On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary

A Expansion

For the duration of this section, we will write u ⁢ ( s ) := u ^ t ⁢ ( s ) , p := ξ ⁢ ( t ) , g ~ ⁢ ( s ) := g ~ p , s , and d ⁢ μ ⁢ [ u , g ~ ] := d ⁢ μ f u * ⁢ g ~ . We consider the map

and prove that z ⁢ ( 0 ) = z ′ ⁢ ( 0 ) = 0 and z ′′ ⁢ ( 0 ) = - 3 ⁢ ∂ i ⁡ H S ⁢ ( p ) . This implies the identities for y t , i ⁢ ( s ) = - s ⁢ z ⁢ ( s ) .

First, we note that, as W ⁢ [ 0 , δ ] = 0 , H ⁢ [ 0 , δ ] = 2 and ∇ ⁡ C i ⁢ [ 0 , δ ] = - 3 2 ⁢ π ⁢ ω i , we may conclude z ⁢ ( 0 ) = 0 and

As u ′ ⁢ ( 0 ) is even,

is also even and the first integral vanishes. For the second integral, we recall Lemma 6.1 to obtain that D 2 ⁢ W ⁢ [ 0 , δ ] ⁢ g ~ ′ ⁢ ( 0 ) is an even function and we deduce that the second integral also vanishes. It remains to compute z ′′ ⁢ ( 0 ) . Again using W ⁢ [ 0 , δ ] = 0 , we get

(A.1)

We claim that the first line vanishes. First, note that by Lemma 6.1 the integrand in the first line is an odd function. We now need to check that the derivative of the measure at s = 0 is even^[1]. For this we use the following two formulas from [2, Lemma 1^[2] and the proof of Lemma 7]:

Inserting g ~ ′ ⁢ ( 0 ) from the proof of Lemma 6.1, we see that

are both even. Thus the first line in equation (A.1) vanishes and we must only compute the last line. By definition,

As g ~ ⁢ ( s ) is the pullback of a flat metric, we may drop the last term. Next, we note that h 0 ⁢ [ 0 , δ ] = 0 , and hence D 1 ⁢ | h 0 | 2 ⁢ [ 0 , δ ] = 0 and similarly D 2 ⁢ | h 0 | 2 ⁢ [ 0 , δ ] = 0 . Using H ⁢ [ 0 , δ ] = 2 , we get

for arbitrary φ, q. Combining these considerations, we deduce

Using [2, Lemma 7] and the standard result for the variation of H, we get

We apply (A.4) to q = g ~ ′′ ⁢ ( 0 ) . Denoting the second fundamental form of S in the chart f ⁢ [ p , ⋅ ] by h i ⁢ j , we can use formula (2.2) to get ( * -entries are to be inferred from symmetry)

Inserting into (A.4) and repeatedly using tr 𝕊 2 ⁡ A = tr ℝ 3 ⁡ A - A ⁢ ( ω , ω ) gives

Next, we must consider

To evaluate this integral, we must exploit the boundary condition B 0 ⁢ [ u ⁢ ( s ) , g ~ ⁢ ( s ) ] = 0 and differentiate twice.

Denote arbitrary even functions from 𝕊 + 2 to ℝ by E. Once Lemma A.1 is proven we may combine it with Lemma 6.1 to get

(A.7)

Combining both components of equation (A.7) gives

Inserting into equation (A.6) and combining with equation (A.5) yields

Multiplying with - 3 2 ⁢ π and recalling equation (A.2) shows z ′′ ⁢ ( 0 ) = - 3 ⁢ ∂ i ⁡ H S ⁢ ( p ) . It remains to prove Lemma A.1.

