The Willmore flow with prescribed isoperimetric ratio

We introduce a non-local $L^2$-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.


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 [6,13], general relativity [12], nonlinear elasticity [11] and image restoration [10].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 f : Σ → R 3 of a closed oriented surface Σ, its Willmore energy is defined by Here µ = µ f denotes the area measure induced by the pull-back of the Euclidean metric g f := f * ⟨•, •⟩, and H = H f := ⟨ ⃗ H f , ν f ⟩ denotes the (scalar) mean curvature with respect to ν = ν f : Σ → S 2 , the unique unit normal along f induced by the chosen orientation on Σ, see (2.1) below.A related quantity is the umbilic Willmore energy, given by W 0 (f ) := ˆΣ|A 0 | 2 dµ, where A 0 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 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 R 3 .By [43,Theorem 7.2.2],we have W(f ) ≥ 4π with equality if and only if Σ = S 2 and f : S 2 → R 3 parametrizes a round sphere.The isoperimetric ratio of an immersion f : Σ → R 3 is defined as the quotient 3 , where (1.2) F. RUPP A(f ) := ˆΣ dµ and V(f ) := − 1 3 ˆΣ⟨f, ν⟩ dµ denote the area and the algebraic volume enclosed by f (Σ), respectively.Here, the normalizing constant is chosen such that by the isoperimetric inequality we always have I(f ) =: σ ∈ [0, 1] with σ = 1 if and only if Σ = S 2 and f : S 2 → R 3 parametrizes 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 Σ g with prescribed isoperimetric ratio, i.e. the minimization problem naturally arises in mathematical biology in the Canham-Helfrich model [6,13] with zero spontaneous curvature and has already been studied mathematically in [38,17,33].While the genus zero case was solved in [38], the results in [17,33] combined with recent findings in [37] and [19] show that the infimum in (1.3) is always attained for any g ∈ N 0 and σ ∈ (0, 1); and satisfies β g (σ) < 8π.The energy threshold 8π also plays an important role in the analysis of the Willmore energy, since by the famous Li-Yau inequality [26], any immersion f of a compact surface with W(f ) < 8π has to be embedded.A sufficiently smooth minimizer in (1.3) is a Helfrich immersion, i.e. a solution to the Euler-Lagrange equation where ∆ = ∆ g f denotes the Laplace-Beltrami operator on (Σ, g f ).In [30], solutions to (1.4) with small umbilic Willmore energy have been classified, depending on the sign of the Lagrange-multipliers λ 1 and λ 2 .We observe that for λ 1 , λ 2 ∈ R fixed, (1.4) is also the Euler-Lagrange equation of the Helfrich energy given by H λ1,λ2 (f ) := W 0 (f ) + λ 1 A(f ) + λ 2 V(f ), (1.5) where the energy either penalizes or favors large area or volume, depending on the sign of λ 1 and λ 2 , respectively.The L 2 -gradient flow of the Willmore energy was introduced and studied by Kuwert and Schätzle in their seminal works [21,20,22].Their methods are very robust and allow to handle also other situations, such as the surface diffusion flow [32,42] and the Willmore flow of tori of revolution [9].The locally constrained Helfrich flow, i.e. the L 2 -gradient flow for the energy (1.5), and its asymptotic behavior have been studied in [31,4], 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 (1.3).To this end, we introduce the Willmore flow with prescribed isoperimetric ratio, which decreases W as fast as possible while keeping I(f ) ≡ I(f 0 ) = σ fixed.This yields the evolution equation where the Lagrange multiplier λ := λ(t) := λ(f t ) depends on f t := f (t, •) and is given by In (2.9) below we will justify the particular choice of λ, which yields that I is actually preserved along a solution of (1.6)-(1.7).Definition 1.1.Let σ ∈ (0, 1), T > 0 and let Σ g denote a connected, oriented and closed surface with genus g ∈ N 0 .A smooth family of immersions f : [0, T ) × Σ g → R 3 satisfying (1.6) with λ as in (1.7) and I(f ) ≡ σ is called a σ-isoperimetric Willmore flow with initial datum f 0 := f (0, •).
