Eﬃcient Computation of Minimal Surfaces Using Exterior Calculus,

Hodge Theory and Geometric Measure Theory

September 17, 2024

Figure 1: Computing the area minimizing surface of two rings

Abstract

This seminar report discusses the work of Wang and Chern

(2021), which presents an algorithm to eﬃciently compute an

area minimizing surface bounded by a given closed boundary

curve. It applies concepts of diﬀerentiable manifolds and

geometric measure theory. Importantly, the mathematical

rigor provides insight into restricting the solution space while

doing the calculation in a periodic domain. This allows a

signiﬁcant speedup compared to traditional techniques.

1 Introduction

Computing an area-minimizing surface is a classical ques-

tion in computational diﬀerential geometry. The obvious

application is to model soap ﬁlms, but it is also commonly

used when modelling structures in nature or design.

We will ﬁnd out that even though the idea is rather straight

forward, developing a precise mathematical language re-

quires some eﬀort, especially with regard to the acceleration

of the algorithm. The goal of this report is to give a con-

cise but nevertheless complete view of all the mathematical

methods that result in the optimization problem. We will

ﬁrst describe a beautiful generalization of vector calculus

regarding smooth manifolds in Sections 3.1 to 3.3. Then we

will explore an unexpected connection of these generalized

vector ﬁelds to the holes in the smooth manifolds in Sec-

tion 3.4. After that, we will discover surprising features of

area-minimizing surfaces in Section 3.5.

Next, we will apply all developed methods in the algorithm

in Section 4. We will see that the extensive mathematical

study allows the algorithm to be accelerated substantively

compared to traditional techniques. The presented results

will be analyzed in Section 5. We will give some suggestions

for further research in Section 6, which which will be

followed by a ﬁnal conclusion.

2 Related Work

Ever since the problem was posed by Lagrange in 1762,

the study of area-minimizing surfaces has developed into

a vast ﬁeld. After their existence was proved by Douglas

(1931) and Radó (1930), Federer (1969) developed a general

framework called geometric measure theory, which general-

izes diﬀerential geometry through measure theory. It is this

theory that will be used in this report.

Douglas (1931) pioneered a numerical method which

evolves an initial surface. Another prominent implementa-

tion of this approach was given by Brakke (1992). These

methods, however, are limited in their solution to the topol-

ogy of the initial surface and therefore are unsuited for a

general algorithm.

Instead, geometric measure theory allows a general frame-

work, which was ﬁrst discretized by Sullivan (1990). A more

recent result was given by Brezis and Mironescu (2019),

who connected geometric measure theory to optimal trans-

port. The focus of the report at hand lies on Wang and

Chern (2021), who have reformulated the problem in order

to eﬃciently compute minimal surfaces by changing the

domain.

3 Fundamentals

In this seminar report we will try to develop an understanding

of the concepts without giving rigorous proofs. We assume

knowledge of simplicial complexes, vector calculus and

basic topology.

3.1 Exterior Algebra

Before we develop a mathematical language for the descrip-

tion and measurement of surfaces, we ﬁrst must be able to

describe generalized volumes, i.e. lines, planes, volumes

et cetera. We will later span ﬁelds of these to show how

to do calculus with these generalized volumes. This entire

section is based on Lee (2012) and Morgan (2016), who can

be consulted for a more rigorous approach. Another good

source for a ﬁrst read is Crane (2023), which has also been

the inspiration for the ﬁgures.

3.1.1 k-Vectors

Figure 2:

𝑘

-vector with

𝑘 = 1, 2, 3

. Note that the object is

only deﬁned by its measure and orientation, therefore the

wedge product of diﬀerent vectors can yield the same object.

Similar to a vector representing an oriented 1-plane and an

orientation, a

𝑘

-vector represents an oriented

𝑘

-plane through

the origin with a certain magnitude usually measuring its

area or volume. To be more precise, we can multiply

𝑘

ordinary vectors by using the wedge product to obtain a

𝑘

-vector. This wedge product is multilinear, i.e. we have for

any vectors v

, v

, u

∈ R

𝑛

and any scalar 𝑐 ∈ R

(𝑐v

) ∧ v

= v

∧ (𝑐v

) = 𝑐(v

∧ v

+ v

) ∧ (u

+ u

) = u

∧ u

+ u

∧ v

+ v

∧ u

+ v

∧ v

and alternating, i.e. we have

∧ v

= −v

∧ v

Figure 3: The wedge product is alternating, meaning

∧

= −v

∧ v

. It is 0 if and only if the vectors are linearly

dependent.

We call the space of these objects a

𝑘

-vector space and

denote it as

𝑘

𝑛

, which turns out to be a vector space of

dimension



𝑛

𝑘



For example, we ﬁx

𝑛 = 3

and denote the basis of

, b

. A possible basis of

𝑘

is now given as follows

with b

= b

∧ b

• Λ

{

}

• Λ

{

, b

}

• Λ

{

, b

}

• Λ

{

123

}

Other 2-vectors, for example

, are just linear combinations

of the basis vectors as seen in Figure 4 and Figure 5.

1 · (b

∧ b

)

−0.8 · (b

∧ b

)

−0.6 · (b

∧ b

)

Figure 4: Basis decomposi-

tion of a 2-vector

Figure 5: k-vector addition

satisﬁes associativity and

distributivity, e.g.

= b

∧ (b

+ b

3.1.2 k-Forms

How do we measure these new objects, i.e. how do we

deﬁne some well behaving function mapping

𝑘

𝑛

→ R

For

𝑘 = 1

we can just ﬁx any vector

v ∈ R

𝑛

to deﬁne a

function

𝜑

: Λ

→ R := x ↦→ ⟨u, x⟩

for any given inner

product. These functions are called forms, covectors or

dual vectors. Note that these maps are linear by design, i.e.

𝜑

) + 𝜑

) = 𝜑

+ u

)

and

𝑡 · 𝜑

(u) = 𝜑

(𝑡 · u)

for any

𝑡 ∈ R

. Moreover, the space of forms itself is a

function vector space by the natural operators given by

(𝜑

+ 𝜑

)(x) = 𝜑

(x) + 𝜑

(x)

and

(𝑡 · 𝜑

)(x) = 𝜑

(𝑡 · x)

for

any x ∈ R

𝑛

As we only require a vector space with an inner product,

we can construct the same dual space for

𝑘

𝑛

. Recall

that the determinant of a matrix reﬂects the area of the

parallelogram of the vectors which constitute the matrix.

Therefore, the natural inner product to choose is

⟨u

∧ . . . ∧ u

𝑘

, v

∧ . . . ∧ v

𝑘

⟩ = det(⟨u

𝑖

, v

𝑗

⟩), (1)

with the determinant of the

𝑘 × 𝑘

matrix with entries at

index

𝑖

and

𝑗

. Following the construction above result in

the

𝑘

-form space, denoted by

𝑘

𝑛

which consists of the

functions we search for.

Figure 6: An 1-form: the

two vectors in blue yield the

same value here.

Figure 7: Geometrically, a

2-form (red) measures how

well a projected 2-vector

lines up.

This description gives rise to a nice duality: Fix any

basis in

𝑘

𝑛

and its dual in

𝑘

𝑛

as constructed above.

Since any

𝑘

-vector is a linear combination of this basis, it

is also a multilinear map

𝑘

𝑛

→ R

𝑛

. Due to this duality

there is a natural isomorphism, which is expressed by the

operator

♯ : Λ

𝑘

𝑛

→ Λ

𝑘

𝑛

mapping

𝑘

-forms to

𝑘

-vectors

and ♭ : Λ

𝑘

𝑛

→ Λ

𝑘

𝑛

mapping 𝑘-vectors to 𝑘-forms.

