THE WEYL CURVATURE TENSOR of HYPERSURFACES UNDER the MEAN CURVATURE FLOW 1. Introduction

Ø Ñ

MathematicalDOI: 10.2478/tmmp-2020-0023 Publications Tatra Mt. Math. Publ. 76 (2020), 143–156

THE WEYL CURVATURE TENSOR OF HYPERSURFACES UNDER THE MEAN CURVATURE FLOW

Ghodrat Moazzaf — Esmaiel Abedi ∗

Azarbaijan Shahid Madani University, Tabriz, IRAN

ABSTRACT. In this paper, we study the evolution of the Weyl curvature tensor W of hypersurfaces in Rn+1 under the mean curvature flow. We find a bound for the Weyl curvature tensor of hypersurfaces during the evolution in terms of time. As a consequence, we suppose that the initial hypersurface is conformally flat, i.e., W =0att = 0 and then we find an upper estimate for W during the evolution in terms of time.

1. Introduction

n+1 Let ϕ0 : M −→ R be a smooth immersion from an n−dimensional mani- n+1 fold M to R . The mean curvature flow of ϕ0 is defined as a family of smooth n+1 immersions, ϕt : M −→ R for t ∈ [0,T) such that setting ϕ(p, t)=ϕp(t) the map ϕ : M × [0,T) −→ Rn+1 is a smooth solution of the following PDE ∂ ϕ(p, t)=H(p, t)ν(p, t) , ∂t (1) ϕ(p, 0) = ϕ0(p) , where H(p, t)andν(p, t) are the mean curvature and the unit normal of the hypersurfaces Mt = ϕ(·,t)(M), respectively. Notice that the field H(p, t)ν(p, t) is independent of the sign of the normal vector field ν(p, t). The mean curvature flow was first introduced in 1956 by Mullins in [8] and independently by Brakke in 1978 from the viewpoint of geometric measure the-

ory [1]. Since then, this ﬂow has been widely studied. It has been studied in the © 2020 Mathematical Institute, Slovak Academy of Sciences. 2010 M a t h e m a t i c s Subject Classification: 53C44. K e y w o r d s: Evolution equations, maximum principle, mean curvature ﬂow, second fundamental form, Weyl curvature tensor. Licensed under the Creative Commons Attribution-NC-ND4.0 International Public License. ∗Corresponding author: [email protected]

143 MOAZZAF GHODRAT—ABEDI ESMAIEL smooth setting by Huisken in 1984. He showed in [5] that if the initial hypersurface is convex and closed then the flow has an unique solution in [0,T), where T is the maximal time where the flow exists, and as t → T , the hypersurfaces con- verge to a point. He also studied the singularity of mean curvature flow when the initial hypersurface of the flow is non-convex [6]. One of the interesting subjects in mean curvature flow is studying how geometric quantities and their properties change when the hypersurface evolves under the flow. For example, if the initial hypersurface is compact, then it remains compact during the evolution too, or if the initial hypersurface is compact and its mean curvature H is nonnegative, then H will be positive under the mean curvature flow [5,6]. The Weyl curvature tensor has been studied in other flows like the Ricci flow. For a Riemannian manifold (M n,g) the Ricci flow is defined by ∂ g = −2R , (2) ∂t ij ij where Rij is the Ricci curvature of M. For example, it has been shown in [7] that if the Weyl curvature tensor remains identically zero then either the Ricci tensor is proportional to the metric or it has an eigenvalue of multiplicity (n−1) and another of multiplicity 1. The evolution equation of Weyl curvature tensor of hypersurfaces under the mean curvature flow is more complicated than 2, because the evolution of metric tensor of the hypersurface under the mean curvature flow has one more term, it is ∂ g = −2Hh = −2R − 2hkh . (3) ∂t ij ij ij i kj In this paper, we present an estimate of the evolution of the Weyl curvature tensor of hypersurfaces in a Euclidean space under the mean curvature flow and its application under the assumption of conformal flatness of the initial hypersurface. The Weyl curvature tensor W is used in general relativity. Closely related is the Ricci curvature tensor whose behaviour with respect to the Ricci flow played the central role in the famous solution of the geometrization conjecture in dimension 3 by Perelman, Hamilton and others. The main results are Theorem 4.1 and Corollary 4.2 . Theorem 4.1. says that if the initial hypersurface has the Weyl curvature tensor bounded by some constant M, then the evolving hypersurfaces also have the Weyl curvature tensor bounded by a specific formula involving M and some other constants. Corollary 4.2 is then the special case when M = 0. The proof of Theorem 4.1 is based on the so-called “Maximum principle” recalled here as Theorem 2.3 . Roughly, it says that if the variation of a function u (such as |W |) can be appropriately estimated in terms of the Laplacian, the gradient and a locally Lipschitz function F, then function u is estimated by the solution of an initial value problem defined by F. Hence, the method of the proof is to estimate the variation of the Weyl curvature tensor achieved in Corollary 3.8 and to obtain the estimate for the

