Geometric Invariants of the Time-Slice of the Mean Curvature Flow

Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, Vol.10, (2021), Issue 1, pp.15-31

GEOMETRIC INVARIANTS OF THE TIME-SLICE OF THE MEAN CURVATURE FLOW

FRANCK HOUENOU, CARLOS OGOUYANDJOU, LEONARD TODJIHOUNDE AND JOEL TOSSA

Abstract. Let Σ be a submanifold of a Riemannian manifold M. The evolution equations of some extrinsic geometric quantities such as the norm of the second fundamental form, the norm of the mean curvature, etc., have been computed along the mean curvature flow and then used to investigate either the long-time existence of the floworsome extrinsic geometrical properties (minimality, totally geodesic, umbilicity, etc.) of the limiting submanifold (see e.g. [1, 2, 4, 5]). But few results are known about the intrinsic geometry of the evolving submanifold along the flow [3]. In the present work, assuming that the mean curvature flow exists, we compute for a submanifold of a Riemannian manifold, the evolution equations of the Riemannian curvature, the Ricci curvature and the scalar curvature which are intrinsic geometrical invariants of the submanifold. From the evolution equation of the scalar curvature, we derived that the graphical mean curvature flow favours positive scalar curvature ; a well known result in thisarea.

1. Introduction and motivations The celebrated theory of J. F. Nash on isometric immersion of a Riemannian manifold into a suitable Euclidean space gives a very important and effective motivation to view each Riemannian manifold as a submanifold in an Euclidean space. Since ever, the problem of discovering simple and sharp relationships between intrinsic and extrinsic invariants of a Riemannian submanifold becomes one of the most fundamental problems in submanifolds theory. Extrinsic and intrinsic geometries are the often used two approaches to study the Riemannian geometry of a submanifold. In terms of geometrics quantities, these two approaches are equivalent since Riemannian geometry deals with the geometrical elements such as metric, curvatures, connections, etc. For the extrinsic approach, the study is carried out in reference to the geometry of the ambient manifold but any reference to an- other manifold is needed for the intrinsic approach [7]. The main intrinsic invariants are the scalar curvature and the Ricci curvature while the main extrinsic one are the second fundamental form and its relatives such as the mean curvature, the shape operator etc. Talking about geometric flow, where the purpose is to describe the evolution ofthege- ometry of manifold under some flow, the widely known flows are the mean curvature flow and the Ricci flow for their interest and applications in mathematic and physic. Roughly

2010 Mathematics Subject Classification. 53C17 ; 53C44. Key words and phrases. Sub-Riemannian geometry, Geometric evolution equations.

15 Franck Houenou, Carlos Ogouyandjou and Leonard Todjihounde speaking, the motion by the mean curvature is a process under which a submanifold de- forms in the direction of its mean curvature vector. Essentially, the purpose in mean curvature flow is to study the behavior of a “moving” submanifold inside some ambient manifold ; thus the flow mainly describes its extrinsic geometry whereas the Ricci flowis based on the intrinsic geometry. Therefore, these properties turn out to be important to study when one submits a submanifold to the mean curvature flow in a Ricci flow back- ground. In [3], the author computed the evolution equations of the intrinsic curvatures along the mean curvature flow of an embedded submanifold in a higher-dimensional flat space ; he also proved that the evolution equation of the scalar curvature of a Riemannian submanifold is similar to the Ricci flow. Later on he deduced that negativity rather than positivity of the scalar curvature is preserved, what does no longer hold according to our understanding. So, the present work compute the evolution equations of the curvatures for a submanifold in a (non necessarily) flat Riemannian manifold. These computations generalize those in [3]. Contrarily to the result in [3], this work proved that the positivity of the scalar curvature is preserved along the flow : a well known result in the area. The mean curvature flow equation is a quasi-parabolic equation and it is well known that there is no theory which trivially guarantees the long time existence of the solution. However, from Huisken seminal work [2], one can prove using the monotonicity formula and the blow-up analysis, the long time existence for suitable initial data. The argument is based on the geometric structure of the ambient manifold ; thus, proving the long time existence heavily relies on the curvature. Therefore, computing such evolution equations is very illuminating. In the sequel, we assemble in Section 2 some useful materials for the study of the geometry of Riemannian submanifold along the mean curvature flow by recalling the evolution equation of the induced first fundamental form, of a frame on the normal bundle and the time derivative of the mean curvature vector. Thereafter, we compute in Section 3 the evolution equation of the (0, 4)-tensor Riemann curvature and derive those of the Ricci and the scalar curvatures. In Section, 4 we deal with the special case of constant curvature of the ambient manifold where we rewrite the evolution equation of the scalar curvature of the evolving submanifold.

2. Preliminaries The motion by the mean curvature of a submanifold Σ in a Riemannian manifold M is a one parameter family of submanifolds Σ , for some time interval containing , t t∈I I 0 which admits a parametrization F(t) ≡ Ft : Σ −→ Σt ⊂ M over a fixed (the initial) submanifold Σ = Σ with velocity equals to the mean curvature vector; namely 0 ( ∂ ∂ Ft(P) = H ( ) t Ft P (2.1) F0(P) = P, ∀P ∈ Σ where stands for the mean curvature vector at ( ). HFt(P) Ft P Let Σn ⊂ Mm be a n-dimensional submanifold of a Riemannian manifold M of dimension m and F : Σ × [0, T) → M, a mean curvature flow of Σ in M. Choose a local orthonormal { ··· ··· } Σ { ··· } frame e1, , en, en+1, , em of M along t such that e1, , en consists of tangent Σ ··· Σ vectors to t whereas en+1, , em are in the normal bundle over t. In the sequel we will use the following indexes convention : 1 6 i, j, k, l, ··· 6 n, and n + 1 6 α, β, γ 6 m,

16 Geometric invariants of the time-slice of the mean curvature flow i.e lowercase of Latin (resp. Greek) letters from 1 to n (resp. from n + 1 to m). We also use Einstein summation convention for repeated indexes. {∂ } If we denote by i 16i6n the partial derivatives with respect to the local coordinates in Σ and by ⟨·, ·⟩ ≡ g the metric on M, the induced metric on Σ is given by

