Abstract
We introduce a non-local L2-gradient flow for the Willmore energy of immersed surfaces which preserves the isoperimetric ratio. For spherical initial data with energy below an explicit threshold, we show long-time existence and convergence to a Helfrich immersion. This is in sharp contrast to the locally constrained flow, where finite time singularities occur.
2020 Mathematics Subject Classification:
1 Introduction and main results
Finding the shape which encloses the maximal volume among surfaces of prescribed area is certainly one of the oldest and yet most prominent problems in mathematics and goes back to the legend of the foundation of Carthage. Since then generations of mathematicians have been studying isoperimetric problems, aiming to find the best possible shape in all kinds of settings. It turns out that—by the isoperimetric inequality—the optimal configuration in Euclidean space is given by a round sphere.
Likewise, the round spheres are the absolute minimizers for the Willmore energy, a functional measuring the bending of an immersed surface with various applications also beyond geometry, for instance in the study of biological membranes [Citation1, Citation2], general relativity [Citation3], nonlinear elasticity [Citation4] and image restoration [Citation5].
Note that the round spheres describe the optimal shape in both situations. In this article, we will study their relation using a gradient flow approach.
For an immersion of a closed oriented surface Σ, its Willmore energy is defined by
Here denotes the area measure induced by the pull-back of the Euclidean metric , and denotes the (scalar) mean curvature with respect to , the unique unit normal along f induced by the chosen orientation on Σ, see Equation(2.1)(2.1) (2.1) . A related quantity is the umbilic Willmore energy, given by where A0 denotes the trace-free part of the second fundamental form. As a consequence of the Gauss–Bonnet theorem, these two energies are equivalent from a variational point of view, since for a surface with fixed genus g, we have (1.1) (1.1)
Both energies are not only geometric, i.e. invariant under diffeomorphisms on Σ, but—remarkably—also conformally invariant, i.e. invariant with respect to smooth Möbius transformations of . By [Citation6, Theorem 7.2.2], we have with equality if and only if and parameterizes a round sphere.
The isoperimetric ratio of an immersion is defined as the quotient (1.2) (1.2) denote the area and the algebraic volume enclosed by , respectively. Here, the normalizing constant is chosen such that by the isoperimetric inequality we always have with σ = 1 if and only if and parameterizes a round sphere. Critical points of the isoperimetric ratio—or equivalently, critical points of the volume functional with prescribed area—are precisely the CMC-surfaces, i.e. the surfaces with constant mean curvature, which form an important generalization of minimal surfaces and naturally arise in the modeling of soap bubbles.
The problem of minimizing the Willmore energy among all immersions of a genus g surface Σg with prescribed isoperimetric ratio, i.e. the minimization problem (1.3) (1.3) naturally arises in mathematical biology in the Canham–Helfrich model [Citation1, Citation2] with zero spontaneous curvature and has already been studied mathematically in [Citation7–9]. While the genus zero case was solved in [Citation7], the results in [Citation8, Citation9] combined with recent findings in [Citation10] and [Citation11] show that the infimum in Equation(1.3)(1.3) (1.3) is always attained for any and ; and satisfies . The energy threshold also plays an important role in the analysis of the Willmore energy, since by the famous Li–Yau inequality [Citation12], any immersion f of a compact surface with has to be embedded.
A sufficiently smooth minimizer in Equation(1.3)(1.3) (1.3) is a Helfrich immersion, i.e. a solution to the Euler–Lagrange equation (1.4) (1.4) where denotes the Laplace–Beltrami operator on . In [Citation13], solutions to Equation(1.4)(1.4) (1.4) with small umbilic Willmore energy have been classified, depending on the sign of the Lagrange-multipliers λ1 and λ2. We observe that for fixed, Equation(1.4)(1.4) (1.4) is also the Euler–Lagrange equation of the Helfrich energy given by (1.5) (1.5) where the energy either penalizes or favors large area or volume, depending on the sign of λ1 and λ2, respectively.
The L2-gradient flow of the Willmore energy was introduced and studied by Kuwert and Schätzle in their seminal works [Citation14–16].
Their methods are very robust and allow to handle also other situations, such as the surface diffusion flow [Citation17, Citation18] and the Willmore flow of tori of revolution [Citation19]. The locally constrained Helfrich flow, i.e. the L2-gradient flow for the energy Equation(1.5)(1.5) (1.5) , and its asymptotic behavior have been studied in [Citation20, Citation21], where it was shown that finite time singularities must occur below a certain energy threshold. However, this flow does not preserve the isoperimetric ratio.
The goal of this article is to discuss a dynamic version of the minimization problem Equation(1.3)(1.3) (1.3) . To this end, we introduce the Willmore flow with prescribed isoperimetric ratio, which decreases as fast as possible while keeping fixed. This yields the evolution equation (1.6) (1.6) where the Lagrange multiplier depends on and is given by (1.7) (1.7)
In Equation(2.9)(2.9) (2.9) we will justify the particular choice of λ, which yields that is actually preserved along a solution of Equation(1.6)(1.6) (1.6) –Equation(1.7)(1.7) (1.7) .
Definition 1.1.
Let and let Σg denote a connected, oriented and closed surface with genus . A smooth family of immersions satisfying Equation(1.6)(1.6) (1.6) with λ as in Equation(1.7)(1.7) (1.7) and is called a σ-isoperimetric Willmore flow with initial datum .
Stationary solutions of the flow Equation(1.6)(1.6) (1.6) –Equation(1.7)(1.7) (1.7) are solutions to the Helfrich Equationequation (1.4)(1.4) (1.4) for and . Conversely, any Helfrich immersion is also a stationary solution to Equation(1.6)(1.6) (1.6) –Equation(1.7)(1.7) (1.7) , see Lemma 2.6.
However, as the Lagrange multiplier λ defined in Equation(1.7)(1.7) (1.7) depends on the solution, the isoperimetric flow Equation(1.6)(1.6) (1.6) substantially differs from the L2-gradient flow of the Helfrich energy Equation(1.5)(1.5) (1.5) , where the parameters λ1 and λ2 are fixed numbers and chosen a priori. On the analytic side, the integral nature of the Lagrange multiplier makes the evolution Equationequation (1.6)(1.6) (1.6) a non-local, quasilinear, degenerate parabolic PDE of 4th order. Also geometrically, the constraint causes new difficulties, as we cannot control the area and the volume independently along the flow (as in [Citation21], for instance), but only the isoperimetric ratio .
The Willmore flow with a constraint on either the area or the enclosed volume has been studied in [Citation22] and a recent article by the author [Citation23]. However, the situation here is fundamentally different and several new challenges arise.
First, if only the area or the volume is prescribed (and nonzero), constrained critical points of the corresponding variational problem are in fact Willmore immersions, i.e. solutions of Equation(1.4)(1.4) (1.4) with , due to the scaling invariance of the Willmore energy. Although still an active field of research, the classification of these Willmore immersions is much better understood than that of general solutions of Equation(1.4)(1.4) (1.4) and a crucial ingredient in classifying the blow-ups in [Citation23]. Second, in [Citation23] the different scaling of the energy and constraint has been used to represent the Lagrange multiplier in a way that allows for good a priori estimates. This neat trick is clearly not available for the flow Equation(1.6)(1.6) (1.6) –Equation(1.7)(1.7) (1.7) . Third, unlike in [Citation23], the Lagrange multiplier has a much more complicated algebraic structure and cannot be treated as a lower order term.
