Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.


Introduction
In the past few decades, many interesting properties of eigenvalues of some self-adjoint elliptic operators such as the usual Laplace operator (also called Laplace-Beltrami operator), the p-Laplace operator (also called p-Laplacian), the biharmonic operator and so on have been investigated in fixed Riemannian metrics (see [1][2][3]). Motivated by the work of Perelman [4] and Cao [5], research on eigenvalues of the Laplace operator and some other deformations related to the Laplace operator such as p-Laplacian and Witten-Laplacian under various geometric flows such as the Ricci flow, the mean curvature flow (MCF), the Yamabe flow and the Gaussian curvature flow has always been an active area in the study of geometry and topology of manifolds during these years.
Some results associated with eigenvalue problems have been attained under the MCF and deformations of the MCF. For instance, Zhao [6] considered a compact, strictly convex two-dimensional surface without boundary smoothly immersed in 3 and proved that the first eigenvalue of the Laplace operator is nonincreasing along the unnormalized powers of the MCF if the initial two-dimensional surface is totally umbilical. Subsequently, Zhao [7] obtained that the first eigenvalue of the p-Laplace operator is nondecreasing along the unnormalized powers of the mth MCF under some assumptions on the mean curvature and second fundamental form of initial given closed manifold. Furthermore, Zhao [8] also proved that the first eigenvalue of the p-Laplace operator is increasing along the unnormalized powers of the MCF under similar assumptions on mean curvatures. Mao [9] derived that the first eigenvalues of the Laplace and the p-Laplace operators are monotonic under the forced MCF by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term and an almost-umbilical pinching condition on the second fundamental form of the initial hypersurface. After going through these results, we wish to obtain some other conclusions for the case of eigenvalue problems related to the Laplace operator and the p-Laplace operator along the MCF and some other deformations associated with MCF, for example, the volume preserving MCF (see [10,11]), the area preserving MCF (see [12]), and more generally, MCF with a prescribed forcing term (see [9,[13][14][15]) etc. In this paper, motivated by Mao's work [9], we consider the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator evolving along the forced MCF under more general pinching conditions imposed on the second fundamental form of the initial hypersurface.

Preliminaries
In this section, for the reader's convenience, we would like to give a sketch of the eigenvalue problem and recall some basic knowledge about the forced MCF.
The Laplace operator is defined as For a compact Riemannian manifold M without boundary, Δ is a self-adjoint operator, then it has discrete eigenvalues according to the spectral theory in functional analysis. Obviously, the smallest eigenvalue in the problem is zero and the corresponding eigenfunctions should be constant functions. By Rayleigh's theorem and extreme principle, the first nonzero closed eigenvalue ( ) λ M 1 (λ 1 for short) can be defined by denotes the Sobolev space given by the completion of ( ) Now let's recall some facts of the p-Laplace operator ( < < ∞ p 1 ). The p-Laplace operator is a natural generalization of the Laplace operator for the fact that the p-Laplace operator is the Laplace operator when = p 2. The p-Laplacian eigenvalue problem concerns the partial differential equation: where u Δ p is given by , is the outer unit normal vector of the hypersurface M t at ( ) F x t , . The equation system (2.1) is also called unnormalized MCF (see [16,17] for more details) for which Huisken [18] showed that (2.1) is actually a quasilinear parabolic system with a smooth solution at least on some short time interval and also proved that the surfaces stay convex and contract to a point in finite time if the initial surface is convex. Li et al. [14] considered a more general MCF with a forcing term in the direction of its position vector which is called to be the forced MCF defined by the following evolution equation: where ( ) κ t is a continuous function which depends only on t, ( ) H x t , is the mean curvature of the hypersurface , is the outer unit normal vector of the hypersurface M t at ( ) F x t , . They showed that this parabolic equation can be converted to a strictly parabolic equation and thus has a smooth solution on a maximal time interval [ ) T 0, for some > T 0 by the standard theory of parabolic equations. In fact, they have proved that different forcing term will lead to different maximal time interval. The forced MCF (2.2) is an extension of the MCF (see [9,  For the forced MCF (2.2), the evolution for g ij , h ij and H (see [14]) are given by Note that for the Laplace operator, many papers have pointed out that its differentiability under geometric flows from the perturbation theory (see [19] for the details), but we are not clear whether the first eigenvalue or the corresponding eigenfunction of the p-Laplace operator is C 1 -differentiable under the forced flow (2.2) for ≠ p 2 till now. So we cannot use Ma's trick [20] to derive the monotonicity of the first eigenvalue of the p-Laplace operator. Fortunately, we can follow the arguments of Cao [5] and Wu [21]. Specifically, let M t be an n-dimensional closed Riemannian manifold, and ( ) g t be a smooth solution of the forced flow (2.2) on the time interval [ )( > ) T T 0, 0 . We now define a general smooth function ( ) λ u t , p 1, as follows: then we can check that ( ) u x t , defined here satisfies (2.7). Therefore, there exists a smooth function ( ) , 0 is the eigenfunction for the first eigenvalue ( ) λ t p 1, 0 of the p-Laplace operator, then we have that 3 Evolution equations for the first eigenvalue of the Laplace operator and the p-Laplace operator In this section, by applying the aforementioned evolution equations, we obtain the following result.