144 THE WEYL CURVATURE TENSOR

Weyl tensor itself by the above principle. The estimate of the variation takes several pages due to the fact that all the components of the tensor have to be estimated. Many notions in classical diﬀerential geometry are used such as the Riemannian metric, Ricci tensor, the second fundamental form, ∗-product and various technical manipulations of tensors. Since for all surfaces and three dimensional hypersurface W is zero, it will be assumed n ≥ 4.

2. Preliminaries and the geometry of hypersurfaces

In this section some deﬁnitions and formulas for hypersurfaces in Rn+1 are introduced (see [5]). Throughout the paper, we consider n−dimensional complete hypersurfaces immersed in Rn+1, that is, for the pair (M,ϕ) M is an n−dimensional smooth manifold with empty boundary and ϕ : M −→ Rn+1 is a smooth immersion. Also, the Euclidean metric in Rn+1 is denoted by , . The metric g on M is obtained by pulling back the Euclidean metric by ϕ, i.e., ∂ ∂ ∂ϕ ∂ϕ gij = g , = , , ∂xi ∂xj ∂xi ∂xj for 1 ≤ i, j ≤ n. The inverse metric and the area element of M are denoted by √ ij −1 g =(gij) and g = detgij, respectively. The metric g of M extended to tensors is given by

j1r1 jlrl i1...ik s1 ... sk g(T,S)=gi1s1 ... giksk g ... g Tj1 ... jl Sr1...rl , where T and S are two (k, l)-tensors, and the norm of a tensor is given by |T | = g(T,T). The induced covariant derivative on (M,g) of a vector ﬁeld is

i ∇ i ∂X i k j X = +ΓjkX , ∂xj

i where Γjk are the Christoﬀel symbols that means i 1 il ∂ ∂ − ∂ Γjk = g gkl + gjl gji i, j, k =1, 2, ..., n. 2 ∂xj ∂xk ∂xl

145 MOAZZAF GHODRAT—ABEDI ESMAIEL

The gradient of a function f and the divergence of vector field X at a point p ∈ M are defined, respectively, as following g ∇f(p),w =dfp(w), for all ∈ ∇ ∇ i ∂ i i k w Tp(M) and divX = tr X = iX = X +ΓikX . ∂xi The Laplacian ΔT of a tensor T is defined as ij ΔT = g ∇i∇jT.

The second fundamental form A = hij of M is the symmetric 2-form deﬁned as ∂2ϕ ∂ν ∂ϕ hij = ν, = − , , ∂xi∂xj ∂xi ∂xj where ν is the unit normal vector at every point of M. The mean curvature H ij is the trace of A,thatis,H = g hij. We also have 2 ij kl |A| = g g hikhjl. The Gauss-Weingarten relations are 2 ∂ ϕ k ∂ϕ ∂ − ls ∂ϕ =Γij k + hijν, v = hjlg . ∂xi∂xj ∂x ∂xj ∂xs The Riemann curvature tensor, Ricci tensor and the scalar curvature for hypersurfaces can be expressed by means of the second fundamental form as follows

Rijkl = hikhjl − hilhjk, (4) kl lk Ricij = g Rijkl = Hhij − hilg hkj, (5) ij 2 2 r = g Ricij = H −|A| . (6) The Codazzi equations