Stationary solutions of the flow (1.6)-(1.7)are solutions to the Helfrich equation (1.4) for λ 1 = 3 A(f ) λ and λ 2 = − 2 V(f ) λ. Conversely, any Helfrich immersion is also a stationary solution to (1.6)-(1.7),see Lemma 2.6 below.However, as the Lagrange multiplier λ defined in (1.7) depends on the solution, the isoperimetric flow (1.6) substantially differs from the L 2 -gradient flow of the Helfrich energy (1.5), where the parameters λ 1 and λ 2 are fixed numbers and chosen a priori.On the analytic side, the integral nature of the Lagrange multiplier makes the evolution equation (1.6) a non-local, quasilinear, degenerate parabolic PDE of 4 th order.Also geometrically, the constraint I(f ) ≡ σ causes new difficulties, as we cannot control the area and the volume independently along the flow (as in [4], for instance), but only the isoperimetric ratio I.The Willmore flow with a constraint on either the area or the enclosed volume has been studied in [16] and a recent article by the author [36].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 (1.4) with λ 1 = λ 2 = 0, 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 (1.4) and a crucial ingredient in classifying the blow-ups in [36].Second, in [36] 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 (1.6)-(1.7).Third, unlike in [36], 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 f 0 : S 2 → R 3 be a smooth immersion with I(f 0 ) = σ ∈ (0, 1) and such that W(f 0 ) ≤ min 4π σ , 8π .Then there exists a unique σ-isoperimetric Willmore flow with initial datum f 0 .This flow exists for all times and, as t → ∞, it converges smoothly after reparametrization to a Helfrich immersion f ∞ with I(f ∞ ) = σ solving (1.4) with λ 1 ̸ = 0 and λ 2 ̸ = 0.This shows a fundamentally different behavior of the isoperimetric Willmore flow and the Helfrich flow, where finite time singularities occur, cf.[4,31].Consequently, despite its new analytic challenges, the introduction of the non-local Lagrange multiplier has a regularizing effect on the gradient flow, see also [15] for a related result for the mean curvature flow.The 4π σ -threshold in Theorem 1.2 is motivated by the following simple application of the triangle inequality in L 2 (dµ).With I(f ) = σ and (1.2), we have This estimate bounds the denominator in (1.7) from below if W(f 0 ) < 4π σ .Moreover, it allows to control the Lagrange multiplier in the crucial estimates by essentially lower order quantities, see Section 4. We highlight that the assumption in Theorem 1.2 is not an implicit smallness of the initial energy, cf.[20,42], but the threshold is explicitly given, although very little is known about minimizers and critical points of (1.3).Moreover, as σ ↗ 1, the interval of admissible initial energies in Theorem 1.2 becomes arbitrarily small.This seems plausible, since if σ = 1, f 0 is a round sphere and the denominator in (1.7) vanishes.Thus, it is a priori unclear whether there exists an admissible immersion f 0 in Theorem 1.2 if σ ∈ ( 1 2 , 1) -in fact, this is equivalent to the condition β 0 (σ) ≤ 4π σ .In Theorem 7.1 below, we will prove β 0 (σ) < 4π σ for σ ∈ (0, 1), which is asymptotically sharp as σ ↗ 1, and consequently the existence of a suitable f 0 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 [3,9].The proof of Theorem 1.2 is based on the methods developed by Kuwert-Schätzle for the Willmore flow [21,20,22].Under a non-concentration assumption on the curvature, we use localized energy estimates to control the evolution, see Section 3 below.However, as in [36], these estimates depend on certain L p -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 Section 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 [20], see Section 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 Lojasiewicz-Simon inequality in the spirit of [7], combined with recent progress on this inequality in the presence of constraints [34].Due to the rigidity of the blow-up, we can follow the inversion strategy in [22] relying on the classification of compact Willmore spheres [5] to exclude the second case.This last step is also where we crucially make use of the assumption Σ g = S 2 .In the case of higher genus, a classification result for Willmore surfaces as in [5] 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 g ≥ 1, we establish the following remarkable dichotomy result.
