Equivariant index bound for min–max free boundary minimal surfaces

Given a three-dimensional Riemannian manifold with boundary and a finite group of orientation-preserving isometries of this manifold, we prove that the equivariant index of a free boundary minimal surface obtained via an equivariant min–max procedure á la Simon–Smith with n-parameters is bounded above by n.


Introduction
The aim of this paper is to show that, through an equivariant min-max procedure à la Simon-Smith [Smi82] and Colding-De Lellis [CDL03] in a three-dimensional manifold with boundary, it is possible to obtain a free boundary minimal surface with equivariant index bounded above by the number of parameters of the sweepouts.
The min-max theory developed by Simon-Smith and Colding-De Lellis differs from the one by Almgren-Pitts [Alm65], [Pit81] and Marques-Neves [MN14], [MN17] since, in the former one, stronger regularity and convergence conditions are imposed on the sweepouts (cf.[MN14, Section 2.11]) and the ambient dimension is assumed to be equal to three.This makes the Simon-Smith approach less flexible; indeed, note for example that the sweepout constructed by Marques and Neves in [MN14] to prove the Willmore conjecture does not satisfy the Simon-Smith assumptions.
On the other hand, the stronger regularity makes proofs easier and enables to obtain a partial control on the topology of the resulting surface.This has been an advantage of the Simon-Smith approach when one wants to construct a minimal surface with a certain topological type.Indeed, the only way we know so far to obtain some information on the topology of the resulting minimal hypersurface in the Almgren-Pitts min-max theory is to first control the index and then use that the index bounds the topology (see for example [ACS18b], [Max18], [Son19] and [ACS18c] for the free boundary case), although such bounds (even when effective) are far from sharp.
Luckily, many proofs working in the Almgren-Pitts setting can be adapted to the Simon-Smith one.This is also the case for the upper bound on the index, proved in [MN16] for the Almgren-Pitts min-max theory (see also [MN21], and [GLWZ21] for the free boundary setting).Indeed, this paper consists in adapting those arguments to our "smoother" setting (see [MN16, Section 1.3]), but with the additional presence of a group of isometries imposed to the min-max procedure.What we obtain is an upper bound on the equivariant index of the surface resulting from this equivariant min-max procedure.Let us now describe precisely the statement of this result.To this purpose, we first give some preliminary definitions.
1.1.Setting and definitions.The equivariant min-max theory considered in this paper has been mostly developed by Ketover in [Ket16a] for the closed case and in [Ket16b] for the free boundary one, as proposed by Pitts-Rubinstein in [PR88].One of its primary goals is to construct new families of minimal surfaces in S 3 and of free boundary minimal surfaces in B 3 .Indeed, encoding the right symmetry group in the min-max procedure allows us to produce surfaces with fully controlled topology, as done by the author in joint work with Carlotto and Schulz in [CFS20].
Our setting is a three-dimensional compact Riemannian manifold (M 3 , γ) with strictly mean convex boundary and with a finite group G of orientation-preserving isometries of M .
Remark 1.1.We assume the manifold M to have strictly mean convex boundary because we want it to satisfy the following property.
(P): If Σ 2 ⊂ M is a smooth, connected, complete (possibly noncompact), embedded surface with zero mean curvature which meets the boundary of the ambient manifold orthogonally along its own boundary, then ∂Σ = Σ ∩ ∂M .
Here we ask for more, namely, the strict mean convexity of the boundary of the ambient manifold (which implies (P)), because we want it to be an open property with respect to the class of smooth metrics.
Remark 1.2.Li in [Li15] studies the free boundary min-max problem without curvature assumptions on the ambient manifold M .However, as remarked in [CF20, Appendix A] (see also [CF20,pp. 5]), there are problems in defining the Morse index for free boundary minimal surfaces that are not properly embedded in M .Moreover, the compactness results in [ACS18a], which we use for example in Appendix B, need property (P).That is another good reason why we require (P) to hold in this paper.
Remark 1.3.Here we assume the group G to be finite, but it is due mentioning that the equivariant min-max theory has been developed also in the case of a compact connected Lie group with cohomogeneity not 0 or 2 in [Liu21] and in the Almgren-Pitts setting with a compact Lie group of cohomogeneity greater or equal than 3 in [Wan20].
In what follows, we denote by I n = [0, 1] n the product of n-copies of the unit interval.Moreover, when we say that Σ m is a hypersurface (or surface if m = 2) in an ambient manifold M m+1 , we mean that Σ is smooth, complete and properly embedded, i.e., ∂Σ = Σ ∩ ∂M .Definition 1.4.A family {Σ t } t∈I n of subsets of a three-dimensional Riemannian manifold M is said to be a generalized family of surfaces if there are a finite subset T of I n and a finite set of points P of M such that: (i) t → Σ t is continuous in the sense of varifolds; (ii) Σ t is a surface for every t ∈ T ∪ ∂I n ; (iii) for t ∈ T \ ∂I n , Σ t is a surface in M \ P .
Moreover, the generalized family {Σ t } t∈I n is said to be smooth if it holds also that: (iv) t → Σ t is smooth for t ∈ T ∪ ∂I n ; (v) for t ∈ T \ ∂I n , Σ s → Σ t smoothly in M \ P as s → t.
Finally, we say that {Σ t } t∈I n is a G-sweepout if it is a smooth generalized family of surfaces and Σ t is G-equivariant for all t ∈ I n , i.e., h(Σ t ) = Σ t for all h ∈ G and t ∈ I n .
Remark 1.5.Slightly different variations of this definition are given in several references (see for example [CDL03, Definition 1.2], [DLP10, Definition 0.5], [CGK18, pp. 2836]).In order to have the regularity and the index bound on the resulting surface, it is sufficient to consider generalized families of surfaces, however smoothness is needed for the genus bound (see [DLP10, Definition 0.5]).
Definition 1.6.We say that a smooth map Φ : Then the min-max width of Π is defined as Remark 1.8.Note that here we consider isotopies Φ such that Φ t : M → M is a diffeomorphism of the ambient manifold.This choice does not coincide with the definition in [Li15], where outer isotopies are considered (see also Remark 1.2).
Remark 1.9.Note that the uniform bound on the "bad points" (the points belonging to P in Definition 1.4), which is required in [CDL03, Remark 1.3] (see also [DLP10, Remark 0.2]), is trivially satisfied for the G-saturation of a G-sweepout.
1.2.Main result.We obtain the following equivariant min-max theorem, which is the combination of regularity, genus bound and equivariant index bound, the latter being the main object of this paper.Indeed, the regularity and the genus bound were addressed in [Ket16b, Theorem 3.2] (see also [Ket16a,Theorem 1.3], where Ketover dealt with the closed case, for which many arguments are similar).
