A survey on Inverse mean curvature flow in ROSSes

Abstract In this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.


Introduction
In the last decades geometric ows have been intensively studied in many contexts. Inverse mean curvature ow is a way to deform hypersurfaces in the normal direction with speed proportional to the inverse of the mean curvature. More precisely, x a Riemannian manifold (M,ḡ), called ambient manifold and a smooth hypersurface F ∶ M →M. The inverse mean curvature ow with initial datum F is a smooth one-parameter family of smooth immersions F ∶ M × [ , T) →M such that where H is the mean curvature of F t = F(⋅, t) and ν is the outward unit normal vector of F t . The problem is well posed if the initial datum is mean convex, that is H > everywhere on M = F (M). It can be checked that (1) is a geometric ow in the sense that it does not depend on the parametrization of the hypersurface. Then a solution of the inverse mean curvature ow will be often expressed as the family of immersed hypersurfaces (M t (M, t).
The inverse mean curvature ow is the leading example in the class of expanding ows, hence, if the ow can be de ned for any positive time, then it explores the structure at in nity of the ambient space. It has many applications in General relativity, for example it was the main tool in proving the Penrose inequality in the celebrated paper [HI].
On the other hand, a very proli c and interesting research eld is the study of real hypersurfaces in complex and quaternionic space forms. They are examples of rank one symmetric spaces, or ROSSes for short. In this survey we restrict our attention to the evolution of star-shaped hypersurfaces: an hypersurface of a Riemannian manifold is said star-shaped if there exists a point such that every geodesic passing from this point intersects the hypersurface in two distinct point in a trasversal way. In particular, any star-shaped hypersurface is an embedding of a sphere of suitable dimension.
The aim of this survey is to give a review on the state of art on the evolution by inverse mean curvature ow of star-shaped hypersurfaces in rank one symmetric spaces. We start reviewing classical results in (real) space forms [Ge1,Ge2,Ur] and the recent progresses in this subject: the fundamental remark of [HW] in case of the hyperbolic space and generalization to complex and quaternionic space forms of negative curvature [Pi2,Pi3].
The behaviour of the ow depends on the initial datum, of course, but the geometry of the ambient manifold has a crucial role too. One of the goal of this work is to show the interactions between the ow and the geometry of the ambient manifolds. We will discuss analogies and di erences, in results and techniques, between the cases mentioned above. Moreover we try to introduce an uni ed approach as far as possible. The main similarities are that, in every case, star-shapedness and mean convexity are proprieties preserved by the ow, and that the ow can be de ned for any positive time. In this way, the family of induced metrics on the evolving hypersurface produce a family of Riemannian metrics on the sphere of suitable dimension.
A rst natural question is then to understand if such family converges to some nice limit when time goes to in nity. At this level we nd the rst important in uence of the geometry of the ambient manifolds: in case of real space forms, we have a convergence to a Riemannian limit, but, considering the case of the other ROSSes, the family of metric degenerates in some special directions and the limit is only sub-Riemannian.
Once we have clarify the convergence, a second question is to classify the possible limits in terms of their curvature: in particular authors tried to identify conditions on the initial data in order to ensure a limit with constant scalar curvature. When the limit is Riemannian it is clear what we mean with curvature, however an interesting phenomenon appears: if the ambient manifold is the Euclidean space the limit is always the round metric on the sphere, when the ambient manifold is the hyperbolic space we have in nite examples of not round limit. It is the main result of [HW]. We describe the strategy for the construction of non round limits in Section 5.1. For hypersurfaces in the other ROSSes, as we said, the limit is only sub-Riemannian and the usual notion of curvature does not work, see Remark 3.5 below. The Webster and the quaternionic contact curvature were proposed in [Pi2,Pi3] as the right notion of curvature to use in this context. We introduced these notions and their main properties in Sections 4.2 and 4.3. Moreover, with a strategy inspired by [HW] is possible to see that in this setting too there are examples which produce limits with non constant scalar curvature.
The same kind of problem has been studied also in other ambient manifolds: for example rotationally symmetric in [Di] and asymptomatically hyperbolic in [Ne] and warped products [Sc, Zh]. Evolution of starshaped hypersurfaces in ROSSes was studied for di erent purpose in [KS] too.