These obstructions are the reason for a new energy threshold in the following main result on global existence and convergence.
Theorem 1.2.
Let be a smooth immersion with and such that . Then there exists a unique σ-isoperimetric Willmore flow with initial datum f0. This flow exists for all times and, as , it converges smoothly after reparametrization to a Helfrich immersion with solving Equation(1.4)(1.4) (1.4) with and .
This shows a fundamentally different behavior of the isoperimetric Willmore flow and the Helfrich flow, where finite time singularities occur, cf. [Citation20, Citation21]. Consequently, despite its new analytic challenges, the introduction of the non-local Lagrange multiplier has a regularizing effect on the gradient flow, see also [Citation24] for a related result for the mean curvature flow.
The -threshold in Theorem 1.2 is motivated by the following simple application of the triangle inequality in . With and Equation(1.2)(1.2) (1.2) , we have
This estimate bounds the denominator in Equation(1.7)(1.7) (1.7) from below if . Moreover, it allows to control the Lagrange multiplier in the crucial estimates by essentially lower order quantities, see Sections 4.
We highlight that the assumption in Theorem 1.2 is not an implicit smallness of the initial energy, cf. [Citation15, Citation18], but the threshold is explicitly given, although very little is known about minimizers and critical points of Equation(1.3)(1.3) (1.3) . Moreover, as , the interval of admissible initial energies in Theorem 1.2 becomes arbitrarily small. This seems plausible, since if σ = 1, f0 is a round sphere and the denominator in Equation(1.7)(1.7) (1.7) vanishes. Thus, it is a priori unclear whether there exists an admissible immersion f0 in Theorem 1.2 if —in fact, this is equivalent to the condition . In Theorem 7.1, we will prove for , which is asymptotically sharp as , and consequently the existence of a suitable f0 follows. We also point out that it is unknown if the energy threshold in Theorem 1.2 is optimal as it is for the classical Willmore flow [Citation19, Citation25].
The proof of Theorem 1.2 is based on the methods developed by Kuwert–Schätzle for the Willmore flow [Citation14–16]. Under a non-concentration assumption on the curvature, we use localized energy estimates to control the evolution, see Sections 3. However, as in [Citation23], these estimates depend on certain Lp-type bounds on λ. The key ingredient of this paper is that for locally small curvature and if the initial energy is below the threshold of Theorem 1.2, the Lagrange multiplier can be absorbed in the estimates, see Sections 4, in particular Lemmas 4.1 and 4.2. This is an essential observation, which we can use to prove a lower bound on the lifespan and to construct a blow-up limit in the spirit of [Citation15], see Sections 5. Using the control over the Lagrange multiplier in the energy regime of Theorem 1.2, we deduce a crucial rigidity result: either the blowup is a compact Helfrich immersion or a Willmore immersion, see Proposition 5.4. In the first case, we conclude global existence and convergence by an argument based on the Łojasiewicz–Simon inequality in the spirit of [Citation26], combined with recent progress on this inequality in the presence of constraints [Citation27]. Due to the rigidity of the blow-up, we can follow the inversion strategy in [Citation16] relying on the classification of compact Willmore spheres [Citation28] to exclude the second case.
This last step is also where we crucially make use of the assumption . In the case of higher genus, a classification result for Willmore surfaces as in [Citation28] is currently lacking. Even if such a classification were available, a precise comprehension of the behavior under inversion would be indispensable to extend the argument beyond the spherical case. However, since the blow-up analysis is also available if , we establish the following remarkable dichotomy result.
Corollary 1.3.
Let , let Σ be a closed, oriented and connected surface and suppose that is a maximal σ-isoperimetric Willmore flow such that . Then there exist and such that the sequence of immersions converges, as , smoothly on compact subsets of after reparametrization to a proper Helfrich immersion where is a complete surface without boundary. Moreover
if is compact, then and, as , the flow f converges smoothly after reparametrization to a Helfrich immersion as .
if is not compact, then is a Willmore immersion.
Hence, under the above assumptions, in the singular case (b) the influence of the (non-local) constraint vanishes after rescaling as and the purely local term in Equation(1.6)(1.6) (1.6) , coming from the Willmore functional, dominates.
We now outline the structure of this article. After a brief review of the most relevant analytic and geometric background in Sections 2, we start our analysis by carefully computing and estimating a localized version of the energy decay in Sections 3. In Sections 4, we control the Lagrange multiplier in the energy regime of Theorem 1.2 which then enables us to construct a blow-up limit in Sections 5. Finally, in Sections 6 we prove our convergence result, Theorem 1.2, and Corollary 1.3 before we show Theorem 7.1 in Sections 7, yielding that the set of admissible initial data in Theorem 1.2 is always non-empty.
2 Preliminaries
In this section, we will briefly review the geometric and analytic background and prove some first properties of the flow Equation(1.6)(1.6) (1.6) , see also [Citation29] for a more detailed discussion.
2.1 Geometric and analytic background
In the following, Σg always denotes an abstract compact, connected and oriented surface of genus without boundary.
An immersion induces the pullback metric on Σg, which in local coordinates is given by where denotes the Euclidean metric. The chosen orientation on Σg determines a unique smooth unit normal field along f, which in local coordinates in the orientation is given by (2.1) (2.1)
We will always work with this unit normal vector field.
The (scalar) second fundamental form of f is then given by and the mean curvature and the tracefree part of the second fundamental form are defined as where . Important relations are (2.2) (2.2) where K denotes the Gauss curvature. Consequently, using Equation(1.1)(1.1) (1.1) , we find (2.3) (2.3)
The Levi-Civita connection induced by the metric gf extends uniquely to a connection on tensors, which we also denote by . For an orthonormal basis of the tangent space, the Codazzi–Mainardi equations yield (2.4) (2.4) cf. [Citation15, (5)].
Clearly, potential singularities for the flow Equation(1.6)(1.6) (1.6) occur if becomes zero or if the denominator in Equation(1.7)(1.7) (1.7) vanishes. Note that in the latter case , thus f is a constant mean curvature immersion.
Lemma 2.1.
Let and let be an immersion with . Then
;
if g = 0, i.e. , or if f is an embedding, then . In particular, the denominator in Equation(1.7)(1.7) (1.7) is nonzero.
Proof.
The first statement follows immediately from the definition of . For (ii), we assume by contradiction that , so is an immersion with constant mean curvature. If , then f has to parameterize a round sphere by a result of Hopf [Citation30, Theorem 2.1, Chapter VI]. In the second case, f has to parameterize a round sphere by the famous theorem of Aleksandrov [Citation31]. In both cases this contradicts . □
Despite its geometric degeneracy, Equation(1.6)(1.6) (1.6) is still a parabolic equation. Thus, starting with a smooth nonsingular initial datum, it is possible to prove the following short-time existence result in similar fashion as it is outlined in [Citation32, Chapter 4, Proposition 2.1], after observing that we can integrate by parts in Equation(1.7)(1.7) (1.7) so that the numerator of the Lagrange-multiplier contains no second order derivatives of A any more.
