Existence and Regularity of Spheres Minimising the Canham–Helfrich Energy

Mondino, Andrea; Scharrer, Christian

doi:10.1007/s00205-020-01497-4

Cited by 20 publications

(43 citation statements)

References 34 publications

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“…Nevertheless, some progress has been made: Existence of Helfrich surfaces with prescribed surface area near the sphere has been achieved in [24] by examining the corresponding L 2 -flow. This result has been extendend to a general existence result for spherical Helfrich immersions with prescribed area and enclosed volume in [29] by variational parametric methods. The axisymmetric case has been handled independently in [5] and [4].…”

Section: Introductionmentioning

confidence: 78%

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Lower semicontinuity for the Helfrich problem

Eichmann

2020

Ann Glob Anal Geom

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We minimise the Canham–Helfrich energy in the class of closed immersions with prescribed genus, surface area, and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false under varifold convergence by a counter example by Große-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of $${{\mathbb {R}}}^3$$ R 3 .

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Section: Introductionmentioning

confidence: 78%

“…geometric measure theory. In the case of spherical topology this has been achieved with a parametric approach in [29,Thm. 3.3].…”

Section: Introductionmentioning

confidence: 99%

Lower semicontinuity for the Helfrich problem

Eichmann

2020

Ann Glob Anal Geom

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“…The constant c 0 is referred to as spontaneous curvature. Existence of minimisers in the class of (possibly branched and bubbled) spheres was proven by Mondino-Scharrer [MS20a]. Existence and regularity for minimisers with higher genus remains an open problem.…”

Section: Theoremmentioning

confidence: 99%

Embedded Delaunay tori and their Willmore energy

Scharrer

2021

Preprint

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A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below 8π. Combining such a strict inequality with previous works by Keller-Mondino-Rivière and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Moreover, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Furthermore, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case.

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“…A recent application of these methods to the minimization of a Willmore-type energy among disk type surfaces with clamped boundary data and constrained area is contained in [8]. Another recent application is [25].…”

Section: Introductionmentioning

confidence: 99%

On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves

Pozzetta

2021

ESAIM: COCV

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For a smooth closed embedded planar curve, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus greater than 1 having the given curve as boundary, without any prescription on the conormal. By general lower bound estimates, in case the curve is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and we calculate the infimum. Then we study the case in which the genus is 1 and the competitors are restricted to a suitable class of varifolds including embedded surfaces, and we prove that the non-existence of minimizers implies a lower bound on the infimum; therefore we use such criterion in order to explicitly ﬁnd an inﬁnite family of curves for which such problem does have minimizers in the corresponding class of varifolds.

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Existence and Regularity of Spheres Minimising the Canham–Helfrich Energy

Cited by 20 publications

References 34 publications

Lower semicontinuity for the Helfrich problem

Lower semicontinuity for the Helfrich problem

Embedded Delaunay tori and their Willmore energy

On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves

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