DSpace at VNU: p-harmonic l-forms on Riemannian manifolds with a weighted Poincare inequality

DSpace at VNU: p-harmonic l-forms on Riemannian manifolds with a weighted Poincare inequality tài liệu, giáo án, bài giả...

Nonlinear Analysis 150 (2017) 138–150 Contents lists available at ScienceDirect Nonlinear Analysis www.elsevier.com/locate/na p-harmonic ℓ-forms on Riemannian manifolds with a weighted Poincar´e inequality Nguyen Thac Dung Department of Mathematics, Mechanics, and Informatics (MIM), Hanoi University of Sciences (HUS-VNU), Vietnam National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam article info Article history: Received June 2016 Accepted 10 November 2016 Communicated by Enzo Mitidieri MSC: 53C24 53C21 Keywords: Flat normal bundle p-harmonic ℓ-forms The second fundamental form Weighted Poincar e inequality Weitzenbă ock curvature operator abstract Given a Riemannian manifold with a weighted Poincar´ e inequality, in this paper, we will show some vanishing type theorems for p-harmonic ℓ-forms on such a manifold We also prove a vanishing result on submanifolds in Euclidean space with flat normal bundle Our results can be considered as generalizations of the work of Lam, Li–Wang, Lin, and Vieira (see Lam (2008), Li and Wang (2001), Lin (2015), Vieira (2016)) Moreover, we also prove a vanishing and splitting type theorem for p-harmonic functions on manifolds with Spin (9) holonomy provided a (p, p, λ)Sobolev type inequality which can be considered as a general Poincar´ e inequality holds true © 2016 Elsevier Ltd All rights reserved Introduction Let (M n , g) be a Riemannian manifold of dimension n and ρ ∈ C(M ) be a positive function on M We say that M has a weighted Poincar´e inequality, if   ρϕ2 ≤ |∇ϕ|2 (1.1) M C0∞ (M ) M holds true for any smooth function ϕ ∈ with compact support in M The positive function ρ is called the weighted function It is easy to see that if the bottom of the spectrum of Laplacian λ1 (M ) is positive then M satisfies a weighted Poincar´e inequality with ρ ≡ λ1 Here λ1 (M ) can be characterized by variational principle   | ∇ϕ | ∞ M λ1 (M ) = inf : ϕ ∈ C0 (M ) ϕ2 M E-mail address: dungmath@gmail.com http://dx.doi.org/10.1016/j.na.2016.11.008 0362-546X/© 2016 Elsevier Ltd All rights reserved N.T Dung / Nonlinear Analysis 150 (2017) 138–150 139 When M satisfies a weighted Poincar´e inequality then M has many interesting properties concerning topology and geometry It is worth to notice that weighted Poincar´e inequalities not only generalize the first eigenvalue of the Laplacian, but also appear naturally in other PDE and geometric problems For example, λ1 (M ) is related to the problem of finding the best constant in the inequality ∥u∥L2 ≤ C∥∇u∥L2 obtained by the continuous embedding W01,2 → L2 (M ) It is also well known that a stable minimal hypersurface satisfies a weighted Poincar´e inequality with the weight function ρ = |A| + Ric(ν, ν) where A is the second fundamental form and Ric(ν, ν) is the Ricci curvature of the ambient space in the normal