Annali di Matematica (2017) 196:1489–1511 DOI 10.1007/s10231-016-0625-0

p-Harmonic functions and connectedness at infinity of complete submanifolds in a Riemannian manifold Nguyen Thac Dung1 · Keomkyo Seo2

Received: 20 January 2016 / Accepted: 4 November 2016 / Published online: 19 November 2016 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Abstract In this paper, we study the connectedness at infinity of complete submanifolds by using the theory of p-harmonic function. For lower-dimensional cases, we prove that if M is a complete orientable noncompact hypersurface in Rn+1 and if δ-stability inequality holds on M, then M has only one p-nonparabolic end. It is also proved that if M n is a complete noncompact submanifold in Rn+k with sufficiently small L n norm of the traceless second fundamental form, then M has only one p-nonparabolic end. Moreover, we obtain a lower bound of the fundamental tone of the p Laplace operator on complete submanifolds in a Riemannian manifold. Keywords p-Harmonic function · p-Nonparabolicity · δ-Stability · The first eigenvalue · Connectedness at infinity Mathematics Subject Classification 53C24 · 53C21

1 Introduction Let M n be an n-dimensional complete orientable hypersurface in a complete Riemannian manifold N of nonnegative sectional curvature. When M is minimal in N , M is called δ-stable if any compactly supported Lipschitz function ϕ on M satisfies

B

Keomkyo Seo [email protected] http://sites.google.com/site/keomkyo/ Nguyen Thac Dung [email protected]

1

Department of Mathematics, Mechanics, and Informatics (MIM), Hanoi University of Sciences (HUS-VNU), Vietnam National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

2

Department of Mathematics, Sookmyung Women’s University, Cheongpa-ro 47-gil 100, Yongsan-ku, Seoul 04310, Korea

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δ

 

  2 |A| + Ric(ν, ν) ϕ dv ≤ |∇ϕ|2 dv, 2

M

(1.1)

M

where ν denotes the unit normal vector of M, Ric(ν, ν) denotes the Ricci curvature of N in the ν direction, |A|2 denotes the square length of the second fundamental form A, and dv denotes the volume form for the induced metric on M. Obviously, δ1 -stability implies δ2 -stability for 0 < δ2 < δ1 ≤ 1. In particular, when δ = 1, M is said to be stable. The L 2 harmonic function theory has played an important role in the study of complete orientable δ-stable minimal hypersurfaces. For instance, Palmer [26] proved that if M is a complete orientable minimal hypersurface in Rn+1 and if there exists a codimension one cycle separating M, then M is unstable by applying the nonexistence of L 2 harmonic 1form on such M. Miyaoka [20] gave the nonexistence of L 2 harmonic 1-forms on a complete orientable noncompact stable minimal hypersurface in a nonnegatively curved manifold. Cao et al. [4] proved that an n(≥