UNSTABLE WILLMORE SURFACES OF REVOLUTION SUBJECT TO NATURAL BOUNDARY CONDITIONS

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1 UNSTABLE WILLMORE SURFACES OF REVOLUTION SUBJECT TO NATURAL BOUNDARY CONDITIONS ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER Abstract. In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.. Introduction Given a smooth immersed surface f : Σ R 3, the Willmore functional is defined by Wf = H dµ g, where g is the Riemannian metric induced by the standard Euclidean inner product, in R 3 along the pull-back of f, dµ g = det g dx is the induced surface element, H = κ + κ / denotes the mean curvature, and κ, κ are the principal curvatures of f. An important feature of the Willmore functional is its invariance under the full Möbius group of R 3. See [5, 6], [8, Section 7.3] and Weiner [7]. A critical point of the Willmore functional is called a Willmore surface and is a solution of the Willmore equation H + HH K = on Σ where denotes the Laplace-Beltrami operator on f and K = κ κ is the Gauss curvature. In the literature there are many results concerning the existence and regularity of Willmore surfaces. The case where f is a closed immersed surface has been well-studied, and we only mention here the papers [, 7, 6, 7,, 4]. For the case where f is an immersed surface with boundary much less is known. In this setting one must supplement with appropriate boundary conditions. A discussion of the range of possibilities can be found in Nitsche [9]. One possible choice is the Dirichlet boundary conditions where the boundary of the surface and the tangent bundle at the boundary are prescribed. Another is the natural or Navier boundary conditions. In this case the position and the mean curvature H are prescribed as boundary data. The latter boundary condition arises naturally from the formula for the first variation see 4 below. The Navier Willmore boundary value problem is that which we shall study in this paper. For both types of boundary conditions, existence results obtained Mathematics Subject Classification. 35J4 and 35B38 and 58E99 and 49J45 and 49Q. Σ

2 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER through perturbative methods can be found in [9, ]. Schätzle in [3] proves the existence of Willmore surfaces satisfying Dirichlet boundary conditions in S n. These are embedded if their Willmore energy is small enough. Uniqueness theorems without symmetry assumptions are due to Palmer [] and the first author [9]. So far other existence results are known only in the class of embedded surfaces of revolution. It is interesting to study this special case in order to understand the possible qualitative behavior of the solutions and which phenomena may occur. Existence results for Willmore surfaces of revolution subject to Dirichlet and Navier boundary conditions have been obtained in [,, 3] and [, 3, ] respectively. Before stating our new result we must first fix some notation. Given a smooth and positive function u : [, ] R we consider the surface of revolution Γu = f u [, ] [, π], where fu is the embedding associated to u defined by f u : x, ϕ x, ux cos ϕ, ux sin ϕ, x [, ], ϕ [, π]. The surface Γu is said to be generated by u. For such a surface the mean curvature and the Gauss curvature are given by Hx = H[u]x = u x + u x + 3 3/ ux, and + u x u x Kx = K[u]x = ux + u x, respectively. The Willmore functional reads Wu := Wf u = π u x + u x + 3/ ux ux + u x dx. + u x Given α >, let us denote by S α the set of positive, symmetric functions in H, with boundary value α; that is, S α = {u H, ux >, ux = u x, x [, ] and u± = α}. The first variation of the Willmore functional W at u S α H 4, in the direction of a function ϕ H, H, is given by cf. [, Lemma 6] 4 d ] dt Wu + tϕ t= = π [H uϕ π uϕ H + HH K dx. + u Critical points then solve the following Navier boundary value problem: 5 6 H + HH K = in,, u± = α, H± =.

3 UNSTABLE WILLMORE SURFACES OF REVOLUTION 3 Note that 5 is equivalent to cf. [, Section.] ux d ux + u x dx + u x H x 7 + Hx u x + + u x 3 ux =, x,. + u x By also expressing H in terms of u one observes that 5 is a highly nonlinear ODE of fourth order. Furthermore, the boundary conditions H± = are nonlinear as well. Particular solutions of 5-6 are given by the family of catenaries u b x = b coshbx, as long as the equation α = u b ± = coshb has a positive solution. There is a critical b boundary value α below which catenaries cease to exist, given by α = coshy inf y, y = coshb b.588 with b = satisfying coshb = b sinhb. For α > α the equation α = coshb has precisely two solutions < b b α < b < b α <, and since H[u b ] H[u b ], the functions u b and u b are solutions of 5-6. As Wu and Wu b = Wu b =, these functions are global minimisers of W in the class S α. It is natural to expect the existence of a third critical point of W between the two catenaries. By carrying out a bifurcation analysis it was shown in [] that b, u b is the unique bifurcation point for 5-6 on the branch b b, u b in the class of symmetric solutions; in particular one has: Theorem. [, Theorem.3]. There exists an ε > such that for α α, α +ε the boundary value problem 5-6 has at least three solutions in S α. One of these is not a minimal surface. In order to extend this result for larger values of α it is natural to first attempt a straightforward adaptation of classical mountain pass theory. The min-max characterisation of the Palais-Smale limit would then automatically imply instability. Such a strategy was successfully carried out by Struwe see [5] and the references therein to prove the existence of unstable solutions to the Plateau problem via an adaptation of classical Ljusternik-Schnirelmann theory. Palais-Smale sequences for the Willmore functional have been recently investigated in [4], where it is proven that local Palais- Smale sequences that are uniformly bounded in W, W, possess a convergent subsequence. The limit is shown to satisfy a system of differential equations which, under a smallness condition on the energy, is equivalent to the constrained Willmore equation. The indication from [4] is thus that for a Palais-Smale approach to be successful, one typically requires some kind of energy assumption. By following a different approach we are able to avoid any smallness assumption in our case and obtain the following global result. Theorem.. For every α > α the boundary value problem 5-6 has, in addition to the two catenaries u b and u b, a third smooth solution u S α which is unstable in the sense that u is not a local minimum of W in S α. Furthermore