Rank one symmetric spaces
The theory of symmetric spaces is very rich and well known. They can be seen in many ways, but in the present paper we privilege the Riemannian point of view. A classical introduction to these manifolds is Helgason's book [He]. Here we just recall some basic facts, focusing mostly on the small class of ROSSes.

De nition 2.1. A Riemannian manifold (M, g) (or simply M if it is clear what is the metric) is said symmetric
if for every point p ∈ M there exists an isometry ϕ p of (M, g) (2) p is an isolated xed point of ϕ p . Such ϕ p is called symmetry at p. We say that M is locally symmetric if conditions (1) holds just in a neighbourhood of p.
Obviously any symmetric manifold is also locally symmetric. It can be proved that the contrary is true if M is simply connected. Symmetric spaces have many important properties. We list just few of them. (M, g) be a symmetric space, then the following hold: (1) for every p ∈ M and every geodesic γ such that γ( ) = p we have ϕ p ○ γ(t) = γ(−t);

Proposition 2.2. Let
(2) M is complete; (3) M is homogeneous, i.e. the isometry group acts transitively on M; (4) Let ∇ be the Levi-Civita connection of g and R its Riemann curvature tensor, then ∇R = .
De nition 2.3. Let M be a symmetric space, a totally geodesic submanifold of M is said at if it is isometric to the Euclidean space and it is said maximal if it is not contained properly in any other at submanifold of M. It can be checked that all such maximal at submanifolds have the same dimension. We call this number the rank of M.
The simplest symmetric spaces is, of course, the Euclidean space R n . Trivially its rank is n. In general the rank is always greater or equal to because geodesic curves can be considered as at submanifolds of dimension . The rank one symmetric spaces is a well known class of manifolds. They are nitely many and they can be divided into compact and non-compact. They can be characterized in many ways. For example we have The rst class if often denoted with the acronym CROSSes and it is composed by the round sphere S n , and the projective spaces KP n , where K is one of the following algebras: the real R, the complex C, the quaternionic H or the octonion numbers O. In general n can be any integer, except to the last case where only n = is allowed. OP is called the Cayley plane. An excellent introduction to these spaces can be found in Chapter 3 of [Be].
In the present paper we focus mostly on non-compact rank one symmetric spaces. In the following, they will be called simply rank one symmetric spaces or with the acronym ROSSes. As suggested by Proposition 2.4, in the class of ROSSes we nd the hyperbolic analogous of the CROSSes.

