OPTIMAL ISOPERIMETRIC INEQUALITIES FOR SURFACES IN ANY CODIMENSION IN CARTAN-HADAMARD MANIFOLDS

Let (M, g) be simply connected, complete, with non-positive sectional curvatures, and Σ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in M . Let S be an area minimising integral 3-current (resp. flat chain mod 2) such that ∂S = Σ. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from Σ, to show that S satisfies the optimal Euclidean isoperimetric inequality: 6 √ πM[S] ≤ (M[Σ]). We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by −κ < 0 and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in L.


Introduction
The classic Euclidean isoperimetric inequality states that for any bounded open set Ω ⊂ R n+1 with sufficiently regular boundary it holds that |Ω| ≤ γ(n + 1)|∂Ω| If equality in (1.3) is attained, then Σ is a smooth embedded 2-sphere, has unit density, its mean curvature vector has constant length and Σ is totally umbilic. Furthermore, it bounds a totally geodesic embedded 3-ball S, with the mean curvature vector of Σ proportional to the unit conormal of S at every point in Σ. S is isometric to a geodesic ball in the 3-dimensional model space such that the mean curvature of the boundary coincides with the one of Σ.
A central ingredient in the proof is an optimal lower bound for the Willmore energy of an integral 2-varifold with compact support.

