On the Morse Index of Branched Willmore Spheres in $3$-Space

We develop a general method to compute the Morse index of branched Willmore spheres and show that the Morse index is equal to the index of certain matrix whose dimension is equal to the number of ends of the dual minimal surface. As a corollary, we find that for all immersed Willmore spheres $\vec{\Phi}:S^2\rightarrow \mathbb{R}^3$ such that $W(\vec{\Phi})=4\pi n$, we have $\mathrm{Ind}_{W}(\vec{\Phi})\leq n-1$


Introduction
It was proposed by Tristan Rivière in [33] to study the topology of immersions of surfaces into Euclidean space by means of a quasi-Morse function (say L ). Fix a closed surface M 2 and let Imm(M 2 , R n ) be the space of smooth immersions Φ : M 2 → R n . We look for a Lagrangian L : Imm(M 2 , R n ) → R satisfying the following properties for all Φ : M 2 → R n : (1) L ( Φ + c) = L ( Φ) for all c ∈ R n (translation invariance) Indeed, an immersion does not change geometrically when one translates, rotates or dilates it. Now, assume that n = 3. To an immersed surface one can attach two natural quantities: the principal curvatures κ 1 , κ 2 (introduced by Euler in 1760 [11]) which are the maximum and the minimum of the curvature of normal section of the surface at a given point. Then we define the mean curvature H and Gauss curvature K (introduced by Meusnier in 1776 [21]) by Thanks to the third property, L must be a quadratic expression of the principal curvatures (see also [28] for a more general study of conformal invariants of Euclidean space), which says that up to scaling for some λ ∈ R, where g = g Φ = Φ * g R 3 . Thanks to Gauss-Bonnet theorem, is a constant independent of the immersion. Therefore, up to constants, the only non-trivial such quasi-Morse function is which is generally denoted by L = W and is called the Willmore energy.
This Lagrangian actually first appeared in the work of Germain and Poisson in 1811 and 1814 respectively in their work about elasticity ( [13], [31]). It was considered by many geometers in the following years, including in important work of Navier ([29]). For more information on the history in which these considerations of elasticity emerged, we refer to the comprehensive work of Todhunter ([43]). Poisson was the first one to obtain the correct Euler-Lagrange equation, more than 100 years before Blaschke and Thomsen, who attributed it to Schadow in 1922 ( [42], [4]). He also found in 1814 the first version of Gauss-Bonnet theorem, and his student Rodrigues computed the following year the exact constant 4π for ellipsoids, but unfortunately made a sign mistake and found 8π for tori ([35], [34]). The famous memoir of Gauss on the subject of the curvature of surfaces appeared only in 1827 ( [12]), and Gauss-Bonnet in a published form in 1848 ( [5]).
This Lagrangian only reappeared in 1965 in Willmore's work who proposed the famous conjecture about minimisers of the Willmore energy for tori ( [44]), which was finally proved in 2012 by Marques-Neves ( [20]).
In higher codimension, we can also define the Willmore energy as follows where H is the mean curvature vector (the half-trace of the second fundamental form). It has the fundamental property of being invariant under conformal transformations (of ambient space). In particular, as minimal surfaces ( H = 0) are absolute minimisers, inversions of complete minimal surfaces with finite total curvature are Willmore surfaces (though they may have branch points in general). Furthermore, Bryant showed that all immersions of the sphere in R 3 are inversions of complete minimal surfaces with embedded planar ends ( [7]). Now, a basic problem that we can address is to try to understand the following quantities : let γ ∈ π k (Imm(M 2 , R n )) be a non-zero class (of regular homotopy of immersions) and let Then one would like to understand if we can estimate these numbers and get some information on the critical immersions realising them (if this is possible to realise the width of these min-max problems).
The first non-trivial number is given as follows : let M 2 = S 2 , n = 3, and γ ∈ π 1 (Imm(S 2 , R n )) ≃ Z × Z 2 be a non-trivial class (Smale,[40]). Then we define 1] W ( Φ t ). (1.1) By the work of Smale, the space of immersions from the round sphere S 2 in three-space R 3 is pathconnected (π 0 (Imm(S 2 , R 3 )) = {0}), we have where Ω is the set of path { Φ t } t∈[0,1] ⊂ Imm(S 2 , R 3 ) such that Φ 0 = ι and Φ 1 = −ι, where ι : S 2 → R 3 is the standard embedding of the round sphere. These two min-max widths are equal since the Froissart-Morin eversion generates π 1 (Imm(S 2 , R 3 )) (see [33]). We will explain in the following what can be said about this problem in general and show a path to determine (1.1) and find which immersions may realise it. In relationship with these quantities, Kusner proposed the following conjecture.
Conjecture (Kusner, 1980's [17]). We have β 0 = 16π, and an optimal path is given by a Willmore gradient flow starting from the inversion of Bryant's minimal surface with 4 embedded ends.
Here, recall that a Willmore surface Φ : Σ → R n has no first residue if for all path γ around a branch point p of Φ (which does not contain or intersect other branch points) We refer to [32], [1] and [26] for more information on this quantity.
This theorem shows that the previous conjecture should be interpreted as follows.

Main results
If Ψ : Σ → R 3 is a branched Willmore sphere, we write for all normal admissible variations v = v n ∈ E Ψ (Σ, R 3 ) (see Section 3 for a precise definition) the quadratic form of the second derivative of the Willmore energy W at Ψ. Then we define the Willmore Morse index as the maximum dimension of sub-vector spaces of E Ψ (Σ, R 3 ) on which Q Ψ is negative definite.
Furthermore, if Ψ is a smooth immersion then m = 0 and we have where Ind Λ( Ψ) is the number of negative eigenvalues of Λ( Ψ).
In general, we can obtain a general bound which generalised [22] to the case of branched Willmore surfaces.
Theorem C. Let Ψ : Σ → R 3 be a branched Willmore surface and assume that Ψ is the inversion of a complete minimal surface with finite total curvature Φ : Σ \ {p 1 , · · · , p n } → R 3 . Then there a universal symmetric matrix Λ = Λ( Ψ) = {λ i,j } 1≤i,j≤n such that for all smooth v ∈ W 2,2 (Σ) and admissible normal for some v 0 ∈ W 2,2 (Σ) such that v 0 (p i ) = 0 for all 1 ≤ i ≤ n. In particular, we have Remark. For true branched immersions with ends of multiplicity at most 2, we have by showing that λ i,i = 0 for ends of multiplicity 2 in (2.2). See Section 10 for the proof (Theorem 10.7).
We can generalise Theorem A to the branched case at the price of getting a possibly weaker bound.

