key: cord-0553188-wz8tivbs
authors: Hirsch, Jonas; Kusner, Rob; Mader-Baumdicker, Elena
title: Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions
date: 2021-10-27
journal: nan
DOI: nan
sha: 035cbffa14a5adf7c969f607489d81b22e07cdf6
doc_id: 553188
cord_uid: wz8tivbs

We study complete minimal surfaces in $mathbb{R}^n$ with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy $mathcal{W}: =frac{1}{4} int|vec H|^2$. In codimension one, we prove that the $mathcal{W}$-Morse index for any inverted minimal sphere with $m$ such ends is exactly $m-3=frac{mathcal{W}}{4pi}-3$, completing previous work. We consider several geometric properties - for example, the property that all $m$ asymptotic planes meet at a single point - of these minimal surfaces and explore their relation to the $mathcal{W}$-Morse index of their inverted surfaces.

We explore the connection between geometric properties of complete minimal surfaces with embedded planar ends in R n and the Willmore Morse index of their inverted surfaces. The Morse index is the dimension of the maximal subspace of variations that locally decrease an energy to second order: in our case, the Willmore bending energy. Crucial to this connection is the Möbius invariance of the Willmore energy.

We begin with a short review of Willmore surfaces -that is, surfaces stationary for the Willmore energy -and their relation to complete minimal surfaces with finite total curvature. Given an immersed, closed surface f : Σ → R n , we define the Willmore bending energy as

where the mean curvature vector H is the trace of the second fundamental form of the immersion f . The symmetry group preserving W consists of the Möbius group of all conformal diffeomorphisms of the ambient space R n ∪ ∞ = S n , the group generated by inversions in spheres (see, for example [4, 43] ). A quantitative version of this invariance was pointed out by the second author [16] : composing f with the inversion ϕ x (y) = x + R 2 y−x |y−x| 2 yields

where θ(x) = ♯{f −1 (x)} is the number of preimages of x. Equality (1) has several consequences. The most important one for us is the following: consider a complete minimal surface with finite total curvature in R n ; then inversion about a point x ∈ R n off the surface yields a Willmore surface with a possible singularity at x. In fact, as proven by Bryant [5] in R 3 , the resulting Willmore surface can be extended to a smooth immersion across the singularity if and only if all ends of the minimal surface are embedded and planar -meaning each end is asymptotic to a single plane at infinity. The other possibility for an embedded end is that it has logarithmic growth at infinity [41] . In this case, the corresponding inverted surface is only of regularity C 1,α for every α < 1, but it is not C 1,1 (see [20, 21] ). More literature about the smoothness and extendability of the Willmore equation can be found in [22, 38, 3] .

The topology of Σ is important for the problem of classifying Willmore surfaces. Bryant proved [5] the remarkable fact that any compact genus zero Willmore surface arises as the inversion of a complete minimal surface with embedded planar ends. This is of course not true for higher genus surfaces: a counterexample is the Clifford torus, a minimal surface in S 3 whose stereographic projection to R 3 is a Willmore surface. It is in fact the minimizer of the W functional among tori in R 3 -subject of the long-standing Willmore conjecture -proven only recently by Marques and Neves [25, 26] . Since the Clifford torus is embedded, an inverted Clifford torus can have at most one end. It cannot be an inversion of a complete minimal torus with embedded planar ends in R 3 . (In fact, up to homothety, the end of any embedded surface after inversion at a non-umbilic point is asymptotic to the graph of the biharmonic function cos 2θ over the punctured plane in polar coordinates; this was the key observation in the first proof of existence for W-minimizers with higher genus [17, 2] .)

The result of Bryant [5] implies that the bending energies of Willmore spheres in R 3 are quantized: each such surface has W(f ) = 4πm, where m ∈ N is the number of embedded planar ends of the corresponding minimal surface. Not all numbers are allowed: there are no Willmore spheres for m = 2, 3, 5, 7 [6] , but there are Willmore spheres with W(f ) = 4πm with m = 2k for all k ∈ N \ {1}; there are also examples with m = 2k + 1 for k ≥ 4 (see the recent work of Heller [11] ). The number m = 1 corresponds to a plane which is the inversion of the round sphere, and the next possible number is m = 4.

In [6, Section 5] , Bryant analyzes the family of Willmore spheres (up to Möbius transformation of the ambient space) with W = 16π: it turns out that there is a four-parameter family of these, whose moduli space is studied in [19] via the spinor representation. Two examples among them were already known: the Rosenberg-Toubiana surface [40] , and the second author's explicit parametrization [15] of the minimal surface which gives rise after inversion to the Morin surface [35] . This Morin surface has a 4-fold orientation reversing symmetry, meaning that (after an appropriate rotation in R 3 ) it can be rotated by π 2 around the x 3 -axis obtaining the same surface with the opposite orientation, making it a good half-way model for a sphere eversion: the remarkable fact that a round sphere can be turned inside out without creasing or tearing.