4 4 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER Figure. Numerically calculated unstable solutions between the two catenaries for α = 5 left and α = right. See also Remark 3.7. i H[u] < in, ; ii u and H[u] are strictly convex in [, ]; iii u b x < ux < u b x for x,. To our knowledge, this is the first existence result for unstable Willmore surfaces satisfying prescribed boundary conditions. A sufficient condition for unstable closed Willmore surfaces is given in [] together with some examples. See also [8]. The proof of Theorem. essentially consists of two steps. In the first step, we show that the Willmore equation is equivalent to certain singular first order ODEs for the mean curvature H[u]. Theorem.3. Let u : [, ] R be a smooth, positive, symmetric function. Then u is a solution of 5 if and only if for all x [, ] 8 H xux + u x ux xu x Hxuxx + uxu x Hx + u x ux =. + u x Furthermore, if u is a solution of 5 then there exists a constant C such that for all x [, ] 9 H xuxu x uxhx + u x + u x + Hx + u x = C. The converse holds provided u is not identically constant on [, ]. The proof of this result will be given in Section and relies on the scale and translation invariance of the Willmore functional. By evaluating 8 at x = one readily confirms that a smooth, symmetric solution of 5 with u± = ±u ± satisfies H± = or H± = u + u. This observation allows us in the second step to obtain a solution of 5-6 by constructing a symmetric solution of 5 with u± = α, u ± = ±α in such a way that the case H± = α is excluded. In this manner we are led to consider +α

5 UNSTABLE WILLMORE SURFACES OF REVOLUTION 5 a particular Dirichlet boundary value problem see 5 which is accessible to the direct method of the calculus of variations. We approach this problem in Section 3 by minimising W over the sets Ñ α,β := {u S α C, [, ] : u = β, u x α for all x [, ]} for β > α. The desired solution u is then obtained as the limit of minimisers when β α. Perhaps surprisingly, the constraint u α ensures that u is convex which in turn implies that H± =. At the same time this method yields the instability of our solution by exploiting the strict monotonicity of the map β inf Wv. v Ñα,β Such monotonicity properties were used in [] to prove the existence of solutions to the Navier boundary value problem for α < α. The qualitative properties i iii are finally obtained by combining the minimising property of u with 8 and 9. Acknowledgements The authors thank Friedhelm Schieweck for his help with the numerical experiments. Financial support for Glen Wheeler from the Alexander-von-Humboldt Stiftung and for Anna Dall Acqua and Klaus Deckelnick from the Deutsche Forschungsgemeinschaft DFG for the project Randwertprobleme für Willmoreflächen - Analysis, Numerik und Numerische Analysis DE 6/5. is gratefully acknowledged.. Proof of Theorem.3 An important feature of the Willmore functional is its invariance with respect to conformal transformations. It is natural to expect that this large class of invariances yields extra information at the boundary, in the spirit of the Pohozaev identity. In the case under consideration, we can obtain quite a lot of additional information. For this purpose it is useful to derive the first variation of W. Given a compact surface Σ with boundary and a sufficiently smooth immersion f : Σ R 3 we denote by Wf := H + HH K the Euler-Lagrange operator of W in L at f, and use φ := φ, ν ν, with ν the exterior unit normal vectorfield, to denote the normal projection of a vectorfield φ. We also use the notation H := Hν to denote the mean curvature vector. Lemma. [5, Theorem.]. Let Σ be a compact surface with boundary, t R, and δ >. For a smooth variation h : Σ t δ, t + δ R 3 with φ = t h we have d Wh, t = Wh, t, φ + ω dµ g. dt Here, given a frame {τ i } i= on T Σ, the one-form ω is defined by ωτ i = φ, ih i φ, H H φ, i h, and ω = Σ g ij i ωτ j where i is the derivative along τ i, and i is the covariant derivative along τ i. Choosing h in according to the invariances of the Willmore functional allows us now to carry out the i,j=