Theorem 1.10.Let (M 3 , γ) be a three-dimensional Riemannian manifold with strictly mean convex boundary and let G be a finite group of orientation-preserving isometries of M .Let {Σ t } t∈I n be a G-sweepout and let Π be its G-saturation.Assume that Then there exist a minimizing sequence {{Σ j t } t∈I n } j∈N ⊂ Π (i.e., lim j→∞ sup t∈I n H 2 (Σ j t ) = W Π ) and a sequence {t j } j∈N ⊂ I n such that {Σ j = Σ j t j } j∈N is a min-max sequence (i.e., lim j→∞ H 2 (Σ j ) = W Π ) converging in the sense of varifolds to Ξ := k i=1 m i Ξ i , where Ξ i are disjoint free boundary minimal surfaces and m i are positive integers.
Moreover, the G-equivariant index of the support spt and the following genus bound holds where O is the set of indices i ∈ {1, . . ., k} such that Ξ i is orientable and N is the set of indices i ∈ {1, . . ., k} such that Ξ i is nonorientable.
The precise definition of equivariant index is given in Section 3, but it is just what one would expect, i.e., the maximal dimension of a linear subspace of the G-equivariant vector fields where the second variation of the area functional is negative definite.
Remark 1.11.The assumptions that M has dimension 3 and that G consists of orientation-preserving isometries are required in [Ket16a] and [Ket16b], which we use for the regularity and genus bound in Proposition 5.9 (see also Remark 5.10).In this paper, such assumptions are used only in Section 4. Elsewhere the arguments work for any finite group of isometries G and any ambient dimension 3 ≤ m + 1 ≤ 7.
Remark 1.12.The equivariant index estimate in Theorem 1.10 implies the analogous result in the closed case, since, if we repeat the proofs forgetting about the boundary, everything works in the same way (actually more easily).
1.3.Applications.As a direct application, we deduce that the family of free boundary minimal surfaces in B 3 constructed in [CFS20] have dihedral equivariant index equal to one.
Remark 1.14.Note that the phrase "we can assume" in the statement is due to the fact that in Theorem 1.10 we prove the existence of a surface with a bound on the equivariant index, but we do not show that every surface obtained from the min-max procedure has this property.Therefore, Theorem 1.10 has to be applied in place of the min-max theorem used at the beginning of Section 4 in [CFS20].
Remark 1.15.Even if, for simplicity, we state the result only for the surfaces M g constructed in [CFS20, Theorem 1.1], Theorem 1.10 can be applied to any surface obtained via an equivariant minmax procedure fitting the framework defined above.See Remark 8.2 for further details on this matter.
Besides its own interest, Theorem 1.13 is meant to be a first step toward the computation of the (nonequivariant) Morse index of the family of surfaces M g (or possibly any another family arising from an equivariant min-max procedure, even with multiple parameters in play).This would be particularly relevant, as the only free boundary minimal surfaces in B 3 for which we know the index are the equatorial disc (with index 1) and the critical catenoid (with index 4, see [Dev19], [SZ19], [Tra20]).The idea is that the equivariant index gives information on the index of any of the isometric parts (which are 2(g + 1) in the case of M g ) of the equivariant surface.This problem will be addressed elsewhere.
Acknowledgements.The author wishes to thank her supervisor Alessandro Carlotto for many useful comments and discussions, and Tongrui Wang for helpful observations on the first version of this manuscript.This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 947923).

Basic notation
Let (M m+1 , γ) be a compact Riemannian manifold with boundary and let Σ m ⊂ M be a hypersurface.Then we denote by: • X(M ) the set of vector fields on M tangent to ∂M , namely X ∈ X(M ) if and only if X(x) ∈ T x ∂M for all x ∈ ∂M .In what follows, we consider (tangent) vector fields in X(M ) often omitting the adjective "tangent".• X ⊥ ∈ Γ(N Σ) the normal component to Σ of a vector X ∈ X(M ), where Γ(N Σ) denotes the sections of the normal bundle of Σ. • D the connection on M , ∇ the induced connection on Σ and ∇ ⊥ the induced connection on the normal bundle of Σ. • Ric M the Ricci curvature of M .
• η the outward unit conormal vector field to ∂M (which coincides with η along ∂Σ when Σ satisfies the free boundary property).
second fundamental form of Σ ⊂ M and |A| 2 its Hilbert-Schmidt norm.

Equivariant spectrum
In this section (M m+1 , γ) denotes a compact Riemannian manifold with boundary and G denotes a finite group of isometries of M .Definition 3.1.A (tangent) vector field X ∈ X(M ) is G-equivariant if h * X = X for all h ∈ G.We denote by X G (M ) the set of all G-equivariant vector fields.
G (M ) the set of 2-dimensional G-equivariant varifolds supported in M , endowed with the weak topology.
Recall that V 2 G (M ) is metrizable and we denote by F a metric metrizing it (see [Pit81,pp. 66] or [MN14, pp.703]).Moreover, given a varifold V in M , we denote by V the Radon measure in M associated with V .
Definition 3.4.Given a varifold V in M and a vector field X ∈ X(M ), we denote by δV (X) the first variation of the area of V along X, namely where Φ : [0, +∞) × M → M is the flow generated by X.
Remark 3.5.By the principle of symmetric criticality by Palais (see [Pal79], or [Ket16a, Lemma 3.8] for the result in this setting), we have that V ∈ V 2 G (M ) is G-stationary if and only if it is stationary.Now, let Σ m ⊂ M be a compact G-equivariant hypersurface and let Φ : [0, +∞) × M → M be the flow associated to a G-equivariant vector field X ∈ X G (M ), then we have that where η is the outward unit conormal vector field to ∂Σ.Moreover, if Σ is stationary (namely a free boundary minimal hypersurface), we have where Let Γ G (N Σ) denote the sections of the normal bundle of Σ obtained as restriction to Σ of Gequivariant vector fields in M .Then, the G-equivariant (Morse) index ind G (Σ) of Σ is defined as the maximal dimension of a linear subspace of Γ G (N Σ) where Q Σ is negative definite.
Remark 3.6.Note that, given a G-equivariant submanifold Σ, any G-equivariant vector field defined along Σ can be extended to a G-equivariant vector field on the ambient manifold M .Indeed, we can first extend it to a vector field X ∈ X(M ) (not necessarily G-equivariant) and then define 3.1.Equivariant index for two-sided hypersurfaces.If Σ m ⊂ M m+1 is two-sided, given a Gequivariant section Y ∈ Γ G (N Σ) of the normal bundle, we can write it as Y = uν, where u ∈ C ∞ (Σ) and ν is a choice of unit normal to Σ.Then, the second variation of the volume of Σ along Y is given by where L Σ := ∆ + Ric M (ν, ν) + |A| 2 is the Jacobi operator associated to Σ. Now let us further assume that Σ is connected.Note that, if Y = uν is G-equivariant, for all h ∈ G we must have and Σ is connected (thus the sign does not depend on the point on Σ).Note that sgn Σ (h 1 • h 2 ) = sgn Σ (h 1 ) sgn Σ (h 2 ).This leads to the following definition.
