Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces

Lanbao Hou; Feng Du; Jing Mao; Chuanxi Wu

doi:10.1515/math-2021-0100

Article Open Access

Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces

Lanbao Hou , Feng Du , Jing Mao and Chuanxi Wu

Published/Copyright: October 7, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we study the eigenvalue problem of poly-drifting Laplacian on complete smooth metric measure space ( M , ⟨ , ⟩ , e − ϕ d v ) , with nonnegative weighted Ricci curvature Ric ϕ ≥ 0 for some ϕ ∈ C 2 ( M ) , which is uniformly bounded from above, and successfully obtain several universal inequalities of this eigenvalue problem.

Keywords: eigenvalues; universal inequalities; poly-drifting Laplacian; smooth metric measure space; weighted Ricci curvature

MSC 2010: 35P15; 53C20; 53C42

1 Introduction

A smooth metric measure space (SMMS) is actually a Riemannian manifold equipped with some measure which is conformal to the usual Riemannian measure. More precisely, for a given complete n -dimensional Riemannian manifold ( M , ⟨ , ⟩ ) with metric ⟨ , ⟩ , the triple ( M , ⟨ , ⟩ , e − ϕ d v ) is called a SMMS, where ϕ is a smooth real-valued function on M and d v is the Riemannian volume element related to ⟨ , ⟩ (sometimes, we also call d v the volume density). On a SMMS ( M , ⟨ , ⟩ , e − ϕ d v ) , we can define the so-called ∞ -Bakry-Émery Ricci tensor Ric ϕ by

Ric ϕ = Ric + Hess ϕ ,

which sometimes is also called the weighted Ricci curvature. Here Ric , Hess are the Ricci curvature and the Hessian operator on M , respectively.

The SMMS is not a trivial generalization of the corresponding Riemannian manifold, and actually, some classical results in the Riemannian geometry cannot be extended trivially to the case of SMMSs. For example, when Ric ϕ is bounded from below, Myers’s theorem, Bishop-Gromov’s volume comparison, Cheeger-Gromoll’s splitting theorem and Abresch-Gromoll excess estimate cannot hold as the Riemannian case. On the other hand, the equation Ric ϕ = κ ⟨ , ⟩ for some constant κ is called the gradient Ricci soliton equation, which plays an important role in the study of the Ricci flow. If κ = 0 , κ > 0 or κ < 0 , the gradient Ricci soliton ( M , ⟨ , ⟩ , e − ϕ d ν , κ ) is said to be steady, shrinking or expanding, respectively – for more details about Ricci solitons, we refer to [1]. One can define the so-called drifting Laplacian (or the weighted Laplacian) L ϕ as follows:

L ϕ ≔ Δ − ⟨ ∇ ϕ , ∇ ⋅ ⟩ ,

where ∇ , Δ are the gradient and the Laplace operators, respectively. When ϕ = 1 4 ∣ x ∣ 2 , then the drifting Laplacian L ϕ is the operator L ≔ Δ − 1 2 ⟨ x , ∇ ( ⋅ ) ⟩ , which was introduced by Colding and Minicozzi [2] to study self-shrinker hypersurfaces. Some interesting results for eigenvalues of the drifting Laplacian can be found in [3,4, 5,6,7] and references therein.

Let Ω be a bounded domain in an n -dimensional complete SMMS ( M , ⟨ , ⟩ , e − ϕ d v ) . In this paper, we consider the following eigenvalue problem:

(1.1) ( − L ϕ ) m u = Λ u in Ω , u = ∂ u ∂ ν = ⋯ = ∂ m − 1 u ∂ ν m − 1 = 0 on ∂ Ω ,

where ν denotes the outward unit normal vector field of ∂ Ω , and m is an arbitrary positive integer. We know that L ϕ m is self-adjoint on the space of functions

F = f ∈ C m + 2 ( Ω ) ∩ C m + 1 ( ∂ Ω ) : f ∣ ∂ Ω = ∂ f ∂ ν ∂ Ω = ⋯ = ∂ m − 1 f ∂ ν m − 1 ∂ Ω = 0

with respect to the inner product

⟨ ⟨ f , g ⟩ ⟩ = ∫ Ω f g d μ ,

so the eigenvalue problem (1.1) has a discrete spectrum whose elements (i.e., eigenvalues) can be listed increasingly as follows:

