Rigidity for critical points in the Lévy-Gromov inequality

The Lévy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics corresponding to the Lévy-Gromov inequality and prove that, in two-dimensions, this criticality condition is quite rigid, as it characterizes round spheres and projective planes.

This is the spirit of the celebrated Lévy-Gromov inequality [5,Appendix C]: if Ric g ≥ K g for some constant K > 0, then where (S, g S ) is the standard n-dimensional sphere with Ricci curvature equal to K (see also [1,7] for the generalization to the case K ≤ 0 and diameter bounded above, and [2] for the extension to non-smooth spaces). Having in mind the relation between the Euclidean isoperimetric theorem (balls are the only volume-constrained minimizers of perimeter) and Alexandrov's rigidity theorem (balls are the only volume-constrained critical points of perimeter), in this note we ask what can be said about critical points in the variational problem corresponding to the Lévy-Gromov inequality, and, at least in dimension two, we prove a full rigidity theorem. Our terminology will be as follows. The Lévy-Gromov functional on a Riemannian manifold (M, g) at volume fraction v ∈ (0, 1) is defined as We denote with M M the space of Riemannian metrics over M and, given K ∈ R, we consider the family M M,K metrics on M with Ricci tensor bounded below by K , and the family M M,K ,g of metrics in M M,K that are conformal to a given metric g, i.e. we set Endowing M M with the C 2 -topology, we notice that both M M,K and M M,K ,g have non-empty boundary. A natural definition of critical point associated to the Levy-Gromov inequality is then the following: we say that g is a critical isoperimetric metric (with constant K ) if g ∈ M M,K and the following holds: When, in the above definition, M M,K ,g is considered in place of M M,K , we say that g is a conformally-critical isoperimetric metric. The question we pose is what degree of rigidity can be expected for conformally critical isoperimetric metrics.
A first remark is that no metric can be conformally-critical with constant K ≤ 0. Indeed, let us recall that ifĝ = e 2u g for some u ∈ C 2 (M), then where Hess g u denotes the Hessian of u (with respect to the Levi-Civita connection of g) and g u = g i j (Hess g u) i j is the Laplace-Beltrami operator with respect to g applied to u. In particular, if we pick u = log λ for some λ > 0 and Ric g ≥ K g, then Ricĝ = Ric g ≥ K g = K λ −2ĝ . Given that K ≤ 0, we have Ricĝ ≥ Kĝ for every λ 2 ≥ 1. Since Vĝ( ) = λ n V g ( ) and Aĝ(∂ ) = λ n−1 A g (∂ ) for every ⊂ M, we also have , and thus, setting . From now on we shall thus take K > 0. In dimension n = 2 (where one simply has Ric g = K g g, K g denoting the Gauss curvature of g) it turns out that the apparently very weak notion of conformally-critical isoperimetric metric implies the maximal degree of rigidity one could expect: is a two-dimensional closed Riemannian manifold and K > 0, then g is a conformallycritical isoperimetric metric with constant K if and only if (M, g) is either a sphere or the real projective plane with K g = K .
We now present the proof of Theorem 1. For the sake of clarity we work in dimension n until the last step of the argument. We also notice that we shall use conformal-criticality only on a sequence of volumes v h → 0 + , and thus that we end up proving a slightly stronger statement than Theorem 1.