We give a short introduction to this phenomenon and explain why a good half-way model is important for sphere eversions. The existence of a sphere eversion was proven by Smale [42] : the round sphere in R 3 with a given orientation is regularly homotopic to the round sphere with the opposite orientation, where two immersions f 0 , f 1 : Σ → R n are regulary homotopic if there is a C 1 −path of immersions F : Σ × [0, 1] → R n such that F (·, 0) = f 0 ,F (·, 1) = f 1 which induces a homotopy of the tangent bundles. Following Smale's discovery, many mathematicians contributed to this field [10, 1, 36, 34] . In particular, it was shown by Banchoff and Max that along any eversion the path of immersed spheres must develop a quadruple point [27] : there is an immersion where a point in the image has four preimages. It follows from an inequality of Li and Yau [24] that a closed surface with a quadruple point has W ≥ 16π. Morin (based on suggestions of Froissart, [35] ) found a surface with one quadruple point and the orientation-reversing four-fold symmetry. This symmetry is very important for constructing a sphere eversion, since a deformation from that surface to a round sphere starting in the direction vn (n is a unit normal of the closed surface) joined with the opposite deformation in −vn will automatically give a sphere eversion: a surface which does this job is the sought-for half-way model! In the early 1980s the second author had the idea (see [18] ) of a sphere eversion such that at each step of the deformation the immersions have the least possible Willmore energy: an optimal sphere eversion. Such an eversion needs a suitable half-way model with W = 16π energy, perturbing that surface so that the Willmore energy decreases and then starts the Willmore flow, that is, the gradient flow of W. So with a discretization of the second author's W-stationary Morin surface half-way model as the starting place, using the Brakke Evolver the authors of [9, 8] were able to compute an optimal (discretized) animation of the sphere eversion that Morin and Petit had described [35] . It remains an open problem whether, starting from the (suitably perturbed) Morin surface with W = 16π, the Willmore flow will lead to the round sphere, and thus produce a smooth optimal sphere eversion: this is sometimes called the 16π conjecture.

To start (half of) the optimal sphere eversion from the Morin surface halfway model, one first needs to perturb the Morin surface such that the Willmore energy decreases. The authors of [9, 8] computed numerically that there is a one-dimensional space of variations of the Morin surface decreasing the Willmore energy W to second order [9, Section 3]: that is, the Morin surface has Morse index one. This numerically indicated result was recently proven [12] by the first and third authors, who establish the following formula for the W-index of a Willmore sphere in R 3 :

where m is the number of planar ends of the corresponding minimal surface X : S 2 \ {p 1 , ..., p m } → R 3 , and d := span{ν(p 1 ), ..., ν(p m )} is the dimension of the span of the Gauß map of X at the ends; they also show d = 3 for m = 4, and 2 ≤ m ≤ 3 for general m.

One of the main results of this article is to prove d = 3 for complete minimal spheres in general. Our first result is the following, see Section 2: Theorem 1.1. Let X : S 2 \ {p 1 , ..., p m } → R 3 be a complete minimal sphere with embedded planar ends that is conformally parametrized over S 2 \{p 1 , ..., p m } where p 1 , ..., p m ∈ S 2 , m > 1, correspond to the ends of the surface. Let ν : S 2 → S 2 ⊂ R 3 be the holomorphic extension of the Gauß map of X. Then the asymptotic normals satisfy d := dim span{ν(p 1 ), ..., ν(p m )} = 3.

(

As a consequence [12] , the Morse index of the inverted surface ψ :

In [12] , the authors were able to draw conclusions for the Morse index of a Willmore sphere by studying area-Jacobi fields -functions in the kernel of the second order elliptic operator corresponding to the second variation of the area functional. Certain questions related to the geometry of complete minimal surfaces with embedded planar ends arose that are of independent interest. One such question is whether the asymptotic planes of the complete minimal surface always meet at a single point in R 3 . We answer this question partly in Section 3. In [15] , the second author constructs a family of symmetric minimal surfaces with m = 2p embedded planar ends. We call these surfaces minimal flowers. Using the spinor representation of minimal surfaces we prove the following:

.., p m } → R 3 be a complete minimal sphere with embedded planar ends that lies in the S 1 × SO(3, C)-orbit (see Section 3) of a minimal flower. Then all the asymptotic planes of the ends meet at one point.

Note that, in the case of four-ended surfaces, the whole moduli space of complete minimal spheres with embedded planar ends consists of the S 1 ×SO(3, C)orbit of the inverted Morin surface [19] . With this property in hand, we can simplify the proof [12] that each Willmore sphere with W = 16π has W-Morse index one. In Section 3, we also give a weaker condition than in [12] for a complete minimal surface with embedded planar ends of arbitrary genus to have W-Morse index at least one after inversion.