Theorem 1.4.
Let (M n , g) ∈ CH(n, −κ), −κ ≤ 0, n ≥ 3 and Σ 2 be an integral 2-varifold in M with compact support and that the first variation of Σ is summable in L 2 (μ). Then where H is the weak mean curvature of Σ. If equality is attained, then Σ is a smooth embedded 2-sphere, has density one, the mean curvature vector has constant length and Σ is totally umbilic. Furthermore, it bounds a totally geodesic embedded 3-ball S, with the mean curvature vector of Σ proportional to the unit conormal of S at every point in Σ. S is isometric to a geodesic ball in the 3-dimensional model space such that the mean curvature of the boundary coincides with the one of Σ.
This estimate for n = 3 and κ = 0 appeared already in [Sch08, Lemma 6.7]. In Euclidean space the estimate can be found in work of Simon [Sim93], and follows rather directly from the usual calculations leading to the monotonicity formula. The characterisation of the equality case in an Euclidean ambient is given by Lamm-Schätzle [LS14] together with a stability result. For smooth surfaces in an Euclidean ambient space this is the well known Li-Yau estimate, [LY82]. For sufficiently regular surfaces in codimension one which are outward minimising, an analogous estimate following from the Gauss-Bonnet formula is central in the argument of Kleiner [Kle92] (see also the alternative proof of Ritoré [Rit05], which does not require the condition of outward minimising).
Remark 1.5. For (M, g) ∈ CH(n, 0) and Σ an integral m-varifold in M , one can use the variant of the Michael-Simon Sobolev inequality [MS73] for Riemannian manifolds by Hoffman-Spruck [HS,HS74] (which carries over to the setting of integral varifolds) to get an estimate where C(m) depends only on m. This constant is not optimal, but the proof of Theorem 1.2 carries over to any dimension and codimension, yielding a non-optimal inequality for integral currents or flat chains mod 2 as in (1.2) with a constant only depending on m. Alternatively, restricting to an open, precompact set K ⊂ M , a 258 F. SCHULZE GAFA direct comparison with Euclidean space gives a non-optimal isoperimetric inequality, where the constant depends on (M, g) and K, see Lemma A.2.
We give a first outline of the idea of the proof of Theorem 1.2 for κ = 0. Assume n ≥ 3, (M n , g) ∈ CH(n, 0) and that Σ is an orientable, closed, smooth, 2-dimensional submanifold of M . Let (Σ t ) 0≤t<T be its smooth evolution by mean curvature flow with Σ 0 = Σ. Assume further that there exists a smooth family (S t ) 0≤t<T (1.5) of minimal 3-dimensional (immersed) submanifolds in M such that ∂S t = Σ t , and let X t be the variation vectorfield along (S t ) 0≤t<T . The first variation formula then implies that where n is the unit conormal of S t along ∂S t = Σ t and H is the mean curvature vector of Σ t . Similar to [Sch08], we consider the isoperimetric difference and compute, using (1.4) in the second line, and thus d dt I t ≤ 0. If the flow (Σ t ) 0≤t<T and the family (S t ) 0≤t<T exists long enough such that lim t→T |S t | = 0, this shows that But in general it can't be expected that the flow does not develop singularities before the spanning volume goes zero. It is also not clear why a sufficiently regular family (Σ t ) 0≤t<T should exist. To be able to evolve through singularities we would like to work with a weak solution of mean curvature flow, in our case the most suitable one seems to be a Brakke flow. But there are only very little regularity results for higher codimension, even sudden vanishing is possible. Furthermore, it is not clear to us how to construct a sufficiently regular family of spanning minimal surfaces such that the above monotonicity calculation can be performed.
To circumvent this problem we work with Ilmanen's elliptic regularisation scheme [Ilm94]. In this work Ilmanen combines the elliptic regularisation approach of Evans-Spruck [ES91] in codimension one with the moving varifold solutions of Brakke [Bra78] to construct Brakke flow solutions with special properties. Treating all surfaces as if they were smooth and avoiding some of the technical details, we give an overview of the argument to prove Theorem 1.2 for κ = 0.
Let Σ 0 ⊂ M be an integral 2-current with compact support such that ∂Σ 0 = 0. We consider Σ 0 ⊂ M ×{0} ⊂ M ×R and denote by z the coordinate in the additional R-direction and τ the corresponding unit vector. Ilmanen's elliptic regularisation scheme yields a sequence ε i > 0, ε i → 0, a sequence of integral 3-currents P i such that ∂P i = Σ 0 , which yield translating solutions to mean curvature flow in M × R via Let {μ i t } t∈R be the corresponding family of Radon measures. This sequence of flows converges as i → ∞ to a limiting Brakke flow {μ t } t≥0 which is invariant in zdirection, starting at Σ 0 ×R. The Brakke flow {μ t } t≥0 starting at Σ 0 is then obtained via slicing {μ i t } t≥0 at height z = const. Additionally, the sequence where μ Tt is the mass measure associated to the slice T t of T at height z = t. The current T is called the undercurrent of the flow {μ i t } t≥0 . Treating the z-direction as time, it can be helpful to think of T as the space-time track of the flow {μ t } t≥0 , after taking into account possible cancellations. Furthermore for all t > 0 as i → ∞, where π : M × R → M is the projection on the first factor. We choose S 0 , a mass-minimising integral 3-current with ∂S 0 = Σ and S i massminimising integral 4-currents in M × R such that This family will serve as a family of minimal surfaces approximating the family (1.5) considered in the smooth monotonicity calculation. Note that the variation 260 F. SCHULZE GAFA vectorfield of this family is just given by X = −ε −1 i τ , which makes the monotonicity calculation for (1.8) feasible.
Let l > 1. We choose ϕ l ∈ C 2 c (R) such that 0 ≤ ϕ l ≤ 1/l with ϕ l = 1/l on [2, l + 2], ϕ l = 0 on (0, ∞)\[1, l + 3]. We define the approximate area and volume by The averaging function ϕ l takes into account that in the limit i → ∞, P i (t) becomes vertical, and thus A i t approximates μ t (M ). For t fixed and i → ∞ we expect that S i (t) has a similar behaviour and thus V i t approximates the measure of a family as in (1.5).
In Euclidean space shrinking spheres with radius R(t) = √ R 2 − 2mt act as barriers for integral m-Brakke flows from the inside and from the outside. Using the properties of the Hessian of the distance function to a point p in a Cartan-Hadamard manifold M n one can show that this remains true as barriers from the outside, and thus the flow {μ i t } t≥0 has a finite maximal existence time To see that one can use the future space-time track of the flow as a competitor: motivated by the fact that an estimate for the volume traced out by a mean curvature flow is given by the L 1 -norm in time of the mean curvature vector, and the natural estimate Noting that ∂(π(T ∩ {z ≥ t})) = π(T t ) and recalling (1.6) we can use π(T ∩ {z ≥ t}) × R, up to a small error, as a competitor to S i (t) to achieve (1.7). For the monotonicity calculation we consider the approximate isoperimetric difference and show that this quantity is monotone in the limit as l → ∞ and i → ∞ between t 0 = 0 and 0 < t 1 < T max . To see this we show that the error terms in the time derivative of (1.8) are controllable and combine the property that P i (t) becomes vertical with the estimate (1.4) and the lower semicontinuity of the L 2 -norm of the mean curvature. Together with (1.7) this yield that Structure of the paper. In section 2 we recall Ilmanen's elliptic regularisation scheme [Ilm94] and show the improved approximation (1.6). The barrier argument and a comparison principle due to B. White yield the estimate on the maximal existence time. We also prove a positive lower estimate on the maximal existence time for the limiting Brakke flow.
An essential ingredient in controlling the error terms when showing the almost monotonicity of the approximate isoperimetric difference is to know that as i → ∞. To achieve this we first assume that S 0 is the unique mass-minimising current spanning Σ 0 . Using this assumption, we show in section 3 that (1.9) holds. We later show that by perturbing Σ 0 slightly we can assume that Σ 0 bounds only one mass-minimising current. We also give uniform local area bounds for S i . In section 4 we prove (1.7).
In section 5 we compute the time derivative of the approximate isoperimetric difference and show that the error terms are controllable in the limit i → ∞. We use a lower semi-continuity argument together with (1.4) to prove Theorem 1.2. We also show that we can treat the case of equality, Theorem 1.3, using the characterisation of equality in Theorem 1.4.
In section 6 we prove Theorem 1.4. In the "Appendix" we collect several results needed in the prequel. We show that the mass minimising currents S i are strongly stationary and that there is a non-optimal isoperimetric inequality in any dimension and codimension in a Cartan-Hadamard manifold. Furthermore, we recall White's avoidance principle for Brakke flows and show how unique continuation for minimal surfaces in any codimension follows from work of Kazdan.

