A note on singularities in ﬁnite time for the L 2 gradient ﬂow of the functional

. This work investigates the formation of singularities under the steepest descent L 2 -gradient ﬂow of W λ 1 ,λ 2 with zero spontaneous curvature, i.e., the sum of the Willmore energy, λ 1 times the area, and λ 2 times the signed volume of an immersed closed surface without boundary in R 3 . We show that in the case that λ 1 > 1 and λ 2 = 0, any immersion develops singularities in ﬁnite time under this ﬂow. If λ 1 > 0 and λ 2 > 0, embedded closed surfaces with energy less than and positive volume evolve singularities in ﬁnite time. If in this case the initial surface is a topological sphere and the initial energy is less than 8 π , the ﬂow shrinks to a round point in ﬁnite time. We furthermore discuss similar results for the case that λ 2 is negative. These results strengthen the ones of McCoy and Wheeler (Commun Anal Geom 24(4):843–886, 2016). For λ 1 > 0 and λ 2 ≥ 0, they showed that embedded closed spheres with positive volume and energy close to 4 π , i.e., close to the Willmore energy of a round sphere, converge to round points in ﬁnite time.


Introduction
In [6], Helfrich suggested the functional The author was supported through FWF Grant P 29487 "The gradient flow of curvature energies". for immersions f : Σ → R 3 of two-dimensional compact connected surfaces Σ without boundary to study lipid bilayers.
Here, H f = 1 2 (κ 1 + κ 2 ) denotes the mean curvature and μ f the surface measure on Σ induced by f . The constant H 0 ∈ R is called spontaneous curvature and vol( f ) denotes the signed inclosed volume given by where φ * (dvol) denotes the pull-back of the standard volume form dvol = dx 1 Helfrich found that these energies are well suited to explain the characteristic shape of red blood cells: the shape of a biconcave disk. In honor of his work, the functional W H 0 λ 1 ,λ 2 is now called Helfrich functional.
In this article, we will restrict our attention to the special case of zero spontaneous curvature H 0 . We only consider (1.1) in which case the first summand is the Willmore energy We only deal with this case since the Willmore functional is scale invariant-indeed already Blaschke [1] observed that it is invariant under Möbius transformations that leave the surface bounded. Willmore proved that the Willmore energy is always greater or equal to 4π with equality only for a parametrized round sphere. Helfrich calculated the L 2 -gradient of W H 0 λ 1 ,λ 2 . It is known to be equal to 3) where ν f denotes the unit normal along f , K f the Gauß curvature, and Δ f the Laplace-Beltrami operator. In the case of zero spontaneous curvature, this reads as (1.4) We will consider smooth families of smooth immersions f t : Σ → R 3 , t ∈ [0, T ) of a compact surface Σ without boundary that are solutions to the steepest L 2 -gradient flow of the Helfrich functional with zero spontaneous curvature H 0 = 0, i.e., that solve (1.5) Note that such a family of immersions satisfies the equality Let us state the following short time existence theorem which is an immediate consequence of Theorem 1.1 in [14] the proof of which was based on the Lions-Lax-Milgram theorem and an analysis of the linearized problem by Polden [7]. THEOREM 1.1. (cf. [14]) Suppose f 0 : Σ → R 3 is a compact smoothly immersed surface without boundary. There exists a unique maximal smooth family of immersions For the case of the gradient flow of Σ |H f − H 0 | 2 dμ f constrained to immersions of fixed length and volume, short time existence was proven by Kohsaka and Nagasawa [8]. Note that short time existence of the flow above can for example also be derived using analytic semigroups as carried out for the Willmore flow and the surface diffusion flow in [5,16]. Liu [12] could bound the lifespan of the flow from below if there is only a small quantum of energy within balls of a given scale thus extending the corresponding result of Kuwert and Schätzle for the Willmore flow [10].
The following two theorems summarize the results of this article. We set for λ 2 = 0 (1.7) Then the following theorem on existence of finite time singularities holds. THEOREM 1.2. Let us assume that λ 1 > 0 and let Σ be a smooth compact manifold without boundary.
(i) If λ 2 = 0, the locally constrained Willmore flow (1.5) starting with any initial immersed surface f 0 : Σ → R 3 forms a singularity in finite time. (ii) If λ 2 = 0, the flow starting with an embedding f 0 : This substantially improves the result of McCoy and Wheeler [18] who only considered the case λ 2 ≥ 0, vol( f 0 ) ≥ 0 and had to assume that W λ 1 ,λ 2 ( f 0 ) < 4π + ε for a universal constant ε > 0, in principle calculable but not further specified. Furthermore our method of proof is based on the different scaling of the three summands that make up the energy and does not rely on sophisticated energy estimates as the proof of the corresponding result by McCoy and Wheeler. Similar effects were for example used in [4] to get control over the elastic flow constrained to fixed length.