3.1.3 Hodge Duality

We now turn to a diﬀerent kind of duality between diﬀerent

types of

𝑘

-vectors. In

there is an obvious connection

between any plane and its normal, i.e. between any

-vector

and an

-vector. In

there is an obvious connection between

any vector and its orthogonal counterpart. Therefore, in

general we search for a linear operator which reﬂects the

orthogonal complement. The ﬁrst clue is that since



𝑛

𝑘





𝑛

𝑛−𝑘



, this operator must be a map

★ : Λ

𝑘

𝑛

→ Λ

𝑛−𝑘

𝑛

The second clue is that we require some sort of orthogonality,

for which we turn to the inner product deﬁned in Equation (1).

Formally, let

u ∈ Λ

𝑘

𝑛

v ∈ Λ

𝑘

𝑛

and

★v ∈ Λ

𝑛−𝑘

𝑛

Then the hodge star operator is uniquely deﬁned by

u ∧ ★v = ⟨u, v⟩ · (b

∧ . . . ∧ b

𝑛

)

where

∧ . . . ∧ b

𝑛

is the unit

𝑛

-vector with volume

. So

we assert that the resulting

𝑛

-vector on the left matches the

unit volume depending on how well it matches up.

Figure 8: In

the hodge

star of a 2-vector is the nor-

mal of the plane (red) with

equal magnitude. Wedging

this together with any other

2-vector (green) yields the

same volume as the unit 3-

vector scaled by the inner

product.

Figure 9: If the inner prod-

uct of

and

yields 0, then

★v

must result in a vector

linear dependent on v.

It turns out that this deﬁnition also ensures the alternating

property, i.e. for any

u ∈ Λ

𝑘

𝑛

the hodge star operator

satisﬁes

★ ★ u = (−1)

𝑘 (𝑛−𝑘 )

Additionally, because of the duality between

𝑘

𝑛

and

𝑘

𝑛

the hodge star operator is also deﬁned for

𝑘

-forms so

that ★ maps Λ

𝑘

𝑛

→ Λ

𝑛−𝑘

𝑛

Figure 10: In

, since

(−1)

1·(3−1)

= 1

, we have

★ ★ u = u

and there is no

further rotation.

Figure 11: In

, since

(−1)

1·(2−1)

= −1

, the alter-

nating property reﬂects half

a rotation if we apply the

hodge star operator twice.

3.2 Exterior Calculus

While Exterior Algebra is the language of generalized vol-

umes, Exterior Calculus generalizes vector calculus to de-

scribe the behavior of ﬁelds composed of these. Furthermore,

it generalizes the underlying space to smooth manifolds,

which can be visualized by attaching a k-vector to each point.

Since an intensive study of this topic is not possible in this

report, we refer to Lee (2012), Morgan (2016) and Crane

(2023), who this section is based on. We require general

knowledge on smooth manifolds and denote by

𝑀

a compact

oriented Riemannian Manifold in this section.

3.2.1 Smooth k-Vector Fields on Smooth Manifolds

A tangent vector space deﬁned at each point

𝑝 ∈ R

𝑛

, denoted

𝑇

𝑝

𝑛

, is itself an

𝑛

-dimensional vector space. In fact,

𝑇

𝑝

𝑛

and

𝑛

are identical and therefore any basis in

𝑛

also a basis in every tangent space. We often consider the

disjoint union of all tangent spaces called the tangent bundle,

𝑇R

𝑛

= ⨿

𝑝∈R

𝑛

𝑇

𝑝

𝑛

. Similarly, we can attach a dual

vector space

𝑇

★

𝑝

𝑛

to each point and deﬁne the cotangent

bundle to be 𝑇

★

𝑛

= ⨿

𝑝∈R

𝑛

𝑇

★

𝑝

𝑛

From this view, vector ﬁelds are continuous maps

𝑋 :

𝑛

→ 𝑇R

𝑛

with

𝑋

𝑝

∈ 𝑇

𝑝

𝑛

. For these we denote the

special space of smooth vector ﬁelds with

𝔛

. Likewise,

diﬀerential forms are continuous maps

𝑋 : R

𝑛

→ 𝑇

★

𝑛

Finally, we denote the space of smooth diﬀerential forms

with R

𝑛

→ Λ

𝑘

𝑇R

𝑛

with Ω

𝑘

Figure 12: A smooth 2-form ﬁeld can be applied to a smooth

2-vector ﬁeld at each point to yield a number. Here, we

visualize a slice in

with unit volumes and forms. If they

overlap well, this will results in 1 (red), if not, this will

results in 0 (blue).

These ﬁelds generalize well to smooth

𝑛

-manifolds which

are spaces that locally look like Euclidean space

𝑛

. How-

ever, the previously used identiﬁcation with the original

space is lost, which makes a rigorous deﬁnition more diﬃ-

cult. Speciﬁcally, it is not possible to describe the resemb-

lence to Euclidean space in a globally applicable manner,

as illustrated in Figure 13. Instead, a collection of charts is

employed, each mapping a portion of the manifold. These

charts are designed to be smoothly compatible with one an-

other wherever they overlap, such that we can deﬁne smooth

functions

𝑓 : 𝑀 → R

, the space of which we denote by

𝐶

∞

(𝑀).

The concept of tangent spaces for manifolds is now con-

structed through a generalization of directional derivatives in

Euclidean space. Recall the fact that the directional deriva-

tive is a linear operator which at a point a is deﬁned for any

tangent vector 𝑣

𝑎

𝐷

𝑣

𝑎

: 𝐶

∞

𝑛

) → R := 𝑓 ↦→

𝑛



𝑖=1

𝑣

𝑖

· (𝜕

𝑖

𝑓 )(𝑎),

where

𝜕

𝑖

𝑓

denotes the partial derivative of

𝑓

with respect

to the

𝑖

’th coordinate. The space of all such operators turns

out not only to be a function vector space, but also to be

isomorphic to the Euclidean tangent bundle deﬁned above!

We will generalize the fact that this operator satisﬁes the

product rule, i.e. for all 𝑓 , 𝑔 ∈ 𝐶

∞

𝑛

) we have

𝐷

𝑣

𝑎

( 𝑓 · 𝑔) = 𝑓 ( 𝑝) · 𝐷

𝑣

𝑎

(𝑔) + 𝑔( 𝑝) · 𝐷

𝑣

𝑎

( 𝑓 ).

Now, the tangent space to a smooth manifold, denoted by

𝑇

𝑝

𝑀

, is similary deﬁned as the set of all linear functions

𝑣 : 𝐶

∞

(𝑀) → R

, which satisfy for all

𝑓 , 𝑔 ∈ 𝐶

∞

(𝑀)

at any

𝑝 ∈ 𝑀:

𝑣( 𝑓 · 𝑔) = 𝑓 (𝑝) · 𝑣(𝑔) + 𝑔( 𝑝) · 𝑣( 𝑓 ).

On an

𝑛

-dimensional smooth manifold this space turns out

to be an

𝑛

-dimensional vector space, which behaves as one

would expect. Consequently, we generalize the deﬁnitions of

a tangent bundle to

𝑇 𝑀 = ⨿

𝑝∈ 𝑀

𝑇

𝑝

𝑀

, a cotangent bundle

𝑇

★

𝑀

, a vector ﬁeld to

𝑋 : 𝑀 → 𝑇 𝑀

, the space of smooth

vector ﬁelds to

𝔛

and the space of smooth diﬀerential k-form

𝑘

: 𝑀 → Λ

𝑘

𝑇 𝑀

. Finally, we can extend linear maps

of Euclidean space, such as the Hodge star operator, to

diﬀerential forms by applying it to the tangent spaces of all

points.