direction For further discussion on this topic, we refer to [12,15,18,20,23] and the references there in On the other hand, suppose that M is a complete noncompact oriented Riemannian manifold of dimension n At a point x ∈ M , let {ω1 , , ωn } be a positively oriented orthonormal coframe on Tx∗ (M ), for ℓ ≥ 1, the Hodge star operator is given by ∗(ωi1 ∧ · · · ∧ ωiℓ ) = ωj1 ∧ · · · ∧ ωjn−ℓ , where j1 , , jn−ℓ is selected such that {ωi1 , , ωiℓ , ωj1 , , ωjn−ℓ } gives a positive orientation Let d is the exterior differential operator, so its dual operator δ is defined by δ = ∗d ∗ Then the Hodge–Laplace–Beltrami operator ∆ acting on the space of smooth ℓ-forms Ω ℓ (M ) is of form ∆ = −(δd + dδ) In [15], Li studied Sobolev inequality on spaces of harmonic ℓ-forms Then he gave estimates of the bottom of ℓ-spectrum and proved that the space of harmonic ℓ-forms is of finite dimension provided the Ricci curvature bounded from below When M is compact, it is well-known that the space of harmonic ℓ-forms is isomorphic to its ℓ-th de Rham cohomology group This is not true when M is non-compact but the theory of L2 harmonic forms still has some interesting applications For further results, one can refer to [4,5] Later, in [20], the authors investigated spaces of L2 harmonic ℓ-forms H ℓ (L2 (M )) on submanifolds in Euclidean space with flat normal bundle Assuming that the submanifolds are of finite total curvature, Lin showed that the space H ℓ (L2 (M )) has finite dimension Recently, in [23], Vieira obtained vanishing theorems for L2 harmonic forms on complete Riemannian manifolds satisfying a weighted Poincar´e inequality and having a certain lower bound of the curvature His theorems improve Li–Wang’s and Lam’s results Moreover, some applications to study geometric structures of minimal hypersurfaces are also given Therefore, it is very natural for us to study p-harmonic ℓ-forms on Riemannian manifolds with a weighted Poincar´e inequality Recall that an ℓ-form ω on M is said to be p-harmonic (p > 1) if ω satisfies dω = and δ(|ω|p−2 ω) = When p = 2, a p-harmonic ℓ-form is exactly a harmonic ℓ-form Some properties of the space of p-harmonic ℓ-forms are given by X Zhang and Chang–Guo–Sung (see [24,7]) In particular, in [6], Chang–Chen–Wei studied p-harmonic functions with finite Lq energy and proved some vanishing type theorems on Riemannian manifold satisfying a weighted Poincar´e inequality, recently Moreover, Sung–Wang, Dat and the author used theory of p-harmonic functions to show some interesting rigidity properties of Riemannian manifolds with maximal p-spectrum (See also [8,22]) N.T Dung / Nonlinear Analysis 150 (2017) 138–150 140 In this paper, we will prove the following theorem Theorem 1.1 Suppose that M is a Riemannian manifold satisfying the weighted Poincar´e inequality (1.1) with the positive weighted function If the Weitzenbă ock curvature operator K > −aρ, and 4(p−1) a < p2 then every p-harmonic ℓ-form ( ≤ ℓ ≤ n − 2) with finite Lp norm on M is trivial This theorem can be considered as a generalization