Proposition 2.2.
Let be a smooth immersion with and . Then there exist and a unique, non-extendable σ-isoperimetric Willmore flow with initial datum .
If , assumption in Proposition 2.2 follows from by Lemma 2.1 (ii).
2.2 Evolution of geometric quantities
In this subsection, we will briefly review the variations of the relevant geometric quantities and energies.
Lemma 2.3.
[Citation23, Lemma 2.3] Let be a smooth family of immersions with normal velocity . For an orthonormal basis of the tangent space, the geometric quantities induced by f satisfy (2.5) (2.5) (2.6) (2.6) (2.7) (2.7)
As a consequence, we have the following first variation identities, cf. [Citation23, Lemma 2.4].
Proposition 2.4.
Let be an immersion and let . Then we have
Moreover, if , we have
Proof.
Since , and are invariant under orientation-preserving diffeomorphisms of Σg, we only need to consider normal variations, as any tangential variation corresponds to a suitable orientation-preserving family of reparametrizations (see for instance [Citation33, Theorem 17.8]), which leaves the quantities unchanged.
The variation of then follows immediately from Equation(2.5)(2.5) (2.5) . For and consider [Citation23, Lemma 2.4], for instance. The variation of then follows. □
The scaling behavior of the energies yields the following important identities.
Lemma 2.5.
Let be an immersion. Then we have
Proof.
By the scaling invariance of the Willmore energy, we find so Proposition 2.4 yields the claim. For and we may proceed similarly, using the scaling behavior for all . □
This yields that Helfrich immersions are precisely the stationary solutions of Equation(1.6)(1.6) (1.6) –Equation(1.7)(1.7) (1.7) .
Lemma 2.6.
Let be an immersion with and . Then f is a Helfrich immersion if and only if it is a stationary solution to the σ-isoperimetric Willmore flow.
Proof.
The “if” part of the statement is immediate. Suppose f is a Helfrich immersion. We multiply Equation(1.4)(1.4) (1.4) with , integrate and use Lemma 2.5 to conclude
By Lemma 2.1(i) we have . Hence, with , EquationEquation (1.4)(1.4) (1.4) reads (2.8) (2.8)
We have by Proposition 2.4, so by testing Equation(2.8)(2.8) (2.8) with and integrating it follows that λ is given as in Equation(1.7)(1.7) (1.7) , so f is indeed stationary. □
It is not difficult to see that along a solution of Equation(1.6)(1.6) (1.6) with , the isoperimetric ratio is indeed preserved, since by Proposition 2.4, Equation(1.6)(1.6) (1.6) and Equation(1.7)(1.7) (1.7) we have (2.9) (2.9)
On the other hand, the Willmore energy decreases since by Equation(2.9)(2.9) (2.9) (2.10) (2.10)
EquationEquations (2.9)(2.9) (2.9) and Equation(2.10)(2.10) (2.10) are the key features in studying the flow Equation(1.6)(1.6) (1.6) and of vital importance for our further analysis. We highlight two immediate consequences.
Remark 2.7.
The computation in Equation(2.10)(2.10) (2.10) implies that is a strict Lyapunov function along the flow Equation(1.6)(1.6) (1.6) , i.e. is strictly decreasing unless , so f is stationary (by uniqueness of the solution). By Equation(1.1)(1.1) (1.1) , this also holds for .
Since is monotone, the limit exists.
As Equation(1.6)(1.6) (1.6) is a (degenerate) parabolic equation, the scaling behavior in time and space is central in understanding the problem. Therefore, we gather the scaling behavior of some important quantities in the following lemma. The powers appearing in the time integrals below will naturally appear later in our energy estimates, see Proposition 3.3.
Lemma 2.8.
Let be a σ-isoperimetric Willmore flow and let r > 0. Let . Then
is a σ-isoperimetric Willmore flow;
the Lagrange multiplier of satisfies ;
and .
Proof.
Follows from the scaling behavior of the geometric quantities and a direct calculation. □
3 Localized energy estimates
As in [Citation15, Section 3] and [Citation23, Sections 2.3 and 3], we will start our analysis by localizing the energy decay Equation(2.10)(2.10) (2.10) . The main goal of this section is to show that all derivatives of A can be bounded along the flow, if the energy concentration and a suitable time integral involving the Lagrange multiplier are controlled. Note that at this stage, we do not yet need to assume or any restriction on the initial energy.
Lemma 3.1.
Let and let be a σ-isoperimetric Willmore flow. Let and define . Then we have and
Proof.
This computation is very similar to [Citation32, Chapter 4, Lemma 2.8] (see also [Citation15, Section 3]) if one replaces with , so we will focus on the differences. We will use a local orthonormal frame for our computations and find (3.1) (3.1) writing . Moreover, we have (3.2) (3.2)
If we carefully combine the terms with λ in Equation(3.1)(3.1) (3.1) and Equation(3.2)(3.2) (3.2) , the claim follows after integrating by parts, where the terms involving derivatives of H and the factor cancel.
For the second identity, arguing similarly as in [Citation32, Chapter 4, Lemma 2.8] we have
Integrating by parts and using Equation(2.4)(2.4) (2.4) we conclude
Now, using integration by parts and Equation(2.4)(2.4) (2.4) once again, we have
The claim follows. □
We will now carefully estimate the integrals in Lemma 3.1. To this end, we choose a particular class of test functions. Let with and assume for some . Then setting (3.3) (3.3) and note that has compact support in Σg, which is compact, for all , see also [Citation23, Equation(3.1)(3.1) (3.1) ].
For the rest of this article, we denote by C a universal constant with which may change from line to line.
Lemma 3.2.
Let , let be a σ-isoperimetric Willmore flow and let γ be as in Equation(3.3)(3.3) (3.3) . Then we have
Proof.
We have for any by a direct computation in a local orthonormal frame. Hence, using Lemma 3.1 and , cf. Equation(2.2)(2.2) (2.2) , we find
The terms and can be estimated as in [Citation15, Lemma 3.2]. Since we have by Equation(3.3)(3.3) (3.3)
Consequently, we find
Choosing small enough, the claim follows from the estimates above. □
Note that on the right hand side of Lemma 3.2, terms involving the Lagrange multiplier multiplied with powers of A up to 4-th order and even second derivatives of H appear. With the energy, we can only control the L2-norms of H and A. In the following Proposition 3.3 we will close this gap by using higher powers of the Lagrange multiplier, the area and the volume, see also [Citation23, Proposition 3.3]; these powers behave correctly under rescaling, cf. Lemma 2.8. We will combine this with the interpolation techniques from [Citation14, Citation15] to get control on the local -norm of A, in terms of the (localized) Willmore gradient, at least if the L2-norm of A is locally small.
Proposition 3.3.
There exist universal constants with the following property: Let , let be a σ-isoperimetric Willmore flow and let γ be as in Equation(3.3)(3.3) (3.3) . If we have (3.4) (3.4) then at time t we can estimate
Here .
Proof.