Another consequence of the computation for the second variation of W is a statement about what we call the conformal density at infinity. The classical density at infinity of a complete minimal surface is

which is m for a minimal surface with m embedded planar ends due to the monotonicity formula, see for example [13] . We suggest considering a different, more intrinsic conformal density at infinity. We use conformal coordinates with z(p i ) = 0 and X z dz = − a z 2 + Y (z) dz with a ∈ C 3 , a · a = 0, |a| 2 = 2 and Y is holomorphic at each end p i and define D ǫ (p i ) := z −1 (B C ǫ ). Then our computations in Section 4 show that the convergence of the related density quantity is stronger.

With the choice of coordinates made above we have that

Finally, in Section 5, we study the p-equivariant W-Morse index -that is, the maximal dimension for a p-symmetry-invariant subspace of variations decreasing the W to second order -of the minimal flowers with 2p ends described in [15] . In [12] , we already showed that the Morin surface (which is, after inversion, a minimal flower) has equivariant W-Morse index one, and thus the variational direction that decreases W to second order most rapidly must be the 2-symmetric variation. Here, we show that the p-equivariant Willmore Morse index of an inverted minimal flower is always 1 as well:

is a complete, immersed minimal sphere with 2p embedded planar ends. Assume further that Ψ has an orientation reversing 2p-fold rotational symmetry around an axis of symmetry going through 0 = Ψ(p i ) = Ψ(p j ). Under these conditions, there is precisely one variation on Ψ that is p-fold rotational symmetric. (So if p is prime, all other variations destroy the symmetry completely.)

We show in Section 5 the expected property that the eigenfunction of the lowest eigenvalue of the W-Jacobi operator on the Morin surface must have a sign change.

At this point we would like to mention recent work of Alexis Michelat [29, 30] . In [29] , he computed the second variation of the bending energy for a Willmore surface in a general ambient manifold. He also presented an explicit second variation formula for smooth variations of a Willmore immersion in R 3 that arises from an inverted minimal surface in R 3 with embedded planar ends. As a consequence, he proved that the Willmore index of a Willmore sphere in R 3 is at most m (the number of ends of the minimal surface as above). In more recent work [30] , Michelat studies the Morse index of branched Willmore spheres using another approach than ours. We would also like to mention the recent work of Rivière [39, 31] and Michelat [28] which approach the sphere eversion using a different method (involving perturbations of the Willmore functional and min-max methods).

author is funded by the DFG (MA 7559/1-1) and thanks the DFG for the support.

The following observation was already implicitly contained in [12] :

. . , p m } → R n be a complete minimal surface with finite total curvature and ends p i , i = 1, . . . , m. If m < n 2 , then X is contained in some affine R n−1 .

Proof. Let π i be the orthogonal projection onto the tangent space at the end p i . Consider the linear map A := (π 1 , . . . , π m ) that maps R n onto R 2m . If 2m < n, we have ker(A) = {0}. Let v ∈ ker A. Consider the function u := v · X. Since the coordinate functions are harmonic on X(Σ) and v ∈ π ⊥ i for all i, we conclude that u is a bounded harmonic function on Σ and therefore constant.

With a similar argument as used in the proof of Lemma 2.1 it had been shown [12] that for any minimal surface with finite total curvature one has j T p j Σ = T p 1 Σ if and only if X is a (multiple cover of) a plane. Hence for a non-trivial minimal surface the intersection contains at most a line. The above lemma shows that in case of spheres with more than 2 ends the intersection is generically trivial.

Proof. Suppose (6) contains the line, after a rotation we may assume it contains Re n . Furthermore we may identify S 2 with the Riemann sphere C and assume that X is conformally parametrized, i.e.

The assumption that e n ∈ T p j Σ for all j translates to ℑ(ϕ(p j )) = 0 for all j. Note that, in the case of R 3 , we have that

Since the intersection is one-and not two-dimensional there must be at least p j 1 , p j 2 with ϕ(p j 1 ) = ϕ(p j 2 ). Using a Möbius transformation of C we may assume that that j 1 = 1, j 2 = 2 and p 1 = 0, p 2 = ∞. Now we may find a sequence of λ k → 1 such that ℑ(ϕ(λ k p 3 )) = 0 for all k. Hence {(iϕ(λ k p j ), 1) : j = 1, 2, 3} correspond to three different complex planes in C n and so the sequence of minimal spheres

has trivial intersection of the ends and converges to X. This proves the theorem. Here, we use essentially that X has genus 0 so that we do not have to care about periods.

.., p m } → R 3 be a complete minimal sphere with finite total curvature and embedded planar ends at {p 1 , ..., p m }, m > 1. If ν : S 2 → S 2 ⊂ R 3 denotes the holomorphically extended Gauß map of X, then

Proof. Note again that m > 1 implies m ≥ 4 already. In the work [12] it was proven that the W-index of an inverted complete minimal with m embedded planar end is m − d and that d ∈ {2, 3}. If there were a complete minimal sphere X with m embedded planar ends and d = 2, the W-index of its inverted surface would be m − 2. By the procedure described above we know that Ind W = m − 3 on an open and dense set of W-critical spheres with W = 4πm satisfying d = 3. Thus, the existence of such a W-critical sphere X would contradict the lower semi-continuity of the W-index.