Elliptic Regularisation
We employ Ilmanen's elliptic regularisation scheme [Ilm94] to construct a Brakke flow starting at Σ. We recall the construction of Ilmanen, adapted to our setting, and its properties needed in the sequel.
We outline the main steps of the proof. Ilmanen constructs local integral (m+1)currents P ε in M m+k × R that minimize the elliptic translator functional where z is the coordinate in the additional R-direction, subject to the boundary condition The associated Euler-Lagrange equation implies that the family of Radon mea- Ilmanen's compactness theorem for Brakke flows implies that there is a sequence Furthermore, Ilmanen shows thatμ 0 = μ T0×R andμ t is invariant in the z-direction, which yields the desired solution {μ t } t≥0 via slicing.
The integral current T is constructed via considering a subsequential limit of , which can be seen as an approximation to the space-time track of {μ t } t≥0 where now the z-direction is considered as the time direction. Point (iii) above verifies this interpretation.
Recall that for s ≥ 0 we define the following slices by the height function z: and similarly We note the following estimates from [Ilm94].

GAFA OPTIMAL ISOPERIMETRIC INEQUALITIES IN CARTAN-HADAMARD MANIFOLDS 263
In particular in the flat metric distance In particular in the flat metric distance For details see sections 5.1-5.3 in [Ilm94]. One can use the C 1/2 -continuity of (T ε t ) to show the following improved approximation property.
Taking boundaries this yields Note that T εi t = (κ εi ) # P εi t/εi . This implies that Using (2.3) this yields that for t = 0 and s ≥ 0 or t > 0 and any s ∈ R This yields that for any any s ∈ [0, ∞) and thus Furthermore, by (2.2) we have for any 0 ≤ s 1 < s 2 that This yields that ∂ ∂z is H m+1 − a.e. tangential to P . By the coarea-formula this implies that For t > 0, we obtain that for any any s ∈ R and by the same argument as earlier that We will in the following always assume that (M, g) ∈ CH(n, 0). We consider the local integral 3-currents P ε ⊂ M × [0, ∞) constructed in the previous section, such that ∂P ε = Σ 0 . We choose a sequence ε i → 0 such that as in the proof of Theorem We denote the maximal existence time of the constructed Brakke flow {μ t } t≥0 , by Note that by the monotonicity of the total measure we have μ t (M ) > 0 for all t < T max and μ t (M ) = 0 for all t > T max . Under the present restrictions on the geometry of M we obtain an upper bound for the maximal existence time.
Since M is complete and has non-positive sectional curvature we have Consider 0 < α < n and the function u(p, t) = r 2 + 2αt.
Then with the notation as in Theorem A.3 we see that and thus by Theorem A.3 u(x, t) ≤ R 2 on spt μ t . Letting α → n this implies the first two statements. To obtain the height bound observe that by Huisken's monotonicity formula (with a suitable local modification due to the non-flat background) the support of the Brakke flow (μ ε t ) t≥0 converges in Hausdorff distance to the support of (μ t ) t≥0 .
Let S 0 be an area-minimising 3-current in M such that Note that geodesic spheres in M are convex, thus by the convex hull property we have that the support of S 0 is compact. We then also obtain a lower bound on the maximal existence time.