B Solution of the elliptic problem

Alessandroni and Kuwert [2] study the elliptic problem (2.9) for ϕ ∈ 𝒮 ⁢ ( λ , θ ) by making the ansatz

For given p ∈ S , they first derive a solution to the elliptic problem with prescribed barycenter C ⁢ [ ϕ ] = p . This is achieved by applying the diffeomorphism F λ ⁢ [ p , ⋅ ] and studying the equation in ℝ 3 with the background metric g ~ = g ~ p , λ :

It is shown^[3] that the unique solution u ⁢ ( λ ) ∈ C 4 , γ ⁢ ( 𝕊 + 2 ) depends regularly on λ. In fact, it is shown that, for Ω ∈ C m and l = m - 1 ≥ 6 , we have g ~ p , λ ∈ G 0 l ⁢ ( δ ; σ 0 ) for λ ≤ λ 0 ⁢ ( Ω , σ 0 ) and that u ∈ C l - 4 ⁢ ( [ 0 , λ 1 ) , C 4 , γ ⁢ ( 𝕊 + 2 ) ) for some small λ 1 ⁢ ( Ω ) > 0 . Putting φ := u ′ ⁢ ( 0 ) and using Lemma 6.1, we see that

The uniqueness of this linear problem implies that φ is even, and hence ∥ u - ∥ C 4 , γ ≤ C ⁢ λ 2 . Thus ϕ p , λ ∈ 𝒮 - ⁢ ( λ , K ) for suitable K. Finally, in [2, Section 3], a suitable choice for p is derived that makes ϕ p , λ a solution of (2.9).

C Parabolic Schauder theory and regularity of meta maps

The following definitions are taken from [7]. Let Ω ⊂ ℝ n be a bounded domain, let T > 0 and let Ω T := [ 0 , T ] × Ω . For ( l , α ) ∈ ℕ 0 × ℕ 0 n , we abbreviate D l , α ⁢ u ⁢ ( t , x ) := ∂ t l ⁡ D x α ⁢ u ⁢ ( t , x ) . For m ∈ ℕ 0 , let

For u ∈ C m ⁢ ( Ω ¯ T ) and γ ∈ ( 0 , 1 ) , we define the temporal and spatial-Hölder seminorms

and put

Then the parabolic Hölder space C m , γ ⁢ ( Ω ¯ T ) is defined by

We also refer to this space as C m , [ m / 4 ] - , γ , where [ ⋅ ] - denotes the floor function. The definition of boundary spaces such as C 3 , 0 , γ ⁢ ( [ 0 , T ] × ∂ ⁡ Ω ) is analogue and also covered in [7]. Lifting the definitions onto a manifold M works as usual and is, e.g., covered in [7] for the case when M = ∂ ⁡ U for a sufficiently regular domain U.

Improved Schauder estimates.

All Schauder theory except for the decay estimate used, e.g., in Theorem 3.3 are standard and may be derived by following Simon’s scaling argument [17]. To prove the decay, consider a function φ ∈ X T satisfying φ ⁢ ( t , ⋅ ) ∈ X 0 ⊥ (recall equation (6.5)) for all t ∈ [ 0 , T ] and

(C.1) { φ ˙ + 1 2 ⁢ Δ ⁢ ( Δ + 2 ) ⁢ φ = 0 on  ⁢ [ 0 , T ] × 𝕊 + 2 , φ ⁢ ( 0 , ⋅ ) = ψ 0 on  ⁢ { 0 } × 𝕊 + 2 .

Following [2, Lemma 4], we see that, for all u 0 ∈ X 0 ⊥ , we have

∫ 𝕊 + 2 u 0 ⁢ Δ ⁢ ( Δ + 2 ) ⁢ u 0 ⁢ 𝑑 μ 𝕊 2 ≥ λ 2 ⁢ ( λ 2 - 2 ) ⁢ ∫ 𝕊 + 2 u 0 2 = 24 ⁢ ∫ 𝕊 + 2 u 0 2 ⁢ 𝑑 μ 𝕊 2 ,

where λ 2 = 6 is the second non-zero eigenvalue of - Δ . Let μ := 12 and φ μ ⁢ ( t , ω ) := e μ ⁢ t ⁢ φ ⁢ ( t , ω ) . Then φ μ has zero boundary conditions along ∂ ⁡ 𝕊 + 2 and solves

{ φ ˙ μ + 1 2 ⁢ Δ ⁢ ( Δ + 2 ) ⁢ φ μ - μ ⁢ φ μ = 0 on  ⁢ [ 0 , T ] × 𝕊 + 2 , φ μ ⁢ ( 0 , ⋅ ) = ψ 0 on  ⁢ { 0 } × 𝕊 + 2 .