. Real hyperbolic space
The (real) hyperbolic space RH n , or simply H n , is the best known example in the class of ROSSes. Probably, it does not need any presentations. However, we spend some words that will be useful later on. It is the space form of constant negative curvature. Up to a homothety we can consider that its sectional curvature is equal to − . A model for H n is (R n ,ḡ) where, in polar coordinates,ḡ is given bȳ Here and in the following σ will denote always the usual round metric on S n− with constant sectional curvature . The Riemann curvature tensor ofḡ has the following explicit expression: . Complex and the quaternionic hyperbolic space The complex hyperbolic space CH n and the quaternionic hyperbolic space HH n are respectively the complex and the quaternionic analogous of the real hyperbolic space. Like in the case of H n , there are many equivalent models. Let K be one of the eld C or the associative algebra H , and de ne For any z = (z , . . . , z n+ ), w = (w , . . . , w n+ ) ∈ K n+ let us de ne the following K-analogous of the Minkowski metric in R n+ : Since for every z, ⟨z, z⟩ is real, we may de ne the set V − = z ∈ K n+ ⟨z, z⟩ < . Clearly if z ∈ V − , then all the subspace spanned by z belongs to V − . If z ≠ , we call this -dimensional subspace a negative line. KH n is PV − , i.e. the collection of all the negative lines. The product ⟨⋅, ⋅⟩ induces a Riemannian metric on KH n . When K = C it is called Bergman metric. In this way KH n is a K-manifold of dimension n. The structure of K induces the complex structures of KH n . We are interested mostly in its Riemannian structure, so we will consider it as a (real) manifold of dimension an.
In the following we will use another model, precisely like in (2), we wish to introduce polar coordinates. In order to do that, we need to de ne some special metrics on the sphere. Let us consider the sphere S an− canonically embedded in R an and let ν be the outward unit normal vector of this immersion. R an can be identi ed with K n , and on K n we have the usual complex structures: (1) if K = C we have J = J the multiplication by the imaginary unit, in particular J = −id; (2) if K = H we have J , J and J the multiplication by the three quaternionic units, in particular For every ≤ i ≤ a − we can de ne ξ i = J i ν. They are unit vector elds tangent to S an− . Usually they are called Hopf vector elds because they are the vector elds tangent to the bers of the Hopf bration. The distribution V generated by the Hopf vector elds is called the vertical distribution. The distribution H = V ⊥ , where the V ⊥ denotes the orthogonal complement of V with respect to the round metric σ, is called the horizontal distribution.
De nition 2.5. For any µ > , the Berger metric on S an− of parameter µ is the metric e µ obtained with a deformation of σ on V. For every tangent vector X and Y we impose: When µ goes to in nity, the Berger metric degenerates on V and it is well de ned only on H. However the horizontal distribution has the nice property of being bracket generating, i.e. H + [H, H] is the whole tangent space. This implies that we can join any two point of the sphere with horizontal curves (i.e. curves such that at any point the tangent vector belongs to H). Hence a metric de ned only on H is enough to de ne a distance on the sphere: we use the usual de nition length of a curve and of distance between points, but restricted to the class of horizontal curves. This distance is called the Carnot-Caratheodory distance.
De nition 2.6. A metric de ned only on a bracket generating distribution is called sub-Riemannian. The limit lim µ→∞ e µ is called the standard sub Riemannian metric on S an− and it is denoted by σ sR .
Remark 2.7. We introduced a common notation for both the possible value of K. We point out that the two cases produce two di erent deformation and two di erent sub-Riemannian limit: for example V has dimension a − , hence it depends on the choice of K. If not speci ed explicitly, the right interpretation of e µ will be clear from the context. Now we have all the ingredients to describe properly the metric of KH n in polar coordinates: on the underling manifold R an we consider the usual (real) polar coordinates, then the metric is where the Berger metric is de ned according to Remark 2.7. Note that, in this case,ḡ is not a warped product because the metric e cosh (ρ) changes with the radial coordinate too. Its Riemann curvature tensor has the following explicit expression where the J i 's are the complex structure of KH n that, in this model, coincide with the usual one in R an = K n . It follows that the sectional curvature of a plane spanned by two orthonormal vectors X and Y is given bȳ where YK is the space spanned by {J i Y} i≤a− and pr YK is the projection on YK. Then − ≤K ≤ − and it is equal to − (respectively to − ) if and only if X is orthogonal (respectively parallel) to YK. The fact that the sectional curvature of the planes spanned by X and J i X is constant for every X and i is synthesized saying that KH n is a K-space form. Moreover KH n is Einstein with The last element of the set ROSSes is the octonion hyperbolic plane OH . It can be de ned in a similar way using the algebra of octonions instead of K. Since we will not use this space in the sequel we skip a precise de nition of this manifold, it can be found for example in [Pa2]. More detalils and informations about the geometry ROSSes can be found in [Pa1,Pa2].

Inverse mean curvature flow
In (1) we de ned the inverse mean curvature ow. The very rst problem to deal is to understand if this ow has a solution. In local coordinates (1) produces a weakly parabolic PDE. Standard theory ensure the following existence result.
Theorem 3.1. If the initial datum M is closed (i.e. connected, embedded, compact and without boundary) and mean convex (i.e. H > everywhere), then (1) has an unique solution, at least for times t small enough.
A proof of this result which holds for a wide class of ows can be found in Theorem 3.1 of [HP].
The behaviour of such solution depends, of course, on the initial datum. The geometry of the ambient manifold has a crucial role too. As we will see in the next sections, even in the small setting of geodesic spheres in ROSSes we have very di erent behaviour.