Note that we have skipped the discussion of smooth

manifolds with boundary even if they are fundamental to our

algorithm. This was done because the preceding methods,

even though they require special handling, remain largely

applicable. We will return to them in Section 3.3.3.

Figure 13: A sphere does not

globally look like Euclidean

space since there is no way

to uniquely map the top of

the sphere to a tangent space

in a smooth way.

Figure 14: However, we can

deﬁne a smooth atlas of lo-

cally smooth mapping, such

that they are smoothly com-

patible wherever overlap.

3.2.2 The Diﬀerential of a Smooth Map

We can now deﬁne a mapping between tangent spaces of

diﬀerent manifolds, which will relate the tangent spaces on a

chart to the tangent spaces on a smooth manifold. Given any

smooth map

𝐹 : 𝑀 → 𝑁

between two smooth manifolds,

the diﬀerential at each point is a linear map

𝑑𝐹

𝑝

: 𝑇

𝑝

𝑀 → 𝑇

𝐹 ( 𝑝)

𝑁 (2)

such that for any 𝑣 ∈ 𝑇

𝑝

(𝑀) and any 𝑓 ∈ 𝐶

∞

(𝑁)

𝑑𝐹

𝑝

(𝑣)

  

𝐶

∞

( 𝑁 )→R

( 𝑓 ) = 𝑣( 𝑓 ◦ 𝐹



∈𝐶

∞

(𝑀 )

This means we deﬁne

𝑑𝐹

𝑝

so that every pushed forward

tangent vector applied to any smooth function yields the

same value as applying it to any pushed forward function.

Note that in general, this does not map a vector ﬁeld in

𝑀

a vector ﬁeld in 𝑁, since 𝐹 need not be bijective. However,

the diﬀerential ensures that if F is a diﬀeomorphism, i.e. a

smooth bijective map that has a smooth inverse,

𝑑𝐹

𝑝

will be

a isomorphism, i.e. a linear bijective map. In this sense, it is

the best linear approximation to the map near the given point.

Now for every smooth vector ﬁeld

𝑋

𝑀

𝑑𝑓 (𝑋)

again

yields again a smooth vector ﬁeld on N. We also denote the

resulting vector ﬁeld

𝐹

★

𝑋

and call it the pushforward of X

by F.

The isomorphism also gives rise to the fact that the

derivations tangent vectors given by

𝜕

𝑖

𝜑

for any coordinate

chart

𝜑

are a basis for the tangent space, which in turn

leads to the fact that the pushforward in coordinate form

is the Jacobian matrix. Moreover, looking over the global

diﬀerential obtained by stitching the diﬀerentials together

at all points, we do not only have an isomorphism, but a

diﬀeomorphism and (𝑑𝐹)

−1

= 𝑑(𝐹

−1

Similarly, we can deﬁne a linear map on forms going

in the opposite direction such that

𝑑𝐹

★

𝑝

: 𝑇

★

𝐹 ( 𝑝)

𝑁 → 𝑇

★

𝑝

𝑀

This map is called the pullback by F and is deﬁned by the

fact that given 𝜔 ∈ 𝑇

★

𝐹 ( 𝑝)

𝑁 and 𝑣 ∈ 𝑇

𝑝

𝑀 it satisﬁes

𝑑𝐹

★

𝑝

(𝜔)

  

∈𝑇

★

𝑝

𝑀

(𝑣) = 𝜔(𝑑𝐹

𝑝

(𝑣)

  

∈𝑇

𝐹 ( 𝑝)

𝑁

)

3.2.3 Exterior Derivative

We will now generalize the diﬀerential of a smooth function

to diﬀerential k-forms and smooth manifolds. Just as the

diﬀerential gives the direction and the magnitude of fastest

increase for a function, we will ﬁnd an operator giving the

direction and the area of fastest increase for a vector ﬁeld

and so forth.

We will ﬁrst introduce the concepts in Euclidean space to

point out the similarities to the gradient. For a real-valued

function, i.e. for any

𝜔 ∈ Λ

𝑇R

𝑛

, we have the normal

diﬀerential that is at any point deﬁned as

𝑑(𝜔) =

𝑛



𝑖=1

𝜕

𝑖

𝜔 𝛽

𝑖

where

𝜕

𝑖

𝜔

denotes the partial derivative of

𝜔

with respect

to the

𝑖

’th coordinate and

𝛽

𝑖

denotes the

𝑖

’th basis vector at

point p. This is the dual operator of the gradient as can be

seen in the ﬁrst square of Figure 17.

Now, for

𝜔 ∈ Λ

𝑇R

𝑛

, i.e. for diﬀerentiable 1-forms, the

exterior derivative is deﬁned as

𝑑(𝜔) =



𝑖, 𝑗

𝜕

𝑖

𝜔

𝑗

(𝛽

𝑖

∧ 𝛽

𝑗

where

𝜕

𝑖

𝜔

𝑗

denotes the partial derivative of the

𝑗

’th compo-

nent of 𝜔 with respect to the 𝑖’th coordinate.

Figure 15: For

𝜔 = 𝑥 · 𝑦 ∈

)

the exterior deriva-

tive yields

𝑑𝜔 = 1· 𝛽

+1· 𝛽

Figure 16: For

𝜔 = 0 · 𝛽

−

𝑥

· 𝛽

∈ Ω

)

the exterior

derivative yields

𝑑𝜔 = 2𝑥 ·

(𝛽

∧ 𝛽

)

This concept extends to higher

𝑘

for which we omit

the deﬁnitions as the notation quickly becomes unreadable.

You may have noticed the similarities between the exterior

derivative and the gradient. In fact, it only diﬀers in mapping

to diﬀerential

𝑘

-forms instead of

𝑘

-vector ﬁelds, so we have

grad( 𝑓 ) = (♯ ◦ 𝑑 ◦ ♭)( 𝑓 )

for any

𝑓 ∈ Λ

𝑛

) = 𝐶

∞

𝑛

. The

exterior derivative, however, is deﬁned on any

𝜔 ∈ Λ

𝑘

𝑛

)

and therefore we can generalize the gradient to any

𝑘

-vectors!

Furthermore, since

𝑑

is a linear map, we can compose it

with other linear maps such as the hodge star operator, a

concept which will allow us to generalize the curl and the

divergence to

𝑘

-vectors. The following diagram shows the

interplay between some of the concepts we have discussed

so far.

𝑇Λ

d d d

grad grad grad

⟲ ⟲ ⟲

♭

♭ ♭

♭

♯

♯ ♯

♯

★

Figure 17: Commutative property of the exterior derivative

Writing out the deﬁnitions for curl and divergence, it

turns out that given any smooth vector ﬁeld

𝑋 ∈ 𝔛

we have

curl(𝑋) = (♯ ◦ ★ ◦ 𝑑 ◦ ♭)(𝑋)

and

div(𝑋) = (★ ◦ 𝑑 ◦ ★ ◦

♭)(𝑋)

. This means that we can generalize these operators

to any dimension and to any

𝑘

𝑛

. Furthermore, we will

see that this also extends well to smooth manifolds! It is

this ﬂexibility, which makes exterior calculus so powerful.

Putting all deﬁnitions together, we arrive at the following

diagram, in which the squares commute:

𝐶

∞

) 𝔛(R

) 𝐶

∞

)

) Ω

)

grad

curl

div

𝑑 𝑑 𝑑

♭ ★ ◦ ♭

★

⟲ ⟲ ⟲

We will now generalize the exterior derivative to smooth

manifolds by extracting the desired properties from their

behavior in Euclidean space. For instance, the fact that

curl ◦ grad ≡ 0

and

div ◦ curl ≡ 0

in Euclidean space leads

to the requirement that

𝑑 ◦ 𝑑 ≡ 0

in general. Overall, we

deﬁne the exterior derivative to be the unique operator

𝑑 : Ω

𝑘

𝑀 → Ω

𝑘+1

𝑀 (3)

for all 𝑘 that satisfy the following properties:

(i) 𝑑 is linear over R, that is 𝑑(𝜔 + 𝜂) = 𝑑𝜔 + 𝑑𝜂.