Using the assumption and the interpolation inequality in [Citation15, Proposition 2.6] (see also [Citation23, Proposition 3.2]), we have at time
Consequently, from Lemma 3.2, we find for some (3.5) (3.5)
For the first term on the right hand side of Equation(3.5)(3.5) (3.5) , we infer using Young’s inequality (3.6) (3.6)
The second term on the right hand side of Equation(3.5)(3.5) (3.5) can be estimated by using Young’s inequality with p = 4 and and to obtain (3.7) (3.7)
Moreover, we have the estimate using Young’s inequality. Combining this with Equation(3.5)(3.5) (3.5) , Equation(3.6)(3.6) (3.6) , and Equation(3.7)(3.7) (3.7) and choosing sufficiently small, the claim follows. □
Assumption (3.4) means that the second fundamental form is small on the support of γ. Note that this will only be satisfied locally, since by Equation(2.3)(2.3) (2.3) we always have . We will now study the situation, where Equation(3.4)(3.4) (3.4) is satisfied on all balls with a certain radius, yielding a control over the concentration of the Willmore energy in . Following [Citation16] we introduce the following notation.
Definition 3.4.
For a smooth family of immersions , r > 0, we define the curvature concentration function (3.8) (3.8)
Here and in the rest of this article, we follow the notation of [Citation14], i.e. the integrals over balls have to be understood over the preimages under ft.
If denotes the minimal number of balls of radius 1/2 necessary to cover , then (3.9) (3.9)
We now prove an integrated form of Proposition 3.3.
Proposition 3.5.
Let be as in Proposition 3.3. There exist universal constants with the following property: Let , let be a σ-isoperimetric Willmore flow and let be such that (3.10) (3.10)
Then for all and we have
Proof.
Fix . Let be a cutoff function with and . Therefore, is as in Equation(3.3)(3.3) (3.3) with . Moreover, if we take small enough, we have the estimate (3.11) (3.11) as a consequence of Simon’s monotonicity formula [Citation34], see also [Citation23, Lemma 4.1]. Now, since we have by Equation(2.9)(2.9) (2.9) and Equation(1.2)(1.2) (1.2) , we observe (3.12) (3.12) where we used Young’s inequality (with and q = 3). The statement then immediately follows by integrating Proposition 3.3 in time. □
Remark 3.6.
If we directly integrate Proposition 3.3, we have to deal with two terms involving λ, both of whose time integrals behave correctly under parabolic rescaling, cf. Lemma 2.8. The estimate Equation(3.12)(3.12) (3.12) above reveals that if Equation(3.10)(3.10) (3.10) is satisfied, then it suffices to control merely the -term, since (3.13) (3.13)
For the blow-up construction in Sections 5, we will need the following higher order estimates for the flow in the case of non-concentrated curvature, cf. [Citation15, Theorem 3.5], [Citation23, Proposition 3.5].
Proposition 3.7.
Let and let be a σ-isoperimetric Willmore flow. Suppose is chosen such that for some and (3.14) (3.14) where is as in Proposition 3.5. Moreover, assume (3.15) (3.15)
Then for all and we have the local estimates and the global bounds (3.16) (3.16)
In contrast to [Citation15, Theorem 3.5] and [Citation23, Proposition 3.5], we do not only prove local bounds, but also the global L2-control Equation(3.16)(3.16) (3.16) . Note that the global L2-norms could also be estimated by the -bounds and the area. However, this is disadvantageous since the area cannot be controlled along the flow, and in fact is always expected to diverge in the blow-up process, cf. Lemma 6.1. The necessity for the finer estimates leading to Equation(3.16)(3.16) (3.16) is why we give full details on the proof here, even though the argument is very similar to [Citation15, Theorem 3.5].
Proof of Proposition 3.7.
After parabolic rescaling, cf. Lemma 2.8, we may assume ρ = 1. Let and define and . Then, for all using that by Equation(3.14)(3.14) (3.14) , we deduce from Proposition 3.5 (3.17) (3.17)
Moreover, as by Equation(3.14)(3.14) (3.14) we can interpolate by combining [Citation15, Lemma 2.8] and [Citation14, Lemma 4.2] to find where we used the assumptions (3.14), (3.15) and in the last step. Thus, defining and using and Equation(3.15)(3.15) (3.15) , we have the estimate (3.18) (3.18)
Now, we pick with and . Note that Equation(3.3)(3.3) (3.3) is satisfied with a universal , which we do not keep track of. As in [Citation15, Theorem 3.5], we define Lipschitz cutoff functions in time via where and . We also define and for all if m = 0. We note that and (3.19) (3.19)
Furthermore, for we define . Then, by Proposition B.3, using and Equation(3.13)(3.13) (3.13) we have
Therefore, if we define this implies using Equation(3.19)(3.19) (3.19) and (3.20) (3.20)
We now claim that for all and we have (3.21) (3.21)
We proceed by induction on j. For j = 0 we have on . Therefore, we clearly have . Moreover, by Equation(3.17)(3.17) (3.17) we find
For , integrating Equation(3.20)(3.20) (3.20) on and using , we find by the induction hypothesis and since . Using Gronwall’s inequality and estimating the exponential term by Equation(3.18)(3.18) (3.18) , we find
The estimate Equation(3.21)(3.21) (3.21) then follows by using Equation(3.18)(3.18) (3.18) and estimating the double integral via where we also used Equation(3.18)(3.18) (3.18) once again. Now, for the local estimates, we evaluate Equation(3.21)(3.21) (3.21) at and use Equation(3.14)(3.14) (3.14) , and Equation(3.18)(3.18) (3.18) . Recalling by Equation(3.14)(3.14) (3.14) , this yields (3.22) (3.22)
For the global L2-estimate, we observe that Equation(3.21)(3.21) (3.21) is linear in ej and K. Hence, as in [Citation23, Proposition 4.2], we can sum up the local bounds to get (3.23) (3.23) where we used Equation(2.3)(2.3) (2.3) , Equation(2.10)(2.10) (2.10) , and Equation(3.18)(3.18) (3.18) . After renaming T into t, Equation(3.22)(3.22) (3.22) and Equation(3.23)(3.23) (3.23) are precisely the desired L2-estimate for even orders of derivatives. Exactly as in [Citation15, Theorem 3.5], the local and global L2-estimates for follow by interpolation. The -estimate can then be deduced as in [Citation15, Theorem 3.5] as well. □
4 Controlling the Lagrange multiplier
In this section, we will provide some important estimates for the Lagrange multiplier under the assumption that the initial energy is not too large. In contrast to [Citation23], the crucial power of λ is not of lower order when compared to the left hand side of Proposition 3.3. Nevertheless, a first immediate feature of the energy regime from Theorem 1.2 is that we can uniformly bound the denominator of λ from below.
Lemma 4.1.
Let and let be a σ-isoperimetric Willmore flow with . Then
Proof.
This follows from the reverse triangle inequality in , Equation(2.9)(2.9) (2.9) and Equation(2.10)(2.10) (2.10) . □
While the scaling techniques from [Citation23, Lemma 4.3] are not available here, we still get the following key estimate, which gives a control over λ by quantities which will be suitably integrable.
Lemma 4.2.
Let and let be a σ-isoperimetric Willmore flow with . Then, we have
Proof.
