Global existence for the Willmore flow with boundary via Simon’s Li–Yau inequality

Consider immersions u n , u ∈ W 2 , 2 ⁢ ( I , H 2 ) for n ∈ N with u n → u in W 2 , 2 ⁢ ( I ) . Then E ⁢ ( u n ) → E ⁢ ( u ) .

Proof

The claim is due to (A.3) and the compact embedding W 2 , 2 ⁢ ( I ) ↪ C 1 ⁢ ( I , R 2 ) . ∎

For immersions u n , u ∈ W 2 , 2 ⁢ ( I , H 2 ) for n ∈ N with u n ⇀ u in W 2 , 2 ⁢ ( I ) , one has

Proof

After passing to a subsequence, without loss of generality, u n → u in C 1 ⁢ ( I ) and

It is a well-known fact that the product of a uniformly convergent sequence and a sequence that weakly converges in L 2 is again weakly convergent in L 2 to the product of the respective limits. Thus, the claim is a consequence of (A.3) and the weak lower semi-continuity of ∥ ⋅ ∥ L 2 ⁢ ( I ) . ∎

Consider immersions u n : I → H 2 with u n ∈ W 2 , 2 ⁢ ( I ) defined on a compact interval I ⊆ R such that ∥ u n ( 2 ) ∥ L ∞ ⁢ ( I ) → 0 and L R 2 ⁢ ( u n ) ≥ ℓ > 0 . Then lim sup n → ∞ E ⁢ ( u n ) = ∞ .

Proof

Proposition 2.7 in [31] yields the claim for smooth immersions u n . Using [22] to suitably approximate each u n ∈ W 2 , 2 ⁢ ( I ) and using Lemma A.1, one obtains the claim. ∎

Theorem B.2 ([31, Theorem 2.5])

Consider smooth immersions u n : I → H 2 on a compact interval I ⊆ R such that 0 < ℓ ≤ | ∂ x u n | ≤ L < ∞ . If ( | ∂ x u n | ) n ∈ N converges uniformly on 𝐼, u n ⇀ ∗ u ∞ in W 1 , ∞ ⁢ ( I ) and u n → u ∞ uniformly on 𝐼 and

then Z = { u ∞ ( 2 ) = 0 } ∩ I ∘ consists of at most 𝐸 points.

One has that C LY ⁢ ( p , τ ) ≥ T ⁢ ( τ ) .

One has W ⁢ ( V 0 ) + W ⁢ ( V 1 ) = T ⁢ ( τ ) .

Proof

First, consider any τ ∈ S 1 with τ ( 1 ) ≠ 0 and the spherical cut-out M ⊆ S 2 as above with outer co-normal induced by rotating 𝜏 as in the construction above.

Case 1: τ ( 1 ) < 0 . Choosing θ ∈ ( 0 , π ) such that

we can parametrize 𝑀 by the surface of revolution associated to γ ⁢ ( x ) = ( cos ⁡ ( x ) , sin ⁡ ( x ) ) with x ∈ [ 0 , θ ) . Since the mean curvature of S 2 has length 1 everywhere,

Case 2: τ ( 1 ) > 0 . With similar arguments, one also obtains (C.4).

Using that the Willmore energy is invariant with respect to scaling and translations, that − τ 0 induces the outer co-normal of V 0 and that τ 1 induces the outer co-normal of V 1 , (C.4) yields

It holds that

Proof

To this end, simply note that (C.2) applied to S R y , x y 2 , a manifold without boundary, yields, for y = 0 , 1 ,

using that the support of V y is contained in the sphere S R y , x y 2 . ∎

Proof of Proposition C.1

The claim is concluded by combining (C.3) with both lemmata above. ∎

Writing Γ y = { y } × S 1 , one finds that

for y = 0 , 1 , using (1.4), where f c y h is the immersion one obtains by rotating c y h .

Proof of Proposition D.1

Suppose first of all that c y h parametrizes a segment of the profile curve of a Möbius transformed inverted catenoid. Moreover, denote by V c y h the varifold one obtains by rotating c y h and by V c y h cl the associated inverted catenoid without boundary whose support contains the support of V c y h . Using (D.2), one finds θ 2 ⁢ ( μ V c y h cl , h ⁢ e 1 ) ⁢ π = 2 ⁢ π , i.e. the closed inverted catenoid has density 2 at the point h ⁢ e 1 where it intersects the rotation axis. Moreover, as W ⁢ ( V c y h cl ) = 8 ⁢ π , proceeding as in (C.2) yields

In particular,

By (D.2), we have θ 2 ⁢ ( μ V c y h , h ⁢ e 1 ) ⁢ π = π , i.e. the varifold obtained by rotating c y h only has density equal to 1 at the point where it intersects the rotation axis. Therefore, (C.2) applied to V c y h yields (D.3), using (1.4). With the very same argument, one proves the claim also in the case where c y h parametrizes a segment of the profile curve of a sphere.

Finally, if c y h parametrizes the profile curve of a disk, then W ⁢ ( f c y h ) = 0 . Moreover, using (D.2), τ y = ( − 1 ) y ⁢ ( 0 , − 1 ) and p y ( 1 ) = h . Thus, as in (D.1), using (1.4),

and (D.3) also follows in this case. ∎