∇ihjk = ∇jhik = ∇khij show the symmetry properties of the covariant derivative of A. In [3,4] Hamilton used the T ∗ as a tensor which is formed by a sum of terms that each of them obtained by contracting some indices of the pair T and S with the metric gij. One of the important and useful property of ∗−product is |T ∗ S|≤C|T ||S|, where the constant C depends only on the algebraic “structure” of T ∗ S. Let (M,g)beann-dimensional Riemannian manifold, a conformal change of metric g is defined asg ˜ = σg,whereσ is a positive function on M.ARie- mannian manifold (M,g) is called conformally flat if it is a coefficient of the Euclidean metric.

146 THE WEYL CURVATURE TENSOR

The Weyl curvature tensor on (M,g)isa(0, 4)−tensor locally deﬁned by − 1 − − Wijkl = Rijkl − (Rikgjl + Rjlgik Rilgjk Rjkgil)+ n 2 r (g g − g g ). (7) (n − 1)(n − 2) ik jl il jk For manifolds with dimM ≤ 3 the Weyl curvature tensor is zero. The following

theorem is the theorem of Weyl [9]. º ÌÓÖÑ 2.1 A necessary and suﬃcient condition for a Riemannian manifold M to be conformally ﬂat is that W =0.

At the end of this section we state the Maximum principle (see [2, Lemma 2.12]).

º × −→ R ÌÓÖÑ 2.2 (Maximum Principle) Let u : M [0,T) be a smooth ∂u function on a closed manifold satisfying ≤ Δ ( )u + X, ∇u + F (u) ∂t g t and u(x, 0) ≤ c for all x ∈ M,whereg(t) is a 1-parameter family of metrics and F is a locally Lipschitz. Let φ(t) be the solution to the initial value problem ⎧ ⎨ dφ = F (φ), dt ⎩ φ(0) = 0. Then u(x, t) ≤ φ(t) for all x ∈ M and t ∈ [0,T) such that φ(t) exists. It will be assumed that the second fundamental forms therefore |A|2 and |∇A|2 for initial hypersurface are bounded [5].

3. Evolution equations

In this section, the evolution equations for metric, normal vector, second fundamental form and the mean curvature are mentioned. For the proof of these equations see [5]. Moreover, the evolution equations for the Riemannian curvature tensor, Ricci tensor, scalar curvature and the Weyl curvature tensor are

computed. º ÄÑÑ 3.1 The following evolution equations hold for Mt = ϕ(., t)(M) ∂ g = −2Hh , (8) ∂t ij ij ∂ gij =2Hhij, (9) ∂t ∂ ν = −∇H, (10) ∂t ∂ Γi = ∇H ∗ A + H ∗∇A = ∇A ∗ A. (11) ∂t jk

147

MOAZZAF GHODRAT—ABEDI ESMAIEL

ÓÔÓ× Ø ÓÒ º ÈÖ 3.2 The second fundamental form satisﬁes the evolution equation ∂ h =Δh − 2Hh glsh + |A|2h . (12) ∂t ij ij il sj ij And it follows that ∂ hj =Δhj + |A|2hj , (13) ∂t i i i ∂ |A|2 =Δ|A|2 − 2|∇A|2 +2|A|4, (14) ∂t ∂ H =ΔH + |A|2H. (15) ∂t Now we want to compute the evolution of the Riemannian curvature tensor. By using (12) at ﬁrst we compute the evolutions of its terms ∂(h h ) ∂h ∂h ik jl = h ik + h jl ∂t jl ∂t ik ∂t rs 2 = hjl(Δhik − 2Hhirg hsk + |A| hik)

rs 2 + hik(Δhjl − 2Hhjrg hsl + |A| hjl) rs = hjlΔhik + hikΔhjl − 2Hhjlhirg hsk

rs 2 − 2Hhikhjrg hsl +2|A| hikhjl. Now the Laplacian of the product of two tensors yields