We test the evolution Equationequation (1.6)(1.6) (1.6) with the normal ν and integrate to obtain where we used the divergence theorem. We now estimate the prefactor of λ by using the fact that by Equation(2.9)(2.9) (2.9) . By the assumption and Equation(2.10)(2.10) (2.10) this is strictly positive and the claim follows. □
We remark that the existence of with satisfying the assumption is not yet known and—in general—not true. For the case g = 0, this will follow from Theorem 7.1. However, for tori we have by [Citation35], and hence can only hold for . On the other hand, for and arbitrary genus, there exists f0 with since by [Citation11, Theorem 1.2]. We now use Lemma 4.2 to deduce the time integrability Equation(3.15)(3.15) (3.15) for λ in the case of small curvature concentration, which enables us to bound all derivatives of the second fundamental form by Proposition 3.7.
Lemma 4.3.
Let and let be a σ-isoperimetric Willmore flow. Let and let be such that where is as in Proposition 3.5. Then for all we have
Note that by the invariance of the Willmore energy and the isoperimetric ratio, this estimate is preserved under parabolic rescaling, cf. Lemma 2.8.
Proof of Lemma 4.3.
By the assumption we get the local control from Proposition 3.5. As in [Citation23, Proposition 4.2], we can sum up these local bounds to get the global estimate
Now, by Equation(2.3)(2.3) (2.3) , the energy decay Equation(2.10)(2.10) (2.10) and the assumption, we have (4.1) (4.1)
Thus, we obtain the estimate (4.2) (4.2)
By Equation(2.2)(2.2) (2.2) we have . Therefore, using Lemma 4.2 we find (4.3) (4.3) by Cauchy–Schwarz and Equation(4.1)(4.1) (4.1) , where . For the first term in Equation(4.3)(4.3) (4.3) , by Equation(2.10)(2.10) (2.10) and Equation(1.1)(1.1) (1.1) we have For the second term in Equation(4.3)(4.3) (4.3) , we use Equation(3.11)(3.11) (3.11) , Cauchy–Schwarz in time and space, Equation(4.1)(4.1) (4.1) and then Equation(4.2)(4.2) (4.2) to find for every by Young’s inequality, and estimating in the last step. The statement then follows from Equation(4.3)(4.3) (4.3) by taking sufficiently small. □
5 The blow-up and its properties
In this section, we will rescale an isoperimetric Willmore flow as we approach the maximal existence time to obtain a limit immersion. Analyzing the properties of this limit will be the keystone in proving our main result, Theorem 1.2.
5.1 A lower bound on the existence time
As in [Citation14] and [Citation23], the first step is to prove a lower bound on the existence time of an isoperimetric flow which respects the parabolic rescaling in Section 5.2.
To that end, we state a general lifespan result for possible future reference, where the lower bound only depends on the radius of concentration ρ, the isoperimetric ratio σ and the behavior of the L2-norm of near t = 0.
Proposition 5.1.
Let be as in Proposition 3.5. There exist universal constants nd with the following property: Let be an immersion with and . Let f be the σ-isoperimetric Willmore flow with initial datum f0. Assume that
for some ;
there exists with the following property: For any with for all , we have .
Then the maximal existence time of the flow satisfies for some and (5.1) (5.1)
Note that we always have . The crucial insight here is that only the decay behavior of the L2-norm of under the assumption of small concentration allows control on the existence time in a way which transforms correctly under parabolic rescaling.
Proof of Proposition 5.1.
Without loss of generality, we may assume ρ = 1, otherwise we rescale as in Lemma 2.8, see also [Citation23, Proposition 3.5]. Let T denote the maximal existence time of the flow and let to be chosen. We define where is as in Proposition 3.3 and is as in Equation(3.9)(3.9) (3.9) and set for . By compactness of for t < T, the supremum in the definition of in Equation(3.8)(3.8) (3.8) is always attained and the function is continuous with by (a).
For a parameter , to be chosen later, we now define (5.2) (5.2)
By continuity of and (a), we have . For , we have by Equation(5.2)(5.2) (5.2) and the definition of . Hence, by Proposition 3.5 and assumption (b) we find (5.3) (5.3) for all where from Proposition 3.5. Now, if we choose and we find from Equation(5.3)(5.3) (5.3) (5.4) (5.4)
However, if , together with Equation(3.9)(3.9) (3.9) , this implies for all by our choice of . On the other hand, by Equation(5.2)(5.2) (5.2) and continuity, we must have , a contradiction.
Consequently, has to hold. Assume . Then, as before, from Equation(5.4)(5.4) (5.4) and Equation(3.9)(3.9) (3.9) we find by the definition of . As by assumption and by (b), we can apply Proposition 3.7 to conclude that for any we have (5.5) (5.5) and . Consequently, for all we can estimate (5.6) (5.6) using Equation(1.7)(1.7) (1.7) , Cauchy-Schwarz, Equation(2.2)(2.2) (2.2) and Lemma 4.1. Similarly, we find (5.7) (5.7) where we used and Equation(5.5)(5.5) (5.5) . Exactly with the same arguments as in [Citation14, pp. 330–332] (see also [Citation32, Chapter 4, Proof of Theorem 1.1 after Equation(5.8)(5.8) (5.8) ]), we can deduce that f(t) smoothly converges to a smooth immersion f(T) as . By assumption and the energy decay, we infer from Lemma 4.1 that the denominator in Equation(1.7)(1.7) (1.7) is bounded away from zero for all , so f(T) is not a constant mean curvature immersion. By Proposition 2.2, we can then restart the flow with initial datum f(T) which contradicts the maximality of T.
Hence, has to hold. The estimate Equation(5.1)(5.1) (5.1) then follows from Equation(5.4)(5.4) (5.4) and Equation(3.9)(3.9) (3.9) after choosing . □
Together with the integral estimate for the Lagrange multiplier in Lemma 4.3, this now implies the following
Proposition 5.2
(Lifespan bound for small energy gap). Let , let be a maximal σ-isoperimetric Willmore flow such that
;
, where is as in Proposition 5.1;
, where .
Then the maximal existence time is bounded from below by where and for all we have .
Note that the limit in (iii) exists due to Remark 2.7 (ii).
Proof of Proposition 5.2.
We check that the assumptions in Proposition 5.1 are satisfied. Let be as in Proposition 5.1. Assumption (a) of Proposition 5.1 holds true by assumption (ii). We now verify assumption (b) in Proposition 5.1. To that end, let to be chosen and assume that for some we have for all . By (i), we may apply Lemma 4.3 and use (iii) to find the estimate if we choose and small enough. The assumptions of Proposition 5.1 are thus fulfilled and the result follows with . □
5.2 Existence of a blow-up
In this section, we will rescale as we approach the maximal existence time of a σ-isoperimetric Willmore flow with . To that end, let be arbitrary. By translation invariance and Lemma 2.8 for all the flow (5.8) (5.8) is also a σ-isoperimetric Willmore flow with initial datum and maximal existence time . Throughout this section, we will denote all geometric quantities of the flow fj with a subscript j, such as for example. The next lemma guarantees the existence of suitable tj, rj and xj.
Lemma 5.3.
Let and let be a maximal σ-isoperimetric Willmore flow with . Let be as in Proposition 5.2. Then, there exist sequences and such that for all we have
;
for all , where is as in Proposition 5.1;
.
Proof.