The second derivative of the Willmore energy as a renormalised energy
Let Σ be a closed Riemann surface, n ∈ N, p 1 , · · · , p n ∈ Σ be fixed distinct points and Φ : Σ \ {p 1 , · · · , p n } → R 3 be a complete minimal surface with finite total curvature and assume without loss of generality that 0 / ∈ Φ(Σ \ {p 1 , · · · , p n }) ⊂ R 3 . Then the inversion is a compact branched Willmore surface. Now, recall that we defined in [23] a notion of admissible variations of the Willmore energy as the maximum set of variations for which the second derivative of the Willmore energy is well-defined.
Theorem 3.1 ([23]). Let Σ be a closed Riemann surface and let Ψ : Σ → R d be a branched Willmore immersion and let g = Ψ * g R d be the induced metric. Then the second derivative D 2 W ( Ψ) is well-defined at some point w = E Ψ (Σ, R n ) = W 2,2 ∩ W 1,∞ (R n ) ∩ w : w(p) ∈ T Ψ(p) R n for all p ∈ Σ if and only if w ∈ L ∞ (Σ, g 0 ) and L g w ∈ L 2 (Σ, dvol g ), where g 0 is any fixed smooth metric on Σ and L g w = ∆ n g w + A ( w) is the Jacobi operator and A is the Simons operator. We denote by Var( Ψ) this space of admissible variations.
We can now define the Willmore Morse index as follows (see [23]).

Definition 3.2.
Let Σ be a closed Riemann surface and let Φ : Σ → R n be a branched Willmore immersion. Then Willmore index of Φ, denoted by Ind W ( Φ), is equal to the dimension of the maximal sub-vector space V ⊂ E Ψ (Σ, R n ) on which the quadratic form second variation Q Ψ ( · ) = D 2 W ( Φ)( · , · ) is negative definite. Now, thanks to Proposition 4.5 of [22], for all v = v n Ψ ∈ Var( Ψ), we have where u = | Φ| 2 v. In particular, thanks to Stokes theorem, we have In particular, the limit (3.1) exists for all such v ∈ Var( Ψ). Here, the balls B ε (p i ) are fixed following the following definition for some covering (U 1 , · · · , U n ) of {p 1 , · · · , p n } fixed once and for all.

Definition 3.3.
We say that a family of chart domains (U 1 , · · · , U n ) is a covering of This definition is independent of the chart ϕ i : The independence of the chart ϕ i with the above properties is a trivial consequence of Schwarz lemma (se [22] for more details).

Decomposition of the renormalised energy
We fix a Willmore surface Ψ : Σ → R 3 which is the inversion of a complete minimal surface Φ : Σ \ {p 1 , · · · , p n } with finite total curvature. We fix v ∈ W 2,2 (Σ) (such that v = v n Ψ ∈ Var( Ψ)), and as in the introduction, for all ε > 0 small enough, we consider the following minimisation problem inf w∈Eε(pi) where the class of admissible functions is Notice that for an end p j (for some 1 ≤ j ≤ n) of multiplicity m ≥ 1 of a complete minimal surface with finite total curvature Φ : Σ → {p 1 , · · · , p n } → R n , in any complex chart z : for m ≥ 2, while for m = 1 there exists γ 0 ∈ R n such that Therefore, we have up to scaling In particular, we deduce that so L g is not elliptic in a neighbourhood of p j . Therefore, we will have to consider another problem than (4.1).

Recall first by definition of
Therefore, for all 0 < ε < 1, and for all 0 < δ < ε, and 1 ≤ i ≤ n consider the domain We will also write Then L g and L 2 g are strongly elliptic on Σ i ε,δ and have the uniqueness for the Cauchy problem i.e. if L g u = 0 (resp. L 2 g u = 0) and u = 0 on some open U ⊂ Σ i ε,δ , then u = 0 (this fact was first proved in general by J. Simons [39]), thanks to a classical theorem of Smale (see [41] and [8]) there exists 0 < ε 0 such that for all 0 < ε < ε 0 , there exists 0 < δ(ε) < ε such that for all 0 < δ < δ(ε), the operators L g and L 2 g have no kernel on Σ i ε,δ for all 1 ≤ i ≤ n. More precisely, the only solution of each of the two following problems and is the trivial solution u = 0. Therefore, thanks to the Fredholm alternative (see [6], IX.23) for all 1 ≤ i ≤ n and all but finitely many 0 < ε < ε 0 there exists a unique minimiser u i ε,δ of (4.1) such that where u = | Φ| 2 v and L g = ∆ g − 2K g is the Jacobi operator of the minimal surface Φ : Σ\ {p 1 , · · · , p n } → R 3 . In particular, we fix 0 < ε < ε 0 and we assume 0 < δ < δ 0 (ε) < ε. Furthermore, notice that u i ε,δ is the unique solution to the variational problem inf w∈E ε,δ (pi)

Estimate of the singular energy of the minimisers
Recall the definition . Fix some 1 ≤ i ≤ n and assume that p i has multiplicity m i ≥ 1. Then there exists α i > 0 such that where J 2mi pi is the jet of v of order 2m i at p i ∈ Σ. However, thanks to the Sobolev embedding W 2,2 (Σ) ֒→ C 0 (Σ) and the absence of Sobolev embedding W 2,2 (Σ) ֒→ C 1 [22] for an explicit argument and Section 5 for the explicit singular energy associated to minimal surfaces with embedded ends. As v is admissible and v is smooth, there exists γ ∈ R such that The previous expansion (4.6) shows that Therefore, as Φ is conformal and harmonic, we have ∆ g | Φ| 2 = 4 and This expansion implies that Notice that v(p i ) = 0 implies that L g u = O(|z| mi+1 ) and which shows that the limit (3.1) reduces to Therefore, for all v ∈ C ∞ (Σ) such that v = v n Ψ be admissible, we have Therefore, we deduce the first extension of [22] to the case of branched surfaces.
Proof. Write g = Φ * g R 3 be the induced metric on Σ \ {p 1 , · · · , p n }. The preceding argument shows that and where g 0 is a fixed smooth metric on Σ. The estimate (4.9) must be interpreted as follows. If p i corresponds to an end of multiplicity m i ≥ 1 of Φ, then Ψ admits a branch point of multiplicity θ 0 = m i at p i , and (4.9) means that in the chart ϕ i : Then up to a subsequence, we deduce that (up to taking a subsequence) As v k −→ k→∞ v in C 0 (Σ), and by (4.10), we deduce (as v( Furthermore, as by (4.8) ∆v |z| θ0−1 ∈ L 2 (B(0, 1)), (4.12) and ∆v k −→ k→∞ ∆v almost everywhere, we also find by (4.12) and furthermore, by the expansion (4.11), we deduce that Therefore, v k is an admissible variation of Ψ, and by the preceding discussion we have Now, by the strong W 2,2 convergence and as v k is admissible, we have (see for example the explicit formula for Q Ψ in [22] or [23]) Then (4.13) implies that Q Ψ (v) ≥ 0, but notice also that by Fatou lemma This observation concludes the proof of the theorem, as the last equality in (4.7) comes from the Li-Yau inequality ( [18]) and the Jorge-Meeks formula ( [15]).