. Explicit examples: geodesic spheres in ROSSes
In the Euclidean space and in the ROSSes, included the compact ones, the principal curvature of a geodesic sphere are constants that depend only on the radius (this characteristic does not hold, for examples, for other symmetric spaces). In particular the mean curvature is a constant that depends only on the radius too. Together with the uniqueness of the solution, this suggests that, in these ambient manifolds, the evolution of geodesic spheres is a family of geodesic spheres and everything is reduced to an ODE on the radius. Moreover, trivially, every geodesic sphere is a star-shaped hypersurface. In the following examples let M t be a geodesic sphere of radius ρ = ρ(t). Integrating we can easily nd that the explicit solution is Example 3.3.M = H n+ In this space it is well known that the geodesic spheres are totally umbilical hypersurfaces with second fundamental form Then, in this case, the radial function satis es: The explicit solution is the following By (2), it easy to see that the induced metric on the geodesic sphere is Hence, we have that the rescaled induced metricg converges to a Riemannian metricg ∞ which is a constant multiple of σ. Therefore, in this very special case, the curvature of the limit is constant.
Example 3.4.M = KH n with K ∈ {C, H} In these ROSSes the geodesic sphere are not totally umbilical, indeed it is well known that a totally umbilical hypersurface does not exist (Theorem 5.1 in [NR]). A geodesic sphere has distinct principal curvatures: with multiplicity a − and eigenvectors V; with multiplicity a(n − ) and eigenvectors H. (27)

The solution of this ODE cannot be written nicely in terms of elementary functions, but we can see that the ow is de ned for every time and that
From (

. Evolution equations
Starting from the de nition of the ow (1), we can compute the evolution of the main geometric quantities of a hypersurface. The proof of this Lemma is similar to the computation of the analogous equations for the mean curvature ow which can be found in [Hu]. We use the following notations: let g ij be the induced metric, and g ij its inverse; the second fundamental form is denoted with h ij , while the mean curvature is H = h ij g ji . Finally M t denotes the volume.
Lemma 3.6. For any hypersurface in a general ambient manifold the following evolution equations hold: Here and in the following we are using Einstein convention on repeted indices. The operation of raising/lowering the indices is done with respect to the metric: for example h j i = h ik g kj . Moreover the index is reserved to the normal direction ν.
Remark 3.7. Note that, integrating equation ( ), we have that the inverse mean curvature ow is en expanding ow, precisely M t = M e t .
Since we are considering only symmetric ambient space, as seen in Proposition 2.4, we have that∇R = , hence the formulas above can be simpli ed. They can be even simpler because we known the explicit expression of the curvature tensors, see (3) and (6).

The Yamabe problem
A central and classical subject in Geometric Analysis is the Yamabe problem: nd, if they exist, the metrics with constant scalar curvature in a xed conformal class. The detailed explanation of its solution goes beyond the purposes of the present work. Indeed there exist surveys devoted to it, we suggest [LP] and [BM] and the references therein.
Here we want to discuss only the special case of the conformal class of standard metric on the sphere. Where "standard" must be considered in a broad sense: we will discuss also the generalization of the (Riemannian) classical Yamabe problem to the class of CR and quaternionic contact structures on the sphere. The solution of this problem has been the starting point for the construction of the counterexamples in [HW,Pi2,Pi3]. We will describe these counterexamples in Sections 5.1 and 6.1

. The classical case
Let (S n , σ) be the round sphere. We want to understand if there exist functions f such that the conformal changeσ = e f σ produces a metric with constant scalar curvature. It is well known that the sign of the scalar curvature is invariant, hence in our case it should be positive. The answer is given by this result of Obata [Ob] Theorem 4.1. The metricσ has constant scalar curvature, then, up to a constant factor, it is obtained from σ by a conformal di eomorphism on S n .
In this way we have a full characterization of the functions f that realize the desired conformal multiple. We report Lemma 4 of [HW].