Given any , with essentially the same arguments as in [Citation23, Lemma 6.6], one finds a radius such that (5.9) (5.9) where . One then argues as in [Citation16, p. 349] (see also [Citation23, Proposition 6.7]), to prove the existence of and such that choosing , we find that (iii) is satisfied.
Now, the flow fj satisfies by Equation(5.9)(5.9) (5.9) and since . Moreover, by the invariances of the Willmore energy we have for all and
Consequently, for j sufficiently large, we can apply Proposition 5.2, to find that the maximal existence time of the flow fj is bounded from below by which proves (i) and for all by Equation(5.1)(5.1) (5.1) which proves (ii). □
Proposition 5.4
(Existence and properties of the limit immersion). Let and suppose is a maximal σ-isoperimetric Willmore flow with . Let and be as in Lemma 5.3. Then, there exists a complete, orientable surface without boundary and a proper immersion such that, after passing to a subsequence, and
as smoothly on compact subsets of , after reparametrization;
we have and ;
is a Helfrich immersion, i.e. a solution to Equation(1.4)(1.4) (1.4) ;
if , then is a Willmore immersion.
Any Helfrich immersion which arises from the process described above is called a concentration limit. More precisely, we call a blow-up if , a blow-down for and a limit under translation if . Note that by Lemma 5.3 (i) the last two can only occur if .
We highlight that Proposition 5.4 (iv) is particularly remarkable, since it means that under the assumption of diverging area, the constraint vanishes in the concentration limit, see also [Citation23, Theorem 6.2] for a similar rigidity result. This will be essential in the proof of Theorem 1.2.
Proof of Proposition 5.4.
After passing to a subsequence, we may assume in . We have and by Proposition 5.1 and hence by Lemma 5.3 (ii) we find for all . We may thus use Lemma 4.3 to bound for all and for all . Consequently, using Proposition 3.7 we conclude that for any we have (5.10) (5.10) (5.11) (5.11)
Moreover, from Simon’s monotonicity formula, cf. [Citation34, Equation(1.3)(1.3) (1.3) ], for any R > 0 we find
Thus, we may apply the localized version of Langer’s compactness theorem ([Citation15, Theorem 4.2], see also [Citation23, Appendix A]) to the sequence of immersions . After passing to a subsequence, we thus find a proper limit immersion , where is a complete (possibly empty) surface without boundary, diffeomorphisms , where are open sets and , and functions such that we have as well as as for all , so (i) is proven.
Moreover, sending in Lemma 5.3 (iii) and using the smooth convergence on compact subsets, it follows and hence in particular . The second statement in (ii) follows from the scaling invariance and the lower semicontinuity of the Willmore energy with respect to smooth convergence on compact subsets of , see [Citation19, Appendix B] for instance.
Let be arbitrary. Using Equation(5.11)(5.11) (5.11) and arguing as in Equation(5.6)(5.6) (5.6) and Equation(5.7)(5.7) (5.7) , we find (5.12) (5.12) which when combined with Equation(1.6)(1.6) (1.6) and Equation(5.10)(5.10) (5.10) immediately yields (5.13) (5.13)
Now, as a consequence of Lemma B.1, we find (5.14) (5.14) using Equation(5.10)(5.10) (5.10) , Equation(5.11)(5.11) (5.11) , and Equation(5.12)(5.12) (5.12) . Similarly, using Lemma B.2 instead we obtain (5.15) (5.15)
We will now use this to bound the derivative of the Lagrange multiplier. To that end, we observe that using and integration by parts, we find
Note that by Lemma 4.1 the denominator is bounded from below by some . Using Equation(2.5)(2.5) (2.5) , Equation(5.10)(5.10) (5.10) ,(5.11), Equation(5.13)(5.13) (5.13) , Equation(5.14)(5.14) (5.14) , and Equation(5.15)(5.15) (5.15) , by direct computation we find (5.16) (5.16)
Now, using and Equation(2.5)(2.5) (2.5) we infer
Since for , we can apply Equation(3.11)(3.11) (3.11) with ρ = 1 to obtain and hence using Equation(5.16)(5.16) (5.16) , Equation(5.12)(5.12) (5.12) , Equation(5.13)(5.13) (5.13) , and Equation(2.10)(2.10) (2.10) we have
For , we now define the flows and observe that they satisfy the -estimates Equation(5.10)(5.10) (5.10) with instead of Aj and the evolution equation (5.17) (5.17)
As in [Citation23, Proof of Theorem 6.2], the estimates for together with the C1-estimates for and can then be used to deduce that, after passing to a subsequence, the flows converge in for all and all compact to a limit flow . Moreover, we may assume in for all and all compact, where is a smooth normal vector field along for all , as well as
Now, let be a fixed compact set and let be large enough. Then, using Equation(5.17)(5.17) (5.17) , Equation(2.9)(2.9) (2.9) , and Equation(2.10)(2.10) (2.10) we find
In particular, taking and using in for all , we find by Remark 2.7 (ii)
Consequently, we have in for all and compact.
We observe that is a global and smooth normal vector field along and hence is orientable. Setting and using Equation(5.17)(5.17) (5.17) we find so is a Helfrich immersion and (iii) is proven.
For (iv), we now assume as . By Lemma 4.2, for all and , we have by Cauchy–Schwarz (5.18) (5.18) where we estimated for all , using Equation(5.10)(5.10) (5.10) and Equation(2.10)(2.10) (2.10) . Moreover, as a consequence of Equation(2.5)(2.5) (2.5) and Equation(2.10)(2.10) (2.10) , for all we have so that for all and hence the last term on the right hand side of Equation(5.18)(5.18) (5.18) goes to zero as . Since , the first term in Equation(5.18)(5.18) (5.18) converges to zero for . Consequently so that in particular, . Moreover, as , from Equation(1.2)(1.2) (1.2) we obtain so and thus is a Willmore immersion. □
5.3 The constrained Łojasiewicz–Simon inequality
In this section, we establish a Łojasiewicz–Simon inequality [Citation36–38]. While the unconstrained Willmore energy satisfies such an inequality [Citation26], the constraint of fixed isoperimetric ratio requires us to prove a refined estimate. To that end, we rely on the general framework of constrained or refined Łojasiewicz–Simon inequalities on submanifolds of Banach spaces [Citation27], see also [Citation32, Chapter 1, Section 1.2].
Theorem 5.5
(Constrained Łojasiewicz–Simon inequality). Let be a Helfrich immersion with such that . Then, there exist and such that for all immersions with and we have
The proof of this result is very similar to [Citation23, Section 7.1], so we will only provide full details on the differences. Throughout this section we will fix a smooth immersion with . The normal Sobolev spaces along f are for , with . Here, the L2-inner product always has to be understood with respect to the measure μf and denotes the normal projection along f, given by for any vector field X along f.
Let r > 0 be sufficiently small and let
Consider the shifted energies, defined by
Note that this is well-defined, since is an immersion for all with r > 0 small enough, cf. [Citation23, Lemma 7.5 (i)]. The first main ingredient toward proving Theorem 5.5 is the analyticity of the energy and the constraint.
Lemma 5.6.
For r > 0 small enough, the following maps are analytic.
the function ;
the function , where ;
the function ;
the function .
Proof.
