# Characterization of Lagrangian Submanifolds by Geometric Inequalities in Complex Space Forms

Authors: Lamia Saeed Alqahtani
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifoldMn minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere Sn. We also obtain Simons-type inequality for same ambient space form.


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## Body

## 1. Introduction

In the last few years, there has been attentioned to the classification of Lagrangian submanifolds. Lagrangian submanifolds give an impression being of foliations in the cotangent bundle, and Hamilton-Jacobi type leads to the classification via partial differential equation. In differential geometry of submanifolds, theorems which connect the intrinsic and extrinsic curvatures have significant role in physics [1]. Moreover, the notion of second order differential equations (PDEs) has built essential contribution in the analyze problems in fluid mechanics, heat conduction in solids, diffusive transport of chemicals in porous media, and wave propagation in strings and in mechanics or solids. The eigenvalue problems are trying to obtain all possible real λ such that there exists a nontrivial solution φ to the second order partial differential equation (PDEs) Δφ+λφ=0 [2, 3]. On the other hand, the Ricci tensor is involving in the curvature space-time, which finds the degree where matter will incline to converge or diverge in time (via the Raychaudhuri equation). By means of the Einstein field equation, it is also correlated to the matter content of the universe. In Riemannian geometry, on a Riemannian manifold, lower bounds of the Ricci tensor grant one to right geometric and topological understanding with the notion of a constant curvature space form. In Einstein manifold, the Ricci tensor verifies the vacuum Einstein equation, which have been broadly studied in [4]. In this relation, the Ricci flow equation supervises the working out of a given metric to an Einstein metric. Similarly, the eigenvalue problems are fascinating topics in differential geometry which has physical background. Therefore, a distinguished problem in Riemannian geometry is to find isometrics on a given manifold. One of the most interesting geometries of Riemannian manifolds is to characterize complex space form in the framework of Lagrangian submanifold geometry among the classes of compact, connected Riemannian manifolds. Beginning from the originate work of Obata [5], differential equation has become an influential tool in the investigation of geometric analysis. Obata [5] tested characterizing theorem for the standard sphere. A complete manifold Mn,g yields function φ which is nonconstant and gratifying the ordinary differential equation
(1)∇2φ+φg=0,if and only if Mn,g is isometric the sphere Sn. A large scale of observations has been dedicated to this subject, and therefore, characterization of spaces, the Euclidean space ℝn, the Euclidean sphere Sn, and the complex projective space ℂPn are esteemed fields in differential geometry and are studied by a number of authors [2, 6–26]. Similarly, Tashiro [27] has proved that the Euclidean space ℝn is designated through the differential equation ∇2φ=cg, where c is a positive constant. In [28], Lichnerowicz has been classified that the first nonzero eigenvalue μ1 of the Laplacian on a compact manifold Mn,g with Ric≥n−1 is not less than n, while μ1=n, then Mn,g is isometric to the sphere Sn. This means that the Obata’s rigidity theorem could be used to analyze the equality case of Lichnerowicz’s eigenvalue estimates in [28]. Motivated from previously studied and historical development on such characterizations, we give our first result as the following.Theorem 1.