6 6 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER Proof of Theorem.3. We first prove that every smooth, positive, symmetric solution of 5 satisfies 8 and 9. The opposite implications will be discussed together afterwards. Consider the cylinder Σ = [, ] [, π], where we have identified and π, and set f u : Σ R 3 to be the embedding associated to u as in. As u satisfies 5, f u is clearly a Willmore surface and Wf u. Let x, x [, ] with x < x, and consider the cylinder Σ = [x, x ] [, π], where we have again identified and π. Let h : Σ t δ, t + δ R 3 be a family of sufficiently smooth immersions as in Lemma. satisfying h, t = f u. Denote by Σ and by Σ the left and right boundaries of Σ respectively, so that Σ = Σ Σ, with j Σ the circle given by j Σ = {xj } [, π], j =,. We denote by η j the interior unit conormal of j Σ. The embedding induces a global orthonormal frame {τ x, τ ϕ } for the tangent bundle over Σ, where τ x = + u x x, and τ ϕ = ux ϕ. Note that via the embedding this induces the following frame { τx hx, ϕ, t, τ ϕ hx, ϕ, t } = { τ x f u x, ϕ, τ ϕ f u x, ϕ } { =, u x cos ϕ, u x sin ϕ,, sin ϕ, cos ϕ } + u x of the tangent space at x, ux cos ϕ, ux sin ϕ. In these coordinates η ϕ = τ x hx, ϕ, t, and by symmetry η ϕ = τ x hx, ϕ, t. Since h, t = f u is a Willmore surface, Lemma. combined with Stokes theorem note that ω is a -form yields d dt Wh = t=t = = Σ i,j= Σ Σ g ij i ω j dµ g = ω dµ g Σ ωη dµ g + ωτ x dµ g Σ Σ ωη dµ g ωτ x dµ g, where dµ j g is the induced metric on j Σ. Using the explicit expression for τx given above and taking note of the definition of ω from Lemma., we find d dt Wh t=t [ = φ, xh x φ, H H ] φ, x f u + u x dµ g Σ Σ [ φ, xh x φ, H H ] φ, x f u + u x dµ g

7 = π π UNSTABLE WILLMORE SURFACES OF REVOLUTION 7 [ φ, xh x φ, H H ] φ, x f u [ φ, xh x φ, H H φ, x f u ux + u x dϕ x=x ] ux + u x dϕ. x=x In order to proceed we need to choose the direction of the variation φ. It is natural to consider variations parallel to the generators of the Möbius group of R 3, since for such φ the Willmore functional is invariant and the left hand side of is automatically zero. For the purposes of proving 8 and 9 it will be enough to consider the invariance of W under scaling and translation respectively. We begin with the scale invariance of W. Consider the family of immersions hx, ϕ, t = tx, ux cosϕ, ux sinϕ and take t =. Then φx, ϕ = x, ux cos ϕ, ux sin ϕ = f u x, ϕ. The exterior unit normal ν is given by νx, ϕ = + u x u x, cos ϕ, sin ϕ, and xh x, ϕ = for the first term φx, ϕ, xh x, ϕ x Hx, ϕ, νx, ϕ νx, ϕ = H xνx, ϕ. We thus have = H x f u x, ϕ, νx, ϕ = H x xu x ux + u x. Note that the final expression no longer depends on ϕ. Keeping in mind 3, a direct computation simplifies the second term as x φx, ϕ, Hx, ϕ = x Hx xu x + ux + u x = H x xu x ux + u x For the third term we have + H x x + uxu x x + uxu x Hx ux + u x. H x, ϕ φx, ϕ, x f u x, ϕ = Hx x + uxu x. As h is a continuous family of rescalings of f u, the scale invariance of W implies that the left hand side of vanishes. Inserting the equalities computed above allows us to write down the right hand side of explicitly. We obtain H x = π ux + u x ux xu x Hx + u x x + uxu x + Hx x ux + u x x + uxu x. x Clearly, as u = and H = equation 8 holds at x =. Formula 8 follows for x > by taking x = x and x = in, while taking x = x and x = in implies 8 for x <.