∂(h h ) ik jl =Δ(h h ) − 2∇h , ∇h − ∂t ik jl ik jl rs rs 2 2Hhjlhirg hsk − 2Hhikhjrg hsl +2|A| hikhjl. (16) Similarly, for the other components of the Riemannian curvature tensor we can write ∂(h h ) il jk =Δ(h h ) − 2∇h , ∇h − ∂t il jk il jk rs rs 2 2Hhjkhirg hsl − 2Hhilhjrg hsk +2|A| hilhjk. (17)

Hence, we can state the following lemma. º ÄÑÑ 3.3 The evolution equation for the Riemannian tensor R satisﬁes

∂R ijkl =ΔR +2∇h , ∇h −2∇h , ∇h + ∂t ijkl il jk ik jl r − r | |2 2HhkRijlr 2Hhl Rijkr +2A Rijkl. (18)

148 THE WEYL CURVATURE TENSOR

P r o o f. By applying the equations (4), (12), (16) and (17) we have ∂R ∂(h h ) ∂(h h ) ijkl = ik jl − il jk ∂t ∂t ∂t

=Δ(hikhjl) − Δ(hilhjk) − 2∇hik, ∇hjl

rs rs +2∇hil, ∇hjk−2Hhjlhirg hsk +2Hhjkhirg hsl

rs rs 2 − 2Hhikhjrg hsl +2Hhilhjrg hsk +2|A| (hikhjl − hilhjk)

rs =ΔRijkl − 2∇hik, ∇hjl +2∇hil, ∇hjk−2Hhskg (hirhjl − hilhjr)

rs 2 − 2Hhskg (hikhjr − hirhjk)+2|A| (hikhjl − hilhjk). We obtain (18) by replacing all the statements inside the parentheses by the

Riemannian curvature tensor with proper indices in (4).

ÓÔÓ× Ø ÓÒ º ÈÖ 3.4 The evolution equations for the Ricci tensor and the scalar curvature are ∂R ij =ΔR +2∇h ,hl −2∇h , ∇H ∂t ij il j ij (19) | |2 − s +2A Rij 2Hhi Rjs, ∂r =Δr +2|∇A|2 − 2|∇H|2 +2|A|2r. (20) ∂t Proof. From(5)wehave ∂R ∂H ∂h ∂h ∂hl ij = h + H ij − il hl − h j ∂t ∂t ij ∂t ∂t j il ∂t and by applying the formulas in Proposition 3.2 we have

∂R ij =(ΔH + |A|2H)h + H(Δh − 2Hh grsh + |A|2h ) ∂t ij ij ir sj ij − − rs | |2 l − l | |2 l (Δhil 2Hhirg hsl + A hil)hj hil(Δhj + A hj ) | |2 − rs − l = hijΔH + HΔhij +2A Hhij 2Hg hir(Hhsj hslhj ) − l − l − | |2 l hj Δhil hilΔhj 2 A hilhj 2 rs =Δ(Hhij) − 2∇hij, ∇H +2|A| Rij − 2Hg hirRsj − l ∇ l Δ(hilhj )+2 hil,hj . Which concludes the ﬁrst formula. For the second one using (6) we have ∂r ∂ = (H2 −|A|2). ∂t ∂t

149 MOAZZAF GHODRAT—ABEDI ESMAIEL

Now, the equation (14) yields ∂r ∂H =2H − Δ|A|2 +2|∇A|2 − 2|A|4 ∂t ∂t and (15) implies ∂r =2H(ΔH + |A|2H) ∂t − Δ|A|2 +2|∇A|2 − 2|A|4

=ΔH2 − 2|∇H|2 +2|A|2H2

− Δ|A|2 +2|∇A|2 − 2|A|4

=ΔH2 − Δ|A|2

+2|A|2(H2 −|A|2)+2|∇A|2 − 2|∇H|2

=Δr +2|∇A|2 − 2|∇H|2 +2|A|2r.