Remark 4.2.
The proof of the theorem shows in particular that for all admissible variations Therefore, we have Q i ε > 0 for all ε > 0 small enough, as by (3.1) we have In particular, as the metric g is real analytic on all compact subset K ⊂ Σ \ {p 1 , · · · , p n }, we deduce that for all 1 ≤ i ≤ n The following theorem is the analogous of Theorem V.1, 2, 3 [3]. Here, the vortices are already fixed and correspond to the points p 1 , · · · , p n ∈ Σ where the metric of the corresponding minimal surface degenerates. We first obtain an estimate of the singular energy by a geometric argument, and show that the Jacobi operator of the minimiser u i ε,δ is bounded in L 2 away from p i . This will allow us to pass to the limit to a limit function as δ → 0 and ε → 0. Theorem 4.3. Let 0 < ε < ε 0 and 0 < δ < δ(ε) < ε and u i ε,δ be the unique solution of (4.4). Then there exists a non-decreasing function ω : R + → R + which is continuous at 0 and such that ω(0) = 0 (independent of ε and δ) such that and and let Q ε : W 2,2 (Σ ε ) → R be the associated quadratic form. Then we have and the limit is well-defined. Now, fix a cutoff function ρ i ≥ 0 such that Notice in particular that for all 1 ≤ i ≤ n, 0 < ε < ε 0 and 0 < δ < δ(ε), we have by (4.15) Now, if 0 < ε < ε 0 and 0 < δ < δ(ε) < ε define Integrating by parts, we find As we deduce that B ε (u ε,δ , u i ε,δ ) does not contain a quadratic term of the form C ε v 2 (p i ), as the functions u j ε,δ (j = i) are independent of v(p i ). A similar argument applies for B ε (u i ε , u j ε ) (i = j), so we deduce by (4.14) that the only possibility for the limit (4.18) to be finite is that where O(1) is a quantity bounded independently of 0 < ε < ε 0 and 0 < δ < δ(ε). Therefore, combining (4.20) with (4.19), we deduce that for all 0 < δ < δ(ε) < ε where O(1) is a quantity bounded independently of 0 < ε < ε 0 and 0 < δ < δ(ε). Therefore, we deduce that Furthermore, the boundary conditions imply that u i ε, Therefore, (4.21) and (4.22) imply that and as supp (ρ i ) ⊂ B ε0 (p i ), we deduce that Furthermore, as the error terms are continuous in v ∈ W 2,2 (Σ) (such that v = v n Ψ ), we deduce that there exists a modulus of continuity ω = ω Ψ : R + → R + independent of 0 < ε < ε 0 and 0 < δ < δ(ε) < ε (that we can take non-decreasing and continuous at 0) such that This concludes the proof of the theorem.

Indicial roots analysis : case of embedded ends
The following theorem is the analogous of Theorem VI.1 of [3] and Theorem 1 of [2] from Ginzburg-Landau theory.
Furthermore, for all j = i, we have an expansion in U j as and for all l ∈ N,

Remark 4.7.
Although a i,j , b i,j , c i,j and d i,j depends on ε, we remove this explicit dependence for the sake of simplicity of notation.
We make computations as previously in the previously fixed chart ϕ j : U j → B(0, 1) ⊂ C such that ϕ j (p j ) = 0. By [30] (p. 25) the asymptotic expansion of v i ε,δ at 0 depends only on the linearised operator of e λ L g (e λ · ), which is as Φ has embedded ends Therefore, without loss of generality, we can assume that L * L v i ε,δ = 0. Taking polar coordinates (r, θ) centred at the origin, recall that Therefore, we have Projecting to Vect(e ik· ) (where k ∈ Z is fixed), the operator L (resp. L * ) becomes As L * k L k v i ε,δ (k, ·) = 0, and the space of solutions to L * k L k u = 0 is four-dimensional, we only need to find a basis of solutions to L * k L k u = 0 to obtain all possible asymptotic behaviour at the origin. Let α ∈ C fixed, we have so the basis of solutions to L * L k u = 0 is given by In particular, for k = 0, we need to find two other solutions. For k = 0, we have so one easily check that a basis of solutions of L * L u = 0 is given by 1, r 2 , log(r), r 2 log(r) and that furthermore, L 0 (r 2 ) = L 0 r 2 log(r) = 0.
Finally, for |k| = 1, as α + k , α − k = {1, 3} and β + k , β − k = {−1, 1}, we only have three solutions and we need to find an additional one. As Ker(L * 1 ) = r −3 , r −1 , we need to find a solution u such that L u = 0, L u ∈ Ker(L * 1 ) and u / ∈ Span R (r −1 , r, r 3 ). One checks directly that this additional solution is given by u(r) = r log(r), which satisfies indeed Notice that these computations also show that L * L = ∆ 2 , but we did not want to use this result directly to obtain a formally similar proof in the case of ends of higher multiplicity.
Step 2: Estimate on the biharmonic components. Now, recall that Therefore, as for two real-analytic functions ζ 0 : B(0, 1) → R and ζ 1 : Notice that the first two sums do not involve powers in 1/r (this justifies why there are no cross terms between these two sums and the remaining terms). Now fix some 0 < R < 1 such that the "O(1) functions" be bounded by 1/2 (in absolute value) on B R \ B δ (0). Then we have by Parseval identity As the quantity in the left-hand side of (4.31) is bounded independently of 0 < δ < R, we deduce that for all k ≥ 1, and some uniform constant C > 0 Notice that the second estimate follows from the first one as γ 4 and is bounded independently of δ > 0, there exists C > 0 such that Now, we see the next order of singularity is given by log 2 (δ)/δ 2 , so we have Another singular term is but (4.32) implies that (as the γ k are also uniformly bounded) for 0 < δ < 1 small enough. The next singular term is and using (4.34), we deduce that , so we deduce that Finally, the last singular term is 16π|γ| 2 log 1 δ and we deduce that (4.36) Step 3: Estimates on the harmonic components. Now, we have the inequality (from Theorem 4.5) (4.37) Thanks to (4.29) Furthermore, we have Therefore, we have Notice that the second integral involving the square of the radial component of ∆v i ε,δ is bounded, so we can neglect this term. Now, we also have as |a + b + c + d| 2 ≤ 4 |a| 2 + |b| 2 + |c| 2 + |d| 2 for all a, b, c, d ∈ C and by Parseval identity is bounded in δ, and (4.33) imply that Finally, by (4.39), (4.40) and (4.41), we deduce that there exists C > 0 (independent of δ and ε) such that Therefore, by (4.38), (4.39), (4.42), and (4.36) (for the term in As previously, the terms involving positive powers of k are bounded, and so for all k ≥ 2, as (4.37) implies that (4.43) is bounded independently of δ (and ε), we deduce that for some universal constant C (independent of 0 < ε < ε 0 and 0 < δ < δ(ε) < ε).
Step 4: Conclusion and limit as δ → 0. Finally, we deduce from the two previous steps that Thanks if the previous estimates, all coefficients are bounded, and for all fixed (r, Furthermore, as the operator L * g L g is uniformly elliptic on Σ ε for all fixed 0 < ε < ε 0 and thanks to the uniform bound , we deduce that up to a subsequence, there exists Furthermore, as δ → 0, (4.44) implies that where ϕ is real analytic and ϕ(z) = O(|z| 3 ). Finally, by the Weierstrass parametrisation, if Φ has embedded ends, we can assume ( [36]) that up to rotation Φ admits the following expansion for some α > 0 and β ∈ R Therefore, we have and Therefore, we have and this concludes the proof of the theorem.