. CR-Yamabe problem
As noted in Remark 3.5, even in the very special case of a geodesic sphere of CH n , the usual Riemannian curvature is not enough to understand the nature of the limit. In [Pi2] we proposed the Webster curvature as the right notion for investigate the roundness of the limit. It is a notion of CR-geometry, in fact, since CH n is a complex manifold, every real hypersurface has, in a natural way, a CR-structure: the associated -form is just θ(⋅) = g(Jν, ⋅), where ν is the unit normal vector of the hypersurface. The associated horizontal distribution is H = ker θ. On H we have the complex structure J induced by the complex structure of the ambient manifold. Let π be the canonical projection from the tangent space to H and let ξ θ = Jν. We have θ(ξ θ ) = and dθ(ξ θ , ⋅) = . We can extend J to the whole tangent space of the hypersurface requiring that Jξ θ = . For every X, Y ∈ H we can de ne G θ (X, Y) = dθ(X, JY). The metric g θ = πG θ + θ is called Webster metric associated to θ.
Theorem / De nition 4.3. Let θ, J, ξ θ and g θ be as before. There exists an unique linear connection ∇ TW such that ∇ TW J = ∇ TW θ = ∇ TW ξ θ = ∇ TW g θ = and with torsion T of pure type, i.e. for every X, Y ∈ H the torsion satis es: This connection ∇ TW is called the Tanaka-Webster connection associated to θ. The Webster curvature of θ is the curvature de ned in the usual way, but using the Tanaka-Webster connection instead of the Levi-Civita connection. A natural question is the Yamabe problem in this new setting, the so called CR-Yamabe problem: nd the functions f such that the conformal multiple e f θ has constant Webster scalar curvature.
As before, we will focus only on the case of the sphere, what follows can be said with much more generality. We refer to the monograph [DT] for all the details about CR-geometry and the general solution of the CR-Yamabe problem. On S n− we have a standard contact form:θ(⋅) = σ(ξ, ⋅). With respect to this form we have: ξθ = ξ, the Hopf vector eld. Gθ = σ sR and gθ = σ. Obviously the Webster curvature ofθ is constant. A metric of the form e f σ sR can be thought as the restriction to H × H of the Webster metric associated to the -form e fθ , and vice versa. Then we will talk indi erently about the Webster curvature of a sub-Riemannian metric or of a contact form.
The answer of the CR-Yamabe problem on the sphere is given by the following remarkable formula by Jerison and Lee [JL].
Here we are considering the odd-dimensional sphere immersed in C n and the norm and the product are the usual ones in C n .

. qc-Yamabe problem
In the same spirit of the previous subsection, we need to clarify what we mean when we talk about curvature of a sub-Riemannian metric de ned on the horizontal distribution of codimension of S n− . In this context, the analogous of the CR-geometry, is the quaternionic contact-geometry (qc-geometry for short). The notion of qc-structure has been introduced by Biquard in [Bi]. We refer also to the book [IV1] of Ivanov and Vassilev for further details. A qc-structure on a real ( n − )-dimensional manifold (M, g) is a codimension distribution H locally given as the kernel of a -form η = (η , η , η ) with values in R such that for every i = , , dη i H (⋅, ⋅) = g (I i ⋅, ⋅), where the I i 's are three almost complex structure on H satisfying the identities of the imaginary unit quaternions and which are compatible with the metric g. Such η is determined up to the action of SO( ) on R and a conformal factor. Hence H is equipped with a conformal class [g] of quaternionic Hermitian metrics. To every metric in the conformal class of g one can associate a linear connection with torsion preserving the qc-structure called the Biquard connection. It has been de ned by Biquard in [Bi] if n > and by Duchemin in [Du] if n = . They proved that this connection is unique. Using the Biquard connection we can de ne the qc-Ricci tensor as in [Bi]: it is a symmetric tensor and its trace is called the qc-scalar curvature. Once again, with this new notion of curvature, we can de ne in the usual way the qc-Yamabe problem. This problem too has been solved in great generality, see [IV2] for a survey. In our case M = S n− , g = σ sR and for every i η i (⋅) = σ(J i ν, ⋅). In [IMV] Ivanov, Michev and Vassilev fully characterized the solution of the qc-Yamabe problem in the special case of the quaternionic Heisenberg group. As they noticed, with the Cayley transform, we can nd the corresponding solutions on S n− , [σ sR ] . The result is very similar to Lemma 4.4.
Here we are considering the sphere of real codimension one immersed in R n ≡ H n and the norm and the product are the usual ones in H n .