(ii) If 𝜔 ∈ Ω

𝑘

(𝑀) and 𝜂 ∈ Ω

𝑙

(𝑀), then

𝑑(𝜔 ∧ 𝜂) = 𝑑𝜔 ∧ 𝑛 + (−1)

𝑘

𝜔 ∧ 𝑑𝜂.

(iii) 𝑑 ◦ 𝑑 ≡ 0

(iv)

For

𝑓 ∈ Ω

(𝑀) = 𝐶

∞

(𝑀)

𝑑𝑓

is the diﬀerential of

𝑓

given by 𝑑𝑓 (𝑋) = 𝑋 ( 𝑓 ) for any 𝑋.

We call a diﬀerential form

𝜔 ∈ Ω

𝑘

closed if

𝑑𝜔 ≡ 0

and

exact if there exists an

𝜂 ∈ Ω

𝑘−1

such that

𝜔 = 𝑑𝜂

. Note

that since

𝑑 ◦ 𝑑 ≡ 0

, any exact form is closed, but the

converse is in general not true for non-Euclidean smooth

manifolds. In fact, the question of whether a closed form on

a speciﬁc manifold is exact results a surprising connection

to its topology, which is discussed in Section 3.4.

Figure 18: For example in

\ {0}

the diﬀerential 1-form given by

𝜔 =

𝑥𝛽

−𝑦𝛽

𝑥

+𝑦

is closed, which can be shown

by calculation. It is known that a dif-

ferential 1-form is exact if and only if

the line integral of every piecewise

smooth closed curve segment is 0.

This would only be the case if we re-

stricted the domain to

{(𝑥, 𝑦) : 𝑥 > 0}

which would allow

𝜔

to be expressed

as 𝜔 = 𝑑(arctan (𝑦/𝑥)).

3.3 Integration on Smooth Manifolds

This section continues the discussion started in Section 3.3.2.

We give a formal deﬁnition of integrals and their interplay

with the exterior derivative on manifolds with boundary.

3.3.1 Integration of Diﬀerential Forms

Formally, the integral of a diﬀerential

𝑛

-form over

𝐷 ⊆ R

𝑛



𝐷

𝜔 =



𝐷

𝑓 · (𝛽

∧ . . . ∧ 𝛽

𝑛

) =



𝐷

𝑓 𝑑𝑥

. . . 𝑑𝑥

𝑛

since

𝜔

can be written as

𝑓 · (𝛽

∧ . . . ∧ 𝛽

𝑛

)

for the normal

Euclidean basis. The last term is just a multiintegral over a

multidimensional domain.

In order to integrate over a smooth manifold we now inte-

grate over all smooth charts covering the manifold by adding

the integrals in each mapped Euclidean space. However,

we must be careful to blend together the diﬀerent charts

wherever they overlap. This is achived by a partition of unity

denoted by

𝜓

, whose deﬁnition is beyond the scope of this

report. For a chart at index

𝑖

we denote the diﬀeomorphism

to its Euclidean subset by

𝜑

𝑖

and deﬁne the integral over a

smooth manifold by:



𝑀

𝜔 =



𝑖



𝑀

𝜓

𝑖

𝜔 =



𝑖



𝜑 (𝑈

𝑖

)

(𝜑

−1

𝑖

)

★

(𝜓

𝑖

𝜔)

Figure 19: Integral of a diﬀerential 2-form. The Riemannian

metric on the circle induced by one hemisphere is given by

𝜔

𝑔



1 − 𝑥

− 𝑥

𝛽

∧ 𝛽

, which reﬂects the fact that the

outer part gets mapped to more area.

3.3.2 Calculating the Area of a Surface

This subsection deals with the question of how to calculate

the area of a surface in the special case of an immersed

manifold. All geometry is based on an inner product de-

ﬁned on each point called the Riemannian metric, which

is a map

𝑔 : 𝑇

𝑝

𝑀 × 𝑇

𝑝

𝑀 → R

at each point that it is

symmetric and positive deﬁnite. Given a coordinate basis

𝛽

, . . . , 𝛽

𝑛

∈ 𝑇

★

𝑝

𝑀

and two vector

u, v ∈ 𝑇

𝑝

𝑀

𝑔

has a

matrix representation deﬁned at each point as:

𝑔(u, v) =

𝑛



𝑖=1

𝑛



𝑗=1

𝑔

𝑖 𝑗

· (𝛽

𝑖

(u) · 𝛽

𝑗

(v))

For surfaces or more generally for immersed smooth mani-

folds the Riemannian metric is just taken in the surrounding

space. That means, for

𝑆 ∈ 𝑀

and

𝐹 : 𝑆 → 𝑀

the Rieman-

nian metric on S is given by

𝑔

𝑆

(u, v) := 𝑔

𝑀

(𝑑𝐹(u), 𝑑𝐹(v))

This allows us to deﬁne the Riemannian volume form

𝜔

𝑔

∈ Ω

𝑘

to measure the area of any point in

𝑆

. We denote

the coordinate basis form by 𝛽

∧ . . . ∧ 𝛽

𝑛

and deﬁne it as

𝜔

𝑔



det(𝑔

𝑖 𝑗

) · 𝛽

∧ . . . ∧ 𝛽

𝑛

. (4)

Finally, we can now calculate the area of a surface or in

general the volume for any

𝑆

as deﬁned above by integrating

over it

Vol(𝑆) :=



𝑆

𝜔

𝑔

(5)

In fact, since form ﬁelds are so natural to integrate over,

we can therefore deﬁne an inner product for any diﬀerential

2-form 𝜔, 𝜑 ∈ Ω

𝑘

(𝑀) by

⟨𝜔, 𝜑⟩ =



𝑚

⟨𝜔

𝑝

, 𝜑

𝑝

⟩, (6)

where the inner product inside the integral is given by

Equation (1). This induces the normal 2-norm on form ﬁelds

by ||𝜔||



⟨𝜔, 𝜔⟩.

3.3.3 Integration over Manifolds with Boundary

So far we have only studied manifolds without boundary,

which we now generalize. For a manifold with boundary

we distinguish the interior of a manifold denoted by

Int𝑀

from the boundary denoted by

𝜕𝑀

so that every point in

M lies in only one of them. Points in

𝜕𝑀

do not have a

neighborhood in

𝑛

but instead in the closed

𝑛

-dimensional

upper half-space H

𝑛

⊂ R

𝑛

deﬁned as

𝑛

= {(𝑥

, . . . , 𝑥

𝑛

) ∈ R

𝑛

: 𝑥

𝑛

≥ 0}.

Therefore, manifolds with boundary with

𝜕𝑀 ≠ ∅

are

actually not manifolds in the strict sense, but every manifold

is a manifold with boundary.

Furthermore,

Int𝑀

is an open subset of M and an

𝑛

manifold without boundary while

𝜕𝑀

is a closed subset of

M and an

𝑛 − 1

manifold without boundary. That is why we

know that we know that 𝜕𝜕 ≡ 0 always holds.

𝜕

𝜑

𝑈

∈ R

𝑈

∈ H

𝑈

∈ R

Figure 20: A manifold with boundary with one interior and

one boundary chart as well as its boundary with one chart

3.3.4 Stokes’s Theorem

We now turn to the generalization of vector integral calculus.