Indicial roots analysis: case of ends of higher multiplicity
Theorem 4.9. Let 1 ≤ i ≤ n and 1 ≤ j = i ≤ n, and assume that Φ has an end of multiplicity m ≥ 2 for some real-analytic function ϕ ε such that ϕ ε (z) = O(|z| m+2 ). Furthermore, we have an expansion and the c i,j,k,l are almost all zero, that is all but finitely many as j, k ∈ Z and 1 − m ≤ j + k ≤ 0. Proof.
We have the expansion Then we have as Now, denote by L m the elliptic operator with regular singularities (see [30]) and log is harmonic on D 2 \ {0}, we have where L * m is the formal adjoint of L m . As the indicial roots of on operator of the form where b and c are C ∞ and b(x) = O(|x| 2 ) only depends on c(0) and is independent of b ( [30]).
Therefore, the indicial roots of e λ L g e λ e λ L g (e λ · ) , giving all possible asymptotic behaviour of a solution of L 2 g u i ε = 0 in D 2 \ {0} are the same of the indicial roots of the operator L * m L m . Therefore, consider first a solution v of First, recall that Therefore, for all k ∈ N the projection L m,k on Span(e i·k ) of L m is given by and we first look for solutions of the form v(r) = r α for some α ∈ C. We have by a direct computation Now, for k = 0, we need to find two additional solution and one check immediately that r m+1 log(r), r 1−m log(r) are two additional solutions. Furthermore, notice that when |k| = m, Therefore, for |k| = m, we have the basis of solutions r 1−2m , r, r 2m+1 , r log(r).
so we find the additional solution r log(r) when |k| = m. Therefore, we finally get Step 2: Estimate coming from L m v i ε,δ ∈ L 2 . As (4.51) and m ≥ 2, we deduce by the same argument as Theorem 4.6 that the following three integrals are bounded uniformly in ε and δ Furthermore, notice that for all k ∈ Z * , if P k is the projection on Span(e ik · ), then Furthermore, for all α ∈ Z, so we deduce that the coefficients γ 1 0 and γ 2 0 vanish when δ → 0, as |x| (1−m)−2 = |x| −(m+1) / ∈ L 2 (B(0, 1)). Furthermore, thanks to (4.52) and the proof of Theorem 4.6, we deduce that whenever a power α = m then the corresponding coefficient γ j k vanishes as δ → 0. Notice that all powers r m+1+k , r m+1−k , r 1−m+k and r 1−m−k are all distinct, except when |k| = m, where the powers become either r 2m+1 , r, r, r 1−2m or r, r 2m+1 , r 1−2m , r.
Notice also that the coefficient γ in (4.50) also vanishes as γr log(r) e ±mθ / ∈ Ker(L m ) (and using the same argument as in the proof of Theorem 5.4). So we have a remaining coefficient in The last expansion of u i ε follows directly from this estimate using (4.47).
Finally, we obtain in the following theorem the expansion as ε → 0 of the previously obtained function u i ε . Notice the shift of notation for v i ε .
In particular, for all Proof. The first claim on v i ε follows directly from the uniform bound (4.17) and a standard diagonal argument. Furthermore, Finally, the expansion in U j follows from Theorem, as so as ε → 0, by the strong convergence γ i,j,k,l,ε → γ i,j,k,l ∈ C and we get the expected expansion. Finally, the indicial root analysis shows that and this concludes the proof of the theorem.

Remark 4.11.
We emphasize that the variations v i 0 ∈ W 2,2 (Σ) are admissible at a branch point p ∈ Σ of order θ 0 ≥ 1 corresponds to an end p j (for some 1 ≤ j ≤ n) of multiplicity m = θ 0 ≥ 1, and the previous theorem shows that in so these variations are indeed admissible by the discussion in Section 3. For more details on this important technical point, we refer to [23]. Notice that in general, at a branch point of multiplicity m ≥ 2, we have while for m = 1, but v / ∈ C 1,1 (B(0, 1)) in general.

Definition 4.12. For all admissible variation
is the admissible variation of Ψ constructed in Theorem 4.10.

Explicit computation of the the singular energy
First recall the definition of flux of a complete minimal surface.
Definition 5.1. Let Σ be a closed Riemann surface, p 1 , · · · , p n ∈ Σ be fixed points and Φ : Σ → R d be a complete minimal surface with finite total curvature. For all 1 ≤ j ≤ n, we define the flux of Φ at p j by where γ ⊂ Σ \ {p 1 , · · · , p n } is a fixed contour around p j that does not enclosed other points p k for some k = j.
By the Weierstrass parametrisation, we have at an end of multiplicity m ≥ 1 for some and we compute Therefore, we have is a well-defined quantity independent of the chart.
Theorem 5.2. Let Σ a compact Riemann surface, Φ : Σ \ {p 1 , · · · , p n } → R 3 a minimal surface with n embedded ends p 1 , · · · , p n ∈ Σ and exactly m catenoid ends are chosen as in [22] (with respect to a fixed covering U 1 , · · · , U n of p 1 , · · · , p n ), and Proof. We make the computation of the residue at a catenoid ends, as the residue at a planar end will be the same if we simply take the formula by plugging 0 at the place of the catenoid residue. This simple fact will become clear at the end of the proof. there exists where a ∈ R 2 . Furthermore, in the remaining terms, as Q : W 2,2 (Σ) → R is continuous, and by Sobolev embedding theorem, W 2,2 (Σ) ֒→ C 0 (S 2 ) but does not embed in C 1 (Σ), we deduce that Q cannot depend on the higher derivatives of v at p i . Therefore, one only needs to compute and for a planar end, we have the same expression with β j = 0. This translates if u = | Φ| 2 v as and this gives which concludes the proof of the theorem.

Remark 5.3.
We can easily see that the preceding expression if well-defined directly, even is we already know that it is (as this quantity is the second variation of a compact Willmore surface for an admissible direction). Indeed, (1) and this proves by (5.3) that (5.1) makes sense.