⟨∂ ∂ ⟩ gij = i F, jF (2.2) and we write the second fundamental form and the mean curvature vector as follows :

α α A = A eα = hijeα, (2.3) α ij α α H = H eα = traceg A = g hijeα = hiieα (2.4)

α α where = ⟨∇ α⟩ = −⟨∇ α ⟩ = −⟨ ∇ α⟩ = ⟨∇ α⟩ = are the compo- hij ei ej, e ei e , ej ei, ej e ej ei, e hji α nents of the symmetric matrix A . Since Gauss, Ricci and Codazzi-Mainardi equations play a key role in the study of the geometry of submanifold, let us recall them in local coordinates for their use in the sequel. We have :

α α − α α Rijkl = Rijkl + hilhjk hikhjl, (2.5) α − α − hij,k hkj,i = Rαjki (2.6) where R(U, V, X, Y) = g(∇ ∇ X + ∇ ∇ X − ∇ X, Y), U V V U [U,V] α e ∇ ∇ ∇ ∇ − ∇ ∇ α R(X, Y, Z, T) = g( X Y Z + Y X Z [X,Y]Z, T) and hrp,q = g ( eq A) er, ep , e .

Lemma 2.1. [5] Along the mean curvature flow, the induced metric on Σt satisfies :

∂ ∂ = − α α ij = α α (2.7) ∂ gij 2H hij, and ∂ g 2H hij. t t=0 t t=0

In the following lemma, we derive the evolution equation of the frame on the normal β ∂ α β Σ α ⟨ α β⟩ ⟨∇ α β⟩ − α bundle N . Let us denote by b = ∂t e , e = He , e and observe that bβ = b .

Lemma 2.2. The evolution of the normal frame can be written as :

∂ α β β eα = −∇H − H ⟨∇ eβ, eα⟩e + bα eβ (2.8) ∂t ei i

α α where ∇H is the covariant differentiation of H with respect to the induced connection on Σt.

17 Franck Houenou, Carlos Ogouyandjou and Leonard Todjihounde

Proof. A straightforward computation yields D E D E ∂ ∂ ∂ β eα = eα, e e + eα, eβ eβ = ⟨∇ eα, e ⟩e + bα eβ ∂t ∂t i i ∂t H i i −⟨ ∇ ∂ ⟩ β = eα, H i F ei + bα eβ β −⟨ α ∇ ⟩ = e , ei H ei + bα eβ β β −⟨ α ∇ ⟩ = e , ei (H eβ) ei + bα eβ β β β −⟨ α ∇ ∇ ⟩ = e , ( ei H )eβ + H ei eβ ei + bα eβ α β β − ∇ − ⟨ α ∇ ⟩ = ( ei H )ei H e , ei eβ ei + bα eβ −∇ α − β⟨ ∇ ⟩ β = H H eα, e eβ ei + bα eβ. i ∂ ∂ ∂ ∂ ∂ The third equality use the fact that ∂ ∂ F = ∇ ∂ F = ∂ ∂ F = ∇e H. t xi H xi xi t i Lemma 2.3. The time derivative of the mean curvature vector is given by : ∂ α β β α α α α β H = △ H + H h h eα − H ∇H − H H ⟨∇ eβ, eα⟩e . (2.9) ∂t ik jk ei i α Proof. From H = H eα we get : ∂ ∂ α ∂ α α ∂ H = H eα = H eα + H eα ∂t ∂t ∂t ∂t ∂ α ∂ α α ∂ = gijh eα = gijh eα + gijh eα . ∂t ij ∂t ij ij ∂t Using Lemma 2.1 we obtain : ∂ ∂ ∂ ∂ α = ij α = ij α + ij α ∂ H ∂ g hij ∂ g hij g ∂ hij t t t t β β α ij α β β α β β − β α ⟨ β ∇ α⟩ = 2H hijhij + g H,ji H hikhjk + H R ij + hij e , He β β α α β β α β β △ − ij ⟨ ∇ α⟩ = 2H hijhij + H H hijhij + H g Rβijα + H eβ, He α β β α β β △ ij ⟨ ∇ α⟩ = H + H hijhij + H g Rβijα + H eβ, He α β β α β ij β β △ α = H + H hijhij + H g Rβijα + H b . Combining this with Lemma 2.2 gives : ∂ α β β α β ij β β α α α β α β △ β α α α − ∇ − ⟨∇ β α⟩ α β H = H + H h hij + H g R ij + H b e H H H H ei e , e ei + H b e ∂t ij α β β α β ij β β α α α β β α △ β α α α − ∇ − ⟨∇ β α⟩ α = H + H h hij + H g R ij + H b e H H H H ei e , e ei + H bβe ij α β β α β ij α α α β = △ + + β α α − ∇ − ⟨∇ β α⟩ H H hijhij H g R ij e H H H H ei e , e ei. α β β α In the second and the third equality, used have been made of : H bα eβ = H bβeα = β β −H bα eα.

18 Geometric invariants of the time-slice of the mean curvature flow

3. Intrinsic geometry along the mean curvature flow In this section we study the proper geometric objects of the evolving submanifold assuming that the flow exists. First of all, we compute the evolution equation of the Riemannian curvature from which we derive those of the Ricci curvature and the scalar curvature of the moving submanifold along the flow. Let us recall the Riemannian, the Ricci and the scalar curvature in local coordinates

− Rijkl = R(ei, ej, ek, el) = R(ei, ej, ek, el) + A(ei, el), A(ej, ek) A(ei, ek), A(ej, el) , kl Rij = Ric(ei, ej) = g Rkijl, ij S = g Rij. 3.1. The Riemannian curvature equation.