8 8 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER The case of translations gives equation 9 in a similar manner. Let us consider the family of immersions hx, ϕ, t = x, ux cosϕ, ux sinϕ + t,, with t =. Then φx, ϕ =,,. We have for the first term φx, ϕ, x H x, ϕ The second term becomes x φx, ϕ, Hx, ϕ For the third term we have = u xh x + u x. u xhx = x + u x = H xu x + u x Hx ux + u x + Hx. H x, ϕ φx, ϕ, x f u x, ϕ = Hx. As h is a continuous family of translations, the translation invariance of W combined with the above formulae simplify to H xuxu x uxhx + u x + u x + Hx x + u x =. x Since x, x [, ] are arbitrary, it follows that there exists a constant C such that which is 9. H xuxu x uxhx + u x + u x + Hx = C, for x [, ], + u x Let us now show that every solution of 8 satisfies 5. Denoting the left hand side of 8 by L[u]x, a long but straightforward computation shows that 3 = d dx L[u]x = uxux xu x H + HH K x, x [, ]. Let E = {x [, ] ux xu x = }. Clearly, 5 holds at all points x [, ]\E. Due to the smoothness of u, it is sufficient to show that [, ]\E is dense in [, ]. If this were not the case, there would be an open interval x δ, x + δ such that ux xu x = in x δ, x + δ [, ] =: I. This implies ux = ax in I for some constant a, and therefore L[u]x 4a + a in I, which contradicts 8. Finally we show that a solution to 9 that is not identically constant in [, ] satisfies 5. Denoting the left hand side of 9 by M[u]x, a computation shows that 4 = d dx M[u]x = uxu x H + HH K x, x [, ]. Let E = {x [, ] u x = }. Note that E [, ] since u is not identically constant on [, ]. Clearly, 5 is satisfied at x [, ]\E. As above, it is sufficient

9 UNSTABLE WILLMORE SURFACES OF REVOLUTION 9 to show that [, ] \ E is dense in [, ]. If this were not the case, there would be an open interval I E such that there exists x I that is an accumulation point for [, ] \ E. Then by continuity 5 is satisfied at x. On the other hand, since I E there exists a positive constant a such that ux a for x I and therefore H + HH K x = 4a, 3 a contradiction. This finishes the proof of Theorem.3. Remark.. In the proof of Theorem.3 we have used the translation invariance only in the direction,,. In other directions the integrand retains a dependence on the angle ϕ and hence no interesting equation can be derived. This also occurs if we use the invariance with respect to rotations. It seems that we cannot use the invariance with respect to inversions since we cannot write it as a continuous transformation. Remark.3. One can also derive formulas 8 and 9 from 5 by integrating 3 and 4 on intervals x, x. The surprising fact about 3 and 4 is that the multiplication of the Willmore operator by suitable factors enables us to recognise a derivative. However it is difficult to guess the right form of these factors in advance. We have therefore chosen to use the invariances of the Willmore functional to prove 8 and 9 because this approach seems to be more natural and easier to generalise. Theorem.4. Let u : [, ] R be a smooth, positive, symmetric solution of 5 satisfying u = u. Then, H = or H = u + u. Proof. By Theorem.3 u satisfies 8 for all x [, ]. Taking x = in 8 and using that u = u we obtain Hu + u u + u H =, from which the claim follows. Corollary.5. Let α > α and u : [, ] R be a smooth, positive, symmetric, convex solution of the Dirichlet Willmore boundary value problem { H + HH K = in,, 5 u± = α, and u ± = ±α. Then u solves 5-6 and Γu is not a minimal surface. Proof. Since u is convex we have u x and hence H = u + u + 3/ u + u u + u. Theorem.4 then implies that H =. Finally, if Γu were a minimal surface, i.e. H, then [8, Proposition 5.] would imply that u is a catenary. This is impossible due to α > α and the boundary values of u and u.

10 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER In the following proposition we collect some further consequences of Theorem.3. We shall use 6 to prove that H[u] < in,, while we have included 7 because we feel that such a result is of independent interest. Proposition.6. Let u : [, ] R be a smooth, positive, symmetric solution of 5. Then H[u] satisfies 6 u xh x + C x + uxu x =. Furthermore, u is a solution of the second order differential equation 7 ux u x + u x 3/ ux + u x + Cux + x = D, in [, ]. Here C is as in 9 and D is a constant of integration. Proof. The differential equation 6 is obtained from Theorem.3 by observing that 8 can be rewritten as H xuxu x + u x uxhx + u x + Hx x + uxu x + H xux =, + u x and using 9. The second statement in the claim follows from the first using d [ ] ux Hx = ux H x + d ux dx dx + u x together with the fact that x + uxu x = dx d x + ux, integrating once, and using Existence of a convex solution to 5 In view of Corollary.5 we obtain a solution of 5-6 with H by constructing a convex solution of the Dirichlet Willmore boundary value problem 5. For later purposes it is convenient to consider the family of Dirichlet boundary value problems { H + HH K = in,, 8 u± = α, and u ± = β, for boundary slopes β [ α, sinhb α. As in the introduction, we denote by b = b α, b = b α with < b α < b < b α < the two solutions of the equation α = coshb. Using the properties of the function y coshy one sees that b y coshy 9 < α y b α, b α. y For β [ α, sinhb α we have sinhb < β α = coshb < sinhb, b so that in the parameter range we consider the two catenaries u b, u b are not solutions of 8. Next, let us introduce + β α β := arcsinh β.