Renormalised energy identity
In particular we deduce that the index cannot be more that n − 1.
Proof. We fix ε > 0 small enough such that the ball B 2ε (p i ) 1≤i≤n are disjoint, and we define the following symmetric bilinear form B ε : Step 1 : Estimation of Q ε (u ε ). We first remark that Q ε (u ε ) cannot depend on the derivatives of v at p 1 , · · · , p n by Sobolev embedding theorem. Therefore, each time we differentiate v i ε , we know that analogous cancellations as observed by the explicit computations in [22] will actually make these residues vanish as ε → 0. Whenever one of these terms occur, we shall neglect them.
We fix a chart D 2 → B r (p i ). We recall that close to p i , we have Then we deduce by the Dirichlet boundary condition that and and as as we can neglect all terms containing derivatives of v, we can replace v by v(p i ) and replace ∂ ν v by 0, which gives Furthermore, we note that if we obtain finally Now thanks to the asymptotic behaviour of v i ε 1≤i≤n , we know that B ε (u ε , u i ε ), and B ε (u i ε , u j ε ) are bounded terms, so for the energy to be finite, we must have and we get Step 2 : Estimation of B ε (u i ε , u j ε ) for i = j.
For all 1 ≤ i, j ≤ n, and k = i, j we have by Theorem 4.6 and likewise So we need only to consider the boundary integrals for B ε (p i ) and B ε (p j ). We have up to O(ε 3 log ε) error terms by (5.9) so by symmetry, we have From now on, we find useful to use complex notations. Recall the expansion on ∂B ε (p j ) Furthermore, as Furthermore, we have so we deduce that ∆ g u j ε = 4v(p j ) + O(|z| 2 log 2 |z|).
Furthermore, as K g = O(|z| 4 ), we have also , so we recover a weak form of (5.5) (however sufficient for our purpose here). Now we note that Now, notice that for all smooth ϕ : B(0, 1) → R, we have Therefore, we have (as z/z has no radial component) Therefore by symmetry (5.10) We note that these notations imply that for r > 0 small enough, for all 1 ≤ i ≤ n, for all j = i we have on any conformal chart Step 3 : Estimation of B ε (u ε , u i ε ) for 1 ≤ i ≤ n. We note that the boundary conditions imply that for all 1 ≤ i ≤ n, we have on ∂B ε (p i ) (for some where we used ∂ ν γ i z z = 0. As by the Remark 4.8 for all j = i, we have on ∂ ν B ε (p j ) and as on ∂B ε (p i ) where we used by obvious symmetry Therefore for all 1 ≤ i ≤ n, one has and finally which concludes the proof, as the last claim follows from the fact that which concludes the proof of the theorem.
We deduce from the preceding theorem an improvement of theorem Corollary 5.5. For all 1 ≤ i ≤ n, there exists λ i,j ∈ R such that for all j = i, for all 0 < ε < ε 0 , on every complex chart around p j there exists c i,j , d i,j ∈ C and b i,j ∈ R such that

Equality of the Morse index for inversions of minimal surfaces with embedded ends
Theorem 6.1. Let Σ be a closed Riemann surface, Φ : Σ \ {p 1 , · · · , p n } → R 3 be a complete minimal surface with finite total curvature and embedded ends , and Ψ : Σ → R 3 be its inversion. Assume that 0 ≤ m ≤ n is fixed such that p 1 · · · , p m are catenoid ends, while p m+1 , · · · , p n are planar ends, and for
Let v ∈ C 2 (Σ) be such that v(p i ) = 0 and consider u 0 = n i=1 u i 0 = | Φ| 2 n i=1 v i 0 obtained in Theorem 4.10. We assume for simplicity that the end is planar, as the computation for a catenoid end would be identical up to the addition of two extra terms. Recall now that in the chart U i around p i , we have for all j = i Therefore, we find Therefore, one needs to compute the renormalised energy for variations not only C 2 but also of the form given by (6.2). Let We will now compute Q Ψ (v) for the variation v in (6.3). Now, recall that at a planar end there exists β 2 0 > 0 and α 0 ∈ C such that As Φ is minimal, we deduce that Therefore, we have and (as |z| −2 Re (ζ 0 z) = Re (ζ 0 z −1 ) is harmonic) This implies that we have for some λ 0 , λ 1 ∈ C

This implies that
Im Now, we compute Therefore, we obtain and notice the constant (|ζ 0 | 2 − γ 0 ). Finally, we find for some λ 2 , λ 3 ∈ C Gathering (6.5) and (6.6) we obtain as K g = O(|z| 6 ) by planarity of the end (notice the factor 2 in front of the Laplacian) Now, coming back to (6.2), we see that for v 0 written above we have (with β i replace by β 0 ) Therefore, each end will bring this new contribution (6.8) by, so we obtain (as λ i,i = 0) Indeed, let u i ε the solution of (4.4) where v is replaced by v 0 . We notice as L 2 Furthermore, the computations of (6.4) imply that Therefore, we have on ∂B ε (p i ) for all 1 ≤ i ≤ n and finally which shows by the analysis of Theorem 5.4 the announced formula (6.1).

Jacobi fields associated to the Universal Matrix Λ
Theorem 7.1. Let Φ : Σ \ {p 1 , · · · , p n } → R 3 be a complete minimal surface with finite total curvature and embedded ends (not necessarily planar) and Ψ : Σ → R 3 be its inversion. Let {λ i,j } 1≤i,j≤n be the matrix with 0 diagonal of Theorem 5.4, and assume that λ i,j = 0. Then there exists a Jacobi field ω i,j 0 : Σ \ {p 1 , · · · , p n } → R such that ω i,j 0 for all k ∈ {1, · · · , n}, in every complex chart around p i there exists ζ k i,j ∈ C and µ k i,j ∈ R such that where δ i,k is the Kronecker symbol.
Proof. Fix a covering (U 1 , · · · , U n ) such that p i ∈ U i for all 1 ≤ i ≤ n and U i ∩ U j = ∅ for all 1 ≤ i = j ≤ n, and we assume that U i is a domain of chart for all 1 ≤ i ≤ n, i.e. that there exists a complex diffeomorphism f i : Notice that for a function u ∈ C 2,α (Σ \ {p 1 , · · · , p n }) admitting the following expansion in the chart (U i , f i ) for some λ, µ ∈ R and ζ, κ ∈ C. The constant λ does not depends on U i and f i , and µ does not depends on f i as a complex change of chart D 2 → D 2 fixing 0 is a rotation, while the expansion (7.1) is invariant under rotations. Now, assume that n ≥ 2 and fix 1 ≤ i = j ≤ n. For all ε > 0 small enough, there exists thanks to Theorem 4.9 a solution u i,j Furthermore, by the argument of Theorem 5.4, we have (this would also be true for catenoid ends up to an additional singular term in log, notice that v(p i ) = v(p j ) = 1 here) By the previous indicial root analysis of Theorem 4.6, for all k = i, j there exists λ k i,j ∈ R such that we have in f −1 k (B C (0, ε)) ⊂ U k the expansion Recall also that for all 1 ≤ l = k ≤ n, we have the expansion in Notice that these expansions imply that for all l = i, j, k we have on ∂B ε (p l ) the estimate which implies that for all l = i, j, k Now, recall that on ∂B ε (p i ) we have Therefore, we have Therefore, we deduce that we have on ∂B ε (p i ) thanks to the boundary conditions As for all c ∈ C and by symmetry Finally, using the expansion on ∂B ε (p k ) together with the previous estimates (7.2), (7.5) and (7.6), we deduce that By symmetry of λ p,q , we obtain by (7.7) and (7.9) for all i = j and k = i, j Thanks to (7.10), we have for all k = i, j on ∂B ε (p k ) Furthermore, as we have on ∂B ε (p i ) In particular, we have the estimates on ∂B ε (p i ) ∪ ∂B ε (p j ) (by symmetry of the previous estimates) We also easily deduce by the preceding arguments that for all k = i, j Finally, we find by (7.12) and (7.13) Therefore, we have which implies (by the proof of Theorem 4.10 and Fatou lemma) that (up to a subsequence) where ω i,j 0 is a Jacobi field, i.e. L g ω i,j 0 = 0. Furthermore, by (7.10) and (7.11), ω i,j 0 is bounded at p k for all k = i, while in U i (resp. U j ), we have an expansion of the form (7.14) In particular, we have . Furthermore, notice that we have near p j as e 2λ = α i |z| 4 1 + O(|z| 2 ) and K g = O(|z| 4 ) (at an embedded end, not necessarily planar). As L g ω i,j 0 = 0, we deduce from the expansion (7.14) that Therefore, we deduce that The conclusion of the theorem following by replacing ω i,j 0 by −ω i,j 0 .