Theorem 3.1. Along the mean curvature flow, the curvature tensor of Σt satisfies the following equation : ! ∂ α − △ R = ∇ R e , e , e , e − R + h Rα + Rα ∂t gt ijkl H i j k l ijkl,pp ik plp,j ljp,p − α − α α hil Rαpkp,j + Rαkjp,p hjk Rαplp,i + Rαlip,p + hjl Rαpkp,i + Rαkip,p − α α α α 2 hpiRαjkl,p + hpjRiαkl,p + hpkRijαl,p + hpl Rijkα,p α α − β − β − β +hpi hpr Rrjkl 2h Rαβkl 2h Rαjβl 2h Rαjkβ pj pk pl α α β β α α β − − − αβ +hpj hpr Rirkl 2hpkRiαβl 2hpl Riαkβ + hpk hpr Rijrl 2hpl Rij α α α α − α α α α +hplhprRijkr + hilhmpRmkjp hikhmpRmljp + hmphjkRmlip − α α − α α − α α − α α hmphjl Rmkip H hpiRpjkl H hpjRipkl H hpkRijpl − α α β α − β α β α H hpl Rijkp + H hil Rβjkα H hikRβjlα + H hjkRβilα β α −H h Rβ α + Rα Rα + R α Rα + R α Rα + R αRα jl ik jkl ppi i kl ppj ij l ppk ijk ppl β α − β α α β − α β + h hjk h hjl Rβαip + hilh hikh Rβαjp lp kp kp lp α α − α α α α − α α + hlmhjk hkmhjl Rmpip + hilhkm hikhlm Rmpjp − β α β α β α − β α β α β α H hjk h him + h hln H hil h hjm + h hkn lm in km jn β α β α β α β α β α β α +H hjl h him + h hkn + H hik h hjm + h hln km in lm jn β α β α − β α β β α α − α α +hmphim h hjk h hjl + hmph hlmhjk hkmhjl lp kp pi β α α β − α β β β α α − α α +hmphjm hilh hikh + hmph hilhkm hikhlm kp lp pj β α β α α β − β α − α β β α β α − β α +hmphmp h hjk + hilh h hjl hikh + 2h hmp h hjl h hjk li kj ki lj im kp lp β α α β − α β − α α α α +2hjmhmp hikhlp hilhkp 2hil,phjk,p + 2hik,phjl,p. (3.1)

19 Franck Houenou, Carlos Ogouyandjou and Leonard Todjihounde

Proof. We start by computing the (rough) Laplacian of the curvature. ∇ · − ∇ − ∇ ( ep R) ei, ej, ek, el = ep R(ei, ej, ek, el) R( ep ei, ej, ek, el) R(ei, ep ej, ek, el) −R(e , e , ∇ e , e ) − R(e , e , e , ∇ e ) i j ep k l i j k ep l α α ∇ α α = ( ep R) ei, ej, ek, el + hpiR e , ej, ek, el + hpjR ei, e , ek, el α α α α − ⟨ ∇ ⟩ +hpkR ei, ej, e , el + hpl R ei, ej, ek, e A( ep ei, el), A(ej, ek) ⟨ ∇ ⟩ − ⟨ ∇ ⟩ + A( ep ei, ek), A(ej, el) A(ei, el), A( ep ej, ek) ⟨ ∇ ⟩ − ⟨ ∇ ⟩ + A(ei, ek), A( ep ej, el) A(ei, ep el), A(ej, ek) +⟨A(e , e ), A(e , ∇ e )⟩ − ⟨A(e , e ), A(e , ∇ e )⟩ i k j ep l i l j ep k ⟨ ∇ ⟩ ∇ α α − ∇ α α + A(ei, ep ek), A(ej, el) + ep hilhjk ep hikhjl . (3.2)

The last two terms can be computed as follows : ∇ α α ∇ ⟨ ⟩ ⟨∇ ⟩ ⟨ ∇ ⟩ ep hilhjk = ep A(ei, el), A(ej, ek) = ep A(ei, el), A(ej, ek) + A(ei, el), ep A(ej, ek) ⟨ ∇e ∇ ∇ ⟩ = ( ep A)(ei, el) + A( ep ei, el) + A(ei, ep el), A(ej, ek) ⟨ ∇e ∇ ∇ ⟩ + A(ei, el), ( ep A)(ej, ek) + A( ep ej, ek) + A(ej, ep ek) and ∇ α α ∇ ⟨ ⟩ ⟨∇ ⟩ ⟨ ∇ ⟩ ep hikhjl = ep A(ei, ek), A(ej, el) = ep A(ei, ek), A(ej, el) + A(ei, ek), ep A(ej, el) ⟨ ∇e ∇ ∇ ⟩ = ( ep A)(ei, ek) + A( ep ei, ek) + A(ei, ep ek), A(ej, el) ⟨ ∇e ∇ ∇ ⟩ + A(ei, ek), ( ep A)(ej, el) + A( ep ej, el) + A(ej, ep el) . Plugging them back into equation (3.2) gives : α α ∇ ∇ α α ( ep R) ei, ej, ek, el = ( ep R) ei, ej, ek, el + hpiR e , ej, ek, el + hpjR ei, e , ek, el α α e α α ⟨ ∇ ⟩ +hpkR ei, ej, e , el + hpl R ei, ej, ek, e + ( ep A)(ei, el), A(ej, ek) ⟨ ∇e ⟩ − ⟨ ∇e ⟩ + A(ei, el), ( ep A)(ej, ek) ( ep A)(ei, ek), A(ej, el) −⟨ ∇e ⟩ A(ei, ek), ( ep A)(ej, el) . (3.3) Let us abbreviate using the following notations : ∇ ∇ Rijkl,p = ( ep R) ei, ej, ek, el and Rijkl,p = ( ep R) ei, ej, ek, el .