Now we can compute the evolution of the Weyl curvature tensor but at ﬁrst the evolution of its terms in relation (7) are calculated. We have ∂ ∂R ∂g (R g )= ik g + R jl , ∂t ik jl ∂t jl ik ∂t and from (19) and (8) we get ∂ (R g )= Δ(R )+2∇h , ∇hr −2∇h , ∇H ∂t ik jl ik ir k ik | |2 − s +2A Rik 2Hhi Rks gjl

+ Rik(−2Hhjl), since ∇kgij =0,thenΔgij = 0. Therefore,

Δ(Rikgjl)=gjlΔRik, hence we have ∂ (R g )=Δ(R g )+2g ∇h , ∇hr −2g ∇h , ∇H ∂t ik jl ik jl jl ir k jl ik | |2 − s +2A Rikgjl 2Hhi gjlRsk

− 2HhjlRik.

150 THE WEYL CURVATURE TENSOR

So, we imply ∂ (R g + R g − R g − R g ) ∂t ik jl jl ik il jk jk il

=Δ(Rikgjl + Rjlgik − Rilgjk − Rjkgil) ∇ ∇ r ∇ ∇ r− ∇ ∇ r +2gjl hir, hk +2gik hjr, hl 2gjk hir, hl − ∇ ∇ r − ∇ ∇ − ∇ ∇ 2gil hjr, hk 2gjl hik, H 2gik hjl, H 2 +2gjk∇hil, ∇H +2gil∇hjk, ∇H +2|A| Rikgjl | |2 − | |2 − | |2 − s +2A Rjlgik 2 A Rilgjk 2 A Rjkgil 2Hhi gjlRks − s s s 2Hhj gikRls +2Hhi gjkRls +2Hhj gilRks

− 2HhjlRik − 2HhikRjl +2HhjkRil +2HhilRjk

=Δ(Rikgjl + Rjlgik − Rilgjk − Rjkgil) ∇ ∇ r ∇ ∇ r− ∇ ∇ r +2gjl hir, hk +2gik hjr, hl 2gjk hir, hl − ∇ ∇ r − ∇ ∇ − ∇ ∇ 2gil hjr, hk 2gjl hik, H 2gik hjl, H

+2gj∇hil, ∇H +2gil∇hjk, ∇H

2 +2|A| (Rikgjl + Rjlgik − Rilgjk − Rjkgil) − s − − 2Hhi (Rksgjl + Rjlgks Rlsgjk Rjkgls) − s − − 2Hhj (Rksgil + Rilgks Rlsgik Rikgls). Also, by using (8) we can easily obtain ∂ (g g − g g )=−2Hg h − 2Hg h ∂t ik jl il jk jl ik ik jl +2Hgilhjk +2Hgilhjk. Therefore by using (7) and these last two relations, we can compute the evolution of the Weyl curvature tensor ∂W ∂R 1 ∂ ijkl = ijkl − (R g + R g − R g − R g ) ∂t ∂t n − 2 ∂t ik jl jl ik il jk jk il 1 ∂r + (g g − g g ) (n − 1)(n − 2) ∂t ik jl il jk r ∂ + (g g − g g ). (n − 1)(n − 2) ∂t ik jl il jk

151 MOAZZAF GHODRAT—ABEDI ESMAIEL

Now by taking into account these two relations, (14) and the evolution equations for the Ricci curvature and the scalar curvature, we have ∂W ijkl =ΔR +2∇h , ∇h −2∇h , ∇h ∂t ijkl il jk ik jl r − r | |2 +2HhkRijlr 2Hhl Rijkr +2A Rijkl 1 − Δ(R g + R g − R g − R g ) n − 2 ik jl jl ik il jk jk il 1 − 2g ∇h , ∇hr +2g ∇h , ∇hr n − 2 jl ir k ik jr l − ∇ ∇ r− ∇ ∇ r 2gjk hir, hl 2gil hjr, hk