Renormalised energy for ends of arbitrary multiplicity
for some v 0 ∈ W 2,2 (Σ, R) such that v(p j ) = 0 for all 1 ≤ j ≤ n. In particular, we have Proof. As previously, fix a some residues charts (U 1 , · · · , U n ) around p 1 , · · · , p n , and assume that p i has multiplicity m i ≥ 1 for all 1 ≤ i ≤ n, and fix some 1 ≤ i ≤ m. Then the same argument as in Theorem 9.1 shows that there exists α i > 0 and α l i , γ l i ∈ R and β i ∈ R such that for all smooth variation v (if In particular, we deduce that As the limit in (8.1) exists, retaking the notations of Theorem 5.4 we find that for all for some finite constant c, otherwise the limit would be +∞ or −∞ by taking variations non-zero at only one end, as the other contributions are only depend on quadratic expressions v(p i )v(p j ) for i = j, so they cannot involve singular terms which are all involving quadratic expressions v(p i ) 2 . Furthermore, by the Sobolev embedding theorem, we have For the sake of simplicity, we will remove the indices i of the multiplicities m i (1 ≤ i ≤ n).
First notice that for all k = i, j, we have if p k has multiplicity m k = m ≥ 2 (for m = 1 this was already treated previously) As e λ = α 2 k |z| −2(m+1) (1 + O(|z|)), and K g = O(|z| 2(m+1) ). Therefore, we have by the harmonicity of Therefore, we have This implies that As the indices i and j do not play any role, we also have Therefore, we have Therefore, we have as This implies that Notice that the quantity is bounded as ε → 0. Therefore, cancellations occurs as we integrate (8.3). Furthermore, there is a non-trivial contribution coming from (as Finally, recall if u i ε = e λ w i ε that in our fixed chart near p j (if p j has multiplicity m j = m ≥ 2) Re γ j 1 z m+k + γ 2 |z| m+1 + γ 3 |z| m+1 log |z| + O(|z| m+2 ).
Furthermore, we have on ∂B ε (p j ) is bounded, we deduce as previously that there exists ν i,j ∈ R such that Therefore, we deduce that (as we may have also ends of multiplicity 1, there is an additional error in and by symmetry, in i and j, we deduce that . Furthermore, by symmetry of the argument, we deduce that there exists λ 2 i,j ∈ R such that Likewise, we find by the previous argument and the proof of Theorem 5.4 that there exists λ 4 i,j ∈ R such that Combining (8.6) and (8.7), we deduce if Therefore, (8.2) and (8.8) show that and finally This concludes the proof of the theorem.
We deduce as previously the following corollary.

Corollary 8.2.
For all 1 ≤ i ≤ n, there exists λ i,j ∈ R such that for all j = i, for all 0 < ε < ε 0 , on every complex chart around p j there exists c i,j , c i,j,k,l ∈ C u i ε (z) = Re For ends of higher multiplicity m ≥ 2, we do not know a priori if λ i,j = λ i,j , where λ i,j ∈ R is given by Theorem. Nevertheless, the proof of Theorem 6.1 implies the following result.
Proof. As in the proof of Theorem 6.1, it suffices to compute the renormalised energy for variations v of the form (if p i has multiplicity m ≥ 1) v = v(p i ) + Re (γz m ) + · · · + |z| 2m γ log |z| + O(1).
where · · · indicate terms of lower order that |z| 2m log |z| and higher than |z| m . Now notice if u i 0 is the limit of u i ε that Therefore, as v i 0 = | Φ| −2 u i 0 is admissible, we just need to compute if u = u i 0 the additional constant term in ∂Bε(pi) coming from the addition of the log term. As log |z| is harmonic, we have which implies that the additional constant term in Im ∂Bε(pi) 2 (∆ g u + 2K g u) ∂u (8.9) is 2 Im Finally, the proof of Theorem 5.2 shows that if This identity completed the proof of the theorem.

Remark 8.4.
This is very likely that λ i,j = λ i,j , which would give the exact analogous of Theorem 6.1 for higher order ends, but the argument here does not permit to check this. An explicit computation for precise examples permits nevertheless to compute the additional contributions in λ i,j from λ i,j .

Morse index estimate for Willmore spheres in S 4
Recall that we have from [23] we have the formula (valid in D ′ (Σ)) for all weak immersion Φ ∈ E (Σ, R m ) In particular, as d Φ, d w g = −2 H, w , for a minimal surface ( H = 0), we obtain where A ( w) is the Simon's operator. Observe that the sign is different from the Jacobi operator L g of the associated minimal surface, acting on normal sections of the pull-back bundle Φ * T R m as . Specialising further to the codimension 1 case m = 3, as the minimal immersion that we consider is orientable, it is also two-sided and the unit normal furnishes a global trivialisation of the normal bundle so w = w n for some w ∈ W 2,2 (S 2 ) and we get and we recover the computation of [22]. We have the following generalisation of the afore-cited result to S 4 . Theorem 9.1. Let Ψ : S 2 → S 4 be a Willmore sphere, and n ∈ N such that W ( Ψ) = 4πn and assume that Φ is conformally minimal in R 4 . Then we have Ind W ( Ψ) ≤ n.
Proof. First, use some stereographic projection avoiding Ψ(S 2 ) ⊂ S 4 to assume that Ψ : S 2 → R 4 is a Willmore sphere. By Montiel's classification, let Φ : S 2 \ {p 1 , · · · , p n } → R 4 be the complete minimal surface Ψ : S 2 → R 4 is the inversion, which we assume centred at 0 ∈ R 4 up to translation. Thanks to the argument of [22], for all normal variation v ∈ E Ψ (S 2 , T R 4 ), we have Then at every end, we have an expansion (up to translation) Then A 0 , A 0 = 0 and we find that for some α j > 0 Thanks to the Sobolev embedding W 2,2 (S 2 ) → C 0 (S 2 ) and as W 2,2 (S 2 ) does not embed in C 1 (S 2 ) in general, we deduce that for all smooth v ∈ E Ψ (S 2 , T R 4 ), the residue ∂Bε(pj ) only depend on α j , ε > 0 and v(p j ), up to a negligible term as ε → 0 (it cannot depend on higher derivatives of v at p j ). Furthermore, as we can assume v to be a normal variation, we deduce that A 0 , v(p j ) = 0. In particular, we deduce that ∂Bε(pj ) by the same computation as in Theorem 5.2. The rest of the proof follows [22].