Then the equation (3.3) can be rewritten as : ≡ ∇ α α α α Rijkl,p ( ep R) ei, ej, ek, el = Rijkl,p + hpiRαjkl + hpjRiαkl + hpkRijαl + hpl Rijkα α α α α − α α − α α +hil,phjk + hilhjk,p hik,phjl hikhjl,p. (3.4)

20 Geometric invariants of the time-slice of the mean curvature flow

∇ Put Q = ep R. A straightforward computation gives : ∇ eq Q (ei, ej, ek, el) = Qijkl,q = Rijkl,pq · − ∇ − ∇ = eq Q(ei, ej, ek, el) Q eq ei, ej, ek, el Q ei, eq ej, ek, el − ∇ − ∇ Q ei, ej, eq ek, el Q ei, ej, ek, eq el . (3.5) The first term in the right hand side of (3.5) can be expressed as: · · · α · α eq Q(ei, ej, ek, el) = eq Rijkl,p + eq hpiRαjkl + eq hpjRiαkl · α · α · α α +eq hpkRijαl + eq hpl Rijkα + eq hil,phjk · α α − · α α − · α α +eq hilhjk,p eq hik,phjl eq hikhjl,p . Thus ∇ ∇ Rijkl,pq = ( eq ep R) ei, ej, ek, el · · α · α · α = eq Rijkl,p + eq hpiRαjkl + eq hpjRiαkl + eq hpkRijαl · α · α α · α α − · α α +eq hpl Rijkα + eq hil,phjk + eq hilhjk,p eq hik,phjl − · α α − ∇ − ∇ eq hikhjl,p Q eq ei, ej, ek, el Q ei, eq ej, ek, el − ∇ − ∇ Q ei, ej, eq ek, el Q ei, ej, ek, eq el . (3.6) We have : ! · · ∇ eq Rijkl,p = eq ( ep R) ei, ej, ek, el ∇ ∇ ∇ ∇ ∇ ∇ = ( eq ep R) ei, ej, ek, el + ( ep R) eq ei, ej, ek, el + ( ep R) ei, eq ej, ek, el ∇ ∇ ∇ ∇ +( ep R) ei, ej, eq ek, el + ( ep R) ei, ej, ek, eq el and ! · α · ⟨ ⟩ eq hpiRαjkl = eq A(ep, ei), eα R eα, ej, ek, el α − α − α = Rαjkl hpq,i Rαpqi + hpi Rαjkl,q hqrRrjkl ⟨ ∇ ⟩ ⟨ ∇ ⟩⟩ +Rαjkl A( eq ep, ei), eα + A(ep, eq ei), eα ! α α ∇ α ∇ α ∇ +hpi R e , eq ej, ek, el + R e , ej, eq ek, el + R e , ej, ek, eq el where used have been made of Codazzi - Mainardi equation, precisely α − α − hip,q hqp,i = Rαpqi.

21 Franck Houenou, Carlos Ogouyandjou and Leonard Todjihounde

We also have : · α α ∇ α α ∇ ∇e eq hil,phjk = eq hil,phjk = eq ( ep A) ei, el , A(ej, ek) D E ∇e ∇e ∇e ∇ ∇e ∇ = ( eq ep A) ei, el + ( ep A) eq ei, el + ( ep A) ei, eq el , A(ej, ek) D E ∇e ∇e ∇ ∇ + ( ep )A(ei, el), ( eq A) ej, ek + A eq ej, ek + A ej, eq ek D E α α α α ∇e ∇ ∇e ∇ = hil,pqhjk + hil,phjk,q + ( ep A) eq ei, el + ( ep A) ei, eq el , A(ej, ek) D E ∇e ∇ ∇ + ( ep )A(ei, el), A eq ej, ek + A ej, eq ek , (3.7) and similarly interchanging k and l (k ↔ l), we obtain : · α α ∇ α α ∇ ∇e eq hik,phjl = eq hik,phjl = eq ( ep A) ei, ek , A(ej, el) D E α α α α ∇e ∇ ∇e ∇ = hik,pqhjl + hik,phjl,q + ( ep A) eq ei, ek + ( ep A) ei, eq ek , A(ej, el) D E ∇e ∇ ∇ + ( ep )A(ei, ek), A eq ej, el + A ej, eq el . (3.8)

Besides ∇ ∇ ∇ ∇ Q( eq ei, ej, ek, el) = ( ep R) eq ei, ej, ek, el + A( eq ei, ep), eα R eα, ej, ek, el α α α ∇ α ∇ α ∇ α +hpjR eq ei, e , ek, el + hpkR eq ei, ej, e , el + hpl R eq ei, ej, ek, e D E D E ∇e ∇ ∇ ∇e + ( ep A)( eq ei, el), A(ej, ek) + A( eq ei, el), ( ep A)(ej, ek) D E D E − ∇e ∇ − ∇ ∇e ( ep A)( eq ei, ek), A(ej, el) A( eq ei, ek), ( ep A)(ej, el) . (3.9)

Analogously, we compute the remaining terms in the right hand side of equation (3.6). One may choose arbitrary coordinates around a point P ∈ Σ as we did so far, but the equations are quite simpler in normal coordinates. Let us now choose normal coordinates δ Γk system with center P. Therefore gij(P) = ij, gij,k(P) = 0 = ij(P) for all i, j, k and we obtain :

α α α α α − Rijkl,pq = Rijkl,pq + hqiRαjkl,p + hqjRiαkl,p + hqkRijαl,p + hql Rijkα,p + Rαjkl hpq,i Rαpqi α − α β β β α − +hpi Rαjkl,q hqr Rrjkl + h Rαβkl + h Rαjβl + h Rαjkβ + Riαkl hpq,j Rαpqj qj qk ql α − α β β β α − +hpj Riαkl,q hqr Rirkl + h Rβαkl + h Riαβl + h Riαkβ + Rijαl hpq,k Rαpqk qi qk ql α − α β β β α − +hpk Rijαl,q hqr Rijrl + h Rβjαl + h Riβαl + h Rijαβ + Rijkα hpq,l Rαpql qi qj ql α α β β β α α α α − βα +hpl Rijkα,q hqr Rijkr + hqiRβjkα + hqjRiβkα + hqkRij + hil,pqhjk + hil,phjk,q α α α α − α α − α α − α α − α α +hilhjk,pq + hil,qhjk,p hik,pqhjl hik,phjl,q hikhjl,pq hik,qhjl,p. (3.10)

22 Geometric invariants of the time-slice of the mean curvature flow

Taking its trace with respect to the induced metric gives the (rough) Laplacian of the curvature