For the case of the Helfrich flow of spheres, i.e., in the case that Σ = S 2 , we can also show that the flow sub-converges to a round point in the sense that there are times t j → T and a point x ∈ R n such that the rescaled immersions 4π μ( f t j ) 1 2 f t j − x converge to the unit sphere. More precisely, we have This result builds on the construction of a blowup profile in [18]. We show at the end of Sect. 4 that the constant 8π in Theorem 1.3 is sharp.
Let us finish this introduction with an outline of the structure of the remaining article. In Sect. 2, we shortly recapitulate the findings of McCoy and Wheeler on both the existence of critical points of the Helfrich functional and on the formation of singularities in finite time. Section 3 shows how to use the scaling properties of the different summands that make up the energy W λ 1 ,λ 2 to show that the flow (1.5) develops singularities in finite time. In Sect. 4, we recapitulate the construction of a blowup profile at a point singularity from [9,18]. We again use the different scalings of the components to show that this blowup profile must be a Willmore surfaces without assuming any initial bound on the energy of the initial surface (cf. Theorem 4.2). Combining this with the point removability results of Kuwert and Schätzle [11] and the classification result for Willmore spheres in [3], we can show the sub-convergence to round points as stated in Theorem 1.3.

Review of the results of McCoy and Wheeler
In the pioneering paper [17], McCoy and Wheeler completely classified all critical immersions of the functional W H 0 λ 1 ,λ 2 of complete surface without boundary under the assumption that where the constant ε 0 > 0 can in principle be deduced from the constructive nature of their proof. Note that since the Willmore energy of a round sphere is equal to 4π , (2.1) in a certain way says that the distance of f 0 from parametrizing a round sphere is small. They showed THEOREM 2.1. [17, Theorem 1] There is an absolute constant ε 0 > 0 such that the following holds: Suppose that f : Σ → R 3 is a smooth properly immersed complete surface without boundary and Then the following is true: It is a straightforward calculation that spheres of radius r 0 remain spheres under the flow where the radii satisfy So if λ 1 and λ 2 or both nonnegative and at least one of them is different from zero, the solution to (1.5) sub-converges to a round point in finite time. For the case that λ 1 > 0, λ 2 ≥ 0, McCoy and Wheeler could extend the above result to the following statement about solutions to the gradient flow (1.5).

THEOREM 2.2. [18]
There is an ε > 0 such that initial immersions f 0 : Σ → R 3 of a compact manifold without boundary Σ, with positive signed inclosed volume and One of the main ingredients to the proof of their theorem above is the highly nontrivial fact that under the condition (2.1) we have for some c 0 > 0 which they prove using sophisticated energy estimates. We will not make use of this estimate at all. Furthermore, their method did not allow them to treat the case λ 2 < 0 which we can. Theorem 1.2 reduces the gap between the existence or better nonexistence of critical points in Theorem 2.1 and the existence of finite time singularities in Theorem 2.2.

Existence of finite time singularities: an approach based on scaling
Let us now prove the main results of this article on the existence of finite time singularities of the locally constrained Willmore flow. All these results are based on the different scaling behavior of the three terms building W λ 1 ,λ 2 . Apart from this, we will use that, by an inequality of Yau [13], we have So especially W ( f ) < 8π implies that f is an embedding. Peter Topping [20, Lemma 1] showed that This estimate is a sharpened version of [19, Lemma 1.1]. THEOREM 3.1. Let λ 1 > 0 and f 0 : Σ → R 3 be a compact smoothly immersed surface without boundary. Let us further assume that either λ 2 = 0 or λ 2 > 0 and f 0 be an embedding with positive enclosed volume nad W λ 1 ,λ 2 ( f 0 ) ≤ 8π . Then the maximal time of smooth existence T for the Helfrich flow (1.5) with initial data f 0 satisfies Proof. Let us start with an observation for a general immersion f : Σ → R 3 . We consider the dilations around a fixed point p in f (Σ). Note that the Willmore energy stays constant while the area behaves like α 2 μ( f 1 ) and the volume goes like α 3 vol( f 1 ). By the definition of the L 2 gradient, we hence find (3.2) Using the Cauchy-Schwartz inequality together with (3.1), we have Let us now consider the unique maximal solution f t , t ∈ [0, T ), of the Helfrich flow (1.5) with initial data f 0 as in the statement of the theorem. Under the assumption that λ 2 vol( f ) stays nonnegative along the flow, we get from (1.6) and (3.4) Hence the differential inequality This concludes the proof for the case that λ 2 = 0. Let us finally show that in the case λ 2 > 0 every initial embedding with W λ 1 ,λ 2 ( f 0 ) ≤ 8π and positive volume stays embedded and hence vol( f t ) stays nonnegative. Otherwise there would be a first time t 0 at which a self-intersection occurs and hence especially vol( f t 0 ) ≥ 0, μ( f t 0 ) > 0. But then due to the celebrated inequality by Li and Yau [13], we would have W 0,0 ( f t 0 ) ≥ 8π which would imply This is impossible since the energy monotonically decreases in time. This concludes the proof.