Definition 3.7.Given a compact connected Riemannian manifold Σ m , a finite group G of isometries of Σ and a multiplicative function sgn Σ : G → {−1, 1}, we define the spaces The G-equivariant (Morse) index of Σ coincides with the maximal dimension of a subspace of The G-equivariant index equals the number of negative eigenvalues.
Remark 3.9.Note that the assumption of connectedness of Σ is not restrictive.Indeed, if Σ consists of several connected components Σ 1 , . . ., Σ k for some k ≥ 2, then we can consider each connected component separately and ind on the G-equivariant sections Γ G (N Σ) of the normal bundle, admits a discrete spectrum λ 1 ≤ λ 2 ≤ . . .≤ λ k ≤ . . .→ +∞ and the G-equivariant index coincides with the number of negative eigenvalues.This follows from the two-sided case applied to the double cover of Σ.Indeed, let us consider the double cover π : Σ → Σ, then every isometry h ∈ G lifts to two isometries h1 , h2 of Σ, where h1 • h−1 2 is given by the isometric involution i : Σ → Σ associated to the universal cover.We denote by G the finite group generated by these isometries.Every Y ∈ Γ G (N Σ) lifts to a vector field Ỹ ∈ Γ(N Σ) and we can write it as Ỹ = uν, where ν is a global unit normal to Σ and u ∈ C ∞ G ( Σ). Vice versa, for all u ∈ C ∞ G ( Σ), the vector field Y := π * (uν) is well-defined and belongs to Γ G (N Σ).Hence we can just apply Theorem A.3 to Σ and G (as we did in the previous subsection) to obtain the desired properties on the spectrum of problem (3.1).

Free boundary minimal surfaces with bounded equivariant index
In this section, roughly speaking, we prove that a limit of free boundary minimal surfaces with bounded equivariant index satisfies the same bound on the equivariant index.
Throughout the section, we assume (M 3 , γ) to be a three-dimensional compact Riemannian manifold with boundary satisfying property (P), G to be a finite group of orientation-preserving isometries of M and S ⊂ M to be the singular locus of G, namely Moreover, given a free boundary minimal surface Σ 2 ⊂ M and an open subset U ⊂ M , we let µ 1 (Σ∩U ) denote the first eigenvalue of the problem (4.1) of the normal bundle (see Section 3).
Remark 4.1.We stress the fact that we use the letters λ and µ to denote the eigenvalues respectively taking and not taking into account the symmetry group.
Lemma 4.2 (cf.[Ket16a, Proposition 4.6]).In the setting above, let Σ 2 ⊂ M be a G-equivariant free boundary minimal surface in M and fix x ∈ Σ \ S.Then, there exists ε 0 > 0 such that, for all 0 < ε < ε 0 , the G-equivariant subset U ε (x) := h∈G h(B ε (x)) consists of exactly |G| disjoint balls {B ε (h(x))} h∈G , and it holds Proof.First, note that B ε (h(x)) = h(B ε (x)) since every h ∈ G is an isometry.Moreover, if ε 0 is sufficiently small (in particular smaller than half of the distance between x and h(x) for all h ∈ G), In particular, we get that the subset U ε (x) := h∈G h(B ε (x)) consists of exactly |G| disjoint balls.Now, we let ε > 0 be sufficiently small such that Σ ∩ B ε (x) is two-sided.Then, we can write any section and ν is a choice of unit normal, as in Section 3.1.Consider the first nonnegative eigenfunction ) is nonnegative and G-equivariant, we obtain that φ is the first eigenfunction relative to the eigenvalue µ 1 (Σ ∩ U ε (x)), and also relative to λ 1 (Σ ∩ U ε (x)).This implies that the three numbers Theorem 4.3.In the setting above, let {Σ k } k∈N be a sequence of G-equivariant free boundary minimal surfaces in M , with uniformly bounded area, such that ind G (Σ k ) ≤ n for some fixed n ∈ N. Then Σ k converges locally graphically and smoothly, possibly with multiplicity, to a free boundary minimal surface Σ ⊂ M \ (S ∪ Y) in M \ (S ∪ Y), where S ⊂ M is the singular locus of G and Y is a finite subset of M with |Y| ≤ n|G|.Furthermore, if there exists a G-equivariant free boundary minimal surface Remark 4.4.The previous theorem holds even when Σ k is a free boundary minimal surface with respect to a G-equivariant Riemannian metric γ k on M , for all k ∈ N, and the sequence {γ k } k∈N converges smoothly to γ.
Proof.Let us consider the set By Theorems 18 and 19 in [ACS18a], having uniformly bounded area, the surfaces Σ k converge locally graphically and smoothly in M \ (S ∪ Y) with finite multiplicity to a free boundary minimal surface Assume by contradictions that Y contains n ′ := n|G| + 1 distinct points x 1 , . . ., x n ′ ∈ M \ S.Then, there exists ε > 0 sufficiently small, such that the balls |G| ⌉, say U 1 , . . ., U n+1 , of the sets U 1 , . . ., U n ′ are disjoint.In particular, this contradicts the assumption ind Hence, we have that |Y| ≤ n|G|.
Let us now suppose that Σ extends smoothly to Σ in M , and assume by contradiction that ind G (Σ) > n.Then, there exist G-equivariant vector fields X 1 , . . ., X n+1 ∈ Γ G (N Σ) on Σ such that n+1 i=1 a i X i is a negative direction for the second variation of the area of Σ for all (a 1 , . . ., a n+1 ) ∈ R n+1 \ {0}.Since the isometries in G are orientation-preserving, Σ ∩ S consists of a finite union S 0 of isolated points and of a finite union S 1 = (Σ∩S)\S 0 of smooth curve segments (see [CHK00, Theorem 2.4] and [Ket16a, Lemmas 3.3, 3.4 and 3.5]).In particular, by Lemmas 3.4 and 3.5 in [Ket16a], for all x ∈ S 1 there exists h ∈ G such that h(x) = x and h * ν = −ν, where ν is a choice of unit normal to Σ at the point x.Therefore, by equivariance of X 1 , . . ., X n+1 ∈ Γ G (N Σ), we have that X i = 0 on S 1 for all i = 1, . . ., n + 1.As a result, thanks to a standard cutoff argument, we can assume without loss of generality that X 1 , . . ., X n+1 are compactly supported in M \ (S ∪ Y).Indeed, we can first suppose that X 1 , . . ., X n+1 are compactly supported in M \ S 1 , because every continuous function on a surface that is zero along a smooth curve can be approximated in the H 1 -norm with functions that are zero in a neighborhood of such curve (see e.g. the proof of Theorem 2 in [Eva10, Section 5.5]).Secondly, we can suppose that X 1 , . . ., X n+1 are zero in a neighborhood of the finite set S 0 ∪ Y using a standard log-cutoff argument.Observe that all these operations can be made in an equivariant way, because both S 1 and S 0 ∪ Y are G-equivariant.Since Σ k smoothly converges (possibly with multiplicity) to Σ in M \ (S ∪ Y), we thus get that ind G (Σ k ) > n for k sufficiently large, which contradicts our assumption.This proves ind G (Σ) ≤ n, as desired.