LetΨ:Mn⟶M~n4c be a minimal immersion of a compact Lagrangian submanifold into complex space form M~n4c. If the Laplacian of Mn endowed to the first eigenvalue μ1 corresponding eigenfunction φ, then the following inequality holds
(2)∫M∑i=1nh∇φ,ei2dV+c∫M∇φ2dV≥∫M∇2φ2dV,where ∇2φ2 denotes the norm of the Hessian of φ and e1,⋯,en is frame on Mn which is orthonormal. The equality holds if and only if μ1=nc. Besides, if the inequality holds
(3)∫M∇2φ2dV≥∫M∑i=1nh∇φ,ei2dV.
Then, we haveμ1≥cn−1. In particular, if the following inequality satisfying
(4)∫M∇φ2dV≥ncμ1∫M∑i=1nh∇φ,ei2dV.Then, eigenvalue is satisfied μ1≥cn−1.By considered that compact submanifoldMn immersed in the Euclidean sphere Sn+p or Euclidean space ℝn+p, Jiancheng and Zhang [29] derived the Simons-type [30] inequalities about the first eigenvalue μ1 and the squared norm of the second fundamental form S without using the condition that submanifold M is minimal. They also established a lower bound for S if it is constant. Similar results can be found in [4, 31]. Simon’s inequalities and its corollary motivate the mathematicians try to improve the estimate the upper bound of S and study the rigidity of associated submanifolds. As a generalization of Euclidean sphere and Euclidean spaces, we consider a Lagrangian submanifold which minimally immersed into complex space form with constant holomorphic sectional curvature 4c; we obtain our next result as the following.Theorem 2.
LetΨ:Mn⟶M~n4c be a minimal immersion of a compact Lagrangian submanifold into the complex space form M~n4c. If dimKerh=k, then we have
(5)∫MS∇2φ2dV≥n−knc−1nc−μ1n−k−1nc∫M∇φ2dV.
In circumstantial, ifS is constant, then it is equal to
(6)S≥n−knc−1ncn−k−1nc−μ1.A greatly motivated idea of Obata is associated to characterizing sphereSnc through the second-order differential equation (1). By using the techniques of conformal vector field which have prominent appearance in deriving characterizations of spaces but also have high-level geometry in the theory of relativity and mechanics, Deshmukh and Al-Solamy [32] proved that an n-dimensional compact connected Riemannian manifold whose Ricci curvature satisfied the bound 0<Ric≤n−12−nc/μ1c for a constant c and μ1 is the first nonzero eigenvalue of the Laplace operator; then, Mn is isometric to Snc if Mn admitted a nonzero conformal gradient vector field. They also proved that if Mn is Einstein manifold such that Einstein constant μ=n−1c, then Mn is isometric to Snc with c>0 if it is admitted conformal gradient vector field. Taking account of Obata equation (1), Barros et al. [20] shows that a compact gradient almost Ricci soliton Mn,g,∇φ,λ is isometric to a Euclidean sphere whose Ricci tensor is Codazzi and has constant sectional curvature. For more terminology of Obata equation, see [14]. In the sequel, inspired by ideas are developed in [4, 29, 30]. So we give our result.Theorem 3.
LetΨ:Mn⟶M~n4c be a minimal immersion of a compact Lagrangian submanifold into the complex space form M~n4c and φ a first eigenfunction associated to the Laplacian of Mn. Then, we have the following:
(i)
If∇φ on Kerh, then ΨMn is locally geodesic sphere Sn, or ΨMn is isometric to standard sphere Sn(ii)
IfRicMn∇φ,∇φ≥cn−1∇φ2, then ΨMn is isometric to a sphere SnThe paper is organize as follows: In Section2, we recall some preliminary formulas related to our study. Moreover, we also prove a proposition in this section which helps to derive our main results. In Section 3, we give the proofs of our theorems which we proposed in the first section. Finally, in Section 4, we provided some consequences of main results.