, be a complete minimal sphere with finite total curvature and embedded planar ends at {p 1 , ..., p m }, m ≥ 3. If ψ denotes the compact inversion of X -which is a Willmore sphere -we have that

Proof. In [12] it was proven that Ind W (ψ) ≤ km − d where d = (2 + k) − l and l is the dimension of the intersection of the tangent planes at the end, (6) . Since generically we have l = 0 and the index is lower semi-continuous, the result follows.

Remark Using the Weierstrass representation in R 4 it is an exercise to check that every minimal sphere with two planar embedded ends is (up to C-linear change of coordinates on C 2 ∼ = R 4 ) the Whitney sphere

This can be found in [14] . Furthermore, it had been shown in [7, Corollary 5] that the Whitney sphere is the minimizer of the Willmore energy in its homotopy class (see also [32] ), which implies Ind W (ψ W ) = 0. Hence, Corollary 2.4 holds true without the restriction on the number of ends.

Corollary 2.5. Let ψ : S 2 → R 3 be an unbranched Willmore sphere in R 3 that is not the round sphere. Then its Willmore Morse index is

Proof. By work of Bryant [5] we know that a Willmore sphere in R 3 is an inverted complete minimal surface with m ∈ N embedded planar ends and W(ψ) = 4πm, m ≥ 4. We use [12] , where the Morse index of a W-critical sphere ψ is computed to be Ind W (ψ) = m−d for d = dim span{ν(p 1 ), ..., ν(p m )}. Use Corollary 2.3 to get the claimed formula.

Let us call the symmetric complete minimal surfaces described in [15] the minimal flowers. They have m = 2p embedded planar ends and a 2p dihedral symmetry. For p = 2, the inverted minimal flower is an example of a Morin surface [35] . Proof. Suppose X and X * denote the m-fold dihedrally symmetric minimal flower and its conjugate surface, respectively, i.e. the real and imaginary parts of the holomorphic null curve Φ in C that can be found in [15] . We will repeatedly use the fact that the real and imaginary parts of the Ctangent vector to any complex null curve provide a basis for the tangent plane for the minimal surface (and for its conjugate minimal surface) with planar ends. From the m half-turn symmetries of X, we know that all its asymptotic planes pass through the origin of R 3 , so X is spiny. Its conjugate X * has the conjugate symmetries, i.e. reflections in planes through the origin, each of which is perpendicular to corresponding half-turn axis, and so it too has the property that all its asymptotic planes pass through the origin of R 3 . Thus Φ = X + iX * has the property that all its asymptotic complex lines pass through the origin of C 3 . This implies that the entire S 1 associate family of surfaces X t = ℜ(e it Φ) has the property of being spiny. It remains to understand the asymptotic planes of the SO(3, C) orbit of X. Let L be an asymptotic complex line of Φ and M ∈ SO(3, C), then the corresponding asymptotic complex line on MΦ is ML. This implies that the asymptotic complex lines of MΦ pass through the origin of C 3 . Thus for any M ∈ SO(3, C), the minimal surface X M = ℜ(MΦ) has all its asymptotic tangent planes passing through the origin of R 3 .

Corollary 3.3. Let Ψ : S 2 → R 3 be a compact immersed Willmore sphere with W(Ψ) = 16π. Then the inverted minimal surface X is spiny.

Proof. The moduli space of complete minimal spheres with four embedded planar ends consists of the S 1 ×SO(3, C)/SO(3, R) orbit of the inverted Morin surface, see [19] . By the previous theorem they are spiny. Proof. After translation the asymptotic planes of X pass through the origin.

We remind ourselves about the fact that the support function u arises by dilation of X, see also [37, Section 5] . The family X t := (1 + t)X, t ∈ (−ǫ, ǫ) is a smooth deformation of X. We note that 0 = H(t) is the mean curvature of X t . If we define

then 0 = d dt t=0 H(t) = 1 2 Lu, where L is the Jacobi operator of X. Since X t is a deformation that is tangential at the ends p 1 , ..., p m , we get that u(p i ) = 0.

With this geometric property of being spiny in hand we provide another proof of the fact that all Willmore spheres with 16π energy have Willmore Morse index one, see [12] .

For the following statements we need to fix some notation: Let X : Σ \ {p 1 , ..., p m } → R 3 be a complete minimal surface with m embedded planar ends, g = X * δ the pullback metric on Σ. The two-manifold Σ is compact. As in [12, Section 3] we fix local conformal coordinates z in a neighbourhood U around an each end p i such that z(p i ) = 0 ∀i = 1, ..., m. We choose the coordinates in such a way that

where a ∈ C 3 , a 2 = 0, |a| 2 = 2 and Y (z) is holomorphic and bounded. We set D ǫ (p i ) = z −1 (B ǫ ), where B ǫ ⊂ R 2 is the Euclidean ball in R 2 around 0 with radius ǫ.