We start with the gradient theorem which states that over a

curve 𝛾 from 𝑎 to 𝑏 we have:



𝛾

𝑓 (𝑡)𝑑𝑡 = 𝐹 (𝑏) − 𝑓 (𝑎)

Note that

𝛾

is a smooth manifold with boundary of dimension

and interpret

𝑓 = 𝑑𝐹

as a diﬀerential

-form. Next, note

that

{𝑎, 𝑏} = 𝜕𝛾

and interpret

𝐹 = 𝜄

★

𝜕𝛾

𝐹

, which is the

pullback of the inclusion in

𝛾

. Therefore,

𝐹

is an

𝑛 − 1

-form

on a

𝑛 − 1

-manifold. Overall, using

𝑀

instead of

𝛾

and

𝜔 ∈ Ω

𝑛−1

(𝑀) instead of 𝐹 the equation can be written as:



𝑀

𝑑𝜔 =



𝜕𝑀

𝜔 (7)

It turns out that the last equation does not only hold for

smooth

-manifolds but for general smooth manifolds with

boundary! It is know as Stoke’s Theorem and generalizes the

fundamental theorem of calculus and the classical theorems

of vector calculus.

As a side note, Stoke’s Theorem also gives another moti-

vation why

𝑑(𝑑𝜔) = 0

must hold for all

𝜔 ∈ Λ

𝑘

𝑇 𝑀

. Recall

that 𝜕𝜕𝑀 = 0 holds by deﬁnition and see that

0 =



𝜕𝜕𝑀

𝜔 =



𝜕𝑀

𝑑𝜔 =



𝑀

𝑑𝑑𝜔.

Additionally, if

𝑀

is a manifold without boundary and

𝜔 ∈ Ω

𝑛−1

(𝑀), it will immediately follow that



𝑀

𝑑𝜔 = 0.

3.4 Topology

This section introduces Homology Theory, which gives a

precise meaning to the holes of a smooth manifold. It turns

out that there is a relation to the space of diﬀerential forms.

We especially focus on the homology of the torus since

this will become relevant in the algorithm. The ﬁrst two

subsections are based on Lee (2012).

3.4.1 Singular Homology and Cohomology

First, let

𝑘

denote a standard

𝑘

-simplex, i.e.

𝑘

, b

, . . . , b

𝑘

] ∈ R

𝑘

(c.f. Figure 21). The faces are

connected by the 𝑖th face map deﬁned by

𝐹

𝑘

𝑖

: Δ

𝑘−1

↩→ Δ

𝑘

:= [b

, . . .



𝑖

. . . b

𝑘

where



𝑖

is omitted. Now let

𝐶

𝑘

(𝑀)

be the free abelian

group generated by all singular

𝑝

-simplices, i.e. by all

continuous maps

𝜎 : Δ

𝑘

→ 𝑀

. An element of this group is

called a singular

𝑘

-chain and the boundary of it is deﬁned

as:

𝜕

𝑘

: 𝐶

𝑘

(𝑀) → 𝐶

𝑘−1

(𝑀) := 𝜎 ↦→

𝑘



𝑖=0

(−1)

𝑖

𝜎 ◦ 𝐹

𝑘

𝑖

Extending the boundary operator to singular

𝑘

-chains leads

to a chain of singular chains of the form:

. . .

𝜕

𝑘+2

−−−→ 𝐶

𝑘+1

(𝑀)

𝜕

𝑘+1

−−−→ 𝐶

𝑘

𝜕

𝑘

−−→ 𝐶

𝑘−1

(𝑀)

𝜕

𝑘−1

−−−→ . . .

We deﬁne the space of 𝑘−cycles as

𝑍

𝑘

(𝑀) = ker(𝜕

𝑘

) ⊆ 𝐶

𝑘

(𝑀)

and deﬁne the space of exact 𝑘−boundaries as

𝐵

𝑘

(𝑀) = Im(𝜕

𝑘+1

) ⊆ 𝐶

𝑘

(𝑀).

Since

𝜕 ◦ 𝜕 = 0

, we have

𝐵

𝑘

(𝑀) ⊆ 𝑍

𝑘

(𝑀)

, which leads to

the deﬁnition of the

𝑘

singular homology group deﬁned as

is the quotient group:

𝐻

𝑘

(𝑀) =

𝑍

𝑘

(𝑀)

𝐵

𝑘

(𝑀)

ker 𝜕

𝑘

Im𝜕

𝑘+1

(8)

We call an element of this space a homology class denoted

by [𝜎] ∈ 𝐻

𝑘

(𝑀).

Cohomology. Similar to the duality between vectors and

covectors, there exists a sequence of spaces of linear func-

tions of the type

𝐻

𝑘

(𝑀) → R

called singular cohomology

group and denoted by

𝐻

𝑘

(𝑀; R)

. The only relevant part

for us is that a theorem called the universal coeﬃcient theo-

rem states that

𝐻

𝑘

(𝑀; R)

is the actually dual of

𝐻

𝑘

(𝑀)

, so

𝐻

𝑘

(𝑀; R)  𝐻

𝑘

(𝑀).

−1

𝜎

𝐹

Figure 21: Standard p-

simplices and singular

boundary operator

∈ 𝑍

∈ 𝐵

∈ 𝑍

∉ 𝐵

∈ 𝑍

∉ 𝐵

Figure 22: Construction of

𝑇

: The circles used in the

construction cannot be real-

ized as a boundary of a 2-

chain.

Homology of the Torus. The algorithm requires knowl-

edge of the homology groups of the 3-torus, which is deﬁned

𝑛

= S

× . . . × S

. To calculate this group we can apply

Künneth formula, which states that

𝐻

𝑘

(𝑋 × 𝑌; R) 



𝑖+ 𝑗=𝑘

𝐻

𝑖

(𝑋; R) ⊗ 𝐻

𝑗

(𝑌; R). (9)

A classical result of topology is that the homology group of

a 𝑛-sphere is

𝐻

𝑘

𝑛

) 



R if 𝑘 = 0 or 𝑘 = 𝑛

0 otherwise

(10)

and therefore

𝐻

𝑘

𝑛

)  𝐻

𝑘

× . . . × S

)  R

(

𝑛

)

. Note

that the dimension of

𝐻

𝑘

𝑛

)

and

𝐻

𝑛−𝑘

𝑛

)

are equal,

which becomes relevant in Proposition 1 where we establish

a connection to the hodge star operator.

3.4.2 De Rham Cohomology

In Figure 18 the failure of the closed form being exact seems

to derive from the hole. We now formalize this connection

by making use of the deRham cohomology groups

𝐻

𝑘

(𝑀)

which do not only turn out to be topological invariant but

also isomorphic to 𝐻

𝑘

(𝑀)!

We deﬁne the space of closed 𝑘−forms as

𝑘

(𝑀) = ker(𝑑 : Ω

𝑘

(𝑀) → Ω

𝑘+1

(𝑀))

and deﬁne the space of exact 𝑘−forms as

𝑘

(𝑀) = Im(𝑑 : Ω

𝑘

(𝑀) → Ω

𝑘+1

(𝑀)).

The de Rham cohomology group in degree p is now deﬁned

to be quotient vector space

𝐻

𝑘

(𝑀) =

𝑘

(𝑀)

𝑘

(𝑀)

≃ ≃

≃

Figure 23: The de Rham co-

homology groups are homo-

topy invariants, i.e. homo-

topy equivalent manifolds

𝑀 ≃ 𝑁

have isomorphic de

Rham cohomology groups

𝐻

𝑘

(𝑀)  𝐻

𝑘

(𝑁).