Attainment of Initial Spanning Surface
We can w.l.o.g. assume that M[S 0 ] > 0. We will for the moment work with the following Assumption. We assume S 0 ⊂ M is the unique area-minimising 3-current spanning Σ 0 .
We will later verify that in general one can perturb Σ 0 slightly such that the uniqueness assumption is satisfied.
In the remaining part of this section we aim to show that Proof. The proof of the classical monotonicity formula in R n relies on the fact that the position vectorfield X(x, Using that S ε is strongly stationary, see Lemma A.1, one obtains as in the proof of the monotonicity formula, compare [Sim83], that d dρ Together with estimate (2.1) this yields for ρ ≥ 1 and 0 < ε < 1 d dρ Integrating this for 1 ≤ r < R from r to R yields Letting R → ∞ yields the desired estimate.

GAFA OPTIMAL ISOPERIMETRIC INEQUALITIES IN CARTAN-HADAMARD MANIFOLDS 267
By the uniform local area bound we can thus, up to a subsequence, assume that where S is locally mass-minimising and satisfies We will define for a general integral current Q its slice at height t by which is compatible with the convention used by Ilmanen in [Ilm94].
Proof. We consider for t > 0 the slice S t of S at height t as above.
Claim 1. There exists a sequence t j → ∞ and C > 0 such that This follows since by the coarea formula, and the locally uniform area estimates But then by the coarea formula which yields a contradiction for j sufficiently large.

F. SCHULZE GAFA
Since by assumption S 0 is the unique mass-minimising current spanning Σ 0 we obtain that (σ −t l ) # (S t l ) → S 0 in flat norm and in mass. Thus there exists a sequence δ l → 0 such that But again this yields

Vanishing of the Spanning Area at the Final Time
For convenience of notation we will in the following replace a sub-or superscript ε i by i.
Recall that μ i t is the associated Radon measure of P εi (t). We denote with μ S,i t the associated Radon measure of We define the approximate volume by We show that the approximate volume goes to zero as t → T max . Proof. We use T to construct a competitor to S i (t). Recall that and by Lemma 2.3 that By the co-area formula (as in the proof of claim 1 in the proof of Lemma 3.2), there exists a C > 0 and η i ∈ [0, 1] such that for all i Note further that and note that By the uniform mass bounds on S i 1+ηi , S i 2+l+ηi , R i − , R i + given by (4.2), (3.2) together with Lemmas 2.4 and A.2 there exits D i − , D i + such that and a constant C such that We can now assume that η i → η ∈ [0, 1], and thus note and M[Q i ] → 0. Since S i is locally mass minimising we can use as a competitor to get the desired estimate.
Proof. We can estimate, using Lemma 4.1, that for i sufficiently large