Corollary 1.3.Let σ ∈ (0, 1), let Σ be a closed, oriented and connected surface and suppose that Then there exist ĉ ∈ (0, 1), (t j ) j∈N ⊂ [0, T ), t j ↗ T, (r j ) ∈N ⊂ (0, ∞) and (x j ) j∈N ⊂ R 3 such that the sequence of immersions fj := r −1 j f (t j + r 4 j ĉ, •) − x j converges, as j → ∞, smoothly on compact subsets of R 3 after reparametrization to a proper Helfrich immersion f : Σ → R 3 where Σ ̸ = ∅ is a complete surface without boundary.Moreover (a) if Σ is compact, then T = ∞ and, as t → ∞, the flow f converges smoothly after reparametrization to a Helfrich immersion Hence, under the above assumptions, in the singular case (b) the influence of the (non-local) constraint vanishes after rescaling as t → ∞ and the purely local term in (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 Section 2, we start our analysis by carefully computing and estimating a localized version of the energy decay in Section 3. In Section 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 Section 5. Finally, in Section 6 we prove our convergence result, Theorem 1.2, and Corollary 1.3 before we show Theorem 7.1 in Section 7, yielding that the set of admissible initial data in Theorem 1.2 is always non-empty.

Preliminaries
In this section, we will briefly review the geometric and analytic background and prove some first properties of the flow (1.6), see also [23] for a more detailed discussion.
2.1.Geometric and analytic background.In the following, Σ g always denotes an abstract compact, connected and oriented surface of genus g ∈ N 0 without boundary.An immersion f : Σ g → R 3 induces the pullback metric g f = f * ⟨•, •⟩ on Σ g , which in local coordinates is given by where ⟨•, •⟩ denotes the Euclidean metric.The chosen orientation on Σ g determines a unique smooth unit normal field ν : Σ g → S 2 along f , which in local coordinates in the orientation is given by We will always work with this unit normal vector field.
The (scalar) second fundamental form of f is then given by A ij := ⟨∂ i ∂ j f, ν⟩ and the mean curvature and the tracefree part of the second fundamental form are defined as where g ij := (g ij ) −1 .Important relations are where K denotes the Gauss curvature.Consequently, using (1.1), we find The Levi-Civita connection ∇ = ∇ f induced by the metric g f extends uniquely to a connection on tensors, which we also denote by ∇.For an orthonormal basis {e 1 , e 2 } of the tangent space, the Codazzi-Mainardi equations yield (2.4) cf.[20, (5)].Clearly, potential singularities for the flow (1.6) occur if V(f ) becomes zero or if the denominator in (1.7) vanishes.Note that in the latter case H ≡ const, thus f is a constant mean curvature immersion.
Proof.The first statement follows immediately from the definition of I.For (ii), we assume by contradiction that H ≡ const, so f : Σ g → R 3 is an immersion with constant mean curvature.If Σ g = S 2 , then f has to parametrize a round sphere by a result of Hopf [14, Theorem 2.1, Chapter VI].In the second case, f has to parametrize a round sphere by the famous theorem of Aleksandrov [1].In both cases this contradicts σ ̸ = 1.□ Despite its geometric degeneracy, (1.6) is still a parabolic equation.Thus, starting with a smooth non-singular initial datum, it is possible to prove the following short-time existence result in similar fashion as it is outlined in [35, Chapter 4, Proposition 2.1], after observing that we can integrate by parts in (1.7) so that the numerator of the Lagrange-multiplier contains no second order derivatives of A any more.