11 UNSTABLE WILLMORE SURFACES OF REVOLUTION Note that implies that arcsinh β b α, b α so that 9 with y = arcsinh β implies α > α β, β [ α, sinhb α. Existence of solutions to 8 for arbitrary α > and β R has been obtained in [] by minimizing over suitable subsets of H, the hyperbolic Willmore functional uxu x W h u = + u x + + u x dx, 3/ + u x ux which is the elastic energy of the graph of u as a curve in the hyperbolic half-plane. Motivated by the theory developed in [, Section 4..3] we introduce the following set: Ñ α,β := {u S α C, [, ] u = β, u x α for all x [, ]}. As we shall see later, the constraint on u will ensure the convexity of our solution. In addition we have for all u Ñα,β and therefore α ux = x u x dx α x, x [, ]. ux αx, x [, ]. In view of and the fact that β α, we see that the function ūx := b coshbx + α α β, with b = arcsinh β belongs to Ñα,β, so that this set is not empty. Let M α,β := inf Wu. u Ñα,β As observed by Pinkall and Bryant-Griffiths [6, 4], the Willmore functional and the hyperbolic Willmore functional are related by the identity so that in particular Wu = π W hu π 3 Wu = π W β hu + 4π, + β u x + u x u Ñα,β. Hence, on Ñα,β minimising W is equivalent to minimising W h. Furthermore, by Lemma 4.5 in [], in the minimisation process, we can restrict to functions u Ñα,β additionally satisfying 4 u + x > and + u x cosh u x x for x, ]. ux

12 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER Remark 3.. The inequalities in 4 imply ux + 5 x u x, x, ]. arcsinh u x Applying 9 with y = arcsinh u x, this gives 6 u x sinh b αx, where αx = ux, x, ]. x Moreover, note that if the second inequality in 4 is strict, then so are the inequalities 5 and 6. Remark 3.. The second inequality in 4 is strict for minimisers u Ñα,β, β [ α, sinhb α that satisfy 5 cf. [, Proof of Proposition 4.8]. For the convenience of the reader we have provided a proof of this statement in the Appendix. In what follows we shall make use of the following technical result: Lemma 3.3. Let α > α and β α, sinhb α]. For each u Ñα,β satisfying 4 and for each γ [α β, α there exists a symmetric, strictly positive v C, [, ] such that v satisfies 4, v± = γ, v = β, α v x > for x, ] and Wv Wu. In addition, if u satisfies u δ α in, ] then also the function v satisfies v δ α in, ]. Proof. See Corollary 4. and its proof in []. That result is formulated using the hyperbolic Willmore functional W h which can be rewritten in terms of W in view of 3. Theorem 3.4. For α > α and β α, sinhb α there exists a convex function u Ñα,β such that Wu = inf v Ñ α,β Wv. The function u belongs to C [, ] and solves the Dirichlet boundary value problem 8. Furthermore, u satisfies 4 as well as 7 u x β, x [, ]. Proof. The existence and regularity of a minimizer u satisfying 8, 4 and 7 can be found in [, Section 4..3]. It remains to show that u is convex. Combining 7 with we deduce that 8 d ux dx x = x u x ux x β α <, x, ]. x Next, we claim that u is injective on [, ]. If not, there would exist x, x in, ] such that x < x and u x = u x. We can choose x such that u x < u x for all x [, x. Let us consider the function ũ : [, ] R, ũx = ux x x. Clearly ũ± = ux x =: α and ũ = u x =: β. Recalling that u x < u x for x [, x and arguing similarly as in 8 we derive ũ x = u x x u x < ux x = α, x [, ],

13 UNSTABLE WILLMORE SURFACES OF REVOLUTION 3 so that ũ Ñ α, β, and β > α. On the other hand, 6 computed for u at x yields β sinhb α, and so we have β α, sinhb α]. Evaluating 5 at x and taking into account that u x = u x = β we find ux x + β arcsinh β = α β. Combining this estimate with 8 we infer that α = ux x > ux x α β. Moreover since u satisfies 4, the same holds for ũ. Lemma 3.3 with α = α, β = β, γ = ux x, and δ = u x implies that there exists a symmetric, strictly positive function ṽ C, [, ] satisfying 4, ṽ± = ux x, ṽ = u x, ṽ x u x α for x [, ] and Wṽ Wũ. Recalling that u x = u x we deduce that the function { x x ṽ for x [ x, x ], vx := x ux for x [, x x, ], belongs to Ñα,β. Furthermore, Wv = Wṽ + 4π Wũ + 4π = Wu 4π x Hx ux + u x dx x Hx ux + u x dx x x Hx ux + u x dx. Since u is a mimimum of W over Ñα,β we deduce that H on [x, x ]. It is well known that then necessarily u [x,x ] is a piece of a catenary cf. [8, Proposition 5.]. This however contradicts the fact that u x = u x. As a consequence, u must be injective on [, ], so that u is strictly increasing as u < u and we finally infer that u is convex. Next, we need a result concerning the monotonicity behaviour of β M α,β. Lemma 3.5. Let α > α and β, β [ α, sinhb α with β < β. Then M α,β < M α,β. Proof. See the Appendix. Theorem 3.6. Let α > α. Then there exists a convex function u Ñα, α such that Wu = inf v Ñ α, α Wv. The function u belongs to C [, ] and solves the Dirichlet boundary value problem 5. Proof. Let β k k N be a sequence in α, sinhb α with lim k β k = α. By Theorem 3.4, for each k N there exists a convex function u k Ñα,β k such that