We also introduce the notation and some results from the article [37] . In the following, we closely follow [37, Section 5] . We denote the extended Gauß map of X by ν : Σ → S 2 . The Jacobi operator of the area functional on X is Lu = ∆u + |∇ν| 2 u. We "compactify" L by multiplying g with a conformal factor λ ∈ C ∞ (Σ \ {p 1 , ..., p m }), λ > 0, that decays like |z| 4 around each end in the conformal charts:ĝ = λg. We can, for example, take λ so thatĝ agrees with the pullback metric of the inverted surface X |X| 2 (assuming that 0 ∈ image(X)). The decay of the conformal factor was chosen to extendĝ onto Σ, see see (5) in [37] . The Jacobi operator transforms likeL = 1 λ L and it is now a Schrödinger operator on Σ. Let B ⊂ C 2,α (Σ \ {p 1 , ..., p m }) be the space of functions u that have the form u(z) = β i ln |z| +ũ i (z) in the local conformal charts around each end p i , where β i ∈ R andũ i ∈ C 2,α (D ǫ (p i )). There is a natural way to equip B with a norm so that B is a Banach space, see [37, Section 5] . We define

where V ⊥ denotes the L 2 -orthogonal of V with respect toĝ. Note that K is the (classical) space of (bounded) Jacobi fields on X. Pérez and Ros proved the following statements about the defined spaces: Lemma 3.5 (Lemma 5.2 in [37] ). With the above notations, J , K and K 0 satisfy the following properties:

Theorem 3.6. Let X : Σ \ {p 1 , ..., p m } → R 3 be a complete minimal surface with m embedded planar ends and 0 ∈ image(X). If there exists a logarithmically growing Jabobi field u on X, i.e. dim J − dim K > 0. Then the W-index of the closed inverted Willmore surface ψ := X |X| 2 is at least one. Proof. The condition dim J − dim K > 0 means that there is a function u ∈ C 2,α (Σ \ {p 1 , ..., p m }) which satisfies the following properties: Lu = 0 u(z) = β i ln |z| +ũ i (z) locally around p i , for smooth functions u i ∈ C 2,α (D ǫ (p i )) and β i ∈ R ∃i : β i = 0.

We use the notation β := (β 1 , ..., β m ). If we knew that i β i = 0, then a version of the formula for the second variation of the Willmore functional would give us at least one direction that decreases the Willmore energy of the inverted surface X |X| 2 , see [12, Theorem 3.4] .

For that, we would define v := v 0 + u |X| 2 and compute δ 2 W(v, v). But we cannot guarantee the condition i β i = 0 a priori. Therefore, we need to "replace" v by a suitable function constructed out of u, where we "switch the sign" at one end. This was done already in Lemma 4.5 and Corollary 4.6 in [12] . For the convenience of the reader we repeat the proof here. For the logarithmically growing Jacobi field u we construct a function χ with the following properties: χ |X| 2 ∈ W 2,2 (Σ, dµĝ) and

• |∆ g χ| 2 = 16 on Σ;

• around each end p i it has in local conformal coordinates an expansion of the form

for some σ i ∈ {−1, +1} and some smoothχ i ;

• if σ = (σ 1 , . . . , σ m ) then σ · β = 0.

This was proven in [12, Lemma 4.5] . We repeat the most important steps: If i β i = 0, then we are done by defining χ := |X| 2 . If i β i = 0, then we proceed as follows: We can assume β 1 = 0. Then we will construct a function χ with the desired properties which has σ = (−1, 1, ..., 1). Then we have that σ · β = 0. To construct such χ, first find a function w that agrees with −|X| 2 in D ǫ 2 (p 1 ) and with |X| 2 on D ǫ (p 1 ) c by multiplication with an appropriate cut-off function. Then we define η(x) := 4(1 − 2 · 1 Dǫ(p 1 ) )(x) and f (x) := ∆ g w(x) − η(x). The function f was constructed in a way that it is supported in the annulus D ǫ (p 1 ) \ D ǫ 2 (p 1 ) and it is bounded. The same holds for the function 1 λ f , where λ is such thatĝ = λg. By Lax-Milgram on the compact manifold Σ the operator ∆ĝ = 1 λ ∆ g : W 2,2 (Σ, dµĝ) → L 2 (Σ) is onto. Let h ∈ W 2,2 (Σ, dµĝ) be the solution of ∆ĝ = 1 λ f . Then χ := w − h has all the desired properties. Final step: Computation of the second variation Define w t := χ + tu ∈ W 2,2 (Σ, dµĝ) and v t := wt |X| 2 . Note that χ = σ i |X| 2 +χ i and u = β i ln |z| +ũ i for smoothχ i ,ũ i around each end p i . Thus, we use the formula for the second variation into direction v t of the inverted surface ψ := X |X| 2 from Theorem 3.3 in [12] . For every small ǫ > 0 the formula reads

As |∇n| 2 g = −2K g is decaying very fast towards the ends (see (9) in [12, Section 3] ) and as the error term R ǫ converge to zero, we can pass to the limit ǫ → 0. The result is

We have constructed χ so that σ · β = 0 which implies δ 2 W(Ψ)(v t , v t ) < 0 for a suitable t ∈ R.