α − α − α β β β Rijkl,pp = Rijkl,pp + Rαjkl hpp,i Rαppi + hpi 2Rαjkl,p hpr Rrjkl + h Rαβkl + h Rαjβl + h Rαjkβ pj pk pl α − α − α β β β +Riαkl hpp,j Rαppj + hpj 2Riαkl,p hpr Rirkl + h Rβαkl + h Riαβl + h Riαkβ pi pk pl α − α − α β β β +Rijαl hpp,k Rαppk + hpk 2Rijαl,p hpr Rijrl + h Rβjαl + h Riβαl + h Rijαβ pi pj pl α α α β β β − − βα +Rijkα hpp,l Rαppl + hpl 2Rijkα,p hprRijkr + hpiRβjkα + hpjRiβkα + hpkRij α α α α α α − α α − α α − α α +hil,pphjk + 2hil,phjk,p + hilhjk,pp hik,pphjl 2hik,phjl,p hikhjl,pp. (3.11)

The time derivative of the curvature is given by :

! ∂ ∂ R ≡ R e , e , e , e ∂t ijkl ∂t i j k l ! ∂ = R(e , e , e , e ) + A(e , e ), A(e , e ) − A(e , e ), A(e , e ) ∂t i j k l i l j k i k j l ∇ ∇ ∇ = H R ei, ej, ek, el + R ei H, ej, ek, el + R ei, ej H, ek, el ∂ ∂ ∇ ∇ α α − α α +R ei, ej, ek H, el + R ei, ej, ek, el H + hilhjk hikhjl ∂t ∂t ∇ ⟨∇ ⟩ ⟨∇ ⟩ = H R ei, ej, ek, el + R e H, eα eα + e H, ep ep, ej, ek, el i i ⟨∇ α⟩ α ⟨∇ ⟩ ⟨∇ α⟩ α ⟨∇ ⟩ +R ei, ej H, e e + ej H, ep ep, ek, el + R ei, ej, ek H, e e + ek H, ep ep, el ∂ α α ∂ α α ⟨∇ α⟩ α ⟨∇ ⟩ − +R ei, ej, ek, el H, e e + el H, ep ep + hilhjk hikhjl ∂t ∂t ∇ α − α α = H R ei, ej, ek, el + R H,i eα H hpiep, ej, ek, el α − α α α − α α +R ei, H,jeα H hpjep, ek, el + R ei, ej, H,keα H hpkep, el ∂ ∂ α − α α α α − α α +R ei, ej, ek, H,l eα H hplep + hilhjk hikhjl ∂t ∂t ∇ α − α α α − α α = H R ei, ej, ek, el + H,i Rαjkl H hpiRpjkl + H,j Riαkl H hpjRipkl α α α α α α ∂ α α ∂ α α +H R α − H h R + H R α − H h R + h h − h h . (3.12) ,k ij l pk ijpl ,l ijk pl ijkp ∂t il jk ∂t ik jl

23 Franck Houenou, Carlos Ogouyandjou and Leonard Todjihounde

From these, we can write the evolution equation of the curvature as follows : ! ∂ α α α α − △ R = ∇ R e , e , e , e + H Rα − H h R + H R α ∂t gt ijkl H i j k l ,i jkl pi pjkl ,j i kl α α α α α α α α −H h R + H R α − H h R + H R α − H h R pj ipkl ,k ij l pk ijpl ,l ijk pl ijkp − − α − − α − Rijkl,pp Rαjkl hpp,i Rαppi Riαkl hpp,j Rαppj − α − − α − Rijαl hpp,k Rαppk Rijkα hpp,l Rαppl − α − α β β β hpi 2Rαjkl,p hpr Rrjkl + h Rαβkl + h Rαjβl + h Rαjkβ pj pk pl − α − α β β β hpj 2Riαkl,p hpr Rirkl + h Rβαkl + h Riαβl + h Riαkβ pi pk pl − α − α β β β hpk 2Rijαl,p hpr Rijrl + h Rβjαl + h Riβαl + h Rijαβ pi pj pl α α β β β − − βα hpl 2Rijkα,p hpr Rijkr + hpiRβjkα + hpjRiβkα + hpkRij − α α − α α − α α α α α α α α hil,pphjk 2hil,phjk,p hilhjk,pp + hik,pphjl + 2hik,phjl,p + hikhjl,pp ∂ α α ∂ α α + h h − h h (3.13) ∂t il jk ∂t ik jl ! ∂ α α α α α α − △ R = ∇ R e , e , e , e − H h R − H h R − H h R ∂t gt ijkl H i j k l pi pjkl pj ipkl pk ijpl ! ! α α ∂ α α ∂ α α −H h R + − △ h h − − △ h h pl ijkp ∂t gt il jk ∂t gt ik jl − α − α β β β hpi 2Rαjkl,p hpr Rrjkl + 2h Rαβkl + 2h Rαjβl + 2h Rαjkβ pj pk pl − α − α β β hpj 2Riαkl,p hpr Rirkl + 2h Riαβl + 2h Riαkβ pk pl α α β α α − − αβ − − hpk 2Rijαl,p hpr Rijrl + 2hpl Rij hpl 2Rijkα,p hprRijkr − Rijkl,pp + Rαjkl Rαppi + Riαkl Rαppj + Rijαl Rαppk + RijkαRαppl(3.14). Using the evolution equation of the second fundamental form, we get : ! ∂ α α α α β α α α α α − △ h h = −h Rα − h Rα + h h Rβα + h h R + h h R ∂t gt il jk jk lip,p jk plp,i lp jk ip mp jk mlip lm jk mpip − β α β α β α β α − β α β α β α H hjk hlmhim + hinhln + H hjkRβilα H hil hkmhjm + hjnhkn β α β α β α − β α β α β β α α +H hil Rβjkα + hmphimhlphjk himhmphlphjk + hmphlihmphjk β β α α β β α α α α α β − − − βα himhlphmphjk + hmphpihlmhjk hil Rαkjp,p hil Rαpkp,j + hilhkpR jp α α α α α β α β − α β α β +hilhmpRmkjp + hilhkmRmpjp + hilhmphjmhkp hilhjmhmphkp α β β α − α β β α α β β α − α α +hilhmphkjhmp hilhjmhkphmp + hilhmphpjhkm 2hil,phjk,p (3.15)