Statements (i) and (ii) are exactly as in [Citation23, Lemma 7.6 (ii) and (iii)]. By [Citation23, Lemma 7.6 (i) and (iv)], the maps and are analytic and hence so is I by definition of the isoperimetric ratio and since for all . For statement (iv) recall from Proposition 2.4 that for we have
We note that is analytic by [Citation23, Lemma 7.5 (ii)] and is analytic by [Citation26, Lemma 3.2 (iv)]. We have for all and by continuity for r > 0 sufficiently small, since . This implies (iv). □
We now compute the first and second variations of W and I in terms of their H-gradients, see [Citation27, Section 5].
Lemma 5.7.
Let and let r > 0 be sufficiently small. For each , the H-gradients of W and I are given by
Moreover, the Fréchet-derivatives of the H-gradient maps of W and I at u = 0 satisfy
Proof.
For , we have by Proposition 2.4
Similarly, the statement for can be shown. The Fredholm property of follows from Equation(1.1)(1.1) (1.1) and [Citation26, Lemma 3.3 and p. 356]. For the last statement, we observe that for all we have . Hence, using Equation(2.6)(2.6) (2.6) with and Proposition 2.4 we find where we used and by Equation(2.5)(2.5) (2.5) . Since this only involves terms of order two or less in , the claim follows from the Rellich–Kondrachov Theorem, see for instance [Citation39, Theorem 2.34]. □
Proof of Theorem 5.5.
From the assumption , it follows that and hence . As in [Citation23, Proposition 7.4], we can thus apply [Citation27, Corollary 5.2] to deduce that Theorem 5.5 is satisfied in normal directions, i.e. for the functional W with the constraint . With the methods from [Citation26, p. 357], one can then use the invariance of the energies under diffeomorphisms to conclude that Theorem 5.5 holds in all directions. □
As in [Citation26, Lemma 4.1], the Łojasiewicz–Simon inequality yields the following asymptotic stability result, see also [Citation23, Lemma 7.9] and [Citation40, Theorem 2.1] for related results in the context of constrained gradient flows in Hilbert spaces.
Lemma 5.8.
Let be a Helfrich immersion with . Let . Then there exists such that if is a σ-isoperimetric Willmore flow satisfying
for some ;
whenever for diffeomorphisms ;
Note that by Lemma 2.6, fW above is a stationary solution to Equation(1.6)(1.6) (1.6) –Equation(1.7)(1.7) (1.7) . Consequently, the proof of Lemma 5.8 is a straightforward adaptation of [Citation23, Lemma 7.9], applying our Łojasiewicz–Simon inequality in Theorem 5.5 and can be safely omitted. As an important consequence one then finds the following convergence result by following the lines of [Citation26, Section 5] (see also [Citation23, Theorem 7.1]), which yields that in the case where is compact, below the explicit energy threshold no blow-ups or blow-downs may occur.
Theorem 5.9.
Let , let be a maximal σ-isoperimetric Willmore flow with and let be a concentration limit as in Proposition 5.4. If has a compact component and , then is a limit under translation. Moreover, the flow exists for all times, i.e. , and, as , converges smoothly after reparametrization to a Helfrich immersion with .
Proof.
Let and be as in Proposition 5.4. By arguing as in [Citation15, Lemma 4.3], we may assume and, by Proposition 5.4 (i), we hence have smoothly on Σg, where are diffeomorphisms. Let be as in Lemma 5.8. Then for some fixed sufficiently large and any , we may assume . Moreover, the σ-isoperimetric Willmore flow satisfies . Using Equation(2.10)(2.10) (2.10) and the invariance of the Willmore energy, for any , we find from Remark 2.7 (ii) where we used the smooth convergence in the last step. Also note that we have . Thus, by Lemma 5.8 the flow exists globally and, as , converges smoothly after reparametrization by appropriate diffeomorphisms to a Helfrich immersion with , so by Equation(1.1)(1.1) (1.1) . Consequently, and for all we have as smoothly on Σg. It remains to prove . To that end, we choose times for , such that we find as , since . We thus obtain
Consequently, the diameters converge, so as , whence since Σg is compact.
On the other hand, since smoothly and the limit exists, and consequently has to hold. □
6 Convergence for spheres
The goal of this section is to prove Theorem 1.2. To that end, we want to use the fact that compactness of the concentration limit yields convergence of the flow by Theorem 5.9. We first note that the desired compactness follows, if the area along the sequence in Proposition 5.4 remains bounded.
Lemma 6.1.
Let , let be a maximal σ-isoperimetric Willmore flow with and let be as in Proposition 5.4. If , then is compact.
Proof.
By Lemma 5.3 (iii), we have for all , where with fj as in Equation(5.8)(5.8) (5.8) . We now use the diameter bound [Citation34, Lemma 1.1] to estimate , such that using the assumption, the invariances of the Willmore energy and the energy decay Equation(2.10)(2.10) (2.10) , we find . Consequently, there exists such that for all . Letting and using Proposition 5.4 (i) and the definition of smooth convergence on compact subset of , one then easily deduces and then, since is proper, compactness of . □
We will now use Lemma 6.1 and Proposition 5.4 to conclude that if the concentration limit is non-compact, then it is not only a Helfrich, but even a Willmore immersion. In the spherical case, the classification in [Citation28] and the inversion strategy from [Citation16] will then yield a contradiction. Combined with Theorem 5.9, this will prove our main result.
Proof of Theorem 1.2.
Since and , the existence of a unique, non-extendable σ-isoperimetric Willmore flow with initial datum f0 follows from Proposition 2.2 and Lemma 2.1 (ii). Moreover, by Remark 2.7 (i), the Willmore energy strictly decreases unless the flow is stationary, in which case global existence and convergence to a Helfrich immersion follow trivially. Thus, we may assume .
Let be a concentration limit as in Lemma 5.4. If is compact, we find by [Citation15, Lemma 4.3] and long-time existence and convergence follow from Theorem 5.9 and the fact that by Lemma 2.1 (ii).
For the sake of contradiction, we assume that is not compact. Then we may assume by Lemma 6.1. Consequently, by Proposition 5.4 we find that is a Willmore immersion with . The rest of the argument is as in [Citation23, Proof of Theorem 1.2]: Denote by I the inversion in a sphere with radius 1 centered at and let . Then is compact. By [Citation16, Lemma 5.1], is a smooth Willmore sphere with and hence, using Bryant’s classification result [Citation28], has to be a round sphere. Thus, has to be either a plane or a sphere. Since is non-compact by assumption, this yields that has to parametrize a plane, a contradiction to Proposition 5.4 (ii).
Now the limit immersion satisfies and solves Equation(1.4)(1.4) (1.4) for some . It remains to prove . Arguing as in the proof of Lemma 2.6, we infer
Now, by Lemma 2.1 (i) as and also . Consequently, if one of is zero, then so is the other. In this case is a Willmore sphere with . By Bryant’s result [Citation28], it then has to be a round sphere, so , a contradiction and hence . □
Corollary 1.3 is an immediate consequence of the previous results.
Proof of Corollary 1.3.