Remarks 9.2.
(1) In order to understand fully the indices of Willmore spheres in S 4 , one would also have to estimate the index of images of (complex) curves in P 3 coming from the Penrose twistor fibration P 3 → S 4 . (2) The same proof applies to R m for m ≥ 5 without any change, but as not all Willmore spheres are conformally minimal (or images of complex curves of P 3 through the Penrose twistor fibration), this result has little interest ( [19]).
Here, we show as in [22] that there is a well-defined notion of residues at ends of embedded minimal surfaces in arbitrary codimension.
We also compute and we obtain finally by Therefore, we have Therefore, we obtain which implies that Therefore, we obtain and concludes the proof of the Proposition.
10 Explicit renormalised energy for ends of multiplicity 2

Restriction on the Weierstrass parametrisation
Lemma 10.1. Let Σ be a closed Riemann surface, p 1 , · · · , p n ∈ Σ be distinct points and Φ : Σ \ {p 1 , · · · , p n } → R 3 be a complete minimal immersion with finite total curvature and zero flux. Suppose that U ⊂ Σ is a fix chart domain containing some point p i ∈ Σ (1 ≤ i ≤ n fixed), and that end p i has multiplicity m = 2. Let ( A 0 , A 1 ) ∈ C n \ {0} × C n be such that in a chart ϕ : U → D 2 ⊂ C such that ϕ(p i ) = 0 we have the expansion Then A 1 ∈ Span( A 0 ). and Therefore, we have e 2λ = 4β 0 We also compute thanks to (10.4) and (10.6) thanks to (10.4). Therefore, we compute As ∂B(0,ε) z k z l dz z = 2πi ε 2k δ k,l (10.8) where δ k,l is the Kronecker symbol, we directly obtain Im ∂B(0,ε) Now, we will compute the singular residue for an admissible normal variation v = v n ∈ E ′ Ψ (Σ, R 3 ). First assume that v ∈ C 4 (Σ). We see that thanks to the previous section v must admit the following development To simplify the different estimates, we see that it is equivalent to assume that v ∈ C 5 (Σ), so that v = v(p i ) + γ|z| 4 + 2 Re ζ 0 z 2 + ζ 1 z 2 z + ζ 2 z 3 + ζ 3 z 4 + ζ 4 z 3 z + O(|z| 5 ). Now, we see that all functions | Φ| 2 , e 2λ , v have the following general development for some m ∈ Z |z| 2m µ 0 + µ 1 |z| 2 + µ 2 |z| 4 + Re ν 0 z + ν 1 z 2 + ν 2 z 3 + ν 3 z 2 z + ν 4 z 4 + ν 5 z 3 z + O(|z| 5 ) and K g two (but up to O(|z| 3 ) order). Therefore, as the singular residue is a quadratic expression of derivatives of these functions and by (10.8), we see that may assume that ν 2 = ν 3 = ν 4 = ν 5 = 0 in all these developments. By abuse of notation all such coefficients appearing in computations will be taken equal to 0 (a formal notation would be to write an equalities (mod Span C (z 3 , z 3 , z 4 , z 4 , z 3 z, zz 3 )) instead of the equality symbols, but we find it both heavy and unnecessary). With these new conventions, we obtain for some constant β 3 ∈ R. Therefore, we have This implies that there exists some λ j ∈ C such that This finally implies that Im ∂B(0,ε) Here, we see that no constant term occurs. However, we will see that they do occur for other contributions of the residue. Let u : D 2 → R such that Then −∆u = e 2λ K g , and we have Therefore, first compute Furthermore, as −K g = e −2λ ∆u = 4e −2λ ∂ 2 zz u, and e 2λ = 4β 0 |z| 6 e 2u , we have Now, we compute Therefore, we have and Finally, we get as Therefore, we have as 16 Re (α 0 z) 2 = 8|α 0 | 2 |z| 2 + 8 Re Therefore, we have for some λ j ∈ C We find the appearance of the square remarkable. Now, recall that This implies that and as e 2λ = 4β 0 we also have Finally, we have Therefore, Finally, we have Finally, thanks to Lemma 10.1 and Remark 10.3, if n = 3 (recall that n is the ambient dimension if the immersed minimal surface Φ) we have β 1 = |α 0 | 2 , so Furthermore, notice that replacing |α 0 | 2 = β 1 we find by (10.10) We have proved the following result.
We write these coefficients α 2 k (U j , p j ).
Proof. The independence on the chart is clear as change of charts are rotations D 2 → D 2 under which the expression in (10.14) is unchanged as the α 0 , α 1 , α 2 are norms of coefficients scalar products of Φ in the expressed chart, so they are rotationally invariant.
Remark 10.5. Notice that these "residues" are not independent of U .
We will now state most the following theorems for spheres for simplicity.
Then we have for all v ∈ W 2,2 (Σ) ∩ C 4 (Σ) and for all normal We can improve Theorem 8.1 by showing that the diagonal coefficient of the universal matrix vanishes for ends of multiplicity 2.
Theorem 10.7. Let Σ be a closed Riemann surface, Φ : Σ \ {p 1 , · · · , p n } → R 3 be a complete minimal surface with finite total curvature and zero flux and Ψ : Σ → R 3 be a compact branched Willmore surface such that Ψ = ι • Φ and assume that the ends of Φ have multiplicity at most 2. Then there a universal symmetric matrix Λ = Λ( Ψ) = {λ i,j } 1≤i,j≤n with zero diagonal entries such that for all v ∈ W 2,2 (Σ) ∩ C 4 (Σ) and normal (admissible) for some v 0 ∈ W 2,2 (Σ) such that v(p j ) = 0 for all 1 ≤ j ≤ n. In particular, we have and if Ψ : Σ → R 3 is assumed to have no branched points, then Proof. We have already treated the case of embedded ends. Furthermore, as the expansion is universal (i.e. independent of the multiplicity, see the proof of Theorem 5.4), we only need to compute Furthermore, as we know that Q ε (u i ε ) is only a function of ε, α 0 = α 0,i , α 1 = α 1,i , α 2 = α 2,i and v(p i ), we can (by an abuse of notation) replace all terms ∂ ν v by 0 and v by v(p i ). Furthermore, as Furthermore, observe that for all smooth function f : B(0, ε) → R, we Now, recall that the volume form on ∂B(0, ε) is |z| Im (zdz) . Finally, we deduce that for all f, g ∈ C ∞ (B(0, ε)) Therefore, we have By the preceding remarks, we have Now, define w i ε = L g u i ε . Then one checks directly by the expansion (10.18) that for some constant C > 0 independent of ε as w i ε = 4v(p i )+O(|z|) in a conformal annulus around ∂B ε (p i ). In particular, as . Now, let α : Σ → R be a conformal parameter such that g = e 2α g 0 for some constant Gauss curvature metric g 0 of unit volume on Σ. Then we have where V = −2e 2α K g is a real-analytic Schrödinger potential (by the Weierstrass parametrisation for example). In particular, we have in the distributional sense ∆ g0 w i 0 + V w i 0 = 0, in D ′ (Σ \ {p 1 , · · · , p n }). As w i 0 ∈ L 2 (Σ, g 0 ) and V ∈ L ∞ (Σ), we have ∆ g0 w i ε ∈ L 2 (Σ, g 0 ). By an immediate bootstrap argument we obtain w i 0 ∈ C ∞ (Σ). Furthermore, by direct elliptic estimate thanks to Theorem, for almost all ε 0 > 0 small enough and 0 < ε < ε 0 , we have In particular, as w i 0 ∈ C ∞ (Σ), this implies that w i ε admits a Taylor expansion in the annulus (for some ε 0 fixed and small enough) B ε0 (p i ) \ B ε (p i ) of the form (the first term is given by (10.16)) w i ε = 4v(p i ) + γ 0 |z| 2 + γ 1 |z| 4 + 2 Re ζ 0 z + ζ 1 z 2 + ζ 2 z 2 z + ζ 3 z 3 + ζ 4 z 4 + ζ 5 z 3 z + O(|z| 5 ). so that no singular power Re (λz m z n ) for some n < 0 or m < 0 occurs (these coefficients depend a priori on ε but converge when ε → 0 so they are not singular in ε > 0 small enough). Now recall that This implies that Furthermore, a direct computation shows that Therefore, we have and ζ 5 is a function of α j and v(p i ), but this latter fact is of no importance. In particular, we deduce that w i ε reduces to for some unimportant µ j ∈ C. By (10.5), we have e 2λ = 4β 0 Therefore, we have for some ν j ∈ C |w i ε | 2 e 2λ = 4β 0 Now, we know that Otherwise, as u ε and u j ε (for all j = i) are independent of ε > 0, we would obtain in the limit an infinite quantity, although Q Φ (v) is finite, a contradiction. Now, we have by polar coordinates where the last equality comes from (10.21). Therefore, (10.22) gives the two equalities Now, by Cauchy's inequality, and as |α 0 | 2 = β 1 we have This implies that as expected. Now, as the factors in Re (z 3 ), Re (z 4 ) and Re (z 3 z) do not contributes to the renormalised energy in (10.19), we deduce by (10.24) that Now, recall that by (10.7) Therefore, we have (notice that the purely imaginary term cancels) −Re z · ∂ z | Φ| 2 = β 0 |z| 4 2 + 4β 1 |z| 2 + 2 Re 3α 0 z − α 1 z 3 + O(|z| 5 ) .
Therefore, we directly obtain (10.26) and finally by (10.25) and (10.26) so that no constant term occurs.
Remark 10.8. Notice that the absence of diagonal entries is a consequence (and is equivalent) that the minimal surface Φ : Σ \ {p 1 , · · · , p n } → R 3 has zero flux.