The case of the Euclidean and the Hyperbolic space
From the De nition 2.3 of rank, it is evident that the Euclidean space is not in the class of ROSSes. We decided to discuss this case too for two main reasons. First of all the Euclidean space is the simplest ambient manifold, then it is also the rst case studied in literature: many ideas developed in ROSSes nd their origin in the early works of Gerhardt [Ge1] and Urbas [Ur]. The two authors studied independently and in the same time the inverse mean curvature ow of star-shaped hypersurfaces of the Euclidean space. Moreover we choose to describe this case together with the case of the hyperbolic space because both these manifolds can be seen as a warped product of the real line and the round sphere. The inverse mean curvature ow in the hyperbolic space was studied rst by Gerhardt in [Ge2], but recently an uni ed approach for warped products of this kind has been introduced by Scheuer [Sc]. In this paper we prefer to describe this strategy when possible.
We refer also to [Zh] for inverse mean curvature ow in warped products. Studying the evolution of a starshaped hypersurfaces, there are many common features between the two non-compact space forms, but a big di erence appears when we consider the roundness of the limit as shown by Hung and Wang in [HW].
The ambient manifolds considered in [Sc] are warped products of the type for some function α satisfying some reasonable properties (see Assumption 1.1, 1.2 and 1.3 in [Sc]). For α(ρ) = ρ we have the Euclidean space R n+ , while if α(ρ) = sinh(ρ) then we nd the hyperbolic metric (2). In this class of ambient manifolds there are also many other interesting examples, one of them is the de Sitter-Schwarzschild manifold. From this point of view, the results of [Ge1,Ge2,HW,Ur] can be summarized in the following way.
Theorem 5.1. LetM be either the Euclidean space R n+ or the hyperbolic space H n+ . Let M be a star-shaped, closed and mean convex hypersurface inM and let (M t ) t≥ be its evolution by inverse mean curvature ow. Then: (1) M t is star-shaped and mean convex for any time t the ow is de ned; (2) the ow is de ned for any positive time; (3) the second fundamental form converges exponentially to that one of an horosphere: precisely there exists a positive constant c such that (4) there is a function f ∶ S n → R such that As seen in Lemma 4.2, the nature of the function f describes the shape of the limit. We have: (5) ifM = R n+ then f is always constant, henceg ∞ is always round; (6) ifM = H n+ there are in nitely many examples such thatg ∞ has not constant scalar curvature.
The construction of the counterexample in (6) is the remarkable result present in [HW]. In the following of this section we sketch the main point of the proof of Theorem 5.1, focusing mostly on reasons why the di erence between (5) and (6) appears and describing the strategy of Hung and Wang. We start introducing some notations. Fix an auxiliary function ϕ(ρ) such that dϕ dρ = α(ρ) . Fix a basis (Y , . . . , Y n ) tangent on S n , for any i, j we de ne where ∇ σ is the Levi-Civita connection of σ. Analogous notation are used for the derivatives of every function. Then the set of vectors {V i = Y i + ρ i ∂ρ} ≤i≤n is a basis of the tangent space of M t at any time t. In these coordinates we have: where Taking the trace of (13) we have: Starting from the de nition of the ow (1), it easy to check that the radial function satis es the scalar ow: and that the evolution of hypersurfaces is de ned at least as long as (15). We prefer to work with the auxiliary function ϕ, hence its evolution is given by By (14) we have that The rst step is to prove that the hypothesis on the initial datum are "good" properties, in the sense that they are preserved by the ow.

Proposition 5.2. If M is star-shaped, then M t is star-shaped as far the ow exists.
Proof. A hypersurface is star-shaped if the normal vector and the radial vector are never orthogonal in the ambient space. With the notations introduced above we have that this condition is equivalent to the existence of a constant c such that v ≤ + c ⇔ ∇ σ ϕ σ ≤ c.
Let us de ne ω = ∇ σ ϕ σ = ϕ i ϕ i . Using (16), we can compute the evolution equation of ω: let a ij = − ∂F ∂ϕ ij = σ ij v , b i = ∂F ∂ϕ i and, for simplicity of notation, ∇ = ∇ σ , then We can apply the rule for interchanging derivatives: where this time R is the Riemann curvature tensor of σ, i.e. R sijk = σ sj σ ik − σ sk σ ij . Since a ij is symmetric we get: The following inequality holds: −a ij ϕ i ϕ j + a i i ω ≥ . From these estimates we get: A key property is that a ij is positive de nite, hence the evolution of ω is given by a parabolic equation and we can use the maximum principle to estimate ω itself. The reaction term can be estimated in the following way Finally we have that because, as showed in [Di], if A, B and C are symmetric matrices, with A and B positive de nite, then tr(ACBC) ≥ . The thesis follow by the maximum principle.
The following result shows that, in particular, also the mean convexity is preserved by the ow.