24 Geometric invariants of the time-slice of the mean curvature flow

and similarly ! ∂ α α α α β α α α α α − △ h h = −h Rα − h Rα + h h Rβα + h h R + h h R ∂t gt ik jl jl kip,p jl pkp,i kp jl ip mp jl mkip km jl mpip − β α β α β α β α − β α β α β α H hjl hkmhim + hinhkn + H hjl Rβikα H hik hlmhjm + hjnhln β α β α β α − β α β α β β α α +H hikRβjlα + hmphimhkphjl himhmphkphjl + hmphkihmphjl β β α α β β α α α α α β − − − βα himhkphmphjl + hmphpihkmhjl hikRαljp,p hikRαplp,j + hikhlpR jp α α α α α β α β − α β α β +hikhmpRmljp + hikhlmRmpjp + hikhmphjmhlp hikhjmhmphlp α β β α − α β β α α β β α − α α +hikhmphljhmp hikhjmhlphmp + hikhmphpjhlm 2hik,phjl,p. (3.16)

Gathering everything together gives : ! ∂ α α α α α α α α − △ R = ∇ R e , e , e , e − H h R − H h R − H h R − H h R ∂t gt ijkl H i j k l pi pjkl pj ipkl pk ijpl pl ijkp − α − α β β β hpi 2Rαjkl,p hpr Rrjkl + 2h Rαβkl + 2h Rαjβl + 2h Rαjkβ pj pk pl − α − α β β − α α α α hpj 2Riαkl,p hpr Rirkl + 2h Riαβl + 2h Riαkβ 2hil,phjk,p + 2hik,phjl,p pk pl α α β α α − − αβ − − hpk 2Rijαl,p hpr Rijrl + 2hpl Rij hpl 2Rijkα,p hprRijkr − Rijkl,pp + Rαjkl Rαppi + Riαkl Rαppj + Rijαl Rαppk + RijkαRαppl. − α − α β α α α α α h Rα h Rα + h h Rβαip + h h R + h h Rmpip jk lip,p jk plp,i lp jk mp jk mlip lm jk − β α β α β α β α − β α β α β α H hjk hlmhim + hinhln + H hjkRβilα H hil hkmhjm + hjnhkn β α β α β α − β α β α β β α α − β β α α +H hil Rβjkα + hmphimhlphjk himhmphlphjk + hmphlihmphjk himhlphmphjk β β α α α α α β α α − − βα +hmphpihlmhjk hil Rαkjp,p hil Rαpkp,j + hilhkpR jp + hilhmpRmkjp α α α β α β − α β α β α β β α − α β β α +hilhkmRmpjp + hilhmphjmhkp hilhjmhmphkp + hilhmphkjhmp hilhjmhkphmp α β β α α α − β α − α α +h hmph h + h Rα + h Rα h h Rβαip h h R il pj km jl kip,p jl pkp,i kp jl mp jl mkip − α α β α β α β α − β α hkmhjl Rmpip + H hjl h him + h hkn H hjl Rβikα km in β α β α β α − β α − β α β α β α β α +H hik hlmhjm + hjnhln H hikRβjlα hmphimhkphjl + himhmphkphjl − β β α α β β α α − β β α α α α hmphkihmphjl + himhkphmphjl hmphpihkmhjl + hikRαljp,p + hikRαplp,j α β α α α α α β α β α β α β − βα − − − hikhlpR jp hikhmpRmljp hikhlmRmpjp hikhmphjmhlp + hikhjmhmphlp − α β β α α β β α − α β β α hikhmphljhmp + hikhjmhlphmp hikhmphpjhlm (3.17)

this completes the proof of the theorem after simplification.

25 Franck Houenou, Carlos Ogouyandjou and Leonard Todjihounde

3.2. Evolution equation of the Ricci and the scalar curvatures. In the following theorem, we derive the evolution equation of the Ricci curvature from the equation of the Rie- mannian curvature (3.1) by taking its trace with respect to the induced metric g. In the same way, from the evolution equation of the Ricci curvature, we derive the equation of the scalar curvature.

Theorem 3.2. Under the mean curvature flow, the Ricci curvature of the induced metric satisfies:

! ∂ α α α − △ R = ∇ R e , e , e , e − R − h Rα − H Rα − H Rα ∂t gt ij H k i j k kijk,pp ij pkk,p jip,p pjp,i α α − α +hik Rαjkp,p + Rαpjp,k + hkj Rαkip,p + Rαpkp,i 2hpk Rαijk,p + Rkijα,p − α − α − α β β − α 2hpiRkαjk,p 2hpjRkiαk,p hpi 2hpjRkαβk + 2hpk Rkαjβ + Rβαjk hpr Rkrjk α β β α α α − α β − hpk hpj Rαiβk + 3Rβikα + 2hpkR ij 2hprRrijk + hpjhpr Rkirk α α +Rα Rα + R α Rα + R α Rα + R αRα − h h R ijk ppk k jk ppi ki k ppj kij ppk kj mp mkip − α α − α α β α − β α hmphikRmjkp hjmhikRmpkp + h hij h hik Rβαkp kp jp − α β β α − β α α α − α α H h Rβikα + H hijRβkkα H hikRβkjα + H hjm hkjhkm Rmpip jk − α α α − α − β α β α − H hpiRkpjk + hpjRkipk hpkRpijk H H 2h him Rβijα jm β α α β α β β α β α α β − α β +2H hkm hikhjm + hkjhim + hkmhij + 2himhmp hkjhkp H hjp − β α β α β α − α α β β β β hmph h h + h h h h hmph + h hmp km pi kj jp ik ik jm pk kp − β α β α α β β α β α − α α α α hmphmp hjkhik + hkjhki + 2hkmhmphjphik 2H,phij,p + 2hik,phjk,p (3.18) while the scalar curvature evolves as :

! ∂ α − △ S = ∇ R e , e , e , e − R + 2h Rα + Rα ∂t gt H k i i k kiik,pp ik ikp,p pip,k − α β β hpk 8Rαiik,p + h 6Rαβik + 4Rαiβk + 4h Rαiiβ pi pk α β α α +4RαppkRαiik + 2H hip Rβαip + Rβiαp + 6hkihkmRmppi 2 β α β α α β − β α +2H H Rβiiα + hiphip + 2 ∑ ∑ himhmp himhmp α,β,i,p m − β α β α − |∇ |2 |∇e |2 2hmphmphikhik 2 H + 2 A . (3.19)

