Confined elasticae and the buckling of cylindrical shells

Proof of Lemma 1.

First claim. Immediate from scaling properties.

Second claim. See proof of fourth claim. For embedded curves, it also follows from Hopf’s Umlaufsatz

Third claim. This is essentially a convenient way to phrase the second claim.

Fourth claim. Without loss of generality, we assume that t 1 = 0 . We can now decompose the curve γ into segments γ j = γ | [ t j , t j + 1 ] where we identify L = t k + 1 = t 0 modulo L – note that γ j is a curve segment, not a coordinate function. Then, applying Poincaré’s inequality to the coordinate functions γ i j , we find that

Equality is attained for eigenfunctions of the Laplacian. We note that the boundary conditions for γ j are γ j ⁢ ( t j ) = γ j ⁢ ( t j + 1 ) , so

This only leaves the option that γ i j ⁢ ( s ) = α i ⁢ j ⁢ sin ⁡ ( π ⁢ ( s - t j ) t j + 1 - t j ) for all j = 1 , … , k and i = 1 , … , n . At this point, we make two observations.

1. In the situation of the first claim, the curve t 2 = L (i.e. γ may not have any multiple points), and the curve is even C 1 -periodic. That excludes the first eigenfunction and allows only

   γ i ⁢ ( s ) = α i ⁢ cos ⁡ ( 2 ⁢ π ⁢ s L )   or   γ i ⁢ ( s ) = α i ⁢ sin ⁡ ( 2 ⁢ π ⁢ s L )

   with the larger constant 4 ⁢ π 2 which establishes the first claim.

2. A curve γ such that all coordinate functions are multiples of the sin of the same argument cannot be parameterised by arc-length. Hence we conclude that in fact

   ∑ i = 1 n ∫ t j t j + 1 | ( γ i j ) ′ | 2 ⁢ d s < ∑ i = 1 n | t j + 1 - t j | 2 π 2 ⁢ ∫ t j t j + 1 | ( γ i j ) ′′ | 2 ⁢ d s

   for all admissible curves. We conclude that

   C := inf γ ⁢ ( 0 ) = γ ⁢ ( 1 ) = 0 ⁡ 𝒲 ⁢ ( γ ) > π 2 ℋ 1 ⁢ ( γ ) .

From this we obtain

The sum on the right becomes minimal for equi-distant points t j = j - 1 k ⁢ L giving rise to the estimate

Proof of Lemma 3.

The transversal self-crossing of the figure eight is easily excluded when writing the curves locally as graphs and using the intermediate value theorem. The multiply covered circle, which only has tangential self-contact, is slightly harder to exclude.

Any curve γ which is W 2 , 2 - or more generally C 1 -close to an m-fold covered circle can be written as a radial graph

for a 2 ⁢ π ⁢ m -periodic function r which is C 1 -close to the constant 1-function, applying a general statement about writing a surface as a normal graph over a C 1 -close surface. If m > 1 , we consider the shifted function r ~ ⁢ ( s ) = r ⁢ ( s + 2 ⁢ π ) and pick an interval [ a , b ] ⊂ [ 0 , 2 ⁢ π ⁢ m ] such that r ⁢ ( a ) = max ⁡ r , r ⁢ ( b ) = min ⁡ r . By the intermediate value theorem

imply that there exists s ∈ [ a , b ] such that r ~ ⁢ ( s ) - r ⁢ ( s ) = 0 since r , r ~ are continuous. Then

which means that γ is not embedded. ∎

Remark 1.

While we chose an elementary argument, there are more powerful tools that would cover larger classes of curves. Assuming that a curve γ of length L is parametrised by unit speed, we can write γ ′ ⁢ ( s ) = ( cos ⁡ ω ⁢ ( s ) , sin ⁡ ω ⁢ ( s ) ) and compute that the curvature of γ is κ = ω ′ ⁢ ( s ) . It follows that

since γ ′ ⁢ ( 0 ) = γ ′ ⁢ ( L ) . The function ω is called a Gauss representation of γ. The quantity ω ⁢ ( L ) - ω ⁢ ( 0 ) ∈ 2 ⁢ π ⁢ ℤ which measures how often the tangent turns is the Whitney index of the curve.

It is clear that for the circle and all curves C 1 -close to a circle, we have

Since the space of embedded curves is connected (every embedded curve becomes a round circle under curve shortening flow), all embedded curves must have Whitney index 2 ⁢ π , while an m-fold covered circle has Whitney index 2 ⁢ π ⁢ m and any figure eight curve has Whitney index 0.

The same result on the Whitney index of an embedded curve can be obtained by using the Gauss–Bonnet theorem on the disk bounded by the curve γ due to Jordan’s curve theorem instead of the connectedness of the space of embedded curves.