The Monotonicity Calculation
Recall the approximate volume where |∇ϕ l |, |∇ 2 ϕ l | ≤ 2/l. Note that we can further assume that We define the approximate area as We compute Thus we get for 0 ≤ t 1 < t 2 that For the approximate area we get, using (5.1), where we used the uniform local area bounds to estimate the first integral on the right hand side. This implies the estimate Note that by (2.1) this implies that We define the function f κ : da.
Note that f 0 = 1 6 √ π a 3/2 . Let (M t ) 0≤t<T be a smooth mean curvature flow of closed, embedded hypersurfaces in a Cartan-Hadamard manifold (M 3 , g) with sectional curvatures bounded 272 F. SCHULZE GAFA above by −κ. Let V (t) be the enclosed volume and A(t) the area. We then can apply Theorem 1.4 to estimate Thus f κ (A) − V is monotonically decreasing under the flow. Consider the case that M 3 κ is the model space of constant curvature −κ and let M t be the mean curvature flow of geodesic spheres contracting to a point. Then the estimate of Theorem 1.4 holds with equality for all M t and also the above calculation is an equality. Using that in the model space geodesic balls optimize the isoperimetric ratio, we have for all open and bounded U ⊂ M 3 κ , with equality on geodesic balls. We consider the approximate isoperimetric difference where we can estimate, assuming t 2 ≤ T max and using (5.3), This yields the estimate From (5.2), (5.5) we get the estimate Recall that the limiting Brakke flow (μ t ) (0≤t≤Tmax) is invariant in the z-direction, and for a.e. t the measureμ t is 3-rectifiable and carries a weak mean curvature in L 2 . Using Theorem 1.4 we thus see that Lemma 5.1. For any t 1 , t 2 ∈ [0, T max ), t 1 < t 2 , Proof. From Ilmanen's compactness theorem for Brakke flows, we know that for all t ∈ [t 1 , t 2 ] we have and by the lower semicontinuity of the L 2 norm of H and (5.7) that Since the function x 1/2 − (16π + 4κa) − 1 2 x is decreasing for x ≥ 4π + κa we obtain, using (5.8), that Note that by (5.4) there is C ≥ 0 such that for all t ∈ [t 1 , t 2 ] and all i ≥ i 0 . Then the claim follows from Fatou's lemma.
We will now explain how to perturb Σ slightly such that we can assume that the mass minimising 3-current spanning Σ is unique. Let S be any mass minimising 3-current spanning Σ. Note that by Almgren [Alm00], see also De Lellis-Spadaro [LS11, LS15, LS14, LS16, LS16], the interior singular set of S has codimension 2. Note further that the interior regular set R int can have at most countably connected components, since S has finite mass. We denote these components by R j for j ∈ 1, . . . , N where N ∈ N ∪ {∞}. We can pick points p j ∈ R j and radii r j > 0 such that -B rj (p j ) ∩ spt Σ = ∅, -the balls B rj (p j ) are pairwise disjoint, -S L B rj (p j ) is smooth and consists of one single, smooth, connected component, - The above estimates yield that Σ k → Σ ( 5 . 9 ) in flat norm and in mass.
Lemma 5.2. S k is the unique mass minimizing current spanning Σ k .

GAFA OPTIMAL ISOPERIMETRIC INEQUALITIES IN CARTAN-HADAMARD MANIFOLDS 275
Proof. Assume there is another mass minimizing current S spanning Σ k . But then is mass minimising and bounds Σ. The interior singular set of S has again codimension 2. Thus by unique continuation, see section A.4, S has to coincide with S on each connected component R j of R int . Note that S can have no further connected components of its interior regular set, since it has the same mass as S. Thus S = S and S = S k .
Proof of Theorem 1.2. We will present the proof in the case Σ is an integral 2current. The necessary modifications if Σ is a 2-dimensional flat chain mod 2 will be discussed at the end of the proof. We first replace Σ by Σ k such that by Lemma 5.2 S k is the unique area minimising current spanned by Σ k .
As outlined above we use Ilmanen's elliptic regularisation scheme to construct Brakke flows (μ k t ) (0≤t≤T k max ) , starting at Σ k,0 := Σ k , which vanish at a finite time T k max . These flows arise as the slice of the translation invariant flows ( on M × R, obtained as a limit of approximating flows (μ k,i t ) t≥0 . We will for the moment omit the index k. We use the set-up as before. Note that by Lemma 3.2 we have that (5.10) Given ε > 0, we choose t 1 = 0 and T max − δ < t 2 < T max , l > l 0 , where δ > 0 and l 0 are given by Corollary 4.2. By (5.6) and Lemma 5.1 we can estimate lim sup Putting this together we obtain Letting first l → ∞ and then ε → 0 yields the isoperimetric inequality for Σ k . We can now let k → ∞ to obtain the isoperimetric inequality for Σ.
The equality case: In case of equality in (1.3) we have and the monoticity calculation, using Lemma 5.1, yields where (μ k t ) t≥0 is the constructed Brakke flow starting at Σ k . We aim to let k → ∞ and construct a non-vanishing Brakke flow starting at Σ. By Lemma 2.5, there exists δ > 0 and η > 0 such that for all 0 ≤ t < δ and all k sufficiently large. We can thus consider a subsequential limit as k → ∞ and obtain a limiting Brakke flow {μ t } t≥0 which satisfies (5.12) as well. Similarly as in the proof of Lemma 5.1 we obtain Thus for a.e. t ∈ (0, δ), using (5.12), we have |H| 2 dμ t = 16π + 4κμ t (M ).
(5.13) Thus by Theorem 1.4 for a.e. t ∈ (0, δ), μ t is the Radon measure associated to a smooth embedded 2-sphere Σ t with density one, where the mean curvature vector has constant length and Σ t is totally umbilic. Furthermore, it bounds a totally geodesic embedded 3-ball S t , with the mean curvature vector of Σ t proportional to the unit conormal of S t at every point in Σ t . S t is isometric to a geodesic ball in the 3-dimensional model space such that the mean curvature of the boundary coincides with the one of Σ t . Since all the flows {μ k t } t≥0 are unit regular, see [SW16,§ 4], White's local regularity theorem [Whi05], implies that the convergence is smooth for 0 < t < δ and the limiting flow {μ t } 0<t<δ is smooth [and thus the above characterisation of μ t holds for all t ∈ (0, δ)].
The smooth convergence implies that μ k t = μ T k t for any 0 < t < δ and k sufficiently large. Recall that in the flat norm and Σ k → Σ in flat norm. This yields that Σ t → Σ in flat norm. Since Σ t converges smoothly to a limit as t 0 this yields the claimed statement about Σ.
In the case that Σ is a 2-dimensional flat chain mod 2, we work with flat chains mod 2 instead of integral currents. Note that Ilmanen's elliptic regularisation scheme works analogously in this setting. All the other parts of the argument also directly carry over. The only point to note is that the interior regularity of Almgren [Alm00] has to be replaced by the corresponding result for flat chains mod 2 due to Federer [Fed70].