Proof of Theorem 1
Step one: We start recalling that since M is compact, by the direct method, for every v ∈ (0, 1) there exists an isoperimetric region with V g ( ) = v V g (M). By standard density estimates, is an open set of finite perimeter whose topological boundary ∂ is a closed (n − 1)-rectifiable set, characterized by the property that x ∈ ∂ if and only if V g ( ∩ B r (x)) ∈ (0, V g (B r (x))) for every r > 0. (Here and in the following, B r (x) stands of course for the geodesic ball of center x and radius r in M.) Let us denote by the isoperimetric sweep of (M, g), defined as = ∂ : is an isoperimetric region in (M, g) for some v ∈ (0, 1) .
In this step we prove that for every x ∈ and every r > 0 small enough, there exists where we have setĝ We first notice that, by the area formula, where dvol g |∂ is the (n − 1)-dimensional volume form induced by g on ∂ . Similarly, if we let ( X s ) s∈(−ε,ε) denote the flow with initial velocity given by a smooth vector-field X on M, and set ϕ X := g(ν ∂ , X ) (where the inner unit normal ν ∂ to is defined on the reduced boundary of , thus vol g |∂ -a.e. on ∂ ), then, by a classical first variation argument, see [6,Theorem 17.20], there exists a constant λ ∈ R such that The constant λ is the (distributional) mean curvature of ∂ computed in the metric g with respect to inner normal ν ∂ .) The combination of (0.9) and (0.10) thus gives Let us now fix x ∈ ∂ for some isoperimetric region . For every r > 0 we can find a smooth vector field X supported in the geodesic ball B r (x) such that and then set u = w 1 + w 2 + w 3 . We now apply (0.11) and (0.12) with these choices of u and X , to find Let us consider the function F ∈ C 2 ((−ε, ε) × (−ε, ε)) defined by |t|, |s| < ε .
Step two Now assuming that g is a conformally critical isoperimetric metric with constant K > 0, we show that for every x ∈ (the isoperimetric sweep of M), there exists ξ ∈ T x M such that Indeed, if this is not the case, then we can find an isoperimetric region and x ∈ ∂ such that Ric g,y (ξ, ξ ) > K g y (ξ, ξ ) , Depending on x and , we pick r , X and u as in step one. Recall that, in step one, we constructed u so that it was supported in B 2r (x). Therefore, by (0.13), we can entail that for every |t| < ε Ricĝt,u ≥ Kĝ t,u on M.
By definition of conformally-critical isoperimetric metric we find a contradiction with (0.7).
Step three We now let n = 2. By step two, K g ≡ K on the closure of the isoperimetric sweep of M. However, it is well-known (see for example [3,[8][9][10]) that if { h } h∈N is a sequence of isoperimetric regions corresponding to volume fractions v h → 0 + as h → ∞, then { h } h∈N converges in Hausdorff distance to a point x such that By continuity of K g we thus conclude that K = max M K g , and thus K g is constantly equal to K on M. We have thus proved that if g is conformally-critical with constant K , then K g ≡ K , and thus, since K > 0, that either (M, g) is the sphere or the real projective plane.
Step four: We are now left to show that both the sphere and the real projective plane are conformally-critical. In the case of the sphere this is immediate from the Levy-Gromov inequality, so that we are left to check the case of the real projective plane. Without loss of generality we consider the standard projective plane RP 2 endowed with the metric g 0 of constant curvature K = 1 defined as the quotient of the round sphere (S 2 ,g 0 ) of unit radius in R 3 under the antipodal equivalence relation. We denote by : S 2 → RP 2 the projection map.
Let us first recall that on a general compact Riemannian surface (M 2 , g) without boundary, just by considering the complement of each competitor, the isoperimetric profile I (M 2 ,g) is symmetric with respect to v = 1/2. In particular, we shall restrict v to the range v ∈ (0, 1/2]. Moreover, by direct methods and first variation arguments, for every v there exist isoperimetric regions which are necessarily bounded by finitely many curves with constant geodesic curvature. By direct computation, in the case when (M 2 , g) = (RP 2 , g 0 ) and v ∈ [0, 1/2], isoperimetric regions are metric balls which lift into S 2 as pairs of antipodal spherical cups, each spherical cup having volume 2vπ in S 2 .
Now assume by contradiction that there exists v 0 ∈ (0, 1/2] and a curve g (·) : [0, 1] → M RP 2 ,1 starting from the round metric g 0 on RP 2 , such that lim sup Thus we can find t n → 0 + as n → ∞ and isoperimetric regions t n in (RP 2 , g t n ) such that lim sup where 0 ⊂ RP 2 is a metric ball in metric g 0 with V g 0 ( 0 ) = 2v 0 π. Up to extracting a subsequence and up to translations, by standard density estimates, one can assume that t n converges to 0 in Hausdorff distance with respect to the metric g 0 . (In fact, the convergence is smooth, but this is not needed here.) Consider now the lifted metrics on S 2 defined byg t := * (g t ) and observe thatg t ∈ M S 2 ,1 , asg t is locally isometric to g t . Moreover, by construction,g t is invariant under the antipodal map and Vg t (S 2 ) = 2 V g t (RP 2 ). Since v 0 ∈ (0, 1/2] and t n is Hausdorff close to 0 , the lifted set˜ t n := −1 ( t n ) ⊂ S 2 can be written as˜ t n =˜ 1 t n ∪˜ 2 t n wherẽ 1 t n ,˜ 2 t n ⊂ S 2 areg t -isometric sets at positive Hausdorff distance. In particular (0.15) Notice that for t = 0 one has that˜ 0 := −1 ( 0 ) can be written as˜ 0 =˜ 1 0 ∪˜ 2 0 , wherẽ 1 0 and˜ 2 0 are antipodal spherical caps with Vg 0 (˜ 1 0 ) = Vg 0 (˜ 2 0 ) = 2πv 0 . Note that such spherical caps are disjoint and isoperimetric for their own volume in (S 2 ,g 0 ).
The combination of (0.14) and (0.15) then yields lim inf contradicting the classical Levy-Gromov inequality for v = v 0 /2 and K = 1. The proof of step four and then of Theorem 1 is thus complete.

Remark 2
The above argument actually shows more than what is claimed in Theorem 1, and namely that, if n = 2 and g is conformally-critical for L v just for a sequence of values v = v h → 0 + as h → ∞, then K g is constant. Similarly, in step four, we have proved that (RP 2 , g 0 ) is a critical, and not just conformally critical.