𝑐 ∈ 𝑍

(𝑀)



𝑏 ∈ 𝐶

(𝑀)

𝑐

′

∈ 𝑍

(𝑀)

Figure 24: The deRham

homomorphism is well de-

ﬁned because for any

𝑐, 𝑐

′

representing the same ho-

mology class

𝑐 − 𝑐

′

= 𝜕



𝑏



𝑐

𝜔 −



𝑐

′

𝜔 = 0

and if

𝜔 ∈ B

𝑘

(𝑀)

then



𝑐

𝜔 = 0.

The main result of de Rham cohomology is that there is

an isomorphism

𝓁 : 𝐻

𝑘

(𝑀) → 𝐻

𝑘

(𝑀; R)

being called the

deRham homomorphism. For any de Rham cohomology

class

[𝜔] ∈ 𝐻

𝑘

(𝑀)

and

[𝑐] ∈ 𝐻

𝑘

(𝑀; R)

it is uniquely

deﬁned by

𝓁 [𝜔] [𝑐] =



𝑐

𝜔,

where

𝑐

is a smooth

𝑘

-cycle representing the homology class

[𝑐] (cf. Figure 24).

3.4.3 Hodge Theory

If we are given a Riemannian metric, we can study the

diﬀerential forms of a homology class in more detail. This

section is based on Warner (1983) and prepares a connection

between

𝐻

𝑘

(𝑀)

and speciﬁc solutions of a generalization

of Laplace’s equations.

Codiﬀerential. First we deﬁne the codiﬀerential

𝛿

to be

the map

𝛿 : Ω

𝑘

(𝑀) → Ω

𝑘−1

(𝑀) := 𝜔 ↦→ (−1)

𝑛(𝑘+1)+1

(★𝑑★)(𝜔),

so that we obtain the following maps between the spaces:

. . .

𝑘−1

(𝑀) Ω

𝑘

(𝑀) Ω

𝑘+1

(𝑀)

. . .

𝑑

𝑘−2

𝑑

𝑘−1

𝑑

𝑘

𝑑

𝑘+1

𝛿

𝑘−1

𝛿

𝑘

𝛿

𝑘+1

𝛿

𝑘+2

Comparing Figure 17 and observing the orthogonality of

the hodge star operator it follows that the codiﬀerential is the

adjoint of the exterior derivative, i.e. for

𝜑 ∈ Ω

𝑘

(𝑀)

and

𝜔 ∈ Ω

𝑘−1

(𝑀)

it satisﬁes

⟨𝜑, 𝛿𝜔⟩ = ⟨𝑑𝜑, 𝜔⟩

with respect to

the inner product deﬁned in Equation (6).

Laplacian. This then allows the generalization of the

Laplace operator originally deﬁned as

Δ𝑢 = div(grad𝑢)

any diﬀerential 𝑘-forms by the expression

Δ : Ω

𝑘

(𝑀) → Ω

𝑘

(𝑀) := 𝜔 ↦→ 𝑑𝛿𝜔 + 𝛿𝑑𝜔. (11)

We call a

𝑘

-form harmonic if

Δ𝜔 = 0

and denote the space

of all such 𝑘-forms by

ℋ

𝑘

(𝑀) =



𝜔 ∈ Ω

𝑘

(𝑀) : Δ𝜔 = 0



. (12)

From its construction it follows that

is self adjoint, i.e.

⟨Δ𝜑, 𝜔⟩ = ⟨𝜑, Δ𝜔⟩

, which proves that a diﬀerential form

𝜔

is only harmonic if and only if

𝑑𝜔 = 0

and

𝛿𝜔 = 0

. As a

consequence every harmonic form is closed and more specif-

ically it must be constant on a manifold without boundary.

For instance, the constant basis forms

𝛽

, . . . 𝛽

𝑛

∈ Ω

𝑛

)

are harmonic forms.

Hodge Decomposition. The ﬁrst important result of hodge

theory is the hodge decomposition, which states that

𝑘

(𝑀)

can be expressed as

𝑘

(𝑀)  Im(𝑑

𝑘−1

) ⊕ Im(𝛿

𝑘+1

) ⊕ ℋ

𝑘

(𝑀) (13)

Thus, we propose the following fact:

Proposition 1. In the special case of

𝑛

the hodge star

operator ★ : Ω

𝑘

𝑛

) → Ω

𝑛−𝑘

𝑛

) is an isomorphism.

Proof.

By deﬁnition

★

is linear and injective because

★𝜔 = ★𝜂 ⇒ 𝜔 = 𝜂

and it is bijective at each element.

Therefore, it suﬃces to show bijectivity for each component

in Equation (13). This is given if the ranks of the maps

overlap.

Firstly, since

𝐻

𝑘

 R

(

𝑛

𝑘

)

 𝐻

𝑛−𝑘

and

★

maps constant

linear independent forms to constant linear independent

forms, ℋ

𝑘

𝑛

)  ℋ

𝑛−𝑘

𝑛

). For example, ★ : 𝛽

↦→ 𝛽

Next, we show that

Im(𝑑

𝑘−1

)  Im(𝛿

𝑛−𝑘+1

)

by observing

that

Im(𝛿

𝑛−𝑘+1

) = Im(★𝑑

𝑘−1

★)

. The only function chang-

ing the rank is

𝑑

𝑘−1

, which is found in both components.

Similarly, Im(𝛿

𝑘+1

)  Im(𝑑

𝑛−𝑘−1

). □

Hodge Theorem. The second important result is the hodge

theorem, which states that each de Rham cohomology class

[𝜔] ∈ 𝐻

𝑘

(𝑀)

contains a unique harmonic representative.

Therefore,

ℋ

𝑘

(𝑀)  𝐻

𝑘

(𝑀)

. For example, each of con-

stant forms

𝛽

, . . . , 𝛽

𝑛

∈ ℋ

𝑛

)

corresponds to a separate

homology classes [𝛽

], . . . , [𝛽

] ∈ 𝐻

𝑛

Finally, the following chain of isomorphisms associates

each harmonic form with a homology class:

𝐻

𝑘

(𝑀)  𝐻

𝑘

(𝑀; R)  𝐻

𝑘

(𝑀)  ℋ

𝑘

(𝑀)

3.5 Geometric Measure Theory

The following section gives an overview of the mathematical

principles underlying minimal surfaces. Despite the ap-

parent simplicity of the underlying concept, naive attempts

are not well-deﬁned, resulting in a rather complex theory.

Consequently, this report only provides a brief overview of

the subject based on Morgan (2016), from which also all

ﬁgures in this section are taken from.

The general method for ﬁnding a surface of least area with

a given boundary is, ﬁrstly, to take a sequence of surfaces

with areas decreasing to the inﬁmum, secondly, to extract

a convergent subsequence and, ﬁnally, to show that the

limit surface is the desired surface of least area. A naive

approach would be to deﬁne a surface as a mapping of the

unit disc. However, in general the space of such surfaces

is not compact, i.e. it does not contain the limit of any

convergent sequence (cf. Figure 25). In geometric measure

theory this dilemma is solved by being very deliberate in the

way surfaces are measured and deﬁned.

Figure 25: The series depicted here decreases the area to

the inﬁmum, but due to the lack of compactness the limit

surface breaks down.

The area, or more precisely the Hausdorﬀ measure of a

set is calculated by covering it with a ﬁnite number of closed

balls shrinked inﬁnitesimally. Let

𝑎

𝑘

denote the volume of

closed m-unit ball

𝐵

𝑘

and

diam(𝑆)

denote the diameter of

any

𝑆 ⊆ R

𝑛

deﬁned by

diam(𝑆) = sup{|𝑥 − 𝑦| : 𝑥, 𝑦 ∈ 𝑆}

The Hausdorﬀ measure is now deﬁned as

ℋ

𝑘

( 𝐴) = lim

𝛿→0

inf

𝐴⊂∪𝑆

𝑗

diam(𝑆

𝑗

) ≤ 𝛿



𝑎

𝑘



diam(𝑆

𝑗

)



𝑘

and we say that a property of a set is deﬁned almost every-

where if the subset where it is not deﬁned is of Hausdorﬀ

measure 0.