## 2. Preliminaries and Notations

LetM~4c be a complex space form of constant holomorphic sectional curvature 4c and of complex dimension m. Then, the curvature tensor R of M~m4c can be expressed as:
(7)RU,VZ=cgU,ZV−gV,ZU+gU,JZJV−gV,JZU+2gU,JVJZ,for any U,V,Z∈ΓTM~ [7, 33]. An n-dimensional Riemannian submanifold Mn of M~4c is classified as totally real if the standard complex structure J of M~4c maps any tangent space of Mn into the corresponding normal space [34]. In particular, a totally real submanifold is said to be a Lagrangian submanifold if n=m (maximum dimension). Let e1,⋯,en+p becoming an orthogonal frame to Mn; the second fundamental from h to Mn is given by
(8)hei,ej=∑α=1nσijαeα,where σijα=Aαei,ej and Aα denote the shape operator. The Gauss equation for Lagrangian submanifold Mn in a complex space form M~n+p4c in the form of local coordinates is given by
(9)Rjkli=δiiδjj−δijδjic+∑α=1nσikασjlα−σilασjkα.Then, for Ricci curvature(10)Rjiji=δiiδjj−δijδjic+∑α=1pσiiασjjα−σijασjiα.As we assumed thatΨ is an immersion which is minimal, (10) yields
(11)Ricei,ej=n−1cδij−∑α=1nσikασjkα.Let a functionφ:Mn⟶R established on a Riemannian manifold, then the Bochner formula (see, e.g., [2]) given as:
(12)12Δ∇φ2=∇2φ2+RicMn∇φ,∇φ+g∇φ,∇Δφ,where Hessian is denoted by ∇2φ and Ric denotes the Ricci curvature of Mn.Now, we prove a proposition which authorizes to construct the proof of Theorems1 and 2, that is the following:Proposition 4.
LetΨ:Mn⟶M~n4c be an immersion of a compact Lagrangian submanifold into the complex space form M~n+p4c. Let φ be a first eigenfunction endowed to the Laplacian of Mn and Φ is minimal, then
(13)nc−μ1∫M∇φ2dV=∫M∑i=1nB∇φ,ei2dV+c∫M∇φ2dV−∫M∇2φ2dV.
For exceptional, we have(14)∫M∑i=1nB∇φ,ei2dV=∫M∇2φ+μ1nφI2dV+n−1nc−μ1n∫M∇φ2dV,for any orthonormal frame e1,⋯,en tangent to Mn.Proof.
If the identity operator onTM is denoted by I, then we have
(15)∇2φ−tφI2=∇2φ2−2tφΔφ+nt2φ2.
Therefore, ifΔφ+μφ=0, we obtain for any t∈ℝ. The norm of an operator which is given by I2=trII∗. Taking integration in the above equation (15) and from Stokes theorem, we have
(16)∫M∇2φ−tφI2dV=∫M∇2φ2dV+2t+nμ1t2∫M∇φ2dV.
We settingt=−μ1/n in (16), we get
(17)∫M∇2φ2dV=∫M∇2φ+μ1nφI2dV+μ1n∫M∇φ2dV.
On other hand, (11) yields
(18)Ricφiei,φjej=n−1cδijφiφj−∑α=1p∑k=1nσikασjkαφiφj.
Tracing the above equation, we obtain(19)Ric∇φ,∇φ=cn−1∇φ2−∑i=1nB∇φ,ei2.
Let us assume thatΔφ=−μ1φ. Taking integration in Bochner formula and using Stokes theorem, we get
(20)∫M∇2φ2dV+∫MRic∇φ,∇φdV=μ1∫M∇φ2dV.
From (19), we conclude
(21)cn−μ1∫M∇φ2dV=∫M∑i=1nB∇φ,ei2dV+c∫M∇φ2dV−∫M∇2φ2dV.
This is the first result of the proposition. On the other hand, using (17) in the last equality, we obtain
(22)cn−μ1∫M∇φ2dV=∫M∑i=1nh∇φ,ei2dV+c∫M∇φ2dV−∫M∇2φ+μ1nφI2dV−μ1n∫M∇φ2dV.
After some computation, we get(23)∫M∑i=1nh∇φ,ei2dV=∫M∇2φ+μ1nφI2dV+n−1nc−μ1n∫M∇φ2dV.
Now, we have reached the proof of the proposition.Recall the following lemma which set up to eliminate the proof of Theorem2.Lemma 5 [4].
Let a valid symmetric linear operatorT:V⟶V which trace-less defined over a finite dimensional vector space V. If it is diagonalized T, i.e., Tei=μiei and dimKerT=k, they for any j we have
(24)μj2≤n−k−1T2n−k,for any integer k and for an orthonormal basis e1,⋯,en.


## 3. Proof of Main Theorems

We are in the position to prove our main results.


### 3.1. Proof of Theorem1

Let us consider(25)nc≥μ1.Then, we noticed that left-hand side of (13) of Proposition 4 is different from negative. Therefore, the other side also non-negative, we get
(26)∫M∑i=1nh∇φ,ei2dV+c∫M∇φ2dV≥∫M∇2f2dV.Additionally, the equality holds if and only if the following holds(27)μ1=nc.Moreover, we expressed the first equation of Proposition4 in a new form
(28)∫M∇2φ2dV−∫M∑i=1nh∇φ,ei2dV=μ1−cn−1∫M∇φ2dV.If we consider the following(29)∫M∇2φ2dV≥∫M∑i=1nh∇φ,ei2dV.Then, equation (28) implies that
(30)μ1−cn−1≥0.If we notice that(31)∫M∇φ2dV≥ncμ1∫M∑i=1nh∇φ,ei2dV.This implies that(32)∫M∇2φ2dV≥∫M∑i=1nh∇φ,ei2dV.This completes the proof of Theorem1.