Evolution of geometric quantities.
In this subsection, we will briefly review the variations of the relevant geometric quantities and energies.
be a smooth family of immersions with normal velocity ∂ t f = ξν.For an orthonormal basis {e 1 , e 2 } of the tangent space, the geometric quantities induced by f satisfy ∂ t (dµ) = −Hξ dµ, (2.5) As a consequence, we have the following first variation identities, cf.[36,Lemma 2.4].
F. RUPP Proposition 2.4.Let f : Σ g → R 3 be an immersion and let φ ∈ C ∞ (Σ g ; R 3 ).Then we have Moreover, if I(f ) > 0, we have Proof.Since W 0 , A and V are invariant under orientation-preserving diffeomorphisms of Σ g , we only need to consider normal variations, as any tangential variation corresponds to a suitable orientationpreserving family of reparametrizations (see for instance [24,Theorem 17.8]), which leaves the quantities unchanged.The variation of A then follows immediately from (2.5).For W 0 and V consider [36, Lemma 2.4], for instance.The variation of I then follows.□ The scaling behavior of the energies yields the following important identities.
Lemma 2.5.Let f : Σ g → R 3 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 A and V we may proceed similarly, using the scaling behavior This yields that Helfrich immersions are precisely the stationary solutions of (1.6)-(1.7).
Lemma 2.6.Let f : Σ g → R 3 be an immersion with I(f ) = σ ∈ (0, 1) and H f ̸ ≡ const.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 (1.4) with ⟨f, ν⟩, integrate and use Lemma 2.5 to conclude We have ∇I(f ) ̸ = 0 by Proposition 2.4, so by testing (2.8) with ∇I(f )ν and integrating it follows that λ is given as in (1.7), so f is indeed stationary.□ It is not difficult to see that along a solution of (1.6) with I(f ) > 0, the isoperimetric ratio is indeed preserved, since by Proposition 2.4, (1.6) and (1.7) we have On the other hand, the Willmore energy decreases since by (2.9) Equations (2.9) and (2.10) are the key features in studying the flow (1.6) and of vital importance for our further analysis.We highlight two immediate consequences.
Remark 2.7.(i) The computation in (2.10) implies that W 0 is a strict Lyapunov function along the flow (1.6), i.e.W 0 is strictly decreasing unless ∂ t f = 0, so f is stationary (by uniqueness of the solution).By (1.1), this also holds for W. (ii) Since W is monotone, the limit As (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. dt.
Proof.Follows from the scaling behavior of the geometric quantities and a direct calculation.□

Localized energy estimates
As in [20, Section 3] and [36, Section 2.3 and Section 3], we will start our analysis by localizing the energy decay (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 Σ g = S 2 or any restriction on the initial energy.
) and define η := η • f .Then we have and Proof.This computation is very similar to [35, Chapter 4, Lemma 2.8] (see also [20,Section 3]) if one replaces λν with 3λ A(f ) − 2λ V(f ) ν, so we will focus on the differences.We will use a local orthonormal frame {e i (t)} i=1,2 for our computations and find If we carefully combine the terms with λ in (3.1) and (3.2), the claim follows after integrating by parts, where the terms involving derivatives of H and the factor 2λ V(f ) cancel.For the second identity, arguing similarly as in [35, Chapter 4, Lemma 2.8] we have Integrating by parts and using (2.4) we conclude Now, using integration by parts and (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.