0 < Λ 1 ≤ Λ 2 ≤ ⋯ ≤ Λ k ≤ ⋯ ,

where each eigenvalue is repeated according to its multiplicity. Especially, when ϕ is a constant function, problem (1.1) becomes the eigenvalue problem of the poly-harmonic operator on Riemannian manifolds. In recent years, “Universal inequalities” for eigenvalues (i.e., not depending on the domain) of some self-adjoint elliptic operators is a hot research topic in Geometric Analysis. For eigenvalues of the poly-harmonic operator on Riemannian manifolds, some universal inequalities have been obtained in [8,9,10, 11,12] and references therein.

A natural question is:

Could we generalize those universal inequalities for the Laplacian on Riemannian manifolds to the case of drifting Laplacian on SMMSs?

In fact, when m = 1 , Xia and Xu [13] investigated eigenvalues of the Dirichlet problem of the drifting Laplacian on compact manifolds and got some universal inequalities; when m = 2 , Du et al. [14] obtained some universal inequalities of Yang type for eigenvalues of the bi-drifting Laplacian either on a compact Riemannian manifold with boundary (possibly empty) immersed in a Euclidean space, a unit sphere or a projective space, or on bounded domains of complete manifolds supporting some special functions. Moreover, some new results for the eigenvalues of bi-drifting Laplacian can be found in [15,16]. Recently, when m is an arbitrary integer no less than 2, Pereira et al. [17] and Du et al. [18] gave some universal inequalities for eigenvalues of the poly-drifting Laplacian. More results on universal inequalities for eigenvalues of the drifting Laplacian can be seen, e.g., in [19,20,21, 22,23,24].

Based on the existing results mentioned above, in this paper, we will give some universal inequalities for eigenvalues of problem (1.1) when m is an arbitrary integer no less than 2.

First, we have:

Theorem 1.1

Let ( M , ⟨ , ⟩ , e − ϕ d v ) be a complete SMMS with nonnegative weighted Ricci curvature Ric ϕ ≥ 0 for some ϕ ∈ C 2 ( M ) , which is bounded above uniformly on M . Denote by Λ i the ith eigenvalue of problem (1.1). Then, for m ≥ 2 , we have

(1.2) ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ≤ 4 m ( 2 m − 1 ) l 1 2 ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 Λ i m − 1 m 1 2 ∑ i = 1 k ( Λ k + 1 − Λ i ) Λ i 1 m 1 2 ,

where l is the dimension of the Euclidean l -space R l in the splitting of M given in Lemma 2.5.

Remark 1.2

We prefer to give two explanations to show further and clearly our motivation of considering universal inequalities for eigenvalues of weighted operators:

From the work [25] of Wei and Wylie, we know that a SMMS is not necessarily compact when Ric ϕ ≥ λ > 0 , unlike in the case of Riemannian manifolds where such a complete one is compact if its Ricci curvature is bounded from below uniformly by some positive constant. So, it is meaningful to study the geometry of SMMSs.
For what kind of ( M , ⟨ , ⟩ , e − ϕ d v ) , does there exist universal inequalities for the eigenvalues of problem (1.1) for m ≥ 2 ? Actually, Pereira et al. [17] have given some universal inequalities on bounded domains in a Euclidean space or a unit sphere. Du et al. [18] gave some universal inequalities for eigenvalues of the poly-drifting Laplacian on bounded domains in the Gaussian and cylinder shrinking solitons. Comparing with the results in [17,18], here we have given some universal inequalities for the eigenvalues of problem (1.1) in some new SMMSs.

From the inequality (1.2), we can get universal bounds on the ( k + 1 )th eigenvalue in terms of the first k eigenvalues.