14 4 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER Wu k = inf v Ñ α,βk Wv and u k is a solution of 8 with β = β k. Obviously we have the uniform bounds 9 u k x α, u kx α, x [, ], k N, while Lemma 3.5 implies that 3 Wu k = M α,βk < M α, α, k N. Furthermore we show in the Appendix that there exists a constant c α > such that 3 u k x c α, x [, ], k N. As a consequence, u k k N is uniformly bounded in H, so that, after possibly extracting a subsequence, there exists u H, with 3 u k u in H, and u k u in C [, ]. Clearly, u is strictly positive, symmetric, convex and we have u± = α, u = α. It is not difficult to see that one can approximate u in H, by a sequence of functions belonging to Ñα, α so that we infer Wu M α, α. On the other hand, the weak lower semicontinuity of W together with 3 yields so that Wu lim inf k Wu k lim sup Wu k M α, α Wu, k 33 lim k Wu k = Wu = M α, α. Next, a short calculation together with 3 and 33 shows that π u k x + u k x 5 = Wu k + π π Wu + π =. π u kx u x dx u k x + u k x 3 dx π u k xu k xu x dx + π + u k x 5 u x + u x 3 dx ux + u x dx π u k x dx + u k x u k xu x dx + u k x 5 uxu x + u x 5 With the help of 9 and 3 we infer that u k u in L, so that u k k N converges strongly to u in H,. We can use this information in order to establish that u is a smooth solution of 5. Indeed, since u k is a solution of 8 we have W u k, ϕ =, ϕ H,, k N, dx

15 UNSTABLE WILLMORE SURFACES OF REVOLUTION 5 where the first variation of W in direction ϕ H, is given by the formula cf. 9, [] W v, ϕ = π π + π vxv xϕ x dx 5 + v x 5 π vxv xv x ϕ x dx + v x 7 Letting k we deduce that v xϕ x dx vx + v x 3/ v x ϕx ϕx + v x 5 vx + v x W u, ϕ =, ϕ H,, dx. and proceeding as in the proof of Theorem 4, Step in [] we find that u is a smooth solution of 5. Remark 3.7. Let us emphasize that the constraint u α in the definition of Ñ α,β appears to be crucial in order to construct convex solutions. This observation is based on the results of numerical experiments conducted with the algorithm described in [, Section 7], which calculates approximate solutions of 8 via the L -gradient flow of W. The corresponding solutions are therefore in general local minima of W in the class N α,β = {u S α C, [, ] : u = β}. On the left hand side of Figure we display two solutions for α = β = 3, while the pictures on the right hand side show their corresponding mean curvature H. We remark that the lower solution does not belong to Ñ3, 3, and has energy W.47 while the energy of the upper convex solution is W Note also that the values of H± numerically confirm the statement of Theorem.4. Proof of Theorem.. Fix α > α. We already know that the two catenaries x coshb iαx, i =,, b i α are solutions of 5-6. The existence of a third solution which is convex follows immediately by combining Corollary.5 and Theorem 3.6. Furthermore, denoting by u k k N the sequence occuring in the proof of Theorem 3.6 we infer with the help of 3 that Wu k < Wu, k N. Since u k S α and u k u, k in H,, the function u cannot be a local mimimum of W in S α. Proof of i. We claim that the mean curvature H of u is either strictly positive or strictly negative on,. In view of the symmetry of H it is enough to prove the claim on [,. Assume that there exists an x [, such that Hx =. Then there would also exist an x x, such that H x =, and hence by 6 C x + ux u x =. As u x > in, ] it follows that C =, which upon reinsertion in 6 gives that H is a constant function and hence H, which contradicts Corollary.5. Therefore H is of fixed sign on,.

16 6 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER Figure. Numerically computed solutions to 5 left and their mean curvature right in the case α = 3. In order to determine the sign of H we now compute xu Hx + u x x ux dx d = arccosh + u x dx + u x x dx ux = arccosh + α + α α > arccosh + α + α 34 = since the function α arccosh + α +α is strictly increasing. As u Ñα, α, α we have u x α which together with yields 35 xu x ux α for x [, ]. Assume that H > on,. Combining 34 and 35 shows xu Hx + u x x ux dx >,

