ˜1 .
i,τ0 3 ∗ i i,τ0 x1=0
→∞ (cid:12)
(cid:12)
Converging sequence oftangent vectors. Nowwelookattheunittangent vectorsofρ˜1 .
i,τ0 x1=0
(cid:12)
(cid:12)
xn+1
b
b
b
b
b
b
b
b
b
x1
0
FIGURE 4. Theunittangentvectors.
We translate the unit tangent vectors to the origin (see figure 4). By identifying these unit vectors
with points on the sphere, we observe that a subsequence of these points converges and so obtain
a subsequence of tangent vectors that converges. By translating back each tangent vector to its
original position, we find the corresponding subsequence of rescaled generating curves; we call it
ρ˜2 . Now we have found a subsequence of rescaled generating curves, where the points and the
i,τ0
tangentvectorsconverge atx = 0.
1
Convergence in a small ball. As A˜ 1 for every rescaled hypersurface, we can roll a ball of
i
| | ≤
radius 1 over the entire hypersurface in such a way that when it touches any point, it does not
8 SEVVANDIKANDANAARACHCHI
1
√1 δ2
−
1 √1 δ2
(cid:16) − − (cid:17)
δ
FIGURE 5. Theunitballsittingontherescaledhypersurface
intersectthecurveanywhereelse. Thuswehaveagradientboundinasmallballofradiusδ < 1on
thehypersurface.
Nowwelookatthesequence ofcurvesρ˜2 nearx = 0.
i,τ0 1
b 2c5 b 2c5
c
After ∗bc B(0,c∗)(δ)
rotating
x1 x1
δ δ
FIGURE 6. Convergence inasmallball
We rotate each curve around the point on the curve where x = 0, so that the tangent vector at
1
x = 0 is parallel to the x axis(see figure 6). For x < δ, each of these rotated curves has its
1 1 1
| |
gradient bounded by the same constant ρ˜ δ . This gives equicontinuity. Also at x = 0
′ ≤ √1 δ2 1
theheightofeachcurveisboundedby2c(cid:16). Hence−the(cid:17)heightofalltherotatedcurvesisboundedby
5
2c +(1 √1 δ2)for x < δ (seefigure5). Thereforeourcurvesareuniformly bounded. By
5 1
− − | |
Arzela-Ascoli, thereexistsasubsequence thatconverges uniformly. Letuscallthissubsequence of
rotatedcurvesg3 .
i,τ0
ThuswehaveC0 convergence forthesequence g3 inthesmallballB (δ),whichistheball
i,τ0 (0,c∗)
ofradius δ centred at(0,c ). Wecallthecorresponding subsequence before rotation ρ˜3 .Wedif-
i,τ0
∗
ferentiate the sequence g3 , and call it g3 . We note that g3 is bounded when x < δ. As
i,τ0 i′,τ0 (cid:12) i′,τ0(cid:12) | 1|
(cid:12) g′′3(cid:12)
thecurvature oftherescaled surfaces isbounded wehave (cid:12)| |(cid:12) 3 1,giving uniformbounds
(1+(g′3)2)2 ≤
on g 3 for x < δ. Thus we have uniform boundedness and equicontinuity for the sequence
′′ 1
| |
g3 . FromArzela-Ascoli there exists asubsequence thatconverges uniformly. Letuscallitg4 .
i′,τ0 i′,τ0
Wecall the corresponding subsequence before differentiation g4 .Fromthe Theorem ofuniform
i,τ0
ONTHEEXTENSIONOFAXIALLYSYMMETRICVOLUMEFLOWANDMEANCURVATUREFLOW 9
convergence and differentiability we have C1 convergence for the sequence g4 in the small ball
i,τ0
B (δ). Wecallthecorresponding subsequence beforerotation ρ˜4 .
(0,c∗) i,τ0
We note that the subsequence before rotation ρ˜4 converges as well, because the tangent vectors
i,τ0
and the points at x = 0 converge. Therefore the unrotated curves converge in the ball B (δ)
1 (0,c∗)
aswell.