of the work of Li–Wang, Lam, and Vieira (see [17,13,23]) The second vanishing property of this paper is an extension of Lin’s result Our theorem is stated as follows Theorem 1.2 Let M n , n ≥ be a complete non-compact immersed submanifold of Rn+k with flat normal bundle Denote by H the mean curvature vector of M If one of the following conditions 2 n |H| , |A|2 ≤ max{l,n−ℓ} the total curvature ∥A∥n is bounded by ∥A∥2n <  4(p − 1) n2 , max {ℓ, n − ℓ} p CS max {ℓ, n − ℓ} CS  , supM |A| is bounded and the fundamental tone satisfies sup |A|2 p2 λ1 (M ) > max {ℓ, n − ℓ} M 4(p − 1) holds true then every p-harmonic ℓ-form on M is trivial Here the submanifold M is said to have flat normal bundle if the normal connection of M is flat, namely the components of the normal curvature tensor of M are zero On the other hand, it is worth to mention that in [2] (see also [21]), the author considered complete manifolds M with some (p, q, λ)-Sobolev inequality  λ ϕq  pq  ≤ |∇ϕ|p (1.2) for some constant λ and for every ϕ ∈ C0∞ (M ) Here p, q are real numbers satisfying < p ≤ q < ∞ Defining 1,p (M ) by the p-Laplacian of a function u ∈ Wloc ∆p u = div(|∇u|p−2 ∇u) Hence if u ∈ C ∞ (M ) is a p-harmonic function, namely ∆p u = then du is a p-harmonic 1-form Buckley and Koskela noted that when p = q and M is a bounded Euclidean domain then −∆p u = λ|u|p−2 u has a solution u ∈ C01 (M ) The variational principle tells us that   |∇ϕ|p ∞ M  λ ≤ inf (M ) : ϕ ∈ C ϕp M In the case p = 2, λ is the least eigenvalue for the Laplacian Dirichlet problem Therefore, a (p, pλ)Sobolev inequality can be considered as a generalization of the Poincar´e inequality When M is a complete noncompact with Spin(9) holonomy, we will show that if a (p, p, λ)-Sobolev inequality holds true then either M has no p-parabolic end; or M is splitting Let us recall the definition of p-parabolic ends N.T Dung / Nonlinear Analysis 150 (2017) 138–150 141 Definition 1.3 An end E of the Riemannian manifold M is called p-parabolic if for every compact subset K⊂E  capp (K, E) := inf |∇f |p = 0, E where the infimum is taken among all f ∈ p-nonparabolic Cc∞ (E) such that f ≥ on K Otherwise, the end E is called The paper has three sections In Section 2, we will recall some auxiliary lemmas then give proofs of Theorems 1.1 and 1.2 Then we will give an application to study p-harmonic ℓ-forms on submanifolds with a stable condition Finally, in Section 3, we will show that complete noncompact manifolds with Spin(9) holonomy are splitting via the theory of p-harmonic functions p-harmonic ℓ-forms on Riemannian manifolds with a weighted Poincar´e inequality Suppose that M is a complete noncompact Riemannian manifold satisfying a weighted Poincar´e inequality (1.1) Let {e1 , , en } be a local orthonormal frame on M on M with dual coframe {ω1 , , ωn } Given a -form on M , the Weitzenbă ock curvature operator Kℓ acting on ω is defined by Kℓ (ω, ω) = n  ωk ∧ iej R(ek , ej ) j.k=1 Using the Weitzenbă ock curvature operator, we have the following Bochner type formula for ℓ-forms