Proposition 5.3. There are two constant depending only on n and the initial datum such that for any time the ow is de ned
If the ambient manifold is the hyperbolic space, then c can be chosen strictly positive.
Proof. Both the estimates are proved by maximum principle applied on suitable functions. The inequality A ≥ H n holds for any hypersurfaces. Since the ambient manifold has non-positive sectional curvature, by Lemma 3.6, we can deduce that ∂H ∂t ≤

∆H
where c ≥ . It follows by the maximum principle that H is bounded from above.
To prove that H is bounded from below we de ne ψ = v αH e t n = F e t n = ∂ϕ ∂t e t n and we prove that this function is bounded from above. Preceding like in the proof of Proposition 5.2: for some constant c. By the maximum principle we deduce that ψ grows exponentially in the rst case and it is bounded in the second case. By de nition of ψ we have the thesis.
With an other application of the maximum principle is possible to prove that A is uniformly bounded (see for a proof Lemma 3.8 in [Zh]). From the a priori estimates just described, we have that the ow (16) (or equivalently (15)) is uniformly parabolic and it satis es the requirements to apply the regularity results of Krylov-Safonov (par. 5.5 of [Kr]). In this way we have that the ow is de ned for any positive time and the solution is smooth as the initial datum. The proof of part (3) of Theorem 5.1 is another application of maximum principle on the function G = h j i − α ′ α δ j i . We prefer to skip it, the proof can be read in Theorem 4.8 of [Sc].

. The construction of counterexamples in H n+
The warping function has a fundamental role in the possible value of the conformal multiple that appears iñ g ∞ = e f σ. Since in the Euclidean case α ′ is bounded, only the constants can appear. On the other hand, when the ambient manifold is the hyperbolic space α ′ = cosh(ρ) and, since it is unbounded, we have examples where f is not constant. In this case a natural problem is the classi cation of the possible limits according to their curvature. K.P. Hung and M.T. Wang proved in [HW] that the limit is not always round, indeed the constant scalar curvature of the limit is a very special case and there are in nitely many non trivial examples. The way to de ne such hypersurfaces is indirect and uses a notion that was born in general relativity. They de ned for any closed hypersurface M the modi ed Hawiking mass where ○ A = A − H n g is the trace-free part of the second fundamental form of M. This quantity measures the umbilicity of the hypersurface: in fact it is easy to check that ○ A = if and only if the hypersurface is totally umbilical. One of the main property of m H is the following.
Proposition 5.4. [HW]. Let f be a real function on S n , τ a positive number and letM τ be the star-shaped hypersurface de ned by the radial functionρ = τ + f + o( ), then: where we are integrating with the measure induced by σ and the circle means again "trace-free part". The limit of the induced rescaled metric is e f and, by the classi cation of Lemma 4.2, we have that Now we need link the modi ed Hawking mass with the inverse mean curvature ow. By the formulas in Lemma 3.6, we can compute its evolution. In [HW] the following fundamental estimate is proved: Lemma 5.5. There exists a positive constant c such that along the inverse mean curvature ow (M t In other words, we don't know if m H is monotone, but if it decreases, it decreases very slowly. Now we can collect all the results and describe the strategy of Hung and Wang to produce non trivial examples. (1) Fix a positive constant c big enough and choose a function f ∶ S n → R such that (2) Consider the familyM τ de ned in Proposition 5.4 and choose a τ big enough such thatM τ is mean convex, and, by Proposition 5.4, Q(M τ ) ≥ c .