17 UNSTABLE WILLMORE SURFACES OF REVOLUTION 7 which is impossible. Hence Hx < for all x,, which proves i. Proof of ii. We are now in a position to show that u and H are strictly convex. Strict convexity of u follows immediately by solving 3 for u and using Hx < : u x = Hx + u x u x ux >, x [, ]. In order to see that H is strictly convex, we first observe that evaluating 9 at the origin gives C = uh + H <. Differentiating 6 and using 35 which, due to the symmetry of u, is valid in [, ] as well as i, it follows for all x, that H x = C ux x xu ux + u x 3 ux u x ux C + u x + u x ux ux = C + u x + u x 3 Hx + + u x ux ux ux + u x 3 = CHx ux >. Proof of iii. Let us first show that 36 ux > u b x = b coshb x, x,, where b = b α. Since α [ ] u = d ux dx = + α dx + u x by i, and as α < sinhb, this implies u > α + α > α coshb = b = u b, uxu xhx dx < so that 36 holds at the origin. Set x = inf { x [, ] ux = u b x }. Since 36 holds at x =, we have x, ]. Assume that x <. Clearly 37 u x u b x. Since ux = u b x we deduce with the help of coshb x b x = ux x α, so that by 9 b x b or b x b. Since x < and b < b, this in particular implies b x < b. By Theorem 3.6, u is a minimiser in Ñα, α. Remark 3. followed

18 8 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER by Remark 3. thus yields that ux u x coshb x > sinh b = sinh b x = sinhb b x x = u b x by the definition of b and since b x < b. This contradicts 37 so that x =, proving 36 on [,. The claim on, follows by the symmetry of u and u b. Let us finally show that 38 ux < u b x = b coshb x, x,, where b = b α. Set x = sup { x [, ] u x = u b x }. Clearly x < and u x > u b x for x x, ]. Let us assume that x >. We have u x = u b x = sinhb x and ux u b x = u b x u x dx <. x Therefore, ux < u b x = coshb x + u x = x x b x arcsinh u x. On the other hand 5 computed in x yields ux + u x x arcsinh u x, a contradiction. Hence x = which implies that u x > u b x for x, ]. Integration yields 38 on [,, and the symmetry of u and u b gives 38 on,. This completes the proof of Theorem.. Appendix Proof of Remark 3.. If the second inequality in 4 were not strict, then there would exist an x, such that ux + 39 ux = + u x cosh u x x. ux Let Note that ux = u γ x and u γ x = γ cosh γx, where γ = u γx = sinhγx = + u x. ux cosh γx = u x, where we used 39 for the last equality. Since u Ñα,β this implies u γx u γx = u x α for x [, x ]. As a consequence the function { uγ x for x [ x vx :=, x ], ux otherwise, belongs to Ñα,β and Wv Wu. Since u is a minimiser of W in Ñα,β we infer that Wv = Wu and hence H[u] on [ x, x ]. It follows from [8, Proposition 5.] and the boundary values that u u γ on [ x, x ]. Since both u and

19 UNSTABLE WILLMORE SURFACES OF REVOLUTION 9 u γ solve 5 on, we obtain u u γ on [, ] which is in contradiction with β [ α, sinhb α. Proof of Lemma 3.5. Here we essentially follow the argument of Lemma 3. in [] with minor modifications. Let us first prove that 4 Mα,β < M α,β for all β, β α, sinhb α with β < β. We observe that implies α > α β and α > α β. By Theorem 3.4 there exists a convex function u Ñα,β satisfying 4 such that Wu = inf v Ñ α,β Wv. Since u = β > β there is x, with u x = β. The function ũ : [, ] R, ũx := ux x then satisfies 4, ũ± = ux =: α and ũ = β. In x x view of we have α α. Furthermore, since u is convex we infer that ũ x = u x x u x α α in [, ] so that ũ Ñ α,β. Evaluating 6 at x implies β sinhb α. Since β > α α, we therefore have β α, sinhb α]. As observed earlier, α > α β and so α α β, α]. If α > α, Lemma 3.3 yields the existence of a symmetric, strictly positive function v C, [, ] satisfying v± = α, v = β, v α in [, ], 4 and Wv Wũ. If α = α we simply choose v = ũ. In any case we have v Ñα,β and x Wu = 4π Hx ux + u x dx + 4π Hx ux + u x dx x = Wũ + 4π Wv + 4π M α,β + 4π x Hx ux + u x dx x Hx ux + u x dx x Hx ux + u x dx. We observe that H on [x, ], for otherwise u [x,] would have to be a catenary, contradicting the fact that u = α, u = β and. Recalling that Wu = M α,β we infer 4. In order to complete the proof of Lemma 3.5 it is in view of 4 sufficient to show that 4 Mα,β M α, α for all β α, sinhb α. Fix β α, sinhb α and let u Ñα, α be an arbitrary function satisfying 4. Note that implies α > α β. Since u = α > β there is an x, such that u x = β and < u x < u x, x, x. The function ũx = ux x x then satisfies 4, ũ± = ux =: α and ũ = β. In view of we have x α α and therefore ũ x u x = β < α α for x [, ], so that ũ Ñ α,β. Similarly as above we deduce from, 5, 6 that β α, sinhb α] as well as α α > α β. Choosing v = ũ if α = α, and v according to Lemma 3.3 if