Corollary 3.7. Let ψ : S 2 → R 3 be a W-critical sphere with W(ψ) = 16π. Then the W-index of ψ is one.

Proof. After inversion of ψ we get a complete minimal sphere X with 4 embedded planar ends. It was proven by Montiel and Ros [33, Theorem 25 ] that such a surface has area-index 4 and area-nullity 5, i.e. dim K = 5. Corollary 3.3 shows that X and its conjugated minimal surface X * are spiny. We use Lemma 3.4 to see that the support function u = X ·ν of X and the support functionū = X * · ν of X * are linearly independent bounded Jacobi fields on X such that u(p i ) = 0 =ū(p i ), i = 1, ..., m. Note that we have used that X and X * have the same Gauß map which implies thatū is a Jacobi field of X. A way of seeing this is that the family X(t) = ℜ( e it Φ) for the null curve Φ = X + iX * is a deformation of X by complete minimal surfaces. As a consequence,ū = d dt X(t) t=0 · ν = −X * · ν is a Jacob field of X, see [37] . The mentioned deformation X(t) was studied in [40] . Using Lemma 3.5 (by Pérez and Ros) we have that dim K 0 = 2 and dim J − dim K = 6 − 5 = 1.

The previous theorem shows that the W-index of ψ is at least one. In [12, Proposition 1.7] we showed the upper bound Ind W (ψ) ≤ m − d = 1.

Remark The above approach uses the property of being spiny. Theorem 3.2 shows that the minimal flowers and certain deformations of them are spiny. To the best of our knowledge, it is unclear whether all minimal surfaces with embedded planar ends in R 3 are spiny. In the recent work [23] it is shown that there are minimal surfaces with embedded planar ends in R 4 that are not spiny. A minimal sphere with three embedded planar ends is constructed in R 4 where two of the three ends are parallel. This implies that the three asymptotic planes have an empty intersection. This article [23] also contains several interesting existence and non-existence results about minimal surfaces with embedded planar ends.

In the following we would like to explain that the formula for the second variation of an inverted minimal surface with embedded planar ends can be used to show a statement about the density at infinity of that minimal surface. Lemma 4.1. Let X : Σ → R 3 be a complete minimal surface, g = X * δ the pullback metric on Σ, n X the unit normal along X and Ψ = i • X = X |X| 2 (with corresponding normal n Ψ ). Let L g ϕ = ∆ g ϕ − 2K g ϕ be the Jacobi operator (for the area functional) of X. Then we have that

where n k Ψ = n Ψ · e k and n k X = n X · e k . Proof. For i(x) = x |x| 2 and Ψ = i • X the equation

follows from Di(x) = 1 |x| 2 Id − 2 x⊗x |x| 2 . Using (11) we get that

In the following, we compute everything in a conformal parametrization over U ⊂ C. We can choose for example the Weierstrass representation. We have that g = e 2λ δ and L g ϕ = e −2λ ∆ϕ−2K g ϕ for the Euclidean Laplacian ∆. As X is minimal and conformal, we know that ∆X k = 0, L g n k X = 0 and ∆ g |X| 2 = 4. We compute

where we use summation convention for doubly repeated indices. The second term gives

where we used

As ∂ j n X = (∂ j n X ) T = e −2λ ∂ j n X · ∂ m X ∂ m X we get that ∂ j n s X = −e −2λ n X · ∂ m ∂ j X ∂ m X s . We collect all terms and get

Corollary 4.2. Let Σ be a closed two-dimensional manifold and X : Σ \ {p 1 , ..., p m } → R 3 a complete minimal immersion with m embedded planar ends at p 1 , ..., p m . With the choice of the coordinates described in (9) we have that

where Σ ǫ := Σ \ m i=1 D ǫ (p i ). Proof. We shift the surface X such that 0 ∈ X(Σ). Consider the smooth Willmore surface X := X |X| 2 = i • X. Any component of the normal n k Ψ is a smooth W−Jacobi field because the Willmore functional is translation invariant. We use the formula for the second variation of W for a smooth variation proven by Michelat [29, Theorem 4.7] . Note that a term with the notation Res p (X, U) appears in Michelat's formula, see Definition-Proposition 4.6 in [29] . We have scaled our coordinates around an end in such a way that Res p (X, U) = 2, see also Remark after Lemma 3.1 in [12] . We also use Lemma 4.1 to get that

We sum this equation for k = 1, ..., 3 and get that

Remark

. Whereas the monotonicity formula for minimal surfaces implies that

is non-increasing and converges to mπ. Recall that for sufficiently large R 0 > 0, due to our assumption that all m ends are planar and embedded, we have

Each G u i denotes a minimal graph over the ith end π i satisfying the estimates

compare [41, Chapter 2, Proposition 1] . Hence a classical calculation shows that for ǫ << R −1

Thus one can consider Corollary 4.2 as an "improved convergence" estimate, if one allows to deform the euclidean ball B 1 ǫ appropriately.