For the rest of this article, we denote by C a universal constant with 0 < C < ∞ which may change from line to line.Lemma 3.2.Let σ ∈ (0, 1), let f : [0, T ) × Σ g → R 3 be a σ-isoperimetric Willmore flow and let γ be as in (3.3).Then we have ) by a direct computation in a local orthonormal frame.Hence, using Lemma 3.1 and 2), we find The terms ´⟨∇W 0 (f ), ν⟩⟨∇H, ∇γ 4 ⟩ g f dµ and ´⟨∇W 0 (f ), ν⟩⟨∇ 2 γ 4 , A⟩ g f dµ can be estimated as in Consequently, we find Choosing ε > 0 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 L 2 -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 [36,Proposition 3.3]; these powers behave correctly under rescaling, cf.Lemma 2.8.We will combine this with the interpolation techniques from [21], [20] to get control on the local W 2,2 -norm of A, in terms of the (localized) Willmore gradient, at least if the L 2 -norm of A is locally small.Proposition 3.3.There exist universal constants ε 0 , c 0 , C ∈ (0, ∞) with the following property: Let σ ∈ (0, 1), let f : [0, T ) × Σ g → R 3 be a σ-isoperimetric Willmore flow and let γ be as in then at time t we can estimate Here Proof.Using the assumption and the interpolation inequality in [20, Proposition 2.6] (see also [36 Consequently, from Lemma 3.2, we find for some c 0 ∈ (0, ∞) For the first term on the right hand side of (3.5), we infer using Young's inequality The second term on the right hand side of (3.5) can be estimated by using Young's inequality with p = 4 and q = 4 3 and γ ≤ 1 to obtain ´[γ>0] |A| 2 dµ using Young's inequality.Combining this with (3.5),(3.6)and (3.7) and choosing ε > 0 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 (2.3) we always have ´|A| 2 dµ ∈ [8π, 4W(f ) − 8π + 8πg].
We will now study the situation, where (3.4) is satisfied on all balls with a certain radius, yielding a control over the concentration of the Willmore energy in R 3 .Following [22] we introduce the following notation.Here and in the rest of this article, we follow the notation of [21], i.e. the integrals over balls B r (x) ⊂ R 3 have to be understood over the preimages under f t .If Γ > 1 denotes the minimal number of balls of radius 1/2 necessary to cover We now prove an integrated form of Proposition 3.3.Proposition 3.5.Let ε 0 > 0 be as in Proposition 3.3.There exist universal constants ε 1 ∈ (0, ε 0 ), c 0 , C > 0 with the following property: Let σ ∈ (0, 1), let f : [0, T ) × Σ g → R 3 be a σisoperimetric Willmore flow and let ρ > 0 be such that Moreover, if we take ε 1 > 0 small enough, we have the estimate as a consequence of Simon's monotonicity formula [40], see also [36, where we used Young's inequality (with p = 3  2 and q = 3).
For the blow-up construction in Section 5, we will need the following higher order estimates for the flow in the case of non-concentrated curvature, cf.
where ε 1 > 0 is as in Proposition 3.5.Moreover, assume Then for all t ∈ (0, T ), x ∈ R 3 and m ∈ N 0 we have the local estimates and the global bounds In contrast to [20,Theorem 3.5] and [36, Proposition 3.5], we do not only prove local bounds, but also the global L 2 -control (3.16).Note that the global L 2 -norms could also be estimated by the L ∞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 below.The necessity for the finer estimates leading to (3.16) is why we give full details on the proof here, even though the argument is very similar to [20,Theorem 3.5].

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 [36], 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.
Proof.This follows from the reverse triangle inequality in L 2 (dµ), (2.9) and (2.10).□ While the scaling techniques from [36,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 σ ∈ (0, 1) and let f : [0, T ) × Σ g → R 3 be a σ-isoperimetric Willmore flow with W(f 0 ) < 4π σ .Then, we have Proof.We test the evolution equation (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 I(f ) ≡ σ by (2.9).By the assumption and (2.10) this is strictly positive and the claim follows.□ We remark that the existence of f 0 : Σ g → R 3 with I(f 0 ) = σ ∈ (0, 1) satisfying the assumption W(f 0 ) < 4π σ is not yet known and -in general -not true.For the case g = 0, this will follow from Theorem 7.1 below.However, for tori we have W(f 0 ) ≥ 2π 2 by [29], and hence 2π 2 ≤ W(f 0 ) < 4π σ can only hold for σ < 2 π < 1.On the other hand, for σ ∈ (0, 1  2 ] and arbitrary genus, there exists f 0 with W(f 0 ) < 4π σ since β g (σ) < 8π by [19,Theorem 1.2].We now use Lemma 4.2 to deduce the time integrability (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.