Outline of the proof of regularity and genus bound
In this section, we outline the results about the regularity and the genus bound for the surface obtained via a Simon-Smith equivariant min-max procedure.Besides being interesting on its own, we also need to unfold some of those arguments for the proof of Theorem 1.10.Indeed, we do not prove that the surface obtained from the min-max procedure has controlled equivariant index, but we show that it is possible to properly modify the proof in order to obtain a surface with the desired bound on the equivariant index.
We also include some detailed proofs for completeness, even if they are very similar to existing ones.This is because sometimes we need slight variations of existing results (for example here we consider n-dimensional sweepouts) or the proofs are split in many different references.
Definition 5.1.In the setting of Theorem 1.10, given a minimizing sequence of sweepouts Σ j = {Σ j t } t∈I n ∈ for j ∈ N (namely such that lim j→∞ sup t∈I n H 2 (Σ j t ) = W Π ), we define its critical set as 5.1.Min-max sequence converging to a stationary varifold.The first step is to show that there exists a min-max sequence converging to a stationary varifold, as stated in the following proposition.
Proposition 5.2 (cf.[Ket16a, Proposition 3.9]).In the setting of Theorem 1.10, given a minimizing sequence {Λ j } j∈N = {{Λ j t } t∈I n } j∈N , there exists another minimizing sequence {Σ j } j∈N = {{Σ j t } t∈I n } j∈N such that, if {Σ j t j } j∈N is a min-max sequence, then (up to subsequence) Σ j t j converges in the sense of varifolds to a stationary varifold.Moreover, it holds C({Σ j } j∈N ) ⊂ C({Λ j } j∈N ).
Proof.We include the proof for completeness.However, the argument is the same as in the proof of [CDL, Proposition 2.1], given the suitable modifications to the equivariant setting.
We first restrict to the compact space and we denote by V ∞ the subspace of G-stationary varifolds of X , which coincides with the subspace of stationary varifolds by Remark 3.5, and we define Y := X \ V ∞ .Now, for every V ∈ Y, consider a G-equivariant smooth vector field ξ V ∈ X G (M ) such that δV (ξ V ) < 0. Up to rescaling, we can assume without loss of generality that ξ a cover of Y and thus it admits a subordinate partition of unity {ϕ V } V ∈Y .Therefore, for every V ∈ Y, we can define the smooth G-equivariant vector field As a result, the map V → H V defined on Y can be extended to a continuous function X → X G (M ) by setting it identically equal to 0 on V ∞ .Then, for every V ∈ X , the flow Ψ The map τ : Y → [0, 1] is continuous, hence we can redefine H V multiplying it by τ (V ) in Y, i.e., we reset it to be equal to τ (V )H V in Y and equal to 0 elsewhere.Note that this gives a well-defined vector field X ∋ V → H V ∈ X G (M ), because τ is bounded in Y and therefore the newly-defined H V is continuous if we set it equal to 0 in V ∞ .As a result, we also redefine the flow Ψ V and it holds is C 1 in the parameter t; however, it is not necessarily smooth, hence { Σj t } t∈I n is not necessarily contained in Π.Therefore, for all j ∈ N, let us define a smooth map (t, x) → Φ j (t, x) by convolving Φj with a smooth kernel in the parameter t ∈ I n in such a way that (5.1) Note that Φ j (t, •) is G-equivariant, because we are convoluting in the parameter t and Φj (t, •) is Gequivariant.Then, define the G-equivariant surface Σ j t := Φ j (t, Λ j t ) for all j ∈ N and t ∈ I n .By smoothness of the map t → Φ j (t, •), we have that {Σ j t } t∈I n ∈ Π for all j ∈ N.Moreover, by (5.1), it holds that (5.2) lim As a result, using that H 2 ( Σj thus, a posteriori, all the inequalities are equalities; in particular lim sup j∈N sup t∈I n H 2 (Σ j t ) = m 0 .Our aim now is to prove that the minimizing sequence {Σ j } j∈N = {{Σ j t } t∈I n } j∈N has the properties claimed in the statement.Let t j ∈ I n be such that {Σ j t j } j∈N is a min-max sequence.Up to subsequence, Σ j t j converges in the sense of varifolds to a varifold V , which coincides with the limit in the sense of varifolds of Σj t j by (5.2).Possibly passing to a subsequence, we can also assume that Λ j t j converges in the sense of varifolds to some varifold W and, by continuity of the map Ψ, we have that First consider the case when W is not stationary.Then W (M ) = m 0 , but on the other hand it holds which is a contradiction.As a result W is stationary and therefore W = Ψ W (1, •) ♯ W = V , which implies that V is stationary, as desired.Moreover, since Λ j t j → W = V , we also get that C({Σ j } j∈N ) ⊂ C({Λ j } j∈N ).5.2.Min-max sequence G-almost minimizing in annuli.The second step is to prove that we can further assume that the min-max sequence found in the previous section has a certain "regularizing property", namely that of being G-almost minimizing in suitable subsets.The precise statements and definitions are given below.
and a G-equivariant surface Σ ⊂ U , we say that Σ is (G, ε)-almost minimizing in U if there does not exist any G-equivariant isotopy Ψ : [0, 1] → Diff(M ) supported in U such that A sequence {Σ j } j∈N is said to be G-almost minimizing in U if there exists a sequence of positive numbers ε j → 0 such that Σ j is (G, ε j )-almost minimizing in U .
Remark 5.4.Observe that the constants in the definition above differ from the ones in [CDL03, Definition 3.2].This is due to the fact that we are considering n-dimensional sweepouts and thus we need these different constants for the proof of Lemma 5.7.Definition 5.5.We say that a subset An ⊂ M is a G-equivariant annulus if there are x ∈ M and 0 < r < s such that An = g∈G g(An(x, r, s)), where An(x, r, s) = B s (x) \ B r (x) is the open annulus of center x, inner radius r and outer radius s.Note that, for r, s sufficiently small g∈G g(An(x, r, s)) consists of a finite union of disjoint open annuli (note that some annuli in the union could possibly overlap).Given x ∈ M , we denote by AN G r (x) the union of all G-equivariant annuli g∈G g(An(x, r 1 , r 2 )) as the radii r 1 , r 2 vary in the range 0 < r 1 < r 2 ≤ r.