Standard Schauder theory then gives the a priori estimate

(C.2) ∥ φ μ ∥ C T 4 , 1 , γ ≤ C ⁢ ( 𝕊 + 2 , γ ) ⁢ ( ∥ ψ 0 ∥ C 4 , γ + sup t ∈ [ 0 , T ] ⁡ ∥ φ μ ⁢ ( t ) ∥ L 2 ⁢ ( 𝕊 + 2 ) ) .

Using the fact that φ ⁢ ( t , ⋅ ) ∈ X 0 ⊥ for all t ∈ [ 0 , T ] and remembering (C.1), we compute

d d ⁢ t ⁢ 1 2 ⁢ ∫ 𝕊 + 2 φ μ 2 ⁢ 𝑑 μ 𝕊 2 = - 1 2 ⁢ ∫ 𝕊 + 2 φ μ ⁢ Δ ⁢ ( Δ + 2 ) ⁢ φ μ ⁢ 𝑑 μ 𝕊 2 + μ ⁢ ∫ 𝕊 + 2 φ μ 2 ⁢ 𝑑 μ 𝕊 2 ≤ ( 6 ⋅ ( 2 - 6 ) 2 + μ ) ⁢ ∫ 𝕊 + 2 φ μ 2 ⁢ 𝑑 μ 𝕊 2 = 0 .

This implies

sup t ∈ [ 0 , T ] ⁡ ∥ φ μ ⁢ ( t , ⋅ ) ∥ L 2 ⁢ ( 𝕊 + 2 ) 2 = ∥ ψ 0 ∥ L 2 ⁢ ( 𝕊 + 2 ) 2 ,

and therefore

e μ ⁢ T ⁢ ∥ φ ⁢ ( T , ⋅ ) ∥ C 4 , γ = ∥ φ μ ⁢ ( T , ⋅ ) ∥ C 4 , γ ≤ ∥ φ μ ∥ C T 4 , 1 , γ ≤ C ⁢ ( 𝕊 + 2 , γ ) ⁢ ∥ ψ 0 ∥ C 4 , γ .

Recalling μ = 12 , we obtain

∥ φ ⁢ ( T ) ∥ C 4 , γ ≤ C ⁢ ( 𝕊 + 2 , γ ) ⁢ e - 12 ⁢ T ⁢ ∥ ψ 0 ∥ C 4 , γ .

Regularity of meta maps.

Let U be a bounded domain and let V be a bounded convex domain. Put

A := C 0 , γ 4 ⁢ ( [ 0 , T ] , C l ⁢ ( V , ℝ ) ) , B := C 0 , 0 , γ ⁢ ( [ 0 , T ] × U , V ) , C := C 0 , 0 , γ ⁢ ( [ 0 , T ] × U ) .

For g ∈ A and f ∈ B , put ( g ⋄ f ) ⁢ ( t , x ) := g ⁢ ( t , f ⁢ ( t , x ) ) and consider the meta map

T : A × B → C , ( g , f ) ↦ g ⋄ f .

