On Approximations and Homothetic Behavior in Mean Curvature Flow

App oxima ing G aphical Minimal Su aces
Th ough Mean Cu a u e Flow
Sawye Jacobson∗1and Je e y Zhang†2
1Wes on High School
2Wayland High School
Augus 24, 2025
Abs ac
We implemen an algo i hm o app oxima ing minimal su aces
using g aphical mean cu a u e low, le e aging he mono onically
non-inc easing a ea unde such a low. Ou app oach demons a es
well-es ablished esul s in minimal su ace heo y, including he in-
he i ance o plana i y and o a ional symme y om hei Di ichle
bounda y da a. Fu he mo e, we highligh he low’s u ili y by es i-
ma ing minimal su aces wi h complica ed bounda y condi ions, while
o e ing isual insigh s in o hei e olu ion unde he low. The pape
highligh s he unc ionali y o g aphical mean cu a u e low as bo h
a ool o app oxima ion and a geome ic lens o s udying minimal
su aces.
1 In oduc ion
Geome ic lows a e a class o p ocesses ha desc ibe he e olu ion o geo-
me ic objec s in ime. These lows a e o en d i en by in insic o ex insic
cha ac e is ics, and a e o en used o simpli y complex geome ic shapes.
∗ORCID:0009-0009-2775-6127
E-mail:[email p o ec ed]
†ORCID:0009-0004-4551-3368
E-mail:[email p o ec ed]
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One o he mos s udied geome ic lows is he mean cu a u e low, whe e
he ime e olu ion o he su ace a each poin is gi en by
∂M
∂ ⊥
=Hν,
whe e His he mean cu a u e—gi en by hal he ace o he shape ope a o —
and νis an e ol ing Gauss map. I is shown in 2 ha such an e olu ion
yields a mono one non-inc easing a ea, implying ha he su aces end o-
wa d minimali y—a s a iona y poin o he i s a ia ion o a ea—unde
he mean cu a u e low. Thus, i becomes na u al o explo e he use o
mean cu a u e low as a nume ical ool o app oxima ing minimal su aces,
which a e solu ions o he classic Pla eau’s p oblem–su aces ha can be
pa ame ized by
M(x, y, (x, y)),
wi h a g aphically de ined Di ichle bounda y condi ion.
While heo e ical unde pinnings o mean cu a u e low a e well-es ablished,
analy ical solu ions o en emain in ac able due o he na u e o non-linea
pa ial di e en ial equa ions. Howe e , o g aphically de ined su aces, mean
cu a u e low simpli ies o a non-linea pa abolic pa ial di e en ial equa-
ion, sugges ing nume ical app oxima ion as a use ul ool o es ima ing so-
lu ions.
This pape e e ences he wo k o Sawye Jacobson in [1], by p esen ing
he de i ed algo i hm o es ima ing g aphical minimal su aces h ough he
use o mean cu a u e low. We hen p o ide simula ions, demons a ing he
e ec i eness o he nume ical app oach by including scena ios wi h plana
and o a ionally symme ic bounda y condi ions, as well as he es ima ion
o minimal su aces o complex bounda y da a–whe e analy ical solu ions
would emain enigma ic. The esul s p o ide e idence o p e iously es ab-
lished esul s in g aphical minimal su ace heo y a e ue, including he
inhe i ance o plana i y and symme y om a bounda y. They also highligh
he use ulness o g aphical mean cu a u e low in es ima ing minimal su -
aces, while also p o iding a isual ep esen a ion o hei e olu ion h ough-
ou he low.
2 Algo i hm
Take a amily o su aces pa ame e ized by
M(u1, u2, s) = F(u1, u2) + s∂Fs
∂s , νν(u1, u2),
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whe e sis a compac ly suppo ed a ia ional map and νis a uni no mal
ec o . Now, his amily o su aces has a i s a ia ion o a ea gi en by
d
ds[M(s)]s=0
=−ZM∂Fs
∂s , νHdA,
whe e His he mean cu a u e. So, we choose Fssuch ha
∂Fs
∂s , ν=H(Fs),
yielding he mono one non-inc easing
d
ds[M(s)]s=0
=−ZM
H2dA,
wi h equilib ium a H= 0.This mo i a es ou de ini ion o mean cu a u e
low.
De ini ion 2.1 (Mean Cu a u e Flow).Gi en a amily o su aces {M },
he mean cu a u e low is he e olu ion o he su ace by he sys em
(∂M
∂ ⊥=Hν
M0=M.
Now, i was shown in [1] ha a minimal su ace Mpa ame e ized by
M(x, y, (x, y)) mus sa is y he pa ial di e en ial equa ion
∂
∂x x
p1 + |∇ |2!+∂
∂y y
p1 + |∇ |2!= 0.
So, in pai wi h he Di ichle bounda y condi ion, we ge ha a g aphical
solu ion o Pla eau’s p oblem in Ω is gi en by he sys em



di ∇
√1+|∇ |2= 0 in Ω
M(x, y, ) = Γ(x, y) on ∂Ω.
As ou mean cu a u e low is mono one non-inc easing, lows will gene ally
end owa ds minimali y1, implying he use ulness o such a low o es ima e
1A coun e example occu s when a su ace yields singula i ies.
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minimal su aces. In ou g aphical case, he Mean Cu a u e low e olu ion
simpli ies o he sys em