The case of the Complex and Quaternionic Hyperbolic space
The extention of Theorem 5.1 to the other manifolds in ROSSes has been faced by the author of the present survey in [Pi2] for CH n as ambient manifold and in [Pi3] for HH n . Once again there are similarities with the previous cases, but also some new phenomena appear in this context. The most important is that, even after rescaling, the induced metric do not converges to a Riemannian limit but only to a sub-Riemanninan metric de ned of a distribution of codimension a − . The main reason for this di erent behaviour could be found in the fact that the metric (5) is not a warped product, but the metric on the sphere changes with ρ and degenerates at the boundary at in nity of the ambient manifold (i.e. when ρ diverges). Moreover since for both value of K, the "warping" function in (5) has unbounded derivative, it is possible to produce examples which develop limits with non constant scalar curvature: for the motivation explained in Remark 3.5, the roundness of the limit is studied in term of Webster and qc scalar curvature. The precise statement is the following: (2) the ow is de ned for any positive time; (3) let The goal of this section is to describe the main di erences between Theorem 5.1 and Theorem 6.1, in terms of result and techniques used in the proofs. Moreover we give an uni ed approach for both the value of K.
First of all we need to clarify the new hypothesis. S a− can be identify with a group of isometries of (S an− , σ) acting in the following way: S ∶ (γ, z) ∈ S a− × S an− ⊂ K × K n ↦ γz ∈ S an− .
The projection π ∶ z ∈ S an− ↦ [z] = γz γ ∈ S a− ∈ KP n− is the well known Hopf bration. Since star-shaped hypersurfaces are identi ed with their radial function ρ ∶ S an− → R + , we have the following natural de nition.

De nition 6.2. We say that a star-shaped hypersurface in KH n is S a− -invariant if its radial function is invariant by the action (22).
This new hypothesis is required for proving the analogous of Proposition 5.2 in the new setting. Moreovere the symmetries beehive very well with geometric ows. The paper [Pi1] is about their interaction with the mean curvature ow. Lemma 3.1 of [Pi1] can be reproduced with minor modi cation for the inverse mean curvature ow showing: Lemma 6.3. Let M be an S a− -invariant hypersurface in KH n , then M t is S a− -invariant for any time the ow is de ned.
We describe the main geometric quantity for an S a− -invariant star-shaped hypersurface in KH n . Since the ambient space is not isotropic, we have to x a particular basis tangent to the sphere. Let (Y , ⋯, Y an− ) be (2) Moreover if lim τ →∞ Q( M τ ) ≠ , then e f σ sR -the limit of the rescaled metric on M τ -does not have constant (Webster or qc) scalar curvature.
Let us de ne for brevity In order to prove property ( ) we note that if Q(f ) ≠ , then for sure f cannot be constant, but, recalling that f is S a− -invariant, an easy computation shows that f satis es equations (9) (or (10)) if and only if it is constant.
A careful analysis about the evolution of Q says that if it decreases, then it decreases very slowly. In fact we have the following estimate. Now we have all the ingredient to repeat the strategy of [HW] described in Section 6.1 and we can show the existence of counterexamples in KH n too.
(1) Fix a positive constant c big enough and choose an S a− -invariant function f ∶ S an− → R such that Q(f ) ≥ c .
(2) Consider the family of S a− -invariant star-shaped hyperfurfaces M τ de ned by the radial functioñ ρ τ (z) = τ +f (z). Fix a τ big enough such that M τ is mean convex, and, by Proposition 6.6, Q( M τ ) ≥ c . (3) Let the ow start from intial datum M τ . (4) Since c is big enough, Proposition 6.7 ensures that where (M τ t ) t≥ is the evolution of M τ . (5) The thesis follows from Proposition 6.6.
Remark 6.8. The analogous of Theorem 6.1 can be proved in the octonion hyperbolic plane OH too. In this case we have to use S -invariance. The induced rescaled metric converges to a sub-Riemannian metric on S de ned on a distribution of codimension . De ningĤ(ρ) = sinh(ρ) cos + cosh(ρ) sinh(ρ) , it is possible to de ne the analogous of the quantity Q (32) and produce, with the same strategy described above, a non-trivial example. The problem is that, for the best of our knowledge, there is no analogous of the Tanaka-Webster and Biquard connections in this context, hence it is not clear how to interpret the nature of the limit metric in terms of its curvature.