Estimates for some weighted elliptic operators
We fix an integer m ≥ 2. Let ω : R n → R + a measurable function and for all k ∈ N and 1 ≤ p < ∞ define the weighted Sobolev space By the classical Gagliardo-Nirenberg inequality, we have a continuous injection W k,p ω (R m ) ֒→ W k,p (R m ).

Admissible variations for branched Willmore surfaces
Recall ( [23]) that for a branched Willmore surface Ψ : Σ → R 3 we defined the index as the maximal dimension of the space of normal variations v = v n where v ∈ W 2,2 ∩ W 1,∞ (Σ) satisfying the additional conditions |d v| g ∈ L ∞ (Σ), ∆ ⊥ g v ∈ L 2 (Σ, dvol g ). (11.12) such that D 2 W ( Φ)( v, v) < 0. First, the condition |d v| g ∈ L ∞ (Σ) is really necessary to define the weakest notion of index (for continuous paths, see Lemma 3.11 [23]), and we want to show here that the second condition ∆ g v ∈ L 2 (Σ, dvol g ) cannot be relaxed in general. If p ∈ Σ is a branch point of Ψ of multiplicity θ 0 ≥ 2, taking a complex chart z : U ⊂ Σ → C such that z(p) = 0, there exists α > 0 such that e 2λ = 2|∂ z Φ| 2 = α |z| 2θ0−2 (1 + O(|z|)) , and ∆ g v ∈ L 2 (Σ, dvol g ) is equivalent to ∆v |z| θ0−1 ∈ L 2 (D 2 ). (11.13) Notice that this implies if v is smooth that v must have the following Taylor expansion (for some ζ ∈ C) v = v(p) + Re ζz θ0 + O(|z| θ0+1 ) (11.14) If L 2 is replaced by L ∞ in (11.13) we obtain the same expansion (see [26]), with a O(|z| θ0+1 log 2 |z|) error term instead. We will check that restriction to the case of smooth variations for the simplest example of plane with multiplicity m ≥ 1, the expansion (11.14) is the largest space for which the second derivative makes sense. Indeed, thanks to the pointwise conformal invariance of the Willmore energy, and recalling that for all branched immersions Ψ : Σ → R 3 which is the inversion of a complete minimal surface Φ : Σ \ {p 1 , · · · , p n } → R 3 (we state the result in codimension 1, but they would hold in general thanks to the expression of the second derivative of the Gauss curvature obtained in [23]) and then for all admissible normal variation v = v n Ψ such that D 2 W ( Ψ)( v, v) is well-defined, then where g = g Φ = Φ * g R 3 and u = | Φ| 2 v. In particular, if Σ ε is defined as previously (with respect to some arbitrary covering (U 1 , · · · , U n ) as the ends p 1 , · · · , p n ), then by Stokes theorem 2 (∆ g u + 2K g u) ∂u − ∂|du| 2 g (11.15) so the limit on the right-hand side of (11.15) exists and is finite for all admissible variation v as previously. Proof. We first have A 0 , A 0 = 0 as Φ is conformal ( ∂ z Φ, ∂ z Φ = 0), so we have as A 0 , B 0 = 0 the identity so we normalise for convenience 2| A 0 | 2 = 1 and we let γ = | B 0 | 2 > 0. Furthermore, notice that Φ(z), B 0 = | B 0 | 2 so that so Ψ is a sphere of multiplicity 2 of centre B 0 2| B 0 | 2 and radius 1 2| B 0 | .

Admissible smooth variation of the plane of multiplicity 2m
In general, if m ≥ 1 is any fixed integer, A 0 ∈ C 3 \ {0} is such that 2| A 0 | 2 = 1 and Φ : C \ {0} → R 3 is defined by then one checks easily that for all β ∈ R \ {0}, the normal variation v = v n, where v = v(0) + β|z| 2m ρ is not admissible (for some smooth radial cut-off function ρ identically equal to 1 in an open neighbourhood of 0 ∈ C), as and Im ∂B(0,ε) Finally, we have Im ∂B(0,ε)