Surfaces, i.e.

𝑛 − 1

-dimensional sets embedded in

𝑛

manifolds, are ﬁrstly required to be of ﬁnite measurement,

i.e.

ℋ

𝑛−1

(𝑀) < ∞

. Next, they are constructed such that

they are diﬀerentiable almost everywhere and following

Section 3.2.1 they have

𝑛 − 1

-dimensional tangent spaces

deﬁned almost everywhere. This allows diﬀerentiable forms

to be applied almost everywhere. This fact is ensured by the

requirement that a surface is in the union of the images of

countably many functions

𝑓 : R

𝑛

→ R

𝑛−1

such that for each

function

| 𝑓 (𝑥) − 𝑓 (𝑦)| ≤ 𝑐 · |𝑥 − 𝑦|

holds for some constant

𝑐

and any

𝑥, 𝑦 ∈ R

𝑛

. The resulting sets are called rectiﬁable

sets.

Figure 26: The Hausdorﬀ metric

approximates a set by a cover of

little closed balls.

Figure 27: A rectiﬁ-

able set: The only non-

diﬀerential point is the

centre with measure 0.

Rectiﬁable sets of dimension

𝑘

can be associated with

smooth

𝑘

-vector ﬁelds with compact support called recti-

ﬁable currents and denoted by

ℛ

𝑘

: Importantly, currents

can be seen as linear functionals on diﬀerential forms by

integrating these over rectiﬁable sets through the action

𝑆(𝜑) =



𝑆

⟨

−→

𝑆 (𝑥), 𝜑⟩ 𝑑ℋ

𝑘

where

−→

𝑆 (𝑥)

denotes the unit

𝑘

-plane associated with each

tangent space.

This point of view results in a new way to describe the

Hausdorﬀ measure of a rectiﬁable set. In addition to the

Euclidean norm of a k-forms

𝜑 ∈ Λ

𝑘

𝑛

, there exists a mass

norm deﬁned as:

||𝜑||

= sup{𝜑(u) : u is a unit simple m-vector}

This nicely extends to the mass norm of a rectiﬁable current

𝑇

, which turns out to be just the Hausdorﬀ measure of the

associated rectiﬁable set:

||𝑇 ||

= sup{𝑇 (𝜑) : sup ||𝜑(𝑥)||

≤ 1}

The deﬁnition of a boundary of a rectiﬁable set corre-

sponds to the boundary of a smooth manifold with boundary,

hence Stoke’s theorem takes the form

𝜕𝑆(𝜑) = 𝑆(𝑑𝜑). (14)

If the boundary of a rectiﬁable current

𝑆

is again a rectiﬁable

current, we will call

𝑆

an integral current and denote the space

of all integral current by I

𝑚

= {𝑆 ∈ ℛ

𝑚

: 𝜕𝑆 ∈ ℛ

𝑚−1

Using these deﬁnitions, geometric measure theory

achieves some remarkable results. Firstly, it can be shown

that for any

𝑚 − 1

-dimensional rectiﬁable current

𝐵 ∈ ℛ

𝑚−1

𝑛

there exists an

𝑚

-dimensional area-minimizing inte-

gral current

𝑆 ∈ I

𝑚

with

𝜕𝑆 = 𝐵

, such that for any

𝑇

with

𝜕𝑇 = 𝜕𝑆

||𝑆||

≤ ||𝑇 ||

! This is due to the main result of

this theory, which states that

𝑚

is compact. Hence, the limit

in any convergent sequence in

𝑚

results in an element in

𝑚

. Secondly, area-minimizing integral rectiﬁable currents

have zero mean curvature. Thirdly, it can be shown that

𝑚

-dimensional area-minimizing rectiﬁable currents embed-

ded in

𝑚+1

are smooth embedded manifolds, but only for

dimension

𝑚 ≤ 6

! In higher dimension singularities occur

for geometric reasons, so the limit in dimension is not only

a technicality.

Figure 28: While the least-

area disc does not need

to be embedded, an area-

minimizing rectiﬁable cur-

rent is embedded for

𝑛 <

dimensional Euclidean

space.

Figure 29: The area-

minimizing rectiﬁable cur-

rent need not be unique. In

fact there exist conﬁgura-

tions with an inﬁnite amount

of area-minimizing rectiﬁ-

able currents.

4 Methodology

Now, we ﬁnally apply all techniques to develop an optimiza-

tion problem for ﬁnding minimal surfaces. The general

description is that we want to ﬁnd a smooth 2-manifold

𝑆

bounded by a curve 𝐵, i.e. is we want to

minimize Vol(𝑆)

subject to 𝜕𝑆 = 𝐵.

First, we apply geometric measure theory and consider

𝑆 ∈ I

as an integral current and

𝐵 ∈ ℛ

as an rectiﬁable

current. If applicable we will act on

𝑆

and

𝐵

using their

associated diﬀerential k-forms, i.e. by

𝑑𝑆

we mean

(♭◦𝑑◦♯)𝑆

We apply Stoke’s theorem as in Equation (14). Thus,

given any

𝜔 ∈ Ω

)

, the minimization problem can be

written as

minimize

𝑆∈I

||𝑆||

subject to

𝜔∈Ω

)

𝐵(𝜔) = 𝑆(𝑑𝜔).

Since

𝐵(𝜔) = 𝑆(𝑑𝜔) = (𝛿𝑆)(𝜔)

must hold for any

𝜔 ∈

)

, we know that

𝐵 = 𝛿𝑆

. Therefore,

𝑆 = 𝑑𝐵

and the

problem can be written as

minimize

𝑆∈I

||𝑆||

subject to 𝑆 = 𝑑𝐵.

4.1 Periodic Boundary Condition

We will see in Section 4.3 that we can speed up the calculation

if we consider the surface not to be embedded in

but in

. Nevertheless, we somehow must restrict our solution to

𝑛

To do this, we ﬁrst ﬁnd a valid guess

𝑆

∈ R

𝑛

and ﬁnd its

corresponding homology class

[𝑆

] ∈ 𝐻

(T)

. Then, wen

restrict our solution

𝑆

𝑖

at iteration

𝑖

[𝑆

𝑖

] = [𝑆

]

. This

diﬀers from Wang and Chern (2021), who use the dual space,

i.e. 1-currents for the surface and 2-currents for the curve,

which is equal to our formulation as shown in Proposition 1.

Note that this does not restrict the rectiﬁable sets of the

currents to be homotopy equivalent. This is because for any

𝑘

-chains

𝜔, 𝜔

′

∈ 𝐶

𝑘

(𝑀)

the equivalence

[𝜔] = [𝜔

′

]

does

not imply

𝜔 ≃ 𝜔

′

. Instead, it only ensures that

𝜔 − 𝜔

′

= 𝜕𝜂

for some 𝜂 ∈ 𝐶

𝑘+1

(𝑀) (cf. Figure 24).

We have seen in Section 3.4.1 that

𝐻

)  R

and

in Section 3.4.3 that only constant forms are harmonic

forms. As a consequence of the isomorphism

𝐻

𝑘

(𝑀) 

𝐻

𝑘

(𝑀)  𝐻

𝑘

(𝑀)  ℋ

𝑘

(𝑀)

, any linear independent

constant diﬀerential forms

𝛽

, 𝛽

∈ ℋ

𝑛

)

correspond

to a basis in

𝐻

(T)

. Therefore,

[𝑆

] = [𝑆

𝑖

]

can be forced

by ﬁxing ⟨𝑆

, 𝛽

𝑖

⟩ for 𝑖 ∈ {1, 2, 3}.