26 Geometric invariants of the time-slice of the mean curvature flow

Proof. From the definition of the Ricci curvature we have :

! ! ∂ ∂ − △ R = − △ gkl R ∂t gt ij ∂t gt kijl ! α α ∂ = 2H h R + gkl − △ R kl kijl ∂t gt kijl ! α α β β β β ∂ = 2H h R + h h − h h + gkl − △ R (3.20) kl kijl kl ij kj il ∂t gt kijl

Using the evolution equation of the curvature we get :

! ∂ kl α α α g − △ R = ∇ R e , e , e , e − R − h Rα − h Rα − h Rα ∂t gt kijl H k i j k kijk,pp ij kkp,p ij pkp,k kk jip,p α α α α α −h Rα + h Rα + h Rα + h Rα + h Rα kk pjp,i ik jkp,p ik pjp,k kj kip,p kj pkp,i − α − α β β β hpk 2Rαijk,p hpr Rrijk + 2h Rαβjk + 2h Rαiβk + 2h Rαijβ pi pj pk − α − α β β hpi 2Rkαjk,p hpr Rkrjk + 2h Rkαβk + 2h Rkαjβ pj pk − α − α β − α − α hpj 2Rkiαk,p hpr Rkirk + 2hpkRkiαβ hpk 2Rkijα,p hpr Rkijr

+RαijkRαppk + RkαjkRαppi + RkiαkRαppj + RkijαRαppk α α α α − α α − α α +hkkhmpRmjip + hmphijRmkkp hkjhmpRmkip hmphikRmjkp α α α α α α α α −H h R − H h R − H h R − H h R pk pijk pi kpjk pj kipk pk kijp β α − β α α α − α α + h hij h hik Rβαkp + hkmhij hjmhik Rmpkp kp jp α β − α β α α − α α + hkkh hkjh Rβαip + hkkhjm hkjhkm Rmpip jp kp − β α β α β α − − β α β α β α − H hij h hkm + h hkn Rβkkα H hkk h him + h hjn Rβijα km kn jm in β α β α β α − β α β α β α − +H hik h hkm + h hjn Rβkjα + H hkj h him + h hkn Rβikα jm kn km in β α β α − β α β β α α − α α +hmphkm h hij h hik + hmph hkmhij hjmhik kp jp pk β α α β − α β β β α α − α α +hmphim hkkh hkjh + hmph hkkhjm hkjhkm jp kp pi β α β α α β − β α − α β − β α β α +hmphmp hkkhij + hkkhji hjkhik hkjhki 2hkmhmphkphij − α β α β β α β α α β α β − α α α α 2hkkhimhmphjp + 2hkmhmphjphik + 2hkjhimhmphkp 2H,phij,p + 2hik,phjk,p.

27 Franck Houenou, Carlos Ogouyandjou and Leonard Todjihounde

Therefore

! ∂ α α β β β β − △ R = 2H h R + h h − h h + ∇ R e , e , e , e − R ∂t gt ij kl kijl kl ij kj il H k i j k kijk,pp α α − α − α +hik Rαjkp,p + Rαpjp,k + hkj Rαkip,p + Rαpkp,i hijRαkkp,p hijRαpkp,k − α − α − α − α β β H Rαjip,p H Rαpjp,i hpi 2Rkαjk,p hpr Rkrjk + 2h Rkαβk + 2h Rkαjβ pj pk − α − α β − α − α hpj 2Rkiαk,p hpr Rkirk + 2h Rkiαβ hpk 2Rkijα,p hpr Rkijr pk α α β β β − − α β hpk 2Rαijk,p hpr Rrijk + 2hpiRαβjk + 2hpjRαiβk + 2hpkR ij α α α α +Rα Rα + R α Rα + R α Rα + R αRα − 2H h + 2h h ijk ppk k jk ppi ki k ppj kij ppk ,pij,p ik,p jk,p α α − α − α − α − α +H hmpRmjip hpkRpijk hpiRkpjk hpjRkipk hpkRkijp α α − α α − α α β α − β α +hmphijRmkkp hkjhmpRmkip hmphikRmjkp + h hij h hik Rβαkp kp jp α α − α α α β − α β α α − α α + hkmhij hjmhik Rmpkp + H h hkjh Rβαip + H hjm hkjhkm Rmpip jp kp − β α β α − − β α β α − H hij 2h hkm Rβkkα H H 2h him Rβijα km jm β α β α β α − β α β α β α − +H hik h hkm + h hjn Rβkjα + H hkj h him + h hkn Rβikα jm kn km in − β α β α β β α α − α α β α α β − α β hmphkmh hik + hmph 2hkmhij hjmhik + hmphim H h hkjh jp pk jp kp β β α α − α α β α β α − β α − α β +hmph H hjm hkjhkm + hmphmp 2H hij h hik hkjh pi jk ki β α β α − β α β α α β − α β +2hkmhmp hjphik hkphij + 2himhmp hkjhkp H hjp . (3.21)

In the same way, we derive the evolution equation of the scalar curvature :

! ! ∂ ∂ − △ S = − △ gijR ∂t gt ∂t gt ij ! α α ∂ = 2H h R + gij − △ R ij ij ∂t gt ij ! α α β β β β ∂ = 2H h R + H h + h h + gij − △ R . (3.22) ij kijk ij kj ik ∂t gt ij