σ and let ρ > 0 be such that where ε 1 > 0 is as in Proposition 3.5.Then for all τ ∈ [0, T ) 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
for every δ > 0 by Young's inequality, and estimating b = 4π σ − √ K ≤ C(σ) in the last step.The statement then follows from (4.3) by taking δ > 0 sufficiently small.□

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 [21] and [36], 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 below.
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 L 2 -norm of λ A(f ) near t = 0.
(5.1)Note that we always have lim t↘0 ´t 0 λ 2 A 2 dτ = 0.The crucial insight here is that only the decay behavior of the L 2 -norm of λ A under the assumption of small concentration allows control on the existence time in a way which transforms correctly under parabolic rescaling.
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 ε, δ > 0 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 ω > 0 to be chosen and assume that for some t 0 ∈ [0, min{T, ρ 4 ω}] we have κ(t, ρ) < ε 1 for all 0 ≤ t < t 0 .By (i), we may apply Lemma 4.3 and use (iii) to find the estimate ˆt0 if we choose d = d(K, σ, g) > 0 and ω = ω(K, σ, g) > 0 small enough.The assumptions of Proposition 5.1 are thus fulfilled and the result follows with ĉ = ĉ(σ, ω) = ĉ(K, σ, g).□ 5.2.Existence of a blow-up.In this section, we will rescale as we approach the maximal existence time T ∈ (0, ∞] of a σ-isoperimetric Willmore flow f : [0, T ) × Σ g → R 3 with σ ∈ (0, 1).To that end, let (t j ) j∈N ⊂ [0, T ), t j ↗ T, (r j ) j∈N ⊂ (0, ∞), (x j ) j∈N ⊂ R 3 be arbitrary.By translation invariance and Lemma 2.8 for all j ∈ N the flow is also a σ-isoperimetric Willmore flow with initial datum f j (0) = r −1 j (f (t j , •) − x j ) and maximal existence time r −4 j (T − t j ).Throughout this section, we will denote all geometric quantities of the flow f j with a subscript j, such as A j , λ j , κ j , µ j for example.The next lemma guarantees the existence of suitable t j , r j and x j .
Consider the shifted energies, defined by Note that this is well-defined, since f + ϕ is an immersion for all ϕ ∈ Ũ with r > 0 small enough, cf.[36,Lemma 7.5 (i)].The first main ingredient towards 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.
Lemma 5.7.Let H := L 2 (Σ g ; R 3 ) ⊥ and let r > 0 be sufficiently small.For each ϕ ∈ Ũ , 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 ) ⊥ is a Fredholm operator with index zero, Proof.For ϕ, ψ ∈ Ũ , we have by Proposition 2.4 Similarly, the statement for ∇ H I can be shown.The Fredholm property of (∇ H W ) ′ (0) follows from (1.1) and [7,Lemma 3.3 and p. 356].For the last statement, we observe that for all Hence, using (2.6) with ξ = ⟨ν f , ϕ⟩ and Proposition 2.4 we find where we used I(f ) = σ and d dt t=0 ρ f +tϕ = −H f ⟨ν f , ϕ⟩ by (2.5).Since this only involves terms of order two or less in ϕ ∈ W 4,2 (Σ g ; R 3 ) ⊥ , the claim follows from the Rellich-Kondrachov Theorem, see for instance [2,Theorem 2.34].□ Proof of Theorem 5.5.From the assumption H f ̸ ≡ const, it follows that ∇I(f ) ̸ = 0 and hence ∇ H I(0) ̸ = 0.As in [36,Propsition 7.4], we can thus apply [34,Corollary 5.2] to deduce that Theorem 5.5 is satisfied in normal directions, i.e. for the functional W with the constraint I = σ.With the methods from [7, 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 [7, Lemma 4.1], the Lojasiewicz-Simon inequality yields the following asymptotic stability result, see also [36,Lemma 7.9] and [25,Theorem 2.1] for related results in the context of constrained gradient flows in Hilbert spaces.