It is easy to see that T is well defined if l ≥ 1 . Moreover, T is even continuous if l ≥ 2 . As an example, we show that [ g ⋄ f - g ⋄ f ~ ] γ t is small if f ≈ f ~ . Indeed,

Δ := | g ⁢ ( t , f ⁢ ( t , x ) ) - g ⁢ ( t , f ~ ⁢ ( t , x ) ) - g ⁢ ( s , f ⁢ ( s , x ) ) + g ⁢ ( s , f ~ ⁢ ( s , x ) ) |

≤ | g ⁢ ( t , f ⁢ ( t , x ) ) - g ⁢ ( t , f ~ ⁢ ( t , x ) ) - g ⁢ ( t , f ⁢ ( s , x ) ) + g ⁢ ( t , f ~ ⁢ ( s , x ) ) |

+ | g ⁢ ( t , f ⁢ ( s , x ) ) - g ⁢ ( t , f ~ ⁢ ( s , x ) ) - g ⁢ ( s , f ⁢ ( s , x ) ) + g ⁢ ( s , f ~ ⁢ ( s , x ) ) |

≤ | g ( t , λ f ( t , x ) + ( 1 - λ ) f ~ ( t , x ) ) - g ( t , λ f ( s , x ) + ( 1 - λ ) f ~ ( s , x ) ) | λ = 0 λ = 1 |

+ | g ( t , λ f ( s , x ) + ( 1 - λ ) f ~ ( s , x ) ) - g ( s , λ f ( s , x ) + ( 1 - λ ) f ~ ( s , x ) ) | λ = 0 λ = 1 | .

The convexity of V and the chain rule then readily imply

Δ ≤ ( ∥ D 2 ⁢ g ∥ C 0 ⁢ [ f - f ~ ] γ t + ∥ D 2 2 ⁢ g ∥ C 0 ⁢ ∥ f - f ~ ∥ C 0 ⁢ ( [ f ] γ t + [ f ~ ] γ t ) + [ D 2 ⁢ g ] γ t ⁢ ∥ f - f ~ ∥ C 0 ) ⁢ | t - s | γ 4 ,

which shows the continuity of T. Similarly, one shows that for T ∈ C k it suffices if l ≥ 2 + k , as differentiating g ⋄ f with respect to f produces D ⁢ g ⋄ f , which must be continuous in f. Similar arguments allow one to study g ⋄ f ∈ C 4 , 1 , γ ⁢ ( [ 0 , T ] × U ) for f ∈ C 4 , 1 , γ ⁢ ( [ 0 , T ] × U , V ) . The corresponding meta map is of class C k as long as l ≥ 6 + k , as four additional derivatives of g are required.

D The Riemannian barycenter

Let g ~ ∈ G 0 l ⁢ ( δ ; ϵ ) with l ≥ 2 . This section serves the purpose of constructing the Riemannian barycenter of the immersion f u : 𝕊 + 2 → ( ℝ 3 , g ~ ) for small enough u. The analysis follows the construction in [2], but changes the euclidean projection π ℝ 2 used in [2] to the Riemannian projection π ℝ 2 ⁢ [ g ~ , ⋅ ] . We recall D r := { x ∈ ℝ 2 ∣ | x | < r } and Z r := D ¯ r × [ - r , r ] .

We briefly motivate the modified definition of the Riemannian barycenter. For an immersion f u : 𝕊 + 2 → ( ℝ 3 , g ~ ) , we wish to define the Riemannian barycenter x ∈ ℝ 2 by the implicit equation

Note that, by definition, I ∈ T x ⁢ ℝ 3 , and hence it is sensible to demand orthogonality with respect to g ~ x . We may reformulate the condition as g ~ x ⁢ ( x + I - x , ℝ 2 ) = 0 , which implies π ℝ 2 ⁢ [ g ~ , x + I ] = x . Hence we study zeros of the map

which reduces to X ′ (we include the prime to distinguish the map from [2] from the one studied here) from [2] if we replace π ℝ 2 ⁢ [ g ~ , ⋅ ] with π ℝ 2 ⁢ [ δ , ⋅ ] as is used there. In particular, both maps are identical if g ~ = δ is inserted. Repeating the analysis from [2], we get the following theorem.