### 3.2. Proof of Theorem2

Let the second fundamental formT which diagonalized via an orthogonal frame e1,⋯,en, i.e., Tei=kiei, and the angle is denoted by θi between ∇φ and ei. Thus, we find that
(33)h∇φ,ei2=gT∇φ,ei2=g∇φ,Tei2=ki2cos2θi∇φ2.From the virtue of (13) of Proposition 4, we construct
(34)∫M∑i=1nki2cos2θi∇φ2dV=∫M∇2φ2dV+n−1c−μ1∫M∇φ2dV.Implementation Lemma5 to the previous equation to establish
(35)n−k−1n−k∫MS∇φ2dV≥∫M∇2φ2dV+n−1c−μ1∫M∇φ2dV.Let us assume the following inequality(36)∫M∇2φ2dV≥μ1nc∫M∇φ2dV.Plugging above equation into (35), we arrive at
(37)n−k−1n−k∫MS∇φ2dV≥n2c2−ncμ1−nc2+μ1nc∫M∇φ2dV.After some computations, finally, we get(38)∫MS∇2φ2dV≥n−knc−1nc−μ1n−k−1nc∫M∇φ2dV.This completes the proof of Theorem2.


### 3.3. Proof of Theorem3

As recognize a theorem as a result of Obata in [5], a differentiable function φ on Riemannian manifold Mn is satisfied the following ordinary differential equation
(39)∇2φ=−φ,if and only if Mn is isometric to a unit sphere Sn, where ∇2φ is two time derivatives of φ and is called Hessian of φ. As we assumed that
(40)∇φ∈kerhh∇φ,ei=0,1≤i≤n.Then, using equation (14), we get
(41)n−1μ1−ncn∫M∇φ2dV=∫M∇2φ+μ1nφ2dV.The left-hand side of this previous equation is not negative; we summarize that(42)μ1=nc.Therefore, we have(43)∇2φ=−φ.Now using Obata theorem [5], we conclude that ΨMn is isometric to a unit sphere Sn. This completes the proof of first part of Theorem 3.On the other case, if we consider that(44)Ric∇φ,∇φ≥cn−1∇φ2.Follows the equation (19), we find that
(45)∫Mn−1c∇φ2dV≥∑i=1nh∇φ,ei2dV+n−1c∫M∇φ2dV,which implies that
(46)∑i=1nh∇φ,ei2dV≤0.From where we conclude thath∇φ,ei=0, this means that ∇φ∈kerB. Now, we invoke the first case (i) of Theorem 3 we get required result. The proof of Theorem 3 is completed.


## 3.1. Proof of Theorem1

Let us consider(25)nc≥μ1.Then, we noticed that left-hand side of (13) of Proposition 4 is different from negative. Therefore, the other side also non-negative, we get
(26)∫M∑i=1nh∇φ,ei2dV+c∫M∇φ2dV≥∫M∇2f2dV.Additionally, the equality holds if and only if the following holds(27)μ1=nc.Moreover, we expressed the first equation of Proposition4 in a new form
(28)∫M∇2φ2dV−∫M∑i=1nh∇φ,ei2dV=μ1−cn−1∫M∇φ2dV.If we consider the following(29)∫M∇2φ2dV≥∫M∑i=1nh∇φ,ei2dV.Then, equation (28) implies that
(30)μ1−cn−1≥0.If we notice that(31)∫M∇φ2dV≥ncμ1∫M∑i=1nh∇φ,ei2dV.This implies that(32)∫M∇2φ2dV≥∫M∑i=1nh∇φ,ei2dV.This completes the proof of Theorem1.