∂
∂ =p1 + |∇ |2di ∇
√1+|∇ |2in Ω ×(0,∞)
(x, y, ) = Γ(x, y) on ∂Ω×(0,∞)
(x, y, 0) = ρ(x, y) in ¯
Ω×{0}
o some unc ion ρ(x, y) : Ω −→ Rsuch ha ρ= Γ on ∂Ω.
3 Simula ions
Now ha we ha e a way o es ima e g aphical minimal su aces, we aim o
simula e mul iple cases. All simula ions and isualiza ions we e implemen ed
in Ma hema ica.
3.1 Inhe i ance o Plana i y
In [1], i was shown ha g aphical minimal su aces M ⊆ R3wi h bounda y
Γ⊆π, o some plane π, will be such ha M ⊆ π. The e o e, we look
o simula e such cases o M , ∈[0, T).In his simula ion, we use he
ini ial su ace da a (0, x, y) := sin(πx
5) sin(πy
5).We also ix a uni o mly-ze o
Di ichle bounda y condi ion on [0,5] ×[0,5].Then, we le he su ace low,
go e ned by he pa ial di e en ial equa ion in 2. We see, as expec ed, a
smoo hing e ec , as he su ace ends owa ds i s plana bounda y.
(a) Ini ial Da a (b) In e media e Da a (c) Final Da a
As expec ed, M con e ged o a subse o π, wi h ∂M = Γ ⊆π, exhibi ing
he p ope y shown in [1].
3.2 Inhe i ance o Ro a ional Symme y
I was also shown in [1] ha g aphical minimal su aces M ⊆ R3,a o a-
ionally symme ic bounda y Γ,induces o a ional symme y on M.I is
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he e o e in e es ing o simula e such cases. When he bounda y Γ can be
embedded in a plane, his p ope y is i ial as all planes a e symme ic. So,
we ake he non-plana bounda y da a Γ = 0.1(x−2.5)2−0.1(y−2.5)2+0.8,
which ep esen s he bounda y o a hype bolic pa aboloid in R3.We also
mus de ine he ini ial bounda y da a, which we choose o be non-symme ic
o exhibi how he symme y o he bounda y uly imposes such a p ope y
on M.
(0, x, y) := 0.1(x−2.5)2−0.1(y−2.5)2+0.8+0.25 exp −((x−3.0)2+ (y−3.2)2)
+0.2 exp −((x−1.5)2+ (y−2.0)2)
+0.15 exp −((x−4.0)2+ (y−1.0)2).
+0.1 sin(2x) cos(1.2y)+0.05 sin(3y) cos(1.5x).
A e a sho pe iod o ime, we see ha he su ace begins o e ol e owa ds
symme y. Finally, i con e ges o he hype bolic pa aboloid, spanning he
bounda y de ined abo e.
(a) Ini ial Da a (b) In e media e Da a (c) Final Da a
As expec ed, he su ace Mexhibi s wo- old o a ional symme y, jus as
does i s bounda y.
3.3 Complex Bounda y Da a
One o he p ima y applica ions o g aphical mean cu a u e low is o es i-
ma e minimal su aces wi h complica ed bounda y da a. The e o e, we apply
ou algo i hm o simula e such cases unde mul iple bounda y condi ions. So,
we de ine a complica ed bounda y:
0.2 sin(2x) cos(1.5y)+0.15 sin(3y) cos(1.2x)+0.1 exp −((x−2)2+ (y−3)2)
−0.08 exp −((x−3.5)2+ (y−1.5)2)+ 0.05 sin(xy)+0.5,
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wi h ini ial condi ion:
Γ=0.2 sin(2x) cos(1.5y)+0.15 sin(3y) cos(1.2x)+0.1 exp −((x−2)2+ (y−3)2)
−0.08 exp −((x−3.5)2+ (y−1.5)2)+0.05 sin(xy)+0.5+0.2 exp −((x−1.2)2+ (y−1.5)2)
+0.15 exp −((x−3.5)2+ (y−3)2)+0.1 sin(2.5x) cos(1.8y)+0.05 sin(3y) cos(2x)
+0.1 exp −((x−2.5)2+ (y−2.5)2).
This is analy ically di icul o sol e, highligh ing he impo ance o ou al-
go i hm in es ima ing such su aces.
We see ha he su ace lows, e en ually con e ging, gi ing us an es ima e
(a) Ini ial Da a (b) In e media e Da a (c) Final Da a
o he minimal spanning su ace o he gi en bounda y condi ion.
Re e ences
[1] Jacobson, S., On G aphical Minimal Su aces and he Homo he ici y
o Singula Beha io in Mean Cu a u e Flow, Zenodo, h ps: // doi.
o g/ 10. 5281/ zenodo. 16903054 (2025).
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