Theorem 5.1. Let Ψ : S 2 → R 3 be a closed, immersed Willmore sphere such that X := Ψ |Ψ| 2 : S 2 \ {p 1 , ..., p 2p } → R n , p ∈ N, is a complete, immersed minimal sphere with 2p embedded planar ends. Assume further that Ψ has an orientation reversing 2p-fold rotational symmetry around an axis of symmetry going trough 0 = Ψ(p i ) = Ψ(p j ). Under these conditions, there is precisely one variation on Ψ that is p-fold rotational symmetric. Hence I acts as an orthogonal transformation on the eigenspaces of Z. Furthermore, since Z and I commute, we can can diagonalise Z and I simultaneously.

As I is the generator of a cyclic group, I 2p = 1, its eigenvalues are elements in {e i kπ p : k ∈ N}. We denote by E(λ) = ker(Z − λId) the eigenspace of Z corresponding to the eigenvalue λ.

Claim 1: Let σ be a real eigenvalue of I on E(λ), λ < 0, then σ = +1. We clearly must have σ ∈ {−1, 1}. Now the the same argument presented in [12, Proposition 4.7] applies and rules out −1: Due to [12, Proposition 2.6] we know that Ψ t (x) := Ψ(x) + tu(x) ν(x) can not preserve the 2p-fold symmetry. Nonetheless we have that S(Ψ t (x)) = S(Ψ(x)) + tu(x)S(ν(x)) = Ψ(I(x)) − tu(x)ν(I(x)) = Ψ(I(x)) − tσu(I(x))ν(I(x)) .

We conclude that σ = +1 because otherwise we would have S(Ψ t (x)) = Ψ t (I(x)). Additionally we conclude that S 2 (Ψ t (x)) = Ψ t (I 2 (x)).

Claim 2: ker(I − (+1)Id) ∩ λ<0 E(λ) is at most 1 dimensional. Suppose this is not the case. Then we can find two orthogonal eigenvectors u 1 , u 2 of Z in ker(I − (+1)Id). Since S generates the cyclic group i.e. I(p j ) = p j+1 for all j and u i • I = u i , we have that u i (p j ) = u i (p 1 ) for all j. We must have u i (p 1 ) = 0 because otherwise the variation Ψ t := Ψ + tu i ν would not dissolve the point of multiplicity 2p. And this would create the following contradiction:

Let α := u 1 (p 1 ) u 2 (p 1 ) and v := u 1 − αu 2 . By our choice of alpha we have v(p j ) = 0 for all j. But now, by the same argument as above, we find a contradiction:

where we used the variation Ψ t := Ψ + tvν.

Claim 3: If dim(E(λ)), λ < 0, is odd, then ker(I − (+1)id) ∩ E(λ) = {0}. We can diagonalise I on E(λ) over C. Since every complex eigenvalue µ of I comes in pair with its complex conjugateμ there must be at least one real eigenvalue σ. Now the claim follows by Claim 1.

The three claims together imply the proposition because the index of Ψ is 2p − 3, an odd integer. This follows from Corollary 2.5.

i ) It seems to be likely that the lowest eigenvalue of Z on Ψ is p-equivariant. At this point we cannot prove this statement. The above proof shows that finding a simple negative eigenvalue identifies the most symmetric one.

ii ) The existence of a p-equivariant negative variation can also be shown by using representation theory: A standard result from representation theory tells us that every irreducible real representation of a cyclic group is either one-or two-dimensional. If it is one-dimensional, then it is either trivial (meaning gu = u) or it is the "sign" representation (meaning gu = −u). Since the W-index of Ψ is odd, we get that there is at least one one-dimensional irreducible, real representation corresponding to the isometry I. The "sign" representation (i.e. the case u • I = −u) can be ruled out by the same argumentation as in Claim 1 in the proof above.

iii ) The "minimal flowers", see Section 3, found in [15] satisfy the condition of the theorem.

iv ) The proof shows that also other topological types with the symmetries described in the theorem above have one p-symmetric variation if their W-index is odd.

v ) The statement of the theorem above was conjectured in [8] , Section 1.

Another interesting question is whether the variations that produce negative variation -in particular the one(s) corresponding to the lowest eigenvaluehave a sign change. We can answer this question in the following case.

Proposition 5.2. Let Ψ : Σ → R 3 be a Willmore surface that arises as a complete minimal surface with m embedded planar ends X : Σ \ {p 1 , ..., p m } → R 3 . Let v ∈ C ∞ (Σ) be an eigenfunction corresponding to an eigenvalue of the Jacobi operator Z for W satisfying the condition v(p i ) = v 0 for all i = 1, ..., m.

If v is non-negative or non-positive, then λ ≥ 0.