− 2gjl∇hik, ∇H−2gik∇hjl, ∇H +2gjk∇hil, ∇H +2gil∇hjk, ∇H 2|A|2 + (R g + R g − R g − R g ) n − 1 ik jl jl ik il jk jk il 2Hhs − i (R g + R g − R g − R g ) n − 1 ks jl jl ks ls jk jk ls 2Hhs + j (R g + R g − R g − R g ) n − 1 ks il il ks ls ik ik ls 1 + Δ[r(g g − g g )] (n − 1)(n − 2) ik jl il jk 2|∇A|2 − 2|∇H|2 + (g g − g g ) (n − 1)(n − 2) ik jl il jk 2|A|2r + (g g − g g ) (n − 1)(n − 2) ik jl il jk r + (−2Hg h − 2Hg h +2Hg h +2Hg h ), (n − 1)(n − 2) jl ik ik jl il jk jk il in which, these two relations hold

r i r i hkRijlr = hrRkljr,hl Rijkr = hrRklir, (21)

152 THE WEYL CURVATURE TENSOR then ∂W ijkl =ΔW +2|A|2W +2∇h , ∇h −2∇h , ∇h ∂t ijkl ijkl il jk ik jl 1 − 2g ∇h , ∇hr +2g ∇h , ∇hr n − 2 jl ir k ik jr l − ∇ ∇ r− ∇ ∇ r 2gjk hir, hl 2gil hjr, hk

− 2gjl∇hik, ∇H−2gik∇hjl, ∇H +2gjk∇hil, ∇H +2gil∇hjk, ∇H s − s +2Hhi Rkljs 2Hhj Rklis 2Hhs + j (R g + R g − R g − R g ) n − 2 ks il il ks ls ik ik ls 2Hhs − i (R g + R g − R g − R g ) n − 2 ks jl jl ks ls jk jk ls 2|∇A|2 − 2|∇H|2 + (g g − g g ) (n − 1)(n − 2) ik jl il jk r + (−2Hg h − 2Hg h +2Hg h +2Hg h ). (n − 1)(n − 2) jl ik ik jl il jk jk il We can arrange this as ∂W ijkl =ΔW +2|A|2W +2∇h , ∇h −2∇h , ∇h ∂t ijkl ijkl il jk ik jl 1 − 2g ∇h , ∇hr +2g ∇h , ∇hr−2g ∇h , ∇hr n − 2 jl ir k ik jr l jk ir l − 2g ∇h , ∇hr −2g ∇h , ∇H−2g ∇h , ∇H il jr k jl ik ik jl +2gjk∇hil, ∇H +2gil∇hjk, ∇H 1 − 2Hhs R − (R g + R g − R g − R g ) i kljs n − 2 ls jk jk ls ks jl jl ks r + (g g − g g ) (n − 1)(n − 2) jl sk jk sl 1 +2Hhs R − R g + R g − R g − R g j klis n − 2 ks il jl ks ls ik ik ls r + (g g − g g ) (n − 1)(n − 2) ik sl il sk 2(|∇A|2 −|∇H|2) + (g g − g g ). (n − 1)(n − 2) ik jl il jk

The phrases inside the second and the third brackets can be replaced by the Weyl curvature tensor with respect to the proper indices. Hence, we have

153 MOAZZAF GHODRAT—ABEDI ESMAIEL

∂W ijkl =ΔW +2|A|2W +2∇h , ∇h −2∇h , ∇h ∂t ijkl ijkl il jk ik jl 1 − 2g ∇h , ∇hr +2g ∇h , ∇hr−2g ∇h , ∇hr n − 2 jl ir k ik jr l jk ir l − 2g ∇h , ∇hr −2g ∇h , ∇H−2g ∇h , ∇H il jr k jl ik ik jl +2gjk∇hil, ∇H +2gil∇hjk, ∇H s − s +2Hhj wklis 2Hhi wkljs 2(|∇A|2 −|∇H|2) + (g g − g g ). (n − 1)(n − 2) ik jl il jk

Therefore, we can state the following theorem. º ÌÓÖÑ 3.5 The evolution equation for the Weyl curvature tensor is ∂W ijkl =ΔW +2|A|2W +2HhsW − 2HhsW ∂t ijkl ijkl j klis i kljs +2∇hil, ∇hjk−2∇hik, ∇hjl 1 − 2g ∇h , ∇hr n − 2 jl ir k ∇ ∇ r− ∇ ∇ r +2gik hjr, hl 2gjk hir, hl − 2g ∇h , ∇hr −2g ∇h , ∇H−2g ∇h , ∇H il jr k jl ik ik jl +2gjk∇hil, ∇H +2gil∇hjk, ∇H 2(|∇A|2 −|∇H|2) + (g g − g g ). (n − 1)(n − 2) ik jl il jk