Corollary 1.3

Under the assumption of Theorem 1.1, we have

(1.3) Λ k + 1 ≤ 1 k ∑ i = 1 k Λ i + 2 m ( 2 m − 1 ) k 2 l ∑ i = 1 k Λ i l − 1 l ∑ i = 1 k Λ i 1 l + 4 m 2 ( 2 m − 1 ) 2 k 4 l 2 ∑ i = 1 k Λ i l − 1 l 2 ∑ i = 1 k Λ i 1 l 2 − 1 k ∑ i = 1 k Λ i − 1 k ∑ j = 1 k Λ j 2 1 2

and

(1.4) Λ k + 1 ≤ ( l + 2 m ( 2 m − 1 ) ) k l ∑ i = 1 l Λ i + ( l + 2 m ( 2 m − 1 ) ) 2 k 2 l 2 ∑ i = 1 l Λ i 2 − ( l + 4 m ( 2 m − 1 ) ) k l ∑ i = 1 l Λ i − 1 k ∑ j = 1 l Λ j 2 1 2 .

Our last result is the universal inequality for lower order eigenvalues of the problem (1.1).

Theorem 1.4

Under the assumption of Theorem 1.1, we have

(1.5) ∑ i = 1 l ( Λ i + 1 − Λ 1 ) 1 2 ≤ 2 ( 2 m − 1 ) l Λ 1 1 2 .

2 Some useful facts

In this section, we will give some facts which will play an important role in the proofs of our results listed in Section 1. First, we will state the following general inequality for eigenvalues of the poly-drifting Laplacian which were given in [17,18].

Lemma 2.1

([17, Theorem 2.1], [18, Lemma 2.1]) Let ( M , ⟨ , ⟩ , e − ϕ d v ) be an n -dimensional compact SMMS with boundary (possibly empty), and let Λ i be the ith eigenvalue of the following eigenvalue problem:

( − L ϕ ) m u = Λ u in M , u = ∂ u ∂ ν = ⋯ = ∂ m − 1 u ∂ ν m − 1 = 0 on ∂ M ,

and u i be the orthonormal eigenfunction corresponding to Λ i , that is,

( − L ϕ ) m u i = Λ i u i in M , u i = ∂ u i ∂ ν = ⋯ = ∂ m − 1 u i ∂ ν m − 1 = 0 on ∂ M , ∫ M u i u j d μ = δ i j ∀ i , j = 1 , 2 , … ,

where ν is the outward unit normal vector field of ∂ M and d μ = e − ϕ d v . Then:

For any function h ∈ C m + 2 ( M ) ∩ C m + 1 ( ∂ M ) and any positive integer k , we have
(2.1) ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ‖ u i ∇ h ‖ 2 ≤ δ ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ∫ M h u i ( ( − L ϕ ) m ( h u i ) − Λ i h u i ) + ∑ i = 1 k ( Λ k + 1 − Λ i ) δ ⟨ ∇ h , ∇ u i ⟩ + u i L ϕ h 2 2 ,
where δ is any positive constant and ‖ f ‖ 2 = ∫ M f 2 d μ .
If h i ∈ C m + 2 ( M ) ∩ C m + 1 ( ∂ M ) , i ≥ 2 , satisfies ∫ Ω h i u 1 u j + 1 = 0 for 1 ≤ j < i , then for any positive integer i , we have
(2.2) ( Λ i + 1 − Λ 1 ) 1 2 ∫ M ∣ u 1 ∇ h i ∣ 2 d μ ≤ δ ∫ M h i u 1 ( ( − L ϕ ) m ( h i u 1 ) − Λ 1 h i u 1 ) d μ + 1 δ ⟨ ∇ h i , ∇ u 1 ⟩ + 1 2 u 1 L ϕ h i 2 ,
where δ is any positive constant and ‖ f ‖ 2 = ∫ M f 2 d μ .

Lemma 2.2

[17, Lemma 2.2] Let Λ i and u i , i = 1 , 2 , … , be as in Lemma 2.1, then

(2.3) 0 ≤ ∫ M u i ( − L ϕ ) k u i ≤ ( Λ i ) k m , k = 1 , 2 , … , m − 1 .

Lemma 2.3

(Reverse Chebyshev inequality [26]) Suppose { a i } i = 1 k and { b i } i = 1 k are two real sequences with { a i } increasing and { b i } decreasing, then we have

(2.4) ∑ i = 1 k a i b i ≤ 1 k ∑ i = 1 k a i ∑ i = 1 k b i .