Let us contemplate on the proof of Theorem 3.1 a bit further to see what can be saved of it, if we do not assume that λ 2 vol( f ) ≥ 0. Instead, we want to use the different scaling of the surface area and the volume that guarantees that, for μ( f ) small, the surface area dominates the inclosed volume.
The proof of Theorem 3.1 is based on the fact that for any immersion f : which we have estimated by 2λ 1 μ( f ) from below under the assumption that λ 2 vol( f ) ≥ 0. If we just assume that λ 1 > 0, but not λ 2 ≥ 0, we can estimate, |λ 2 | 2 16π , we can still bound this derivative from below by a positive multiple of μ( f ). Combining this with the estimate (3.3), we get We can only use our control of the energy W λ 1 ,λ 2 to bound the surface area of the immersions along the flow. For this, we again use the isoperimetric inequality to show Under the assumption that μ( f ) ≤ ε 1 , we can estimate this further and get Even for the case of negative λ 2 < 0, we can show, with the help of the estimates above, the existence of finite time singularities if both W λ 1 ,λ 2 ( f 0 ) is close to 4π and the area of the initial surface is smaller than ε 1 (λ 1 , λ 2 ).
16π is optimal since, for λ 1 > 0 and λ 2 < 0, any sphere of radius − 2λ 1 λ 2 is a critical point of W λ 1 ,λ 2 and has area In contrast to this, we do not expect ε 2 to be optimal.
With essentially the same technique, we can prove that the condition W λ 1 ,λ 2 ( f 0 ) ≤ 8π in Theorem 3.1 can be weakened to W λ 1 ,λ 2 ( f 0 ) ≤ 8π + min{ε 2 (λ 1 , λ 2 ), 8π }. But we additionally have to assume that μ( f 0 ) is sufficiently small if the inclosed volume has a negative sign. In this case, we will furthermore face an additional problem: Even if we assume that vol( f 0 ) > 0 and the initial surface is embedded, the inequality of Li and Yau does not tell us that the surface stays embedded along the flow.
Instead, we will use that every sphere eversion must have a quadruple point (cf. [15]). So if after some time we have the situation that f t 0 is an embedding with vol f t 0 < 0, then there must have been a quadruple point before t 0 , i.e., due to the inequality of Li and Yau there was a time t ∈ (0, t 0 ) such that This allows us to prove THEOREM 3.4. For λ 1 > 0, λ 2 > 0 let f 0 : Σ → R 3 be an embedding of a compact manifold Σ without boundary satisfying vol( f 0 ) > 0 and (3.10) Then the maximal time of existence T of the Helfrich flow (1.5) with initial data f 0 is finite.