We say that a family of G-equivariant annuli is L-admissible for some positive integer L, if it consists of L G-equivariant annuli An 1 , . . ., An L such that An i = g∈G g(An(x, r i , s i )) for some x ∈ M and r i , s i > 0 with r i+1 > 2s i for all i = 1, . . ., L − 1.
Definition 5.6.We say that a surface Σ ⊂ M is (G, ε)-almost minimizing in an L-admissible family A of G-equivariant annuli if it is (G, ε)-almost minimizing in at least one G-equivariant annulus of A. Moreover, we say that a sequence {Σ j } j∈N is G-almost minimizing in an L-admissible family A of G-equivariant annuli if there exists a sequence Lemma 5.7 (cf.[CGK18, Lemma A.1]).Let {Σ j } j∈N = {{Σ j t } t∈I n } j∈N be a minimizing sequence obtained from Proposition 5.2, then there exists t j ∈ I n for all j ∈ N such that {Σ j t j } j∈N is a G-equivariant min-max sequence that is G-almost minimizing in every L-admissible family of Gequivariant annuli, where L = (3 n ) 3 n .
Proof.Let {Σ j } j∈N be the minimizing sequence given by Proposition 5.2.Fixed any J > 0, we prove that there exist j > J and t j ∈ I n such that Σ j t j is (G, 1/J)-almost minimizing in every L-admissible family of G-equivariant annuli and H 2 (Σ j t j ) ≥ m 0 − 1/J.To this purpose, consider any j > J and define K := {t ∈ I n : H 2 (Σ j t ) ≥ m 0 − 1/J}.Assume that for all t ∈ K there exists an L-admissible family A t = {An 1 t , . . ., An L t } of G-equivariant annuli where Σ j t is not (G, 1/J)-almost minimizing, namely for all t ∈ K and i = 1, . . ., L there exists a G-equivariant isotopy Ψ i − 1/J.By continuity in the sense of varifolds of the map t → Σ j t , for every t ∈ K we can find a neighborhood U t ⊂ I n such that To this purpose, we follow [Pit81, Chapter 4]).For all m ∈ N, let I(1, m) be the CW complex structure, supported on the 1-dimensional interval Then consider the CW complex structure I(n, m) on the n-dimensional interval I n , whose p-cells for 0 ≤ p ≤ n can be written as σ 1 ⊗ σ 2 ⊗ . . .⊗ σ n , where σ 1 , . . ., σ n are cells of the CW complex I(1, m) such that n i=1 dim(σ i ) = p.For every cell σ = σ 1 ⊗ σ 2 ⊗ . . .⊗ σ n ∈ I(n, m), let T (σ) be the open subset of I n defined as T (σ) := T (σ 1 ) × T (σ 2 ) × . . .× T (σ n ), where Observe that Φ t is well-defined since An By arbitrariness of j, this would give lim inf j→+∞ sup t∈I n H 2 ( Σj t ) < m 0 , which is a contradiction.This proves that, for every J > 0, there exists j > J and t j ∈ I n such that Σ j t j is (G, 1/J)-almost minimizing in every L-admissible family of G-equivariant annuli and H 2 (Σ j t j ) ≥ m 0 − 1/J.In particular, we have that {Σ j t j } j∈N is a G-equivariant min-max sequence that is G-almost minimizing in every L-admissible family of G-equivariant annuli.5.3.Regularity and genus bound as a consequence of G-almost minimality.The third and final step is to show that the properties of the min-max sequence obtained in the previous sections are sufficient to prove that this sequence converges to a G-equivariant free boundary minimal surface (possibly with multiplicity) for which the genus bounded in Theorem 1.10 holds.
Proposition 5.9.In the setting of Theorem 1.10, given a G-equivariant min-max sequence {Σ j } j∈N that is G-almost minimizing in every L-admissible family of G-equivariant annuli for L = (3 n ) 3 n (as the one obtained in Lemma 5.7), there exists a G-equivariant function r : M → R + such that (up to subsequence) {Σ j } j∈N is G-almost minimizing in An ∈ AN G r(x) (x) for all x ∈ M .Moreover, {Σ j } j∈N converges in the sense of varifolds as j → +∞ to Ξ = k i=1 m i Ξ i , where Ξ i is a G-equivariant free boundary minimal surface and m i is a positive integer for all i = 1, . . ., k, and the genus bound in Theorem 1.10 holds.
Proof.Thanks to [CGK18, Lemma A.3], we have that there exists a G-equivariant function r : M → R + such that (up to subsequence) {Σ j } j∈N is G-almost minimizing in every An ∈ AN G r(x) (x) for all x ∈ M .Indeed, the only difference in the proof is that, instead of considering annuli An(x, r, s) centered at some point x ∈ M , here we have to consider G-equivariant annuli g∈G g(An(x, r, s)) "centered" at some point x ∈ M .
At this point the proof of the second part of the statement follows using the G-almost minimality of {Σ j } j∈N .Indeed, given a G-equivariant min-max sequence {Σ j } j∈N that is G-almost minimizing in An ∈ AN G r(x) (x) for all x ∈ M , we get that (up to subsequence) Σ j converges as j → +∞ to Ξ, which is a finite union of G-equivariant free boundary minimal surfaces (possibly with multiplicity), and the genus bound in Theorem 1.10 holds.The proof of the regularity of Ξ can be found in [CDL03, Theorem 7.1] for the closed case and in [Li15,Proposition 4.11] for the free boundary case.The proof of the genus bound is instead contained in [DLP10], [Ket19].The adaptations to the equivariant setting are the object of [Ket16a, Section 4, after Proposition 4.12] and [Ket16b, Section 7.2].
Remark 5.10.Observe that the difficulties in the equivariant setting arise around the points in the singular locus of G, i.e., points x ∈ M such that there exists h ∈ G with h(x) = x.Since we required the isometries in G to be orientation-preserving, we have that the limit surface can intersect the singular locus only in a point where, locally, the symmetry group is conjugate to Z k or D k for some k ≥ 2 (see [CHK00, Theorem 2.4] or [Ket16a, Lemma 3.3]).This is the reason why we add the assumption of "orientation-preserving" to the isometries in G, namely to exclude the case in which locally the singular locus is a plane.