## 3.2. Proof of Theorem2

Let the second fundamental formT which diagonalized via an orthogonal frame e1,⋯,en, i.e., Tei=kiei, and the angle is denoted by θi between ∇φ and ei. Thus, we find that
(33)h∇φ,ei2=gT∇φ,ei2=g∇φ,Tei2=ki2cos2θi∇φ2.From the virtue of (13) of Proposition 4, we construct
(34)∫M∑i=1nki2cos2θi∇φ2dV=∫M∇2φ2dV+n−1c−μ1∫M∇φ2dV.Implementation Lemma5 to the previous equation to establish
(35)n−k−1n−k∫MS∇φ2dV≥∫M∇2φ2dV+n−1c−μ1∫M∇φ2dV.Let us assume the following inequality(36)∫M∇2φ2dV≥μ1nc∫M∇φ2dV.Plugging above equation into (35), we arrive at
(37)n−k−1n−k∫MS∇φ2dV≥n2c2−ncμ1−nc2+μ1nc∫M∇φ2dV.After some computations, finally, we get(38)∫MS∇2φ2dV≥n−knc−1nc−μ1n−k−1nc∫M∇φ2dV.This completes the proof of Theorem2.


## 3.3. Proof of Theorem3

As recognize a theorem as a result of Obata in [5], a differentiable function φ on Riemannian manifold Mn is satisfied the following ordinary differential equation
(39)∇2φ=−φ,if and only if Mn is isometric to a unit sphere Sn, where ∇2φ is two time derivatives of φ and is called Hessian of φ. As we assumed that
(40)∇φ∈kerhh∇φ,ei=0,1≤i≤n.Then, using equation (14), we get
(41)n−1μ1−ncn∫M∇φ2dV=∫M∇2φ+μ1nφ2dV.The left-hand side of this previous equation is not negative; we summarize that(42)μ1=nc.Therefore, we have(43)∇2φ=−φ.Now using Obata theorem [5], we conclude that ΨMn is isometric to a unit sphere Sn. This completes the proof of first part of Theorem 3.On the other case, if we consider that(44)Ric∇φ,∇φ≥cn−1∇φ2.Follows the equation (19), we find that
(45)∫Mn−1c∇φ2dV≥∑i=1nh∇φ,ei2dV+n−1c∫M∇φ2dV,which implies that
(46)∑i=1nh∇φ,ei2dV≤0.From where we conclude thath∇φ,ei=0, this means that ∇φ∈kerB. Now, we invoke the first case (i) of Theorem 3 we get required result. The proof of Theorem 3 is completed.


## 4. Some Applications

It is renowned that the complex Euclidean spaceℂn, the complex projective n-space ℂPn4, and complex hyperbolic n-space ℂHn−4 are special cases of a complex space form M~n4c with c=0,1 and c=−1, respectively. Therefore, we define following corollaries for complex projective spaces as consequences of Theorems 1, 2, and 3.Corollary 6.
LetΨ:Mn⟶ℂPn4 be an immersion of a compact Lagrangian submanifold into complex projective space ℂPn4. If the Laplacian of Mn endowed to the first eigenvalue μ1 corresponding eigenfunction φ and Ψ is minimal, then the following inequality holds
(47)∫M∑i=1nh∇φ,ei2dV+∫M∇φ2dV≥∫M∇2φ2dV.
The equality holds if and only ifμ1=n. Furthermore, if the following inequality holds
(48)∫M∇2φ2dV≥∫M∑i=1nh∇φ,ei2dV.
Then, we haveμ1≥n−1. In particular, if the following inequality satisfying
(49)∫M∇φ2dV≥nμ1∫M∑i=1nh∇φ,ei2dV.
Then, eigenvalue is satisfiedμ1≥n−1.Corollary 7.
LetΨ:Mn⟶ℂPn4 be an immersion of a compact Lagrangian submanifold into the complex projective space ℂPn4. If Ψ is minimal and dimKerh=k, then we have
(50)∫MS∇2φ2dV≥n−μ1n−1n−kn−k−1n∫M∇φ2dV.
In especial case, ifS is constant, then we define
(51)S≥n−1n−knn−μ1n−k−1.From Theorem3, we have the following:Corollary 8.
LetΨ:Mn⟶ℂPn4 be an immersion of a compact Lagrangian submanifold into the complex projective space ℂPn4. Assuming that Ψ is minimal and φ be a first eigenfunction endowed to the Laplacian of Mn. Then, we get following:
(i)
If∇φ on Kerh, then ΨMn is locally geodesic sphere Sn, or ΨMn is isometric to standard sphere Sn(ii)
IfRicMn∇φ,∇φ≥n−1∇φ2, then ΨMn is isometric to sphere Sn



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