Lemma 2.1 ([16,23]) Let ω =  I aI ωI be a ℓ-form on M Then ∆|ω|2 = ⟨∆ω, ω⟩ + 2|∇ω|2 + ⟨Eℓ (ω), ω⟩ = ⟨∆ω, ω⟩ + 2|∇ω|2 + 2Kℓ (ω, ω) where Eℓ (ω) = n j,k=1 ωk ∧ iej R(ek , ej )ω Apply the above Bochner formula to the form |ω|p−2 ω we obtain   ∆|ω|2(p−1) = |∇(|ω|p−2 ω)|2 − (δd + dδ)(|ω|p−2 ω), |ω|p−2 ω + Kℓ (|ω|p−2 ω, |ω|p−2 ω)   = |∇(|ω|p−2 ω)|2 − δd(|ω|p−2 ω), |ω|p−2 ω + |ω|2(p−2) Kℓ (ω, ω) where we used ω is p-harmonic in the second equality This can be read as     |ω|p−1 ∆|ω|p−1 = |∇(|ω|p−2 ω)|2 − |∇|ω|p−1 |2 − |ω|p−2 δd(|ω|p−2 ω), ω + |ω|2(p−2) Kℓ (ω, ω) By Kato type inequality |∇(|ω|p−2 ω)|2 ≥ |∇|ω|p−1 |2 (see [3]) and Kℓ ≥ −aρ, this implies   |ω|∆|ω|p−1 ≥ − δd(|ω|p−2 ω), ω − aρ|ω|p (2.3) Now we will give a proof of Theorem 1.1 Proof of Theorem 1.1 Let ϕ ∈ C0∞ (M ), then multiply both sides of (2.3) by ϕ2 then integrate the obtained results, we have      ϕ2 |ω|∆|ω|p−1 ≥ − δd(|ω|p−2 ω), ϕ2 ω − a ρϕ2 |ω|p M M M 142 N.T Dung / Nonlinear Analysis 150 (2017) 138–150 Integration by parts implies        ∇(ϕ2 |ω|), ∇|ω|p−1 ≤ d(|ω|p−2 ω), d(ϕ2 ω) + a ρϕ2 |ω|p M M M    p−2 = d(|ω| ) ∧ ω, d(ϕ ) ∧ ω + a ρϕ2 |ω|p M M       p−2     ≤ d(|ω| ) ∧ ω d(ϕ ) ∧ ω + a ρϕ2 |ω|p M (2.4) M Here we used d(|ω|p−2 ω) = d(|ω|p−2 ) ∧ ω since dω = in the second equality Moreover, we used Schwartz inequality in the last equality Observe that for any ℓ-form α and m-form β, it is proved in [10] that     ℓ+m |α ∧ β| ≤ c|α| · |β|, where c = ℓ Therefore the first term in the right hand side of (2.4) is estimated by         ∇(|ω|p−2 ) |ω| · |d(ϕ2 )||ω| d(|ω|p−2 ) ∧ ω  d(ϕ2 ) ∧ ω  ≤ c2 M M  = 2c |p − 2| ϕ|ω|p−1 |∇ϕ| · |∇(|ω|)| M   c2 |p − 2| p−2 2 ≤ c |p − 2|ε |ω| |∇(|ω|)| ϕ + |ω|p |∇ϕ|2 ε M M (2.5) for any ε > Here we used the elementary inequality 2AB ≤ εA2 + ε−1 B for any A, B ∈ R in the last inequality Since M satisfies the weighted Poincar´e inequality, we can estimate the second term in the right hand side of (2.4) by        2 p 2 p    ρϕ2 |ω|2 = ρ ϕ|ω|  ≤ ∇ ϕ|ω|  M M M     2 2 ≤ (1 + ε) ϕ ∇|ω|p/2  + + |ω|p |∇ϕ|2 ε M M     2 (1 + ε)p2 p−2  ϕ |ω| ∇|ω| + + |ω|p |∇ϕ|2 (2.6) = ε M M for any ε > On the other hand, we compute the left hand side of (2.4) as follows      ∇(ϕ2 |ω|), ∇|ω|p−1 ≥ (p − 1) |ω|p−2 |∇|ω||2 ϕ2 − 2(p − 1) ϕ|ω|p−1 |∇ϕ||∇|ω|| M M M   p−2 2 ≥ (p − 1) |ω| |∇|ω|| ϕ − (p − 1)ε |ω|p−2 |∇|ω||2 ϕ2 M M  p−1 − |ω|p |∇ϕ|2 ε M By (2.5)–(2.7), it turns out that there exist A, B ∈ R such that   A |ω|p−2 |∇|ω|| ϕ2 ≤ B |ω|p |∇ϕ|2 M M (2.7) N.T Dung / Nonlinear Analysis 150 (2017) 138–150 143 where a(1 + ε)p2 A = (1 − ε)(p − 1) − − c2 |p − 2|ε   c2 |p − 2| p − 1 + + B =a 1+ ε ε ε Since a < that 4(p−1) p2 , choosing ε small enough, this implies there exists a positive constant C = C(p, ε) > such  |ω|p−2 |∇|ω||2 ϕ2 ≤ C  M |ω|p |∇ϕ|2 M Now, we choose ϕ ∈ C0∞ (M ) satisfying ≤ ϕ ≤ 1, |∇ϕ| ≤ R2 and  ϕ = on B(o, R) ϕ = on M \ B(o, 2R) where o ∈ M is a fixed point and B(o, R) is the geodesic ball centered at o with radius R > Then the above inequality implies    2  4C  p/2  p−2 = |ω| |∇|ω|| ≤ |ω|p ∇|ω|  p2 B(o,R) R B(o,R) M Letting R → ∞, we conclude that |∇|ω|p/2 | = Hence |ω| is constant, but since |ω| ∈ Lp (M ), it forces |ω| = Therefore ω = The proof is complete Remark 2.1 If we assume p ≥ 2, ℓ = then the refined Kato type inequality in [9] reads   κ |∇(|ω|p−2 ω)|2 ≥ + |∇|ω|p−1 |2 , (p − 1)2 where   (p − 1)2 κ = 1, n−1 Using this refined Kato inequality we can show that if a< 4(p − + κ) p2 then any p-harmonic 1-form with finite Lp norm is trivial Hence, our result is a generalization of Li–Wang’s, Lam’s, Vieira’s and Chang–Chen–Wei’s results [6,17,12,23] Recall that let M n be an n-dimensional submanifold in the (n + k)-dimensional Euclidean space Rn+k M is said to be satisfied a δ-super stability condition for < δ ≤ if   δ |A|2 ϕ2 ≤ |∇ϕ|2 , M M for any compactly supported Lipschitz function ϕ on M In particular, when δ = 1, M is said to be super-stable Note that if k = and δ = then the concept of super stability is the same as the usual definition of stability Corollary 2.2 Let M be an n-dimensional complete immersed minimal submanifold in Rn+k If M is superstable then there is no non-trivial p-harmonic 1-form on M with finite Lp norm provided that √ √ 2(n + n) 2(n − n) Multiplying both sides of the above inequality by ϕ2 then integrating the result, we obtain  ϕ2 |ω|∆|ω|p−1 ≥ −  M   δd(|ω|p−2 ω), ϕ2 ω M Consequently,    ∇(ϕ2 |ω|), ∇|ω|p−1 ≤ M    d(|ω|p−2 ω), d(ϕ2 ω) M Using (2.5) and (2.7), we infer for any ε >  ((p − 1)(1 − ε) − c(p − 2)ε) p−2 ϕ |ω|  |∇|ω|| ≤ M c(p − 2) p − + ε ε  |ω|p |∇ϕ|2 M Choosing ε > small enough, this implies there exist a constant C > such that p2    2 C   |ω|p ∇|ω|p/2  ≤ R M B(o,R) Let R → ∞, we conclude that |ω| is constant Since |ω| ∈ Lp (M ), it turns out that ω is zero By the previous part, we may assume |A|2 > n2 |H|2 max{ℓ,n−ℓ} then by (2.9), we have   n2 ∇(ϕ2 |ω|), ∇|ω|p−1 + ϕ2 |H|2 |ω|p M M     p−2 ≤ δd(|ω| ω), ϕ ω + max {ℓ, n − ℓ}   M |A|2 ϕ2 |ω|p (2.10) M Now, we will estimate the last term of the right hand side of (2.10) By Lemma 2.3 and Hăolder inequality, we have n−2    2n  n  n−2 p 2 p |ω| ϕ |A| ϕ |ω| ≤ ∥A∥n M M     2  p   ≤ CS ∥A∥2n |H|2 |ω|p ϕ2 , ∇ |ω| ϕ  + M M N.T Dung / Nonlinear Analysis 150 (2017) 138–150 146 where CS is the Sobolev constant depending only on n Since    2 2   p p p     ϕ∇|ω| + |ω| ∇ϕ ∇ |ω| ϕ  = M M      p 2 2 ≤ (1 + ε) ϕ ∇|ω|  + + |ω|p |∇ϕ|2 ε Bx (R) M    (1 + ε)p2 ≤ ϕ2 |ω|p−2 |∇|ω||2 + + |ω|p |∇ϕ|2 , ε M M we get  + ε)p2 |A| |ω| ϕ ≤ ϕ2 |ω|p−2 |∇|ω||2 M M    |H|2 |ω|p ϕ2 |ω|p |∇ϕ|2 + CS ∥A∥2n + CS ∥A∥2n + ε M M  p (1 CS ∥A∥2n Combining the above inequality, (2.5), (2.7) and (2.10), we have  A1 |ω|p−2 |∇|ω||2 +  M n2 − max {ℓ, n − ℓ} CS ∥A∥2n  |H|2 |ω|p ϕ2 ≤ B1 M  |ω|p |∇ϕ|2 , M where (1 + ε)p2 A1 := (p − 1)(1 − ε) − c|p − 2|ε − max {ℓ, n − ℓ} CS ∥A∥2n ,   p − c(p − 2) + + B1 := max {ℓ, n − ℓ} CS ∥A∥2n + ε ε ε   4(p−1) n2 Since ∥A∥2n < max{ℓ,n−ℓ}p , choosing ε > small enough, we conclude that there C , max{ℓ,n−ℓ}C S S exist two positive constants C1 , C2 > such that  |ω|p−2 |∇|ω||2 + C1  B(o,R) |H|2 |ω|p ≤ B(o,R) C2 R2  |ω|p M Let R → ∞, we infer |ω| is constant