Lemma 2.4

(Weighted Chebyshev inequality [26]) Let { a i } i = 1 k , { b i } i = 1 k and { c i } i = 1 k be three sequences of non-negative real numbers with { a i } i = 1 k decreasing, { b i } i = 1 k and { c i } i = 1 k increasing. Then the following inequality

(2.5) ∑ i = 1 k a i 2 b i ∑ i = 1 k a i c i ≤ ∑ i = 1 k a i 2 ∑ i = 1 k a i b i c i

holds.

Lemma 2.5

[27, Theorem 1.1] Let ( M , g ) be a complete connected Riemannian manifold with weighted Ricci curvature Ric ϕ ≥ 0 for some ϕ ∈ C 2 ( M ) , which is bounded above uniformly on M . Then it splits isometrically as N × R l , where N is some complete Riemannian manifold without lines and R l is the l -Euclidean space. Furthermore, the function ϕ is constant on each R l in this splitting.

3 Proofs of our results

In this section, we will give the detailed proofs of the main results.

Proof of Theorem 1.1

By Lemma 2.5, we know that ( M , ⟨ , ⟩ , d μ ) splits isometrically as N × R l . Let x ¯ = ( t , x ) be the standard coordinate functions of N × R l , where t ∈ N and x = ( x 1 , … , x l ) ∈ R l . Since ϕ is constant on each R l in this splitting, for α = 1 , 2 , … , l and for any vector field X ∈ Ξ ( M ) , we have

(3.1) L ϕ x α = Δ x α − ⟨ ∇ ϕ , ∇ x α ⟩ = 0 , ∣ ∇ x α ∣ = 1 , Ric ϕ ( ∇ x α , X ) = 0 ,

where Ξ ( M ) denotes the set of smooth vector fields on M . By the Bochner formula (see [5,25]) and using (3.1), for any functions g ∈ C 3 ( Ω ) , where Ω is a bounded domain of M , we have

(3.2) L ϕ ⟨ ∇ x α , ∇ g ⟩ = 2 ⟨ ∇ 2 x α , ∇ 2 g ⟩ + ⟨ ∇ x α , ∇ ( L ϕ g ) ⟩ + ⟨ ∇ g , ∇ ( L ϕ x α ) ⟩ + 2 Ric ϕ ( ∇ x α , ∇ g ) = ⟨ ∇ x α , ∇ ( L ϕ g ) ⟩ .

By L ϕ x α = 0 and (3.2), we can get

(3.3) ( − L ϕ ) m ( x α u i ) = x α ( − L ϕ ) m ( u i ) + 2 m ( − 1 ) m ⟨ ∇ x α , ∇ ( L ϕ m − 1 u i ) ⟩ = Λ i x α u i + 2 m ( − 1 ) m ⟨ ∇ x α , ∇ ( L ϕ m − 1 u i ) ⟩ .

Taking h = x α into (2.1) and summing α from 1 to l , then we infer from

∣ ∇ x α ∣ = 1 , L ϕ x α = 0 , ∑ α = 1 l ⟨ ∇ x α , ∇ u i ⟩ 2 ≤ ∣ ∇ u i ∣ 2

that

(3.4) l ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ≤ ∑ i = 1 k ∑ α = 1 l δ ( Λ k + 1 − Λ i ) 2 ∫ Ω 2 m ( − 1 ) m x α u i ⟨ ∇ x α , ∇ ( L ϕ m − 1 u i ) ⟩ d μ + ∑ i = 1 k ( Λ k + 1 − Λ i ) δ ∫ Ω ∣ ∇ u i ∣ 2 d μ .

Since

( − L ϕ ) m − 1 ( x α u i ) = x α ( − L ϕ ) m − 1 ( u i ) + 2 ( m − 1 ) ( − 1 ) m − 1 ⟨ ∇ x α , ∇ ( L ϕ m − 2 u i ) ⟩ ,

we have

(3.5) ∫ Ω x α u i ⟨ ∇ x α , ∇ ( L ϕ m − 1 u i ) ⟩ d μ = ∫ Ω x α u i L ϕ m − 1 ⟨ ∇ x α , ∇ u i ⟩ d μ = ∫ Ω L ϕ m − 1 ( x α u i ) ⟨ ∇ x α , ∇ u i ⟩ d μ = ∫ Ω ( x α L ϕ m − 1 u i + 2 ( m − 1 ) ⟨ ∇ x α , ∇ ( L ϕ m − 2 u i ) ⟩ ) ⟨ ∇ x α , ∇ u i ⟩ d μ .