Convergence in a compact set. On the subsequence of rotated curves g4 we find another set of
i,τ0
pointsasfollows. Wemarkthepointsontherotated curvesg4 whereǫ < δ suchthatonly
i,τ0 x1= ǫ
afinitenumberofpointsareoutside theballB (δ).Letusfi(cid:12)rstlo±okatthepointsg4 .
(0,c∗) (cid:12) i,τ0 x1= ǫ
(cid:12) −
(cid:12)
b 2c5 b 2c5
b
After
unrotating b
b
b
x1 x1
δ ǫ δ
−
FIGURE 7. Findingthepointsp4
i
These new points and their tangent vectors converge on the unrotated curves ρ˜4 as shown previ-
i,τ0
ously. We call these new points p5 (see figure 7). Now we translate each curve ρ˜4 along the x
i i,τ0 1
such that the point p5 is on the x axis. Then we rotate each curve as before, around p5 such that
i 3 i
thetangent vector ofeach curve isparallel tothex axis. For x < δ,byArzela-Ascoli, wefind
1 1
| |
asubsequence ofg4 thatconverges inC0. Wecallthissubsequence g5 . Aspreviously bycon-
i,τ0 i,τ0
sideringthedifferentiated sequenceg5 wefindanothersubsequence g6 thatconvergence inC1.
i′,τ0 i,τ0
Wecallthecorresponding subsequencebeforerotationρ˜6 . Wenotethattheoriginalsubsequence
i,τ0
ρ˜6 converges.
i,τ0
To summarise, we first found the limiting curve in a small ball B (δ). Then by considering
(0,c∗)
a ball centred on the limiting curve in B (δ), we found another subsequence that converged
(0,c∗)
in both these balls (see figure 8). In this manner, by considering a diagonal subsequence, we can
extend our limiting curve along small balls centred on the limiting curve. Therefore given any
compact set [α,β] [0,γ] where γ > 2c , we repeat the above process, and find a subsequence
5
×
uniformlyconverginginaunionofsmallballs(seefigure9),whichisasubsetofthegivencompact
set. Thissubsequencemaynotbeunique. Wegettheassociatedhypersurfacebyrotatingthiscurve
aroundthex axis.
1
Wenote that α goes to asigoes to .AsH˜ = α 1H and as H(l,t) cfor this theorem
i ∞ ∞ i −i | | ≤
for all l M2 and for all t < T , H˜ goes to 0, as i goes to . As 0 < c h c and
i 2 3
∈ ∞ ≤ ≤
h˜ = α 1h, we have h˜(τ ) going to 0 as well. As d x˜ = (H˜ h˜)ν˜ , we observe that
i −i i 0 dτ i − i − i i
10 SEVVANDIKANDANAARACHCHI
xn+1
b b
b b
b b
b b
x1
FIGURE 8. Convergence intwosmallballs
b
b b
b b
b
b b b b
b b b b b
b b b b b
FIGURE 9. Convergence inacompactset
lim d x˜ = 0. ThusM isastationary solution and therefore independent of τ. Werename
i dτ i τ0
→∞
thisstationary solution M .
The contradiction. The limiting solution M is a catenoid as it is the only axially symmetric min-
imal surface with zero mean curvature. However for large i if we rescale back M , to get an
i,τ0
understanding of the original surface, then we see that the estimate vy c does not hold on that
≤
surface. Thereforeaswewillshowinthenextparagraph wegetacontradiction.
We denote the quantities associated to the catenoid M by a hatˆ. We obtain the catenoid M by
rotating yˆ = c cosh(c 1xˆ ), around the x axis where xˆ is the x coordinate of the limiting
5 −5 1 1 1 1
surfaceM. Astheconvergence isC1,foranyǫ > 0andforanyl M2 andforanyfixedτ we
1 0 0
∈
have
vˆ(l )yˆ(l ) v˜(l ,τ )y˜(l ,τ ) ǫ forlarge i,
0 0 i 0 0 i 0 0 1
| − | ≤
vˆ(l )yˆ(l ) ǫ v˜(l ,τ )y˜(l ,τ ) for i > I , forsomeI N
0 0 1 i 0 0 i 0 0 0 0
− ≤ ∈
Forthecatenoid vˆ= 1+yˆ2 = cosh(c 1xˆ ). Therefore
′ −5 1
p