As X ⁢ [ ⋅ , δ , ⋅ ] = X ′ ⁢ [ ⋅ , δ , ⋅ ] , we conclude the same explicit formula for C ⁢ [ u , δ ] as is given in [2]:

Following [2], we now compute the L 2 -gradient of C i . The only difference in the analysis that is required is to also take the derivative of the projection into account. Let f ⁢ ( ϵ ) be a variation of f u with f ˙ ⁢ ( 0 ) = φ ⁢ ν ~ , where ν ~ is the inner normal of f u . We set

and compute

(D.1)

We investigate the derivative:

(D.2)

Inserting (D.2) into (D.1) and dropping the arguments on I for spatial reasons gives

As the Riemannian barycenter is invariant under reparameterization, the L 2 -gradient of C i is normal along f u . We may therefore multiply the actual L 2 -gradient with ν ~ to obtain a scalar function which we denote by ∇ ⁡ C i ⁢ [ u , g ~ ] . By definition, we then have

The analysis above gives the explicit formula

Specializing to g ~ = δ gives

Note that

by the definition of x = C ⁢ [ u , δ ] . This eliminates the first term, and thus establishes the same formula that is derived in [2]. In particular, we recover

A note on regularity.

We consider ∇ ⁡ C i as a map

∇ ⁡ C i : G T l × C 4 , 1 , γ ⁢ ( [ 0 , T ] × 𝕊 + 2 ) → C 0 , γ 4 ⁢ ( [ 0 , T ] , ℝ 2 )

defined on a neighborhood of ( δ , 0 ) . We use the fact that the map ( x , v , g ~ ) ↦ exp x g ~ ⁡ v ∈ ℝ 3 defined on a suitably large neighborhood of ( 0 , 0 , δ ) ∈ ℝ 2 × ℝ 3 × G 0 l is of class C l - 1 . This is proven in [2, Appendix 4]. Combined with the explicit formula for ∇ ⁡ C i , we get that ∇ ⁡ C i is of class C l - 7 as long as l ≥ 7 .

Application.

We now define C ⁢ [ ϕ p , u λ ] by applying Theorem D.2 to f u with the pullback metric g ~ p , λ . This gives a point x = C ⁢ [ u , g ~ p , λ ] ∈ ℝ 2 and we wish to define

C ⁢ [ ϕ ] := F λ ⁢ [ p , x ] .

We must now prove that this definition does not depend on p and the chosen frame.

Proof.

Abbreviate g ~ := g ~ p , λ . Rewriting the defining equation for x gives

x = C ⁢ [ f u , g ~ ]   if and only if   ∫ 𝕊 + 2 ( exp x g ~ ) - 1 ⁢ ( f u ⁢ ( ω ) ) ⁢ 𝑑 μ f u * ⁢ g ~ ⊥ ℝ 2 ⁢  with respect to  ⁢ g x ,

This, in turn, is equivalent to

(D.3) ∫ 𝕊 + 2 ( exp x g ~ ) - 1 ⁢ ( f u ⁢ ( ω ) ) ⁢ 𝑑 μ f u * ⁢ g ~ = k ⁢ ν ~ ℝ 2 ⁢ ( x )   for some  ⁢ k ∈ ℝ .

We note that F λ ⁢ [ p , ⋅ ] is an isometry up to the scaling factor λ 2 in the definition of g ~ p , λ . Applying the differential ( D ⁢ F λ ⁢ [ p , x ] ) - 1 to (D.3), we obtain

x = C ⁢ [ f u , g ~ ]   if and only if   ( D ⁢ F λ ⁢ [ p , x ] ) - 1 ⁢ [ ∫ 𝕊 + 2 ( exp x g ~ ) - 1 ⁢ ( f u ⁢ ( ω ) ) ⁢ 𝑑 μ f * ⁢ g ~ ] = λ ⁢ k ⁢ N S ⁢ ( F λ ⁢ [ p , x ] ) .