Deformation theorem
In this section we prove that, in the setting of Theorem 1.10, it is possible to modify a minimizing sequence in a way that, so to say, it avoids a given free boundary minimal surface with G-equivariant index greater or equal than n + 1.The deformation theorem, Theorem 6.5, is the analogue of Deformation Theorem A in [MN16].Before presenting it, we recall three lemmas contained in [MN16] that are needed in the proof.Lemma 6.1.Let (M 3 , g) be a three-dimensional Riemannian manifold with strictly mean convex boundary and let G be a finite group of isometries of M .Given a finite union Θ 2 ⊂ M of G-equivariant free boundary minimal surfaces (possibly with multiplicity) with ind G (spt(Θ)) ≥ n + 1, there exist 0 < c 0 < 1, δ > 0 and a smooth family {F v } v∈B n+1 ⊂ Diff G (M ) of G-equivariant diffeomorphisms with: Proof.Using that ind G (spt(Θ)) ≥ n + 1, we can find G-equivariant normal vector fields X 1 , . . ., X n+1 on spt(Θ) such that for all (a 1 , . . ., a n+1 ) = 0 ∈ R n+1 .These vector fields can be extended to G-equivariant vector fields defined in all M (see Remark 3.6).Furthermore recall that, given the flow Φ : [0, +∞) × M → M of a G-equivariant vector field Y , we have that Φ t ∈ Diff G (M ) for all t ∈ [0, +∞) (see Remark 3.3).Given these observations, the proof follows exactly as in [MN16, Proposition 4.3], see also [MN16, Definition 4.1].
The following two lemmas coincide exactly with [MN16, Lemmas 4.5 and 4.4], since the Gequivariance does not play any role in these two results.We report them here for the sake of expository convenience.Lemma 6.2 ([MN16, Lemma 4.5]).In the setting of Lemma 6.1, for every G-equivariant varifold n+1 → B n+1 be the one-parameter flow generated by the vector field as defined also in [MN16,pp. 476].Then, for all 0 < η < 1/4, there is Theorem 6.5 (Deformation theorem).Let {Σ j } j∈N = {{Σ j t } t∈I n } j∈N be a minimizing sequence in the setting of Theorem 1.10.Moreover, assume that (i) Θ 2 is a finite union of G-equivariant free boundary minimal surfaces (possibly with multiplicity) Then there exist ε > 0 and another minimizing sequence for all j sufficiently large.
Proof.As aforementioned, the result is the analogue of Deformation Theorem A in [MN16] and the idea of the proof is to exploit the fact that we have many negative directions for the second variation of the area along Θ, hence it is possible to push the minimizing sequence {Σ j } j∈N away from Θ, keeping the fact that it is a minimizing sequence.Indeed, the sweepout is n-dimensional, while we have n + 1 negative directions.Since spt(Θ) has G-equivariant index greater or equal than n + 1, we can apply Lemma 6.1 and obtain 0 < c 0 < 1, δ > 0 and the family {F v } v∈B n+1 ⊂ Diff G (M ) given by the lemma.Moreover, up to modifying δ and {F v } v∈B n+1 , we can assume that (6.1) Fixed j ∈ N, define the open subset U j ⊂ I n given by Consider the continuous function m j : U j → B n+1 c 0 / √ 10 (0) given by m j (t) = m(Σ j t ), where the function m is defined in Lemma 6.1.Since dim U j = n < n + 1 = dim(B n+1 c 0 / √ 10 (0)), by the transversality theorem given e.g. in [Hir94, Theorem 2.1], there exists mj : U j → B n+1 c 0 / √ 10 (0) such that mj (t) = 0 and | mj (t) − m j (t)| < 2 −j for all t ∈ U j .Hence, consider the function a j : U j → B n+1 2 −j (0) given by a j (t) = m j (t) − mj (t) and note that a j (t) = m j (t) for all t ∈ U j .In particular, we can assume that there is η j > 0 such that |a j (t) − m j (t)| ≥ η j for all t ∈ U j (possibly taking δ, and so U j , smaller).Now, for all t ∈ U j , consider the one-parameter flow {Φ t,j (s, defined in Lemma 6.2 and given by the lemma.Then, given a nonincreasing smooth function ρ : [0, +∞) → [0, 1] that is 1 in [0, 3δ/2] and 0 in [7δ/4, +∞), we define the continuous function and then set is not necessarily smooth but only continuous.Anyway, let us first show that lim j→+∞ sup t∈I n H 2 (Λ j t ) ≤ W Π and that Λ j t ∩B F ε (Θ) = ∅ for all t ∈ I n , for j sufficiently large, where 0 < ε < δ has to be chosen.Later, we will describe a regularization argument to get a sequence of sweepouts with the same properties of {Λ j } j∈N , but also contained in Π.
Observe that H 2 (Λ j t ) = H 2 (Σ j t ) for t ∈ U j and, for t ∈ U j , we have However |ρ(F(Θ, Σ j t ))a j (t)| ≤ 2 −j , which implies that Now let us prove that Λ j t ∩ B F ε (Θ) = ∅ for all t ∈ I n , for j sufficiently large.Up to taking δ > 0 possibly smaller (note that Lemma 6.1 still holds for δ smaller), we can assume that |H 2 ( Θ) − H 2 (Θ)| ≤ c 0 /20 for all Θ ∈ B F 2δ (Θ).Then, let us distinguish three cases: , then we have v j (t) = Φ t,j (T j , a j (t)) and therefore, by Lemma 6.2, it holds where the last inequality holds for j sufficiently large.Hence, it is possible to choose ε > 0 possibly smaller (depending on Θ and c 0 ) such that this implies that F(Θ, Λ j t ) > ε (indeed note that c 0 does not depend on ε).

Proof of the main theorem
We now have all the tools to prove Theorem 1.10.Inspired by the proofs of [MN16, Theorems 6.1 and 1.2], the idea consists in repeatedly applying Theorem 6.5 in order to obtain a minimizing sequence in Π that stays away from the G-equivariant free boundary minimal surfaces with G-equivariant index greater than n.
Proof of Theorem 1.10.First of all, let us assume that the metric γ on the ambient manifold M is contained in B ∞ G , defined in Definition B.1, i.e., it is bumpy.Let us consider the set V n+1 of finite unions (possibly with multiplicity) of G-equivariant free boundary minimal surfaces in M with area W Π and whose supports have G-equivariant index greater or equal than n + 1.We want to prove that there exists a minimizing sequence {Σ j } j∈N ⊂ Π such that C({Σ j } j∈N ) ∩ V n+1 = ∅.First note that, since γ ∈ B ∞ G , the set V n+1 is at most countable thanks to Proposition B.3 (because V n+1 consists of finite unions with integer multiplicities of G-equivariant free boundary minimal surfaces).Therefore, we can write V n+1 = {Θ 1 , Θ 2 , . ..}.Now, the idea is to repeatedly apply Theorem 6.5 in order to avoid all the elements in V n+1 .