and |H||ω| = Since |ω| ∈ Lp (M ), this implies ω is trivial Suppose that supM |A|2 < ∞, the last term of the right hand side can be estimated as follows  2 p sup |A|2  M |∇(|ω|p/2 ϕ)|2 λ1 M     sup |A|2  (1 + ε)p2 M p−2 p ≤ ϕ |ω| |∇|ω|| + + |ω| |∇ϕ| λ1 ε M M |A| ϕ |ω| ≤ M Combining this inequality, (2.5), (2.7) and (2.10), we infer  A2 |ω|p−2 |∇|ω||2 + M n2  |H|2 |ω|p ϕ2 ≤ B2 M  |ω|p |∇ϕ|2 , M where sup |A|2 A2 := (p − 1)(1 − ε) − c|p − 2|ε − max {ℓ, n − ℓ} B2 := max {ℓ, n − ℓ} M λ1 (1 + ε)p2 ,  sup |A|  p − c(p − 2) M 1+ + + λ1 ε ε ε N.T Dung / Nonlinear Analysis 150 (2017) 138–150 147 2 M |A| p Since λ1 > max {ℓ, n − ℓ} sup4(p−1) , choose ε > small enough, we conclude that there exist two positive constants D1 , D2 > such that  |ω| p−2 |∇|ω|| + D1 B(o,R)  D2 |H| |ω| ≤ R B(o,R) p  |ω|p M Let R → ∞, we infer |ω| is constant and |H||ω| = Since |ω| ∈ Lp (M ), this implies ω is trivial The proof is complete It is also worth to note that whenever a refined Kato inequality for p-harmonic ℓ-forms holds true then we can improve the bound of ∥A∥ and λ1 (M ) as in Theorem 1.2 When p = 2, a refined Kato type inequality for harmonic ℓ-forms was given in [3] Let us finish this section by the following remark Remark 2.2 Recall that a Riemannian manifold N is said to have nonnegative isotropic curvature if R1313 + R1414 + R2323 + R2424 − 2R1234 ≥ Furthermore, assume that N has pure curvature tensor, namely for every p ∈ N there is an orthonormal basis {e1 , , en } of the tangent space Tp N such that Rijkl := ⟨R(ei , ej )ek , el ⟩ = whenever the set {i, j, k, l} contains more than two elements Here Rijkl denote the curvature tensors of N It is worth to notice that all 3-manifolds and conformally flat manifolds have pure curvature tensor By [20], we know that if M n be a compact immersed submanifold in N n+k with flat normal bundle, N has pure curvature tensor and nonnegative isotropic curvature then  1 n |H|2 − max {ℓ, n − ℓ} |A|2 Kℓ ≥ Therefore, the results in Theorem 1.2 are valid provided the above conditions on ambient spaces N hold Rigidity of manifolds with spin(9) holonomy Let M be a complete noncompact Riemannian manifold with holonomy group Spin(9) It was proved in [1] that a manifold with holonomy group Spin(9) must be locally symmetric and its universal covering is either the Cayley projective plane or the Cayley hyperbolic space H2O Since M is noncompact, its universal covering is H2O The following theorem is proved in [12] Theorem 3.1 Let M be a locally symmetric space with universal covering H2O then ∆M r ≤ 14 coth 2r + coth r in the sense of distributions Here r(q) is the distance function between q ∈ M and a fixed point o ∈ M Moreover, V (B(R)) ≤ Ce22R , where B(R) stands for the geodesic ball centered at the point o ∈ M with radius R 1,p Recall that for any function u ∈ Wloc (M ) and p > 1, the p-Laplacian operator is defined by ∆p := div(|∇u|p−2 ∇u) 1,p If u is a positive function u ∈ Wloc (M ) such that for any ϕ ∈ W01,p (M ) we have     |∇u|p−2 ∇u, ∇ϕ = λ1,p ϕup−1 M M N.T Dung / Nonlinear