On the other hand, we follow from the divergence theorem that

(3.6) ∫ Ω x α u i ⟨ ∇ x α , ∇ ( L ϕ m − 1 u i ) ⟩ d μ = − ∫ Ω L ϕ m − 1 u i ( u i ∣ ∇ x α ∣ 2 + x α ⟨ ∇ x α , ∇ u i ⟩ ) d μ .

Combining (3.5) and (3.6), one has

(3.7) ∫ Ω x α u i ⟨ ∇ x α , ∇ ( L ϕ m − 1 u i ) ⟩ d μ = ∫ Ω ( m − 1 ) ⟨ ∇ x α , ∇ ( L ϕ m − 2 u i ) ⟩ ⟨ ∇ x α , ∇ u i ⟩ + 1 2 u i L ϕ m − 1 u i ∣ ∇ x α ∣ 2 d μ .

When m = 2 p , p ∈ Z + , let g = x α , α = 1 , 2 , … , l , we have

∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ ≤ ∫ Ω ∣ ∇ g ∣ 2 ∣ ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ∣ 2 d μ

= ∫ Ω ⟨ ∇ g , ∇ ( ( − L ϕ ) p u i ) ⟩ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ d μ = ∫ Ω − ( − L ϕ ) p u i ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ d μ ≤ ∫ Ω ( ( − L ϕ ) p u i ) 2 d μ 1 2 ∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ 1 2 ≤ ∫ Ω u i ( − L ϕ ) 2 p u i d μ 1 2 ∫ Ω ∣ ∇ g ∣ 2 ∣ ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ∣ 2 d μ 1 2 = Λ i 1 2 ∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ 1 2 ,

which implies that

(3.8) ∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ ≤ Λ i .

By (3.8) and Lemma 2.2, we have

(3.9) ∫ Ω ⟨ ∇ g , ∇ ( ( − L ϕ ) m − 2 u i ) ⟩ ⟨ ∇ g , ∇ u i ⟩ d μ = ∫ Ω ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ d μ = ∫ Ω − ( − L ϕ ) p − 1 u i ⟨ ∇ g , ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ d μ ≤ ∫ Ω ( ( − L ϕ ) p − 1 u i ) 2 d μ 1 2 ∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ 1 2 ≤ Λ i m − 2 2 m Λ i 1 2 ≤ Λ i m − 1 m .

When m = 2 p + 1 , p ∈ Z + ,

∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ ≤ ∫ Ω ∣ ∇ g ∣ 2 ∣ ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ∣ 2 d μ = ∫ Ω ⟨ ∇ g , ∇ ( ( − L ϕ ) p u i ) ⟩ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ d μ = ∫ Ω − ( − L ϕ ) p u i ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ d μ ≤ ∫ Ω ( ( − L ϕ ) p u i ) 2 d μ 1 2 ∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ 1 2

≤ ∫ Ω u i ( − L ϕ ) 2 p u i d μ 1 2 ∫ Ω ∣ ∇ g ∣ 2 ∣ ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ∣ 2 d μ 1 2 ≤ Λ i m − 1 2 m ∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ 1 2 ,

which implies that

(3.10) ∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 ≤ Λ i m − 1 m .

By (3.10) and Lemma 2.2, we have

(3.11) ∫ Ω ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 2 u i ) ⟩ ⟨ ∇ g , ∇ u i ⟩ d μ = ∫ Ω ⟨ ∇ g , ∇ ( ( − L ϕ ) p u i ) ⟩ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ d μ = ∫ Ω − ( − L ϕ ) p u i ⟨ ∇ g , ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ d μ ≤ ∫ Ω ( ( − L ϕ ) p u i ) 2 d μ 1 2 ∫ Ω ⟨ ∇ g , ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p − 1 u i ) ⟩ ⟩ 2 d μ 1 2 ≤ Λ i m − 1 2 m ∫ Ω ∣ ∇ g ∣ 2 ∣ ∇ ⟨ ∇ g , ∇ ( ( − L ϕ ) p u i ) ⟩ ∣ 2 d μ 1 2 ≤ Λ i m − 1 m .