This is equivalent to

∫ 𝕊 + 2 ( exp F ⁢ [ p , x ] δ ) - 1 ⁢ ( F λ ⁢ [ p , f u ⁢ ( ω ) ] ) ⁢ 𝑑 μ ϕ * ⁢ δ = λ ⁢ k ⁢ N S ⁢ ( F λ ⁢ [ p , x ] ) ,

which in turn is equivalent to

∫ 𝕊 + 2 ( f u ⁢ ( ω ) - F ⁢ [ p , x ] ) ⁢ 𝑑 μ ϕ * ⁢ δ = λ ⁢ N S ⁢ ( F λ ⁢ [ p , x ] ) .

For λ small enough, we see that x = C ⁢ [ f u , g ~ ] is equivalent to

(D.4) π S ⁢ ∫ 𝕊 + 2 ϕ ⁢ ( ω ) ⁢ 𝑑 μ ϕ * ⁢ δ = F λ ⁢ [ p , x ] ,

where π S is the nearest point projection onto S. As the left-hand side of equation (D.4) is independent of the choices for p and the frame, the right-hand side must be too. ∎

An immediate consequence is that for ϕ = F λ ⁢ [ p , f u ] we have C ⁢ [ ϕ ] = p if and only if C ⁢ [ f u , g ~ p , λ ] = 0 . We close by proving that we may always parameterize ϕ over its barycenter.

Theorem D.3.

There exist λ 0 > 0 , θ 0 > 0 such that for λ ≤ λ 0 each ϕ ∈ S ′ ⁢ ( λ , θ 0 ) may be parameterized over its barycenter C ⁢ [ ϕ ] . That is, there exists a unique graph function u ~ ∈ C 4 , γ ⁢ ( S + 2 ) such that

ϕ = F λ ⁢ [ C ⁢ [ ϕ ] , f u ~ ] .

Proof.

As | φ ⁢ [ p , λ ⁢ x ] | ≤ C ⁢ λ 2 , we derive

F λ ⁢ [ p , Z 2 ] = im ⁡ F λ ⁢ [ p , ⋅ ] ⊃ B 3 2 ⁢ λ ⁢ ( p )

for small enough λ. Next, we have the following claim.

Claim.

There exist ϵ , δ , λ > 0 such that F λ ⁢ [ p , x ] ∈ im ⁡ F λ ⁢ [ q , ⋅ ] for p , q ∈ S and x ∈ Z 2 satisfying d ⁢ ( p , q ) < λ ⁢ δ and | x | ≤ 1 + ϵ .

Proof of the claim. For small enough λ, we have im ⁡ F λ ⁢ [ q , ⋅ ] ⊃ B ( 3 / 2 ) ⁢ λ ⁢ ( q ) . Now if | x | < 1 + ϵ , then

| F λ ⁢ [ p , x ] - q | ≤ | p - q | + λ ⁢ | x | + | φ ⁢ [ q , λ ⁢ x ] | ≤ λ ⁢ δ + λ ⁢ ( 1 + ϵ ) + C ⁢ λ 2 ,

which implies the claim for sufficiently small choices of ϵ, δ and λ.

Now suppose that ϕ ∈ 𝒮 ′ ⁢ ( ϵ , λ ) . Then we may write ϕ = F λ ⁢ [ p , f u ] for some p ∈ S and ∥ u ∥ C 4 , γ < ϵ . Let q ∈ S denote the barycenter of ϕ. By definition, q = C ⁢ [ ϕ ] = F λ ⁢ [ p , C ⁢ [ f u , g ~ p , λ ] ] and, using the estimate from Theorem D.2, we get | C ⁢ [ f u , g ~ p , λ ] | < δ ⁢ ( ϵ , λ ) with δ → 0 as λ , ϵ → 0 . This gives d ⁢ ( p , q ) ≤ C ⁢ λ ⁢ δ ⁢ ( ϵ , λ ) . As | f u ⁢ ( ω ) | < 1 + ϵ , we may use the claim for small enough ϵ and λ to define u ~ by

f u ~ ⁢ ( ω ) = ( F λ ⁢ [ q , ⋅ ] ) - 1 ⁢ ( F λ ⁢ [ p , f u ⁢ ( ω ) ] ) .

The uniqueness is easily established. ∎