Looking at the Hodge decomposition in Equation (13) of

[𝑆

𝑖

] = [𝑆

]

ﬁxes the component in

ℋ

(T)

. Additionally,

𝑆 = 𝑑𝐵

ﬁxes the component in

Im(𝑑

)

. Therefore, the only

ﬂexibility for any valid

𝑆

is in

Im(𝛿

)

and the minimization

problem reduces to

minimize

𝑆∈I

, 𝜔 ∈Ω

)

||𝑆||

subject to 𝛿𝜔 − 𝑆 = 𝑆

We can calculate the coordinates of

[𝑆

]

from the

vector area, which can be calculated by

[𝑆

] =



𝑖

⟨𝑆, 𝛽

𝑖

⟩ · [𝛽

𝑖

] =



𝐵

𝛾 × 𝑑𝛾, (15)

where

𝛽

, 𝛽

∈ ℋ

𝑛

)

denotes an orthonormal basis

and 𝛾 encodes the boundary curve 𝜕𝑆 = {𝛾(𝑡) : 𝑡 ∈ [0, 1)}.

4.2 Discretization

We discretize

in a grid of vertices

𝑉

and denote the space

of all discrete diﬀerential

𝑘

forms on

, i.e. of all maps

𝑉 → 𝑇𝑉

, by



𝑘

. A single vertex can be identiﬁed by a

multiindex deﬁned by

v = (𝑣

, 𝑣

𝑣

)

. Similarly, we denote

the neighboring vertices with regard to the

𝑖

th coordinate by

𝑛

𝑖

(v) = (𝑣

, . . . , 𝑣

𝑖

+ 1, . . . 𝑣

𝑛

)

and

𝑛

−

𝑖

(v) = (𝑣

, . . . , 𝑣

𝑖

−

1, . . . 𝑣

𝑛

Next, we discretize

𝑆

by a discrete diﬀerential

-form

𝑆

The discrete diﬀerentiable

𝑑 :



𝑘

→



𝑘+1

is deﬁned by the

midpoint rule at each vertex v by

𝑑 := 𝜔(v) ↦→



𝑖=1

𝜔(𝑛

𝑖

(v)) − 𝜔(𝑛

−

𝑖

(v)).

Note that

𝑑

is linearly independent in each coordinate and as

such can be seen as three independent matrices. Additionally,

because

𝑑

is the adjoint of the codiﬀerential

𝛿

, the discrete

version of the latter is given by

𝛿 =

𝑑

𝑇

. Consequently, the

discrete Laplacian is given by

Δ : ˆ𝜔 ↦→

𝑑

𝛿 ˆ𝜔 +

𝛿

𝑑 ˆ𝜔.

In the discrete case,

||𝑆||

can be approximated by the

-norm deﬁned by

𝑆||



v∈𝑉

𝑆

Therefore, for any

valid initial guess

(𝑆)

the discrete optimization problem

becomes:

minimize

𝑆∈



, ˆ𝜔∈



𝑆||

subject to

𝛿 ˆ𝜔 −

𝑆 =

𝑆

4.3 Algorithm

The optimization problem is solved numerically using an

accelerated variant of the alternating direction method of

multipliers (ADMM) as described in Goldstein et al. (2014).

The method alternates between minimizing

ˆ𝜔

and

𝑆

enforc-

ing stability via a restart rule. Applying this method is very

technical and beyond the scope of this report, but in short it

requires us to solve two optimization problems.

Firstly, to solve the problem for

ˆ𝜔

results in solving the

equation

Δ ˆ𝜔 = 𝑑(

𝑆 −

𝑆

−

𝜏

𝜆)

, where

𝜏

and

𝜆

are variables

stem from ADMM. This is the major bottleneck for the

algorithm and normal methods would need

𝑂(𝑛

)

time to

solve this in each iteration. However, since we are in a

periodic domain, this step can be accelerated using the

Fourier transform method as described in Chapter 20.4 of

Press (2007). This involves solving a linear equation in

the Fourier space and is thus limited by two Fast Fourier

Transform procedures, which each take

𝑂(𝑛 log 𝑛)

time in

each iteration!

Secondly, to solve the problem for

𝑆

results in solving the

equation

𝑆 = shrink

𝜏

(𝜏𝜆 +

𝛿 ˆ𝜔 +

𝑆

where the shrinkage

is given by

shrink(𝑥)

𝜏

= 𝑥 · max



1 −

𝜏 | 𝑥 |

, 0



Notice that

this equation is linear and independent in each variable and

can thus be solved in 𝑂(𝑛) time.

Finally, we calculate a valid ﬁrst guess

𝑆

∈



. We

begin by solving

𝜙 =

𝐵

to get

𝜙 ∈



, which is tangent

to the boundary. Next, we set

ˆ𝜔

𝑑

𝜙

, so now

ˆ𝜔

∈



satisﬁes

𝑑 ˆ𝜔

𝐵

. We calculate the homology class

[𝑆

]

Equation (15) and calculate the diﬀerence per component

ℎ

𝑖

= [𝑆

]

𝑖

−



𝑚

𝜔

∧ 𝛽

𝑖

. Next, we set

ℎ

𝑖

to each vertex

to get a constant diﬀerentiable

-form

ℎ ∈



. Finally, we

set

𝑆

= 𝜔 + ℎ.

5 Evaluation

Wang and Chern (2021) have tested the algorithm on a

MacBook Pro with 8 cores and 16 GB memory. The

bottleneck is the Fourier transform method as mentioned

above, which requires two Fast Fourier Transform procedures

each taking on average 27 seconds on a

256 × 256 × 256

grid. For a classic example, the algorithm converges roughly

after 15 steps.

Furthermore, Wang and Chern (2021) have applied this

algorithm to the Poisson reconstruction problem to ﬁnd a

watertight surface bounded by the boundary curve. A few

examples are given in Figure 30.

Figure 30: Examples of minimal surfaces

They have discussed additional possible applications.

They have shown that their algorithm could be used for

the general surface reconstruction problem by using the

measured data as an input curve and they have given an

method to interpolate two planar closed curves.

Thus we can say, that Wang and Chern (2021) have

suceeded in signiﬁcantly accelerating existing methods to

compute area-minimizing surfaces.

6 Future Work

A generalization to higher dimensions should be possible if

an equivalent formulation of Equation (15) could be found.

However, an

𝑛 − 1

-dimensional area-minimizing rectiﬁable

current embedded in

𝑛

is shown to be a smooth manifold

only for

𝑛 ≤ 7

as shown in Section 3.5. As the data model

cannot represent singularities, it would be interesting to

observe if this limitation restricted the algorithm and how to

adjust the data representation if necessary.

In Section 4.2 we have approximated the mathematical

objects by discretization. It would be interesting to ﬁnd out

if the limit, i.e. an inﬁnitesimally dense grid, lead again to

the results of geometric measure theory. For example, the

approximation of the mass norm

||𝑆||

by the 2-norm

||𝑆||

could be a possible failure.

The discretization into an even grid in

𝑇

is both a blessing

and a curse. On the one hand, it allows for the acceleration

compared to conventional methods. On the other hand, it

does not allow the discretization to be ﬁner in interesting

regions. It would be interesting to study whether adaptive

ﬁnite elements methods could be applied while retaining the

run-time.

7 Conclusion

We have discussed the method of Wang and Chern (2021)

along with the necessary foundations of diﬀerentiable man-

ifolds and geometric measure theory. Thanks to a precise

foundation of the mathematics involved, this new method

gives correct result and is very eﬃcient compared to tradi-

tional techniques.

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