Evolution of the first eigenvalue of geometric operators under MCF  1521 Proof. Under the same assumptions of the aforementioned proposition, Mao [9] obtained By integration by parts, the last term of (3.2) becomes be the first nonzero closed eigenvalue of the p-Laplace operator on an n-dimensional compact and strictly convex hypersurface be a smooth function defined by (2.6). Then at time = t t 0 , we have Proof. Under the same assumptions of the aforementioned proposition, at time = By integration by parts, the last term of (3.5) becomes Inputting     , so the convexity is equivalent to mean convexity in this case.
Observing the evolution equations for the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator in Section 3, by applying Proposition 3.1, Proposition 3.2 and the aforementioned lemmas, we have the following.    Since the evolving hypersurface M t is strictly convex, by (4.2) and the Cauchy-Schwartz inequality, we have that is, As Mao [9, Theorem 5.1] does, in view of (4.4), by applying the maximum principle to (2.5), we obtain the following estimates for ( ) H x t , of the general n-dimensional compact and strictly convex hypersurface evolving along the forced MCF (2.2), that is, then is nondecreasing under the forced MCF (2.2) for ∈ [ ) t T 0, . Now for the case of the p-Laplace operator, we assume that ( ) u x t , is any smooth function satisfying is a smooth function with respect to t-variable, we have in any sufficiently small neighbourhood of t 0 provided ( ) ≥ Integrating the inequality with respect to t on a sufficiently small time interval [ ] ( ≤ ) t t t t , for ∈ [ ) t T 0, . The assertion that can be derived by the classical Lebesgue's theorem which states that a monotonous continuous function is differentiable almost everywhere. □ , by Theorem 4.5, we infer that the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator are constants under the normalized MCF if the initial hypersurface is a round sphere. In fact, we know that the evolving hypersurface M t preserves the property of being totally umbilical, so if the initial hypersurface is a round sphere, then the evolving hypersurface M t is also a round sphere. As we know, the evolving hypersurface M t preserves the total area, so the evolving hypersurface M t as a round sphere has the same radius as the initial round sphere, then the first eigenvalue is a constant, for ∈ [ ) t T 0, . Moreover, the second assertion of (ii) in Theorem 4.5 coincides with [6, Theorem 1.2] ( = ) In the following, we consider some special cases of Theorem 4.5.
As is shown in the proof of Theorem 4.5, the evolving hypersurface M t satisfies the pinching condition (4.1) for some constants < ≤ ≤ < ε β 0 1 2 , by (4.5), (4.10), we have 1ˆ.   , then we can get the following conclusions sharper than assertions (iii) and (vi) in Theorem 4.5.  Proof. The proof is similar to that of Theorem 4.7. We assume that ( ) u x t , is any smooth function satisfying   , and the coefficient function of the forcing term of the forced MCF by applying some of the conclusions obtained above. In fact, we can consider the compact strictly convex hypersurface M 0 without boundary to be even more general n-dimensional ellipsoid 1 since they also satisfy the pinching condition (4.1).