Proof. Since the eigenfunctions are smooth on Σ, we can use the formula of Michelat [29] for the second variation of W. Using also v(p i ) = v 0 this formula reads ([29, Theorem 4.5] and [12, Remark after Lemma 3.1])

where w = |X| 2 v, and Σ ǫ := Σ \ m i=1 D ǫ (p i ) for our choice of conformal coordinates z, D ǫ (p i ) := z −1 (B ǫ ), see Section 3. Using ∆ g (|X| 2 v 0 ) = 4v 0 and (12) we compute

The term Σǫ ∆ g (|X| 2 (v −v 0 ))dµ g is a boundary term. As we are in the smooth setting, we get that lim ǫ→0 Σǫ ∆ g (|X| 2 (v − v 0 ))dµ g = lim ǫ→0 ∂Σǫ ∂(|X| 2 (v − v 0 )) ∂r = 0.

This can for example be seen by doing a Taylor expansion of v at p i and multiplying this expansion by |X| 2 . By subtracting v 0 from v we eliminated the leading term in |X| 2 (v − v 0 ), which then is only a linearly growing term towards the ends. It was for example shown in [29, Proof of Theorem 4.5] that only the quadratically growing term in the boundary term has a non-vanishing limit when ǫ → 0 in the smooth setting.

Combining the above calculations we get that

For a minimal surface we have that K g ≤ 0, which implies that δ 2 W(Ψ)(v, v) ≥ 0 if v has a sign. In other words, the eigenvalue λ cannot be negative. Proof. The W-index of the Morin surface is one due to either Corollary 2.5 or [12, Theorem 3.5] . Let u ∈ C ∞ (S 2 ) be the eigenfunction corresponding to the negative eigenvalue. Since Z is self-adjoint we can add elements of the kernel of Z (i.e. W-Jacobi fields) to u without changing the second variation: for every v := u + j, j ∈ kernel(Z), we have that δ 2 W(Ψ)(u, u) = δ 2 W(Ψ)(v, v). In [12, Theorem 2.3] we used known W-Jacobi fields on a Willmore spheres -namely those coming from the translation invariance of W -to show the upper bound Ind W (Ψ) ≤ m − d. For the Morin surface, this approach implies that we subtract suitable linear combinations of j = n Ψ , a for vectors a ∈ R 3 in order to arrange for v := u − 3 l=1 n Ψ , a l the property v(p 1 ) = v 0 > 0 and v(p 2 ) = ... = v(p 4 ) = 0. We will now use the symmetry conditions of Ψ. It can be computed that the unit normals of Ψ span a regular tetrahedron. In fact, in the parametrization of the second author from [15] , we get that n Ψ (p 1 ) = (c, 0, d), n Ψ (p 2 ) = (−c, 0, d), .., 4. We use the above Proposition 5 δ 2 W(ṽ,ṽ) ≥ 0 to get a contradiction ifṽ has a sign.

An algebraic halfway model for the eversion of the sphere

Existence of minimizing Willmore surfaces of prescribed genus

Singularity removability at branch points for Willmore surfaces

Vorlesungenüber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie

A duality theorem for Willmore surfaces

Surfaces in conformal geometry

Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form

Computing sphere eversions

The minimax sphere eversion

Arnold Shapiro's eversion of the sphere

Willmore spheres in the 3-sphere revisited

On the Index of Willmore spheres

Embedded minimal surfaces of finite topology

The geometry of the generalized Gauss map

Conformal geometry and complete minimal surfaces

Comparison surfaces for the Willmore problem

Estimates for the biharmonic energy on unbounded planar domains, and the existence of surfaces of every genus that minimize the squared-mean-curvature integral

Willmore bending energy on the space of surfaces

The spinor representations of surfaces in space

Global geometry of extremal surfaces in three space

Removability of point singularities of Willmore surfaces

Branch points of Willmore surfaces

Minimal surfaces in R 4 like the Lagrangian catenoid

A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Min-max theory and the Willmore conjecture

The Willmore conjecture

Every sphere eversion has a quadruple point

Morse Index Estimates of Min-Max Willmore Surfaces

On the Morse index of Willmore spheres in S 3

On the Morse index of branched Willmore spheres in 3-space

The Classification of branched Willmore spheres in the 3-sphere and the 4-sphere

Willmore two-spheres in the four-sphere

Schrödinger operators associated to a holomorphic map

Équations du retournement de la sphère

Le retournement de la sphere

Problématique du retournement de la sphère

The space of properly embedded minimal surfaces with finite total curvature. Indiana Univ

Analysis aspects of Willmore surfaces

Willmore minmax surfaces and the cost of the sphere eversion

Some remarks on deformations of minimal surfaces

Uniqueness, symmetry, and embeddedness of minimal surfaces

A classification of immersions of the two-sphere

Grundlagen der konformen flächentheorie

We would like to thank Karsten Große-Brauckmann for several illuminating discussions throughout the project, and we are also grateful to Alexis Michelat for pointing us to several recent references. The first author was partially supported by the German Science Foundation DFG in context of the Priority Program SPP 2026 "Geometry at Infinity". The second author is grateful to CIRM in Luminy and FIM at ETH Zürich for hosting his visit to Europe in May and June 2019 when this collaboration commenced, and enjoyed the support of KIMS and Coronavirus University during the pandemic. The third