Analysis 150 (2017) 138–150 148 then u is said to be p-eigenfunction with respect to the first eigenvalue λ1,p By Theorem 3.1, we can estimate λ1,p as follows Corollary 3.2 Let M be a locally symmetric space with universal covering H2O If λ1,p is the first eigenvalue with respect to the p-Laplacian then  p 22 λ1,p ≤ p Proof Without loss of generality, we may assume that λ1,p is positive The variational characterization of λ1,p implies that M has infinite volume Hence, by Theorem 0.1 in [2], M is p-nonparabolic, moreover 1/p V (B(R)) ≥ C1 epλ1,p R Note that by Theorem 3.1, the growth of V (B(R)) is of at most e22R Therefore,  p 22 λ1,p ≤ p The proof is complete Now, we will show a splitting property of complete noncompact manifolds with holonomy Spin(9) Theorem 3.3 Let M be a complete noncompact 16-dimensional manifold with holonomy group Spin(9) p Suppose that the first eigenvalue λ1,p achieves the maximal value, namely λ1,p = 22 pp Then either M has no p-parabolic end; or M is a warped product M = R × N , where N is a compact manifold given by a compact quotient of the horosphere of the universal cover of M Proof Assume that M has a p-parabolic end E Let β be the Busemann function associated with a geodesic ray γ contained in E, namely β(q) = lim (t − dist(q, γ(t))) t→∞ The Laplacian comparison theorem in Theorem 3.1 implies that ∆β ≥ −22 Let a = 22 p and define g = eaβ , we compute ∆p g = ∆p (eaβ ) = div(ap−2 ea(p−2)β ∇(eaβ )) = ap−1 div(ea(p−1)β ∇β) = ap−1 (ea(p−1)β ∆β + b(p − 1)ea(p−1)β |∇β|2 ) ≥ ap−1 ea(p−1)β (−ap + a(p − 1)) = −ap g p−1 = −λ1,p g p−1 For any nonnegative compactly supported smooth function ϕ on M , the variational characterization of λ1,p implies   p λ1,p (ϕg) ≤ |∇(ϕg)|p M M On the other hand, integration by parts infers    ϕp g∆p g = − ϕp |∇g|p − p M M M ϕp−1 g ⟨∇ϕ, ∇g⟩ |∇g|p−2 N.T Dung / Nonlinear Analysis 150 (2017) 138–150 149 As in [22], we have that  p |∇(ϕg)|p = |∇ϕ|2 g + 2gϕ ⟨∇ϕ, ∇g⟩ + ϕ2 |∇g|2 ≤ ϕp |∇g|p + pϕg ⟨∇ϕ, ∇g⟩ ϕp−2 |∇g|p−2 + c|∇ϕ|2 g p for some constant c depending on p Therefore, we obtain     ϕp g(∆p g + λ1,p g p−1 ) = λ1,p (ϕg)p − ϕp |∇g|p − p ϕp−1 g ⟨∇ϕ, ∇h⟩ |∇h|p−2 M M M M    p p p ≤ |∇(ϕg)| − ϕ |∇g| − p ϕp−1 g ⟨∇ϕ, ∇h⟩ |∇h|p−2 M M M  ≤c |∇ϕ|2 g p (3.1) M Now, for R > 0, we choose ϕ ∈ C0∞ (M ), ≤ ϕ ≤ such that  1, on B(R) ϕ= on M \ B(2R) and |∇ϕ| ≤ R Then the right hand side of (3.1) can be estimated by    p 22β e22β |∇ϕ| g = |∇ϕ| e ≤ R M M M   4 ≤ e22β + e22β R M ∩(B(2R)\B(R)) R (M \E)∩(B(2R)\B(R)) (3.2) p −22R Hence, the first term in the right Since λ1,p = 22 pp , Theorem 0.1 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noncompact Riemannian. .. Theorems 1.1 and 1.2 Then we will give an application to study p-harmonic ℓ-forms on submanifolds with a stable condition Finally, in Section 3, we will show that complete noncompact manifolds with Spin(9)... for L2 harmonic forms on complete Riemannian manifolds satisfying a weighted Poincar´e inequality and having a certain lower bound of the curvature His theorems improve Li–Wang’s and Lam’s results