Let us consider a minimizing sequence {Σ j } j∈N and apply Theorem 6.5 with Θ = Θ 1 .Then we get that there exist ε 1 > 0, j 1 ∈ N and another minimizing sequence {Σ 1,j } j∈N ⊂ Π such that Σ 1,j t ∩ B F ε 1 (Θ 1 ) = ∅ for all j ≥ j 1 and t ∈ I n .Moreover, we can assume that no Θ k belongs to )); otherwise we apply Theorem 6.5 with Θ = Θ 2 and K = B F ε 1 (Θ 1 ).In both cases, we get ε 2 > 0, j 2 ∈ N and another minimizing sequence {Σ 2,j } j∈N ⊂ Π such that Σ 2,j t ∩ (B = ∅ for all j ≥ j 2 and t ∈ I n .Moreover, we can assume again that no Θ k belongs to ∂B F ε 2 (Θ 2 ).Then we proceed inductively for all Θ k 's and we have two possibilities: • The process ends in finitely many steps.In this case there exist m > 0, a minimizing sequence {Σ m,j } j∈N ⊂ Π, ε 1 , . . ., ε m > 0 and j m ∈ N such that for all j ≥ j m and t ∈ I n and V n+1 ⊂ B F ε 1 (Θ 1 ) ∪ . . .∪ B F εm (Θ m ).• The process continues indefinitely.In this case for all m > 0 there exist a minimizing sequence {Σ m,j } j∈N ⊂ Π, ε m > 0 and j m ∈ N such that Σ m,j t ∩ (B = ∅ for all j ≥ j m and t ∈ I n and no Θ k belongs to ∂B F ε 1 (Θ 1 ) ∪ . . .∪ ∂B F εm (Θ m ).In the first case we define Λ i = Σ m,i , while in the second case we set Λ i = Σ i,l i for all i ∈ N, for some l i ≥ j i such that {Λ i } i∈N ⊂ Π is a minimizing sequence and C({Λ i } i∈N ) ∩ V n+1 = ∅.Hence, we can apply Proposition 5.2, Lemma 5.7 and Proposition 5.9 to conclude the proof in the case of γ ∈ B ∞ G .Now, consider the case of an arbitrary metric γ and let {γ k } k∈N be a sequence of metrics in B ∞ G converging smoothly to γ, which exists because of Theorem B.4. Thanks to the first part of the proof (in particular applying Proposition 5.2 and Lemma 5.7 to the minimizing sequence found in the first part of the proof with respect to γ k ), for every k ∈ N there exist a G-equivariant min-max sequence {Λ (k),j , the width of Π with respect to the metric γ k ) that is G-almost minimizing in every L-admissible family of G-equivariant annuli with L = (3 n ) 3 n and that converges to a finite union Ξ k of G-equivariant free boundary minimal (with respect to γ k ) surfaces (possibly with multiplicity).Moreover, it holds ind G (spt(Ξ k )) ≤ n and H 2 γ k (Ξ k ) = W Π,γ k .Note that W Π,γ k converges to the width W Π = W Π,γ (the proof is the same as in the Almgren-Pitts setting, for which one can see [IMN18, Lemma 2.1]).Hence, since the varifolds Ξ k have uniformly bounded mass, up to subsequence Ξ k converges in the sense of varifolds to a varifold Ξ with mass equal to W Π .Moreover, taking a suitable diagonal subsequence of {Λ (k),j t j } j,k∈N we can obtain a min-max sequence {Λ j } j∈N for Π with respect to γ, converging in the sense of varifolds to Ξ and which is G-almost minimizing in every L-admissible family of G-equivariant annuli.Then, thanks to Proposition 5.9, Ξ is a disjoint union of G-equivariant free boundary minimal surfaces (possibly with multiplicity) and the genus bound in the statement holds.One can look at Figure 1 for a scheme of the argument.
. . .Now note that, thanks to Theorem 4.3 (see also Remark 4.4), up to extracting a further subsequence, we can assume that spt(Ξ k ) converges smoothly (possibly with multiplicity) to a free boundary minimal surface away from the singular locus S and, possibly, from finitely many additional points Y.This free boundary minimal surface coincides with the limit of spt(Ξ k ) in the sense of varifolds, which is spt(Ξ), in M \ (S ∪ Y).In particular, we get that ind G (spt(Ξ)) ≤ n, which concludes the proof.
8. Equivariant index of some families of (free boundary) minimal surfaces In this section, we want to make use of Theorem 1.10 to compute the equivariant index of some families of minimal surface in S 3 and free boundary minimal surfaces in B 3 .Let us start by proving Theorem 1.13.
Proof of Theorem 1.13.First note that, applying Theorem 1.10, we can assume that the surface M g in [CFS20, Theorem 1.1] has D g+1 -equivariant index less or equal than 1.We want to prove that the D g+1 -equivariant index is exactly 1.To do so, it is sufficient to construct a test function on which the associated quadratic form attains a negative value.
Recall that M g contains the horizontal axes of symmetry ξ 1 , . . ., ξ g+1 .Hence, sgn Mg (h) = −1 for all h ∈ D g+1 given from a rotation of angle π around any horizontal axis.Observe that these isometries generate all D g+1 , because the composition of the rotations of angle π around ξ 1 and ξ 2 is equal to the rotation of angle 2π/(g + 1) around ξ 0 .Hence, we can infer the sign of all the elements of the group.In particular, we obtain that the function u (see [Dev19, Lemma 6.1]).This proves that the D g+1 -equivariant index of M g is exactly 1.
Remark 8.1.Note that the function Similarly to what we did for the surfaces M g constructed in [CFS20], we can apply Theorem 1.10 to any other surface obtained via an equivariant min-max procedure.To our knowledge, the known equivariant min-max constructions so far are: • The minimal surfaces in S 3 of [Ket16a, Sections 6.2, 6.3, 6.5, 6.6 and 6.For all these surfaces we get that the equivariant index (with respect to the corresponding symmetry group) is less or equal than 1.Then, one has to find a suitable equivariant test function to get the equality, on a case-by-case basis.However, in most of the cases above the constant function 1 is such a test function.Indeed, the constant 1 is a negative direction for the second variation of the area functional for every minimal surface in S 3 and every free boundary minimal surface in B 3 .Moreover, if the unit normal is equivariant (i.e., h * ν = ν for all h in the symmetry group, where ν is a choice of unit normal), the constant function is also equivariant.
In other cases some more work is needed, as in the proof of Theorem 1.13.This last claim applies for example to the surfaces in B 3 constructed in [Ket16b, Theorem 1.1], for which a good test function is again u(x 1 , x 2 , x 3 ) = x 3 .Remark B.2.Note that we do not require the Jacobi field in the previous definition to be Gequivariant.