Proof. We will give different upper bounds for depending on whether the inclosed volume is nonnegative or not. If vol( f t ) ≥ 0, we can proceed as in the proof of Theorem 3.1 and get Thus To deal with the times t ∈ [0, T ) for which vol( f t ) < 0, we first observe that for times t for which μ( f t ) ≤ ε 1 (λ 1 , λ 2 ) and f t is not an embedding, we have due to (3.6) and Li and Yau's inequality (3.12) Furthermore, we note that if for a time t 0 ∈ [0, T ) we have and for all t with μ( f t ) < ε 1 we also get But this implies that f t 0 is not embedded as otherwise, due to the result of [15], there would be a time t ∈ [0, t 0 ] such that f t has a quadruple point and hence by the inequality of Li and Yau Combining theses two observations, we have shown that for times t 0 ∈ [0, T ) with μ( f t 0 ) ≤ ε 1 and such that μ( f t ) ≤ ε 1 for all t ∈ [0, t 0 ] with vol( f t ) < 0 we have (3.12) Let us use this to show that for all times t ∈ [0, T ] with vol( f t ) < 0, we obtain by a continuity argument. We set where A of course may be the empty set in which case there is nothing to show. As 0 / ∈ A, we have vol( f t ) = 0 for all t ∈ ∂ A and hence f t cannot be an embedding. Thus by (3.12) μ( f a i ) < ε 1 (λ 1 , λ 2 ). 16π . But then the discussion above implies that we have (3.12), especially and thus d dt (3.14) Since μ( f t ) ≤ ε 1 for all t ∈ A, inequality (3.6) holds for all t ∈ A and shows that the right-hand side in (3.5) is bounded away from zero for all t ∈ A. Hence, the estimates (3.11) and (3.14) imply that singularities form in finite time.

Blowup analysis of finite time singularities
For the convenience of the reader, let us briefly repeat the blowup construction at a singularity in [18] which they perform at the beginning of the proof of Theorem 1.4 on page 25. This result extends the construction of a blowup by Kuwert and Schätzle for the Willmore flow [9,11].
and converge smoothly locally to a smooth family of proper immersions in the following sense: We can represent We will call such a family of immersionf a blowup limit in the following. The next theorem shows that possible blowup limits are stationary and parametrize Willmore surfaces. It is an extension of Theorem 4.4 in [18]. Again McCoy and Wheeler have shown the result only under the assumption that the energy of the initial surface is close to the Willmore energy of a sphere 4π . Proof. Using that f satisfies Eq. (1.5) together with and ∂ t f j = r 3 j ∂ t f we get from (1.5) that the f j satisfy Since f j converges tof locally smoothly and r j → 0, this implies As and r j → 0 as j → ∞, we deduce that ∇ L 2 Wf = 0. Combining Theorem 4.2 with the classification of Willmore spheres due to Bryant [3] and the removability of point singularities of Kuwert and Schätzle [11], we get COROLLARY 4.3. If f 0 : S 2 → R 3 is an immersion of a sphere that develops a singularity in finite time under the Helfrich flow (1.5) and lim t→T W ( f t ) < 8π , then the blowup limit from Theorem 4.1 is a round sphere.
Proof. Let us first assume thatΣ is compact. Since then the local convergence of the rescaled solution is in fact global,Σ is a topological sphere. Sof is a Willmore sphere with energy below 8π and thus is parametrizing a round sphere by the classification result of Bryant [3].
We now lead the case thatΣ is not compact to a contradiction as in [11]. We can assume without loss of generality that 0 / ∈f (Σ) sincef is proper. We consider the images of the f j under the inversion on the standard sphere I : which is well defined for large enough j ∈ N. The embeddings f j = I • f j converge locally smoothly to the embedding I •f in R 3 \{0} and due to the Möbius invariance of the Willmore energy The Möbius invariance of the Willmore energy also implies that I •f is a Willmore surface away from 0. Due to the point removability result of Kuwert and Schätzle [11], I •f can be extended to a Willmore sphere of Willmore energy less than 8π . Hence, due to a result of Bryant [3], it must parametrize a round sphere. But this would imply thatf was a plane-which would contradict Σ Af 2 dμf > 0.
Hence,Σ must be compact which concludes the proof.  REMARK 4.5. The constant 8π in Corollary 4.4 above is sharp which can be shown following the lines of argument in [2]. There we showed that surfaces of revolution exist such that the Gauß map of the profile curve has index equal to three of Willmore energy just slightly larger than 8π . Figure 1 illustrates the construction of such a surface of revolution. As surfaces of revolution remain surfaces of revolution under the flow, the blowup limit can impossibly be a sphere, as otherwise the index of the Gauß map of the initial surface must have been ±1. Shrinking the surface if necessary, we see that for Figure 1. The surface shown above is built out of two round spheres painted in blue and a piece of a catenoid, painted in red. The yellow part is used to connect these pieces. One can adapt the parameters such that the Willmore energy of the resulting surface is arbitrary close to 8π (color figure online) every ε > 0 we can find an immersed sphere f : S 2 → R 3 with W λ 1 ,λ 2 ( f ) < 8π + ε that forms a singularity in finite time in such a way that the blowup limit cannot be a round sphere-instead it consists of a finite number of catenoids and planes.