An Optimal Lower Bound on the Willmore Energy
In this section we give the proof of the optimal lower bound on the Willmore energy.
The case κ = 0: Given σ > 0 we choose On the other hand, applying the divergence theorem yields Combining both equations yields we can take the limit σ → 0 to obtain 2π + 2 1 4 for any p 0 such that (6.4) holds.
Inserting this into (6.1) gives Arguing as before, using (6.6), we arrive at for any p 0 such that (6.4) holds.
The equality case for κ = 0: To see that Σ is a smoothly embedded 2-sphere with unit density, we can nearly verbatim follow the argument in [LS14, Proposition 2.1]. We include it for completeness. We first note that by equality in (6.5), since p 0 ∈ spt μ is arbitrary, we have that Σ has unit multiplicity: θ 2 (μ) = 1 on spt μ. (6.8) Furthermore (6.5) gives that where ⊥y denotes the orthogonal projection onto T ⊥ y μ. In particular H(y) ⊥ T y μ for μ-almost all y ∈ spt μ. (6.9) By Fubini's Theorem, for μ-almost all y it holds that H(y) + 4 ∇ r x (y) ⊥y r x (y) for μ-almost all x ∈ spt μ.
Claim. S with its induced metricg is isometric via the exponential map at y 0 to B r0 (r 0 e 3 ) ⊂ R 3 .
Following the proof of (6.5) we see that we have equality in (6.1) with ϕ = 1/r for every point x = y 0 ∈ Σ. Since all geodesics connecting y 0 with other points in x ∈ Σ intersect Σ at x non-tangentially, we have that the ambient sectional curvatures sec g (∇r y0 ∧ V ) = 0, where V is any unit vector tangent to S r := ∂B r (y 0 ) ∩ S for 0 < r < 2r 0 . The same argument gives that the principal curvatures along S of ∂B r (y 0 ) are equal 1/r for 0 < r < 2r 0 and that intrinsically S r is isometric via the exponential map at y 0 to ∂B r (0) ∩ B r0 (r 0 e 3 ) ⊂ R 3 , written in polar coordinates around 0 ∈ R 3 . But the Gauss equations then also show that secg(V ∧ W ) = 0 for any two unit vectors V, W tangent to S r for 0 < r < 2r 0 . This proves the claim. Note that this implies that the mean curvature vector H S (x) of Σ ⊂ S, seen as a submanifold of S has length 2/r 0 for all y ∈ Σ. Since H S (x) = π TxS H(x) the choice of y 0 in (6.10) implies that It remains to show S is totally geodesic. Pick any point x 0 ∈ Σ. By the argument before we have that where we have chosen e 1 , e 2 , e 3 as before. But this implies that any extrinsic geodesic connecting x 0 with x = x 0 ∈ Σ has the same length as the intrinsic geodesic in S connecting both points, and thus they both have to coincide. This shows that S is totally geodesic, which also implies that Σ is totally umbilic in M . The equality case for κ > 0: We can again by scaling assume that κ = 1. The argument is completely analogous to the case κ = 0, the only thing to note is that the equation 2 = 4 sinh(|x|) 2 cosh(|x|) − 2 x 3 |x| describes the boundary of a geodesic sphere with mean curvature 2 in normal coordinates around the south pole in the 3-dimensional model space.