Now we can use the ∗-product to simplify this evolution equation, because we

want to use the Maximum principle. Therefore, we have the following theorem. º ÌÓÖÑ 3.6 The evolution equation for the Weyl curvature tensor can be written as ∂ W =ΔW + A ∗ A ∗ W + A ∗ A + ∇A ∗∇A. (22) ∂t Also by applying (22), we state following theorem.

º | | ÌÓÖÑ 3.7 The variation of W satisﬁes

2 ∂|W | 2 2 2 2 2 ≤ Δ|W | + a1|A| |W | + a2|A| |W | + a3|∇A| |W |, (23) ∂t where a1, a2 and a3 are real numbers depending on the initial hypersurface.

154 THE WEYL CURVATURE TENSOR

Proof. ∂ ∂ ∂W |W |2 = W, W =2 ,W =2ΔW, W ∂t ∂t ∂t + A ∗ A ∗ W, W + A ∗ A, W + ∇A ∗∇A, W =Δ|W |2 − 2|∇W |2 + A ∗ A ∗ W ∗ W + ∇A ∗∇A ∗ W 2 2 2 2 2 ≤ Δ|W | + a1|A| |W | + a2|A| |W | + a3|∇A| |W |.

Since it was supposed that |A| and |∇A| are bounded, we conclude the fol-

lowing corollary.

ÓÐÐ ÖÝ º ÓÖ 3.8 The variation of the norm of the Weyl curvature tensor holds ∂ |W |2 ≤ Δ|W |2 + a|W |2 + b, (24) ∂t where a and b are real numbers depending on the initial hypersurface. Proof. Since |A| and |∇A| are bounded, then the inequality in the previous theorem can be written in the form ∂ 2 2 2 |W | ≤ Δ|W | + b1|W | + b2|W |, ∂t

2 2 y2 on the other hand, for any two real numbers x and y we have x +xy ≤ 2x + 2 , then ∂ 2 2 2 b2 |W | ≤ Δ|W | + b1(|W | + |W |) ∂t b1 2 2 2 b2 ≤ Δ|W | +2b1|W | + 2b1 =Δ|W |2 + a|W |2 + b,

2 b2 where we set a =2b1 and b = . 2b1

4. Weyl curvature tensor estimate

In this section, it is supposed that the Weyl curvature tensor is bounded at time t = 0, then an upper bound for it during the evolution is estimated. Finally, it is assumed that the initial hypersurface is the conformally ﬂat. Then

an estimate for the Weyl curvature tensor is concluded.

º | |≤ ÌÓÖÑ 4.1 Let us have W (p, 0) M.Then b b |W (p, t)|2 ≤ M + eat − for all t ∈ [0,T), (25) a a where a and b are the real numbers depending on the initial hypersurface M.

155 MOAZZAF GHODRAT—ABEDI ESMAIEL

P r o o f. By taking into account the equation (24) and the Maximum principle, the corresponding solution to the ODE is dφ = aφ + b; φ(0) = M. dt It implies dφ =dt, aφ + b where the solution of this diﬀerential equation is b b φ(t)=eat M + − . a a By considering the solution and the Maximum principle, the proof will be com-

pleted.

ÓÐÐ ÖÝ º | | ÓÖ 4.2 If the initial hypersurface is conformally flat, i.e., W (0,p) =0 for every p ∈ M, then under the mean curvature flow we have b b b |W (p, t)|2 ≤ eat − < eat. a a a P r o o f. It is sufficient if in the previous theorem we put M =0.

Acknowledgementº We are very thankful to the referee who helped to improve our paper. REFERENCES

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Received August 18, 2020 Department of Mathematics Azarbaijan Shahid Madani University 53751–71379 Tabriz, IRAN E-mail: [email protected] [email protected]

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