Lemma 5.8.Let f W : Σ g → R 3 be a Helfrich immersion with then the flow exists globally, i.e. we may take T = ∞.Moreover, as t → ∞, it converges smoothly after reparametrization by some diffeomorphisms Note that by Lemma 2.6, f W above is a stationary solution to (1.6)-(1.7).Consequently, the proof of Lemma 5.8 is a straightforward adaptation of [36, Lemma 7.9], applying our Lojasiewicz-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 [7, Section 5] (see also [36,Theorem 7.1]), which yields that in the case where Σ is compact, below the explicit energy threshold no blow-ups or blow-downs may occur.

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 Σ yields convergence of the flow by Theorem 5.9.We first note that the desired compactness follows, if the area along the sequence fj in Proposition 5.4 remains bounded.
Proof.By Lemma 5.3 (iii), we have fj (Σ g ) ∩ B 1 (0) ̸ = ∅ for all j ∈ N, where fj = f j (ĉ, •) with f j as in (5.8).We now use the diameter bound [40, Lemma 1.1] to estimate diam fj (Σ g ) ≤ C A( fj )W( fj ), such that using the assumption, the invariances of the Willmore energy and the energy decay (2.10), we find sup j∈N diam fj (Σ g ) < ∞.Consequently, there exists R ∈ (0, ∞) such that fj (Σ g ) ⊂ B R (0) for all j ∈ N. Letting j → ∞ and using Proposition 5.4 (i) and the definition of smooth convergence on compact subset of R 3 , one then easily deduces f ( Σ) ⊂ B R (0) and then, since f is proper, compactness of Σ. □ We will now use Lemma 6.1 and Proposition 5.4 to conclude that if the concentration limit is noncompact, then it is not only a Helfrich, but even a Willmore immersion.In the spherical case, the classification in [5] and the inversion strategy from [22] will then yield a contradiction.Combined with Theorem 5.9, this will prove our main result.
Proof of Theorem 1.2.Since Σ g = S 2 and σ ∈ (0, 1), the existence of a unique, non-extendable σisoperimetric Willmore flow with initial datum f 0 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 W(f 0 ) < min{ 4π σ , 8π}.Let f : Σ → R 3 be a concentration limit as in Proposition 5.4.If Σ is compact, we find Σ = S 2 by [20, Lemma 4.3] and long-time existence and convergence follow from Theorem 5.9 and the fact that H f ̸ ≡ const by Lemma 2.1 (ii).