By the assumption on the initial energy, Proposition 5.4 yields the existence of a suitable blow-up sequence and a concentration limit with the desired properties. If has constant mean curvature , using Equation(2.2)(2.2) (2.2) EquationEquation (1.4)(1.4) (1.4) reads
If is compact, we conclude and hence also has to be constant. But then has to parametrize a round sphere (see for instance [Citation30, Chapter V.Citation1]), a contradiction to . Therefore, statement (a) follows from Theorem 5.9. If is not compact, we may assume by Lemma 6.1 after passing to a subsequence. In this case, is a Willmore immersion by Proposition 5.4 (iv), yielding statement (b). □
7 An upper bound for β0
In this section, we will prove an upper bound for the minimal Willmore energy of spheres with isoperimetric ratio .
Theorem 7.1.
For every we have .
We remark that this estimate becomes sharp for since . On the other hand for , the statement follows since by [Citation7, Lemma 1] we have for all . We will prove Theorem 7.1 by comparing energy and isoperimetric ratio of an ellipsoid. To that end, for , we define the half-ellipse in the y-z-plane in . By rotating the curve ca around the z-axis we obtain a particular type of ellipsoid, a prolate spheroid. More explicitly, we define
Fortunately, its area, volume and also its Willmore energy can be explicitly computed without the use of elliptic integrals.
Lemma 7.2.
Let . Then we have
;
;
.
Proof.
(i) and (ii) are standard formulas, see for instance [Citation41, Section 4.8]. For (iii), we observe that the mean curvature and the surface element of fa are given by by standard formulas for surfaces of revolution, see for instance [Citation42, Section 3C]. In order to compute the Willmore energy, we thus have to evaluate the integral
Substituting , this integral can then be explicitly computed yielding (iii). □
Proof of Theorem 7.1.
Clearly, we have . Moreover, by Lemma 7.2 and a short computation we have (7.1) (7.1)
An elementary computation yields as and similarly as . Consequently, we have by a continuity argument.
Now, by Equation(7.1)(7.1) (7.1) , we find for all where the function F is negative for by Lemma 7.3. □
Lemma 7.3.
The function defined by satisfies F(a) < 0 for all .
We will prove Lemma 7.3 in Sections A. A quick glimpse at the plot of F in illustrates that the statement of Lemma 7.3 is true. However, a rigorous proof seems to be surprisingly difficult, since the function combines trigonometric functions with polynomials and its graph becomes very flat near .
Acknowledgments
The author would like to thank Anna Dall’Acqua for helpful discussions and comments. In addition, the author is grateful to the referees for their careful reading and their valuable comments on the original manuscript.
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The author reports there are no competing interests to declare.
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A
Proof of Lemma 7.3
This section is devoted to proving Lemma 7.3. The idea is to make a change of variables, such that the problem is equivalently formulated in terms of a polynomial in and . Then, we use the power series representation of the Cosine and Sine functions, to reduce the problem to the question if a certain polynomial has roots in a given interval. This last point can then be discussed by studying the Sturm chain of the polynomial.
Proof of Lemma 7.3.
For we consider the function , so that expanding we find
We observe that F(a) < 0 for all is equivalent to G(x) < 0 for all .
Using the power series expansion of the Cosine and Taylor’s theorem with the Lagrange form of the remainder, for any we infer for some . An induction argument yields , so the remainder has a sign, depending on the parity of N. Hence, denoting , we infer (A.1) (A.1)
By similar arguments, defining and using we find (A.2) (A.2)
We will now use Equation(A.1)(A.1) (A.1) and Equation(A.2)(A.2) (A.2) to estimate G. For , we have
Now, we observe that k(x) is polynomial of degree 27. Using Mathematica, we find that this can be simplified to for the degree 18 polynomial
By substituting , in order to prove G(x) < 0 for it thus suffices to show that for all . To this end, one may compute the Sturm chain of the polynomial p (see [Citation43, Theorem 8.8.15] for instance), to find that there exist no real roots of p in the interval . Consequently, since , we find p(z) < 0 for all , and hence the claim follows. □
B
Higher order evolution
In this section, we will prove a higher order version of Proposition 3.3. To this end, we follow [Citation14, Citation15] and denote by any multilinear form, depending on and ψ in a universal bilinear way, where are tensors on Σg. In particular, we have for a universal constant C > 0 and . Moreover, for and we denote by any term of the type
In addition, for r = 1 we extend this definition by denoting by any contraction of with respect to the metric g.
With this notation, we observe that along an isoperimetric Willmore flow the covariant derivatives of the second fundamental form A also satisfy a 4-th order evolution equation.
Lemma B.1.
Let and let be a σ-isoperimetric Willmore flow. Then for all we have
Proof.
We observe with (B.1) (B.1)
For m = 0, we thus find by Equation(2.7)(2.7) (2.7) where we used as a consequence of Simons’ identity [Citation44]. Assume the statement is true for . By [Citation14, Lemma 2.3] with and the fact that we are in codimension one, we find using Equation(B.1)(B.1) (B.1) in the last step. □
With similar computations as above, one finds the following
Lemma B.2.
Let and let be a σ-isoperimetric Willmore flow. The for all we have
We can now prove the following higher order analogue of Proposition 3.3.
Proposition B.3.
Let , let be a σ-isoperimetric Willmore flow and let γ be as in Equation(3.3)(3.3) (3.3) . Then for all and we have where .
In order to prove Proposition B.3, we first recall the following
Lemma B.4.
[Citation14, Lemma 3.2] Let be a normal variation, . Let be a -tensor satisfying . Then for any and we have where .
Proof of Proposition B.3.
In the following, note that the value of is allowed to change from line to line. We apply Lemma B.4 with by Equation(B.1)(B.1) (B.1) and estimate the terms on the right hand side. Using and Lemma B.1, we thus have (B.2) (B.2)
Moreover, by Equation(3.3)(3.3) (3.3) we find (B.3) (B.3)
We proceed by estimating all the terms involving λ in Equation(B.2)(B.2) (B.2) and Equation(B.3)(B.3) (B.3) . For the first λ-term in Equation(B.2)(B.2) (B.2) , since we find for every
For the second term, we use [Citation14, Corollary 5.5] with k = m, r = 4 to obtain
The last λ-term in Equation(B.2)(B.2) (B.2) can be estimated by [Citation14, Corollary 5.5] with k = m and r = 3, yielding
Now for the first λ-term in Equation(B.3)(B.3) (B.3) , we use Young’s inequality twice to obtain
For the second λ-term in Equation(B.3)(B.3) (B.3) , we can use Young’s inequality with and q = 4 to estimate
Choosing sufficiently small and absorbing, by Lemma B.4, Equation(B.2)(B.2) (B.2) and Equation(B.3)(B.3) (B.3) (B.4) (B.4) where we used Young’s inequality to obtain the correct powers of λ and . Now, all the terms involving λ on the right hand side of Equation(B.4)(B.4) (B.4) are as in the statement. For the second and the last term in Equation(B.4)(B.4) (B.4) , one may proceed exactly as in the proof of [Citation14, Proposition 3.3]. This way, one creates additional terms which can be estimated by for every , using twice the interpolation inequality [Citation14, Corollary 5.3] (which trivially also holds in the case ). The first term on the right hand side of Equation(B.4)(B.4) (B.4) can then be estimated by means of [Citation14, (4.15)]. After choosing small enough and absorbing, the claim follows since and . □