Theorem B.4.The subset B q G defined in Definition B.1 is comeagre in Γ q G .Remark B.5.In [Whi17, Theorem 2.1], White proved a stronger result in the closed case, namely that a generic, G-equivariant, C q Riemannian metric on a smooth closed manifold is bumpy in the following sense: no closed, minimal immersed submanifold has a nontrivial Jacobi field.We decided to state Theorem B.4 only for properly embedded G-equivariant submanifolds, since White's generalization requires more technical work and we do not need it here.However a similar proof should work also in the case with boundary.
Finally denote by π q G : S q G → Γ q G the projector onto the first factor, namely π q (γ, [w]) = γ.Given these definitions, we can now state the main ingredient to prove Theorem B.4.This is a structure theorem, whose first version was discovered by White in [Whi87], stating the relation between critical points of π q and degenerate minimal hypersurfaces that underlies any bumpy metric theorem.
Theorem B.6 (Structure Theorem, cf.[Whi17, Theorem 2.3] and [ACS18a,Theorem 35]).In the setting described above, S q G is a separable C 1 Banach manifold and π q G : S q G → Γ q G is a C 1 Fredholm map of Fredholm index 0. Furthermore, (γ, [w]) is a critical point of π q G if and only if w(Σ) admits a nontrivial G-equivariant Jacobi field.Remark B.7. Theorem 2.3 in [Whi17] is the version of the theorem in the case without boundary, while Theorem 35 in [ACS18a] is the case with boundary but where no group of symmetries is considered.
Proof.As observed by White in the proof of [Whi17, Theorem 2.3], the proof of the Structure Theorem works the same in the equivariant case replacing "metric" with "G-equivariant metric", "functions" with "G-equivariant functions" and working on a G-equivariant hypersurface instead of on any hypersurface.In fact, the reader can follow the proof of the Structure Theorem in the case with boundary contained in [ACS18a, Section 7.2] with these modifications.In particular note that: • Here we used M m+1 for the ambient manifold and Σ m for the embedded hypersurface, instead of N n+1 and M n as in [ACS18a].
• The background metric γ * shall be chosen G-equivariant.This way, the exponential map with respect to γ * induces a diffeomorphism between V r (as defined in [ACS18a, pp.25]) and a G-equivariant open neighborhood of w(Σ).Then a class [w] such that w ∈ C q−1,α (Σ, M ) is a proper embedding with G-equivariant image corresponds to a function u ∈ C q−1,α G (Σ, V ) with G-equivariant symmetry (cf.Definition 3.7).
• Finally, [ACS18a, Proposition 46] and the final proof of Structure Theorem itself (at [ACS18a,pp. 31]) can be modified accordingly to fit the equivariant setting.Just note that the functions in ker L(γ, u), where L(γ, u) is seen as a map between the equivariant spaces, correspond exactly to the G-equivariant Jacobi fields.
Proof of Theorem B.4.Given Theorem B.6, the proof is exactly the same as the proof of Theorem 9 in [ACS18a, Section 7.1] substituting Γ q , S q , B q with Γ q G , S q G , B q G .In fact, the only difference in the proofs is hidden in the application of [Whi17, Lemma 2.6], which is the point where the Structure Theorem (in the different variants: Theorem B.6, [Whi17, Theorem 2.3] or [ACS18a, Theorem 35]) is used.However, White in [Whi17] works in the equivariant setting as we do here, hence the argument there can be applied in our context without changes, since the presence of the boundary does not play a role in this lemma, as observed also in [ACS18a,pp. 24].As said above (in the proof of Theorem B.6), the reader should pay attention to the fact that here we used M m+1 for the ambient manifold and Σ m for the embedded hypersurface, instead of N n+1 and M n as in [ACS18a].
an open cover of K. Then define O to be such an open cover for some m sufficiently large such that 3 −m √ n is less than the Lebesgue number of U , thus O has diameter less than the Lebesgue number of U for all O ∈ O.Then, for all O ∈ O, let A O be an L-admissible family of G-equivariant annuli in M such that there existsU t ∈ U with O ⊂ U t and A O = A t .Now, note that T (σ 1 ) ∩ T (σ 2 ) = ∅ for some σ 1 , σ 2 ∈ I(n,m) if and only if σ 1 and σ 2 are faces (possibly of different dimension) of a cell in I(n, m).We can thus exploit the following proposition to associate to every O = T (σ) ∈ O a G-equivariant annulus An O ∈ A O .Proposition 5.8 ([Pit81, Proposition 4.9]).For every σ ∈ I(n, m), let A(σ) be an L-admissible family of G-equivariant annuli in M with L = (3 n ) 3 n .Then for every σ ∈ I(n, m) it is possible to choose a G-equivariant annulus An(σ) ∈ A(σ) such that An(σ 1 ) ∩ An(σ 2 ) for every σ 1 = σ 2 ∈ I(n, m) that are faces of a cell in I(n, m).Hence, for all O = T (σ) ∈ O, let us define An O := An(σ), where An(σ) is given from Proposition 5.8.This proves the existence of the open cover O of K with properties (i) and (ii) above.Moreover, by definition of U and O, for all O ∈ O there exists a G-equivariant isotopy Ψ

Figure 1 .
Figure 1.Scheme of the convergence argument in the proof of Theorem 1.10.
be a minimizing sequence and define the G-equivariant surface Σj t := Ψ Λ j t(1, Λ j t ) for all j ∈ N and t ∈ I n .Observe that the map (t, x) → Φj (t, x) for all r ∈ U t .Since {U t } t∈K is an open cover of K and K is compact, we can extract a finite subcover U .We now want to find another cover O of K and for every O ∈ O we want to choose a G-equivariant annulus An O such that (i) for all O ∈ O there exists U t ∈ U such that O ⊂ U t and An O = An i t for some i = 1, . . ., L; (ii) for all O 1 , O 2 ∈ O with nonempty intersection we have that An O 1 ∩ An O 2 = ∅.
isotopy with Φ t = id for t ∈ ∂I n .Indeed, note that d(K, ∂I n ) > 0, hence we can choose m possibly larger in such a way that O ∩ ∂I n = ∅ for all O ∈ O. Therefore, we can define Σj t = Φ t (Σ j t ) and we have that { Σj t } t∈I n ∈ Π.Now note that, for every t ∈ K, there exists O ∈ O such that ϕ O (t) = 1, moreover t ∈ O for at most other 2 n − 1 sets O ∈ O.As a result, we can estimate H 2 Note that { Λj t } t∈I n ∈ Π for all j ∈ N.Moreover, sup t∈I n F( Λj ∅ for all t ∈ I n , for j sufficiently large (possibly taking ε > 0 smaller).Finally note that, thanks to (6.1), for all t ∈ U j it holds that Λj .Moreover, for all t ∈ I n \ U j , we have Λj t = Σ j t ∈ K. Hence { Λj t } t∈I n also avoids K and thus satisfies the desired properties. n+1