It follows from (3.9) and (3.11) that

∫ Ω ⟨ ∇ g , ∇ ( ( − L ϕ ) m − 2 u i ) ⟩ ⟨ ∇ g , ∇ u i ⟩ d μ ≤ Λ i m − 1 m .

By the above inequality, we have

(3.12) ∑ α = 1 l ∫ Ω ⟨ ∇ x α , ∇ ( L ϕ m − 2 u i ) ⟩ ⟨ ∇ x α , ∇ u i ⟩ d μ ≤ l Λ i m − 1 m ,

we infer from (3.7), (3.12) and Lemma 2.2 that

(3.13) ∑ α = 1 l ∫ Ω ( − 1 ) m x α u i ⟨ ∇ x α , ∇ ( L ϕ m − 1 u i ) ⟩ d μ = ∑ α = 1 l ∫ Ω ( − 1 ) m ( m − 1 ) ⟨ ∇ x α , ∇ ( L ϕ m − 2 u i ) ⟩ ⟨ ∇ x α , ∇ u i ⟩ + 1 2 u i L ϕ m − 1 u i ∣ ∇ x α ∣ 2 d μ ≤ ( m − 1 ) l Λ i m − 1 m + ∫ Ω l 2 u i ( − L ϕ ) m − 1 u i d μ = m − 1 2 l Λ i m − 1 m .

Also by Lemma 2.2, one has

(3.14) ∫ Ω ∣ ∇ u i ∣ 2 d μ = ∫ Ω u i ( − L ϕ u i ) d μ ≤ Λ i 1 m .

Substituting (3.13) and (3.14) into (3.4), we have

(3.15) l ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ≤ ∑ i = 1 k δ m ( 2 m − 1 ) l ( Λ k + 1 − Λ i ) 2 Λ i m − 1 m + ∑ i = 1 k ( Λ k + 1 − Λ i ) δ Λ i 1 m .

Taking

δ = ∑ i = 1 k ( Λ k + 1 − Λ i ) Λ i 1 m ∑ i = 1 k m ( 2 m − 1 ) l ( Λ k + 1 − Λ i ) 2 Λ i m − 1 m

in (3.15), we have

∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ≤ 4 m ( 2 m − 1 ) l 1 2 ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 Λ i m − 1 m 1 2 ∑ i = 1 k ( Λ k + 1 − Λ i ) Λ i 1 m 1 2 .

This completes the proof of Theorem 1.1.□

Proof of Corollary 1.3

By Lemma 2.3, with { ( Λ k + 1 − Λ i ) } i = 1 k decreasing, Λ i l − 1 l i = 1 k and Λ i 1 l i = 1 k increasing, we have

(3.16) ∑ i = 1 k ( Λ k + 1 − Λ i ) Λ i 1 l ≤ 1 k ∑ i = 1 k ( Λ k + 1 − Λ i ) ∑ i = 1 k Λ i 1 l

and

(3.17) ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 Λ i l − 1 l ≤ 1 k ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ∑ i = 1 k Λ i l − 1 l .

Putting (3.16) and (3.17) into (1.2), we have

∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ≤ 4 m ( 2 m − 1 ) k 2 l ∑ i = 1 k ( Λ k + 1 − Λ i ) ∑ i = 1 k Λ i l − 1 l ∑ i = 1 k Λ i 1 l .

Solving this quadratic polynomial about Λ k + 1 , we have (1.3) directly.

By Lemma 2.4, we have

∑ i = 1 k ( Λ k + 1 − Λ i ) 2 Λ i l − 1 l ∑ i = 1 k ( Λ k + 1 − Λ i ) Λ i 1 l ≤ ∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ∑ i = 1 k ( Λ k + 1 − Λ i ) Λ i .

Together with (1.2), one has

∑ i = 1 k ( Λ k + 1 − Λ i ) 2 ≤ 4 m ( 2 m − 1 ) l ∑ i = 1 k ( Λ k + 1 − Λ i ) Λ i .