28 Geometric invariants of the time-slice of the mean curvature flow

Taking the trace with respect to the induced metric of the evolution equation of the Ricci curvature yields the following : ! ∂ ij α α β β β β α g − △ R = 2H h R + h h − h h + ∇ R e , e , e , e − R − 2H Rα ∂t gt ij kl kiil kl ii ki il H k i i k kiik,pp iip,p − α α α α − α 2H Rαpip,i + 2hik Rαikp,p + Rαpip,k + H 2hmpRmiip 4hpkRpiik − α − α β β hpk 8Rαiik,p 4hprRriik + 4h Rαβik + Rαiβk + 4h Rαiiβ pi pk − α α − α β − α β +4RαppkRαiik hkihmp Rmkip Rmikp + 2 H h hkih Rβαip ip kp α α − α α − β α β α − +2 H him hkihkm Rmpip 2H H 2h him Rβiiα im β α β α − β β α α − α α +2H hik 2h hkm Rβkiα + 2hmph H hkm himhik im pk β α α β − α β β α β α − β α +2hmphim H h hkih + 2hmphmp H H h hik ip kp ik β α β α − α β − α α α α +4hkmhmp hiphik H hkp 2H,p H,p + 2hik,phik,p. (3.23)

Finally we obtain : ! ∂ α − △ S = ∇ R e , e , e , e − R + 2h Rα + Rα ∂t gt H k i i k kiik,pp ik ikp,p pip,k − α β β hpk 8Rαiik,p + h 6Rαβik + 4Rαiβk + 4h Rαiiβ pi pk α β α α +4RαppkRαiik + 2H h Rβαip + Rβiαp + 6hkihkmRmppi ip β α β α β β α α β α − +2H H R ii + hiphip 4hmphpkhimhik β α β α − β α β α − α α α α +4hkmhmphiphik 2hmphmphikhik 2H,p H,p + 2hik,phik,p.

To the author knowledge the long time existence of the mean curvature flow heavily relies on the curvature of the manifold and is proven only when the ambient manifold is (almost) Kaehler-Einstein manifold. Therefore we study the case when the ambient manifold has constant curvature and rewrite the evolution equation of the scalar curvature.

4. Constant curvature case We assume now that M is of constant curvature, say c. The Riemannian curvature of M can be expressed as : − Rijkl = g R(ei, ej)ek, el = c gil gjk gikgjl . (4.1)

29 Franck Houenou, Carlos Ogouyandjou and Leonard Todjihounde

Therefore − Rαβik = c gαkgβi gαigβk = 0, − − Rαiβk = c gαkgiβ gαβgik = cgαβgik − Rαiiβ = c gαβgii gαigiβ = cgαβgii = cngαβ − Rαiik = c gαkgii gαigik = 0 − − Rijjk = c gikgjj gijgjk = c ngik gijgjk Σ Let us remark that under this hypothesis is curvature-invariant i.e. Rαijk = 0, for any i, j, k ∈ {1, ··· , n} and any α ∈ {n + 1, ··· , m}

Proposition 4.1. As long as the mean curvature flow (Σt)t of Σ ,→ M exists, the scalar curvature of Σt satisfies: ! ∂ − △ S > 2c(n − 1) |A|2 + |H|2 (4.2) ∂t gt

Proof. Under the assumption of constant curvature, the equation (3.19) reduces to : ! ∂ α β β − △ S = 4Rα Rα − h h 6Rαβ + 4Rα β + 4h Rα β ∂t gt ppk iik pk pi ik i k pk ii α β α α β α β α +2H hip Rβαip + Rβiαp + 6hkihkmRmppi + 2H H Rβiiα + hiphip − β β α α β α β α − β α β α − α α α α 4hmph h h + 4h h h h 2hmph h h 2H H + 2h h pk im ik km mp ip ik mpik ik ,p ,p ik,p ik,p α β − α β α α − = 4c hpkhpi gαβgik 4nc hpkhpkgαβ + 6c hkihkm ngmi gmpgpi − α β β α β α β α − β β α α 2c H hipgαβgip + 2nc H H gαβ + 2H H hiphip 4hmphpkhimhik β α β α − β α β α − α α α α +4hkmhmphiphik 2hmphmphikhik 2H,p H,p + 2hik,phik,p 2 − | |2 | |2 β α β α α β − β α = 2c(n 1) A + H + 2H H hiphip + 2 ∑ ∑ himhmp himhmp α,β,i,p m 2 − α α − |∇ |2 |∇e |2 2 ∑ ∑ hikhmp 2 H + 2 A α i,k,m,p > 2c(n − 1) |A|2 + |H|2 (4.3) since we have 2 2 α β − β α − α α > 2 ∑ ∑ himhmp himhmp 2 ∑ ∑ hikhmp 0, α,β,i,p m i,k,m,p α β |∇e |2 − |∇ |2 > β α α ⟨ ⟩2 A H 0 and we can write H H hiphip = H, A . Consequently we have the following corollary

30 Geometric invariants of the time-slice of the mean curvature flow

Corollary 4.1. If the ambient manifold has non negative constant curvature then the flow preserves the positivity of the scalar curvature of the evolving submanifold. The consequence follows from the maximum principle. This result is in contrast with the one proved in [3] since the concluding remark does not agree with the commonly known comment asserting that the mean curvature flow favors positive scalar curvature. Acknowledgments. The authors would like to thank Augustin Banyaga and Claudio Ar- rezo for their helpful discussions and CEA-SMIA at IMSP for their financial support. They express their gratitude to the anonymous referee for their great help to improve the work.

References [1] X. Han and J. Li, On symplectic mean curvature flows, Lecture notes. [2] G. Huisken, Asymptotique behavior for singularity of the mean curvature flow, J. Diff. Geom., N. 31(1990) : 285-299. [3] V. Tapia, Evolution of the curvature tensors under mean curvature flow, Revista Colombiana de Matemáticas, N. 43, no. 2 (2009) : 175-185. [4] M.-T. Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codi- mension, Invent. math., N.148, no. 3 (2002) : 525-543. [5] M.-T. Wang, Mean curvature flow of surfaces in four-dimension manifolds, J. Diff. Geom., N.57 (2001) : 301-338 [6] M.-T. Wang, Lectures on mean curvature flows in higher codimensions, Handbook of geometric analysis. N.1 (2008), 525-543, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA. [7] F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York : Springer- Verlag, ISBN 978-0-387-90894-6 (1983).

Faculty of Sciences and Techniques (FAST), Abomey Calavi, University of Abomey Calavi, Benin Repub- lic Institut of Mathematics and Physical Sciences (IMSP), Dangbo, University of Abomey Calavi, Benin Republic

Email address: [email protected], [email protected], [email protected], [email protected]