20 ANNA DALL ACQUA, KLAUS DECKELNICK, AND GLEN WHEELER α > α, we obtain a symmetric, strictly positive function v C, [, ] satisfying v± = α, v = β, v α in [, ], 4 and Wv Wũ. As a result Wu Wũ Wv M α,β, from which we deduce 4. This finishes the proof of Lemma 3.5. Proof of 3. We use the argument from the proof of Lemma 4.9 in []. For the convenience of the reader we reproduce the relevant calculations and adapt them to our particular situation. To begin, Wu k = π u k x + + u k x 3 u k x + u k x u k x + u k x dx π u k x u k dx π x dx + u k x + u k x 3 = π u k x β dx + π k. + u k x + β k Since u k α in [, ] we have u kx u k + αx, x [, ] and therefore π + α u k + αx dx Wu β k k + π M α + β α, α + π, k + α as β k α. Solving this inequality for u k and recalling that u k is convex we infer inf u α kx = u k x [,] exp α +α M π α, α + α, which proves 3. References [] M. Bauer and E. Kuwert. Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not., 3: , 3. [] M. Bergner, A. Dall Acqua, and S. Fröhlich. Symmetric Willmore surfaces of revolution satisfying natural boundary conditions. Calc. Var. Partial Differential Equations, 393-4:36 378,. [3] M. Bergner, A. Dall Acqua, and S. Fröhlich. Willmore surfaces of revolution with two prescribed boundary circles. J. Geom. Anal.,. On-line first. [4] Y. Bernard and T. Riviere. Local Palais-Smale sequences for the Willmore functional. Arxiv preprint arxiv:94.36, 9. [5] W. Blaschke. Vorlesungen über Differentialgeometrie. Vol. I-III, 99. [6] R. Bryant and P. Griffiths. Reduction for constrained variational problems and k ds. Amer. J. Math., 83:55 57, 986. [7] R.L. Bryant. A duality theorem for Willmore surfaces. J. Differential Geom., :3 53, 984. [8] B. Dacorogna. Introduction to the Calculus of Variations. Imperial College Press, 4. [9] A. Dall Acqua. Uniqueness for the homogeneous Dirichlet Willmore boundary value problem. Preprint Nr. 6/, Universität Magdeburg,. [] A. Dall Acqua, K. Deckelnick, and H.-Ch. Grunau. Classical solutions to the Dirichlet problem for Willmore surfaces of revolution. Adv. Calc. Var., 4: , 8. [] A. Dall Acqua, S. Fröhlich, H.-Ch. Grunau, and F. Schieweck. Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data. Adv. Calc. Var., 4: 8,.

21 UNSTABLE WILLMORE SURFACES OF REVOLUTION [] K. Deckelnick and H.-Ch. Grunau. A Navier boundary value problem for Willmore surfaces of revolution. Analysis, 93:9 58, 9. [3] H.-Ch. Grunau. The asymptotic shape of a boundary layer of symmetric Willmore surfaces of revolution. Preprint Nr. 9/, Universität Magdeburg,. [4] U. Hertrich-Jeromin and U. Pinkall. Ein Beweis der Willmoreschen Vermutung für Kanaltori. J. Reine Angew. Math., 43: 34, 99. [5] E. Kuwert. The Willmore functional. Lectures held at ETH in 7. [6] E. Kuwert and Y. Li. W, -conformal immersions of a closed Riemann surface into R n. Arxiv preprint arxiv:7.3967,. [7] E. Kuwert and R. Schätzle. Closed surfaces with bounds on their Willmore energy. Arxiv preprint arxiv:9.586,. [8] J. Langer and D.A. Singer. Curve straightening and a minimax argument for closed elastic curves. Topology, 4:75 88, 985. [9] J.C.C. Nitsche. Boundary value problems for variational integrals involving surface curvatures. Quart. Appl. Math, 5: , 993. [] B. Palmer. The conformal Gauss map and the stability of Willmore surfaces. Ann. Global Anal. Geom., 93:35 37, 99. [] B. Palmer. Uniqueness theorems for Willmore surfaces with fixed and free boundaries. Indiana Univ. Math. J., 494:58 6,. [] T. Rivière. Analysis aspects of Willmore surfaces. Invent. Math., 74: 45, 8. [3] R. Schätzle. The Willmore boundary problem. Calc. Var. Partial Differential Equations, 373-4:75 3,. [4] L. Simon. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom., :8 36, 993. [5] M. Struwe. Plateau s problem and the calculus of variations. Math. Notes 35, Princeton University Press, Princeton, 989. [6] G. Thomsen. Über konforme Geometrie I: Grundlagen der konformen Flächentheorie. German. Abh. Math. Sem. Hamburg, 3:3 56, 93. [7] J.L. Weiner. On a problem of Chen, Willmore, et al. Indiana Univ. Math. J., 7:9 35, 978. [8] T.J. Willmore. Riemannian geometry. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 993. Otto-von-Guericke-Universität, Postfach 4, D-396 Magdeburg address: anna.dallacqua@ovgu.de address: klaus.deckelnick@ovgu.de address: wheeler@ovgu.de