Solving this quadratic polynomial about Λ k + 1 , we have (1.4) directly. This finishes the proof of Corollary 1.3.□

Proof of Theorem 1.4

By Lemma 2.5, we know that M can split into product manifold N × R l . Let x 1 , x 2 , … , x l be the standard coordinate functions of R l . Define an ( l × l ) -matrix B by

B ≔ ( b i j ) ,

where b i j = ∫ Ω x i u 1 u j + 1 . Using the orthogonalization of Gram and Schmidt, one knows that there exist an upper triangle matrix R = ( R i j ) and an orthogonal matrix Q = ( q i j ) such that R = Q B , i.e.,

R i j = ∑ k = 1 l q i k b k j = ∫ Ω ∑ k = 1 l q i k x k u 1 u j + 1 = 0 , for 1 ≤ j < i ≤ n .

Defining y i = ∑ k = 1 l q i k x k , we have ∫ Ω y i u 1 u j + 1 = 0 , where 1 ≤ j < i ≤ l . On the other hand, since Q = ( q i j ) is an orthogonal matrix, we know that y 1 , … , y l are also the standard coordinate functions of R l and ∣ ∇ y i ∣ = 1 , L ϕ y i = 0 . Then taking h i = y i in (2.2) and summing over i from 1 to l , we can obtain

(3.18) ∑ i = 1 l ( Λ i + 1 − Λ 1 ) 1 2 ≤ δ ∑ i = 1 l ∫ M y i u 1 ( ( − L ϕ ) m ( y i u 1 ) − Λ 1 y i u 1 ) d μ + 1 δ ∑ i = 1 l ∥ ⟨ ∇ y i , ∇ u 1 ⟩ ∥ 2 .

Using a similar calculation as in the proof of Theorem 1.1, we have

(3.19) ∑ i = 1 l ∫ Ω y i u 1 ( ( − L ϕ ) m ( y i u 1 ) − Λ 1 y i u 1 ) d μ = ∑ i = 1 l ∫ Ω ( m − 1 ) ⟨ ∇ y i , ∇ ( L ϕ m − 2 u 1 ) ⟩ ⟨ ∇ y i , ∇ u 1 ⟩ + 1 2 u 1 L ϕ m − 1 u 1 ∣ ∇ y i ∣ 2 d μ ≤ ( m − 1 ) l Λ 1 m − 1 m + ∫ Ω l 2 u 1 ( − L ϕ ) m − 1 u 1 d μ = m − 1 2 l Λ 1 m − 1 m

and

(3.20) ∑ i = 1 l ∥ ⟨ ∇ y i , ∇ u 1 ⟩ ∥ 2 ≤ ∫ Ω ∣ ∇ u 1 ∣ 2 d μ = ∫ Ω u 1 − L ϕ u 1 d μ ≤ Λ 1 1 m .

Substituting (3.19) and (3.20) into (3.18), one can obtain

∑ i = 1 l ( Λ i + 1 − Λ 1 ) 1 2 ≤ δ ( 2 m − 1 ) l Λ 1 m − 1 m + 1 δ Λ 1 1 m .

Taking δ = Λ 1 1 m / ( 2 m − 1 ) l Λ 1 m − 1 m 1 / 2 in the above inequality, we have

∑ i = 1 l ( Λ i + 1 − Λ 1 ) 1 2 ≤ 2 ( 2 m − 1 ) l Λ 1 m − 1 m Λ 1 1 m 1 2 = 2 ( 2 m − 1 ) l Λ 1 1 2 ,

which is (1.5) exactly. This completes the proof of Theorem 1.4.□

Funding information: This work was partially supported by the NSF of China (Grant No. 11801496 and 11926352), the Fok Ying-Tung Education Foundation (China), Hubei Key Laboratory of Applied Mathematics (Hubei University), Research Team Project of Jingchu University of Technology (Grant No. TD202006) and Research Project of Jingchu University of Technology (Grant No. YB202010, ZX202002, ZX202006).
Conflict of interest: Authors state no conflict of interest.
Data availability statement: No data, models, or code were generated or used during the study.

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Received: 2021-02-24

Revised: 2021-07-23

Accepted: 2021-08-19

Published Online: 2021-10-07

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2021-0100

Keywords for this article

eigenvalues; universal inequalities; poly-drifting Laplacian; smooth metric measure space; weighted Ricci curvature

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