Characterization of Lagrangian Submanifolds by Geometric Inequalities in Complex Space Forms

In this paper, we give an estimate of the ﬁ rst eigenvalue of the Laplace operator on a Lagrangian submanifold M n minimally immersed in a complex space form. We provide su ﬃ cient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere S n . We also obtain Simons-type inequality for same ambient space form.


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 spacetime, 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 ðM n , gÞ yields function φ which is nonconstant and gratifying the ordinary differential equation if and only if ðM n , gÞ is isometric the sphere S n . A large scale of observations has been dedicated to this subject, and therefore, characterization of spaces, the Euclidean space ℝ n , the Euclidean sphere S n , and the complex projective space ℂP n are esteemed fields in differential geometry and are studied by a number of authors [2,. Similarly, Tashiro [27] has proved that the Euclidean space ℝ n is designated through the differential equation ∇ 2 φ = cg, where c is a positive constant. In [28], Lichnerowicz has been classified that the first nonzero eigenvalue μ 1 of the Laplacian on a compact manifold ðM n , gÞ with Ric ≥ n − 1 is not less than n, while μ 1 = n , then ðM n , gÞ is isometric to the sphere S n . 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. Let Ψ : M n ⟶M n ð4cÞ be a minimal immersion of a compact Lagrangian submanifold into complex space formM n ð4cÞ. If the Laplacian of M n endowed to the first eigenvalue μ 1 corresponding eigenfunction φ, then the following inequality holds where j∇ 2 φj 2 denotes the norm of the Hessian of φ and fe 1 , ⋯,e n g is frame on M n which is orthonormal. The equality holds if and only if μ 1 = nc. Besides, if the inequality holds Then, we have μ 1 ≥ cðn − 1Þ. In particular, if the following inequality satisfying Then, eigenvalue is satisfied μ 1 ≥ cðn − 1Þ.
By considered that compact submanifold M n immersed in the Euclidean sphere S n+p or Euclidean space ℝ n+p , Jiancheng and Zhang [29] derived the Simons-type [30] inequalities about the first eigenvalue μ 1 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.
In circumstantial, if S is constant, then it is equal to A greatly motivated idea of Obata is associated to characterizing sphere S n ðcÞ 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 ≤ ðn − 1Þð2 − nc/μ 1 Þc for a constant c and μ 1 is the first nonzero eigenvalue of the Laplace operator; then, M n is isometric to S n ðcÞ if M n admitted a nonzero conformal gradient vector field. They also proved that if M n is Einstein manifold such that Einstein constant μ = ðn − 1Þc, then M n is isometric to S n ðcÞ 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 ðM n , g,∇φ, λÞ 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. (i) If ∇φ on Ker h, then ΨðM n Þ is locally geodesic sphere S n , or ΨðM n Þ is isometric to standard sphere S n (ii) If Ric M n ð∇φ,∇φÞ ≥ cðn − 1Þj∇φj 2 , then ΨðM n Þ is isometric to a sphere S n The paper is organize as follows: In Section 2, 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.

Preliminaries and Notations
LetMð4cÞ be a complex space form of constant holomorphic sectional curvature 4c and of complex dimension m. Then, the curvature tensor R ofM m ð4cÞ can be expressed as:

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for any U, V, Z ∈ ΓðTMÞ [7,33]. An n-dimensional Riemannian submanifold M n ofMð4cÞ is classified as totally real if the standard complex structure J ofMð4cÞ maps any tangent space of M n 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 fe 1 ,⋯,e n+p g becoming an orthogonal frame to M n ; the second fundamental from h to M n is given by where σ α ij = hA αe i , e j i and A α denote the shape operator. The Gauss equation for Lagrangian submanifold M n in a complex space formM n+p ð4cÞ in the form of local coordinates is given by Then, for Ricci curvature As we assumed that Ψ is an immersion which is minimal, (10) yields Let a function φ : M n ⟶ R established on a Riemannian manifold, then the Bochner formula (see, e.g., [2]) given as: where Hessian is denoted by ∇ 2 φ and Ric denotes the Ricci curvature of M n . Now, we prove a proposition which authorizes to construct the proof of Theorems 1 and 2, that is the following: For exceptional, we have for any orthonormal frame fe 1 ,⋯,e n g tangent to M n .
Proof. If the identity operator on TM is denoted by I, then we have Therefore, if Δφ + μφ = 0, we obtain for any t ∈ ℝ. The norm of an operator which is given by jIj 2 = trðII * Þ. Taking integration in the above equation (15) and from Stokes theorem, we have We setting t = −μ 1 /n in (16), we get On other hand, (11) yields Tracing the above equation, we obtain Let us assume that Δφ = −μ 1 φ. Taking integration in Bochner formula and using Stokes theorem, we get

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From (19), we conclude This is the first result of the proposition. On the other hand, using (17) in the last equality, we obtain After some computation, we get Now, we have reached the proof of the proposition.
Recall the following lemma which set up to eliminate the proof of Theorem 2.
Lemma 5 [4]. Let a valid symmetric linear operator T : V ⟶ V which trace-less defined over a finite dimensional vector space V. If it is diagonalized T, i.e., Te i = μ i e i and dimKerT = k, they for any j we have for any integer k and for an orthonormal basis fe 1 ,⋯,e n g.

Proof of Main Theorems
We are in the position to prove our main results.

Proof of Theorem 1. Let us consider
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 Additionally, the equality holds if and only if the following holds Moreover, we expressed the first equation of Proposition 4 in a new form If we consider the following Then, equation (28) implies that If we notice that This implies that This completes the proof of Theorem 1.

Proof of Theorem 2.
Let the second fundamental form T which diagonalized via an orthogonal frame fe 1 ,⋯,e n g, i.e., Te i = k i e i , and the angle is denoted by θ i between ∇φ and e i . Thus, we find that h ∇φ, e i ð Þ j j 2 = g T∇φ, e i ð Þ 2 = g ∇φ, ð Te i i 2 = k 2 i cos 2 θ i ∇φ j j 2 : From the virtue of (13) of Proposition 4, we construct