For the sake of contradiction, we assume that Σ is not compact.Then we may assume A( fj ) → ∞ by Lemma 6.1.Consequently, by Proposition 5.4 we find that f : Σ → R 3 is a Willmore immersion with W( f ) ≤ W(f 0 ) < 8π.The rest of the argument is as in [36, Proof of Theorem 1.2]: Denote by I the inversion in a sphere with radius 1 centered at x 0 ̸ ∈ f ( Σ) and let Σ := I( f ( Σ)) ∪ {0}.Then Σ is compact.By [22, Lemma 5.1], Σ is a smooth Willmore sphere with W( Σ) < 8π and hence, using Bryant's classification result [5], has to be a round sphere.Thus, f ( Σ) has to be either a plane or a sphere.Since Σ is non-compact by assumption, this yields that f has to parametrize a plane, a contradiction to Proposition 5.4 (ii).Now the limit immersion f ∞ : S 2 → R 3 satisfies W(f ∞ ) ≤ 8π and solves (1.4) for some λ 1 , λ 2 ∈ R. It remains to prove λ 1 , λ 2 ̸ = 0. Arguing as in the proof of Lemma 2.6, we infer Now, V(f ∞ ) ̸ = 0 by Lemma 2.1 (i) as σ ∈ (0, 1) and also A(f ∞ ) > 0. Consequently, if one of λ 1 , λ 2 is zero, then so is the other.In this case f ∞ is a Willmore sphere with W(f ∞ ) ≤ 8π.By Bryant's result [5], it then has to be a round sphere, so I(f ∞ ) = 1, a contradiction and hence λ 1 , λ 2 ̸ = 0. □ 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 f with the desired properties.If f has constant mean curvature Ĥ ≡ c, using (2.2) Equation (1.4) reads If Σ is compact, we conclude Ĥ ≡ c ̸ = 0 and hence K also has to be constant.But then f has to parametrize a round sphere (see for instance [14, Chapter V.1]), a contradiction to I( f ) = σ ∈ (0, 1).Therefore, statement (a) follows from Theorem 5.9.If Σ is not compact, we may assume A( fj ) → ∞ by Lemma 6.1 after passing to a subsequence.In this case, f is a Willmore immersion by Proposition 5.4 (iv), yielding statement (b).□ 7. An upper bound for β 0 In this section, we will prove an upper bound for the minimal Willmore energy of spheres with isoperimetric ratio σ ∈ (0, 1).
Theorem 7.1.For every σ ∈ (0, 1) we have β 0 (σ) < 4π σ .We remark that this estimate becomes sharp for σ ↗ 1 since β 0 (1) = 4π.On the other hand for σ ∈ (0, 1  2 ], the statement follows since by [38, Lemma 1] we have β 0 (σ) < 8π for all σ ∈ (0, 1).We will prove Theorem 7.1 by comparing energy and isoperimetric ratio of an ellipsoid.To that end, for a ∈ (0, 1], we define the half-ellipse c a (t) := (0, a cos t, sin t) T , t in the y-z-plane in R 3 .By rotating the curve c a around the z-axis we obtain a particular type of ellipsoid, a prolate spheroid.More explicitly, we define f a (t, θ) = (a cos t cos θ, a cos t sin θ, sin t) Fortunately, its area, volume and also its Willmore energy can be explicitly computed without the use of elliptic integrals.
Lemma 7.2.Let a ∈ (0, 1).Then we have (i) V(f a ) = 4π 3 a 2 ; (ii) A(f a ) = 2πa a + arcsin Proof.(i) and (ii) are standard formulas, see for instance [44,Section 4.8].For (iii), we observe that the mean curvature and the surface element of f a are given by dµ fa = a cos t a dt.
We will prove Lemma 7.3 in Appendix A below.A quick glimpse at the plot of F in Figure 1 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 F (1) = 0. where we used Young's inequality to obtain the correct powers of λ and ∥A∥ L ∞ ([γ>0]) .Now, all the terms involving λ on the right hand side of (B.4) are as in the statement.For the second and the last term in (B.4), one may proceed exactly as in the proof of [21,Proposition 3.3].This way, one creates additional terms which can be estimated by ˆ|ϕ| 2 γ s−4 dµ + ˆ|∇ϕ| 2 γ s−2 dµ ≤ ε ˆ|∇ 2 ϕ| 2 γ s dµ + C ε ˆ[γ>0] |A| 2 γ s−4−2m dµ, for every ε > 0, using twice the interpolation inequality [21, Corollary 5.3] (which trivially also holds in the case k = m = 0).The first term on the right hand side of (B.4) can then be estimated by means of [21, (4.15)].After choosing ε > 0 small enough and absorbing, the claim follows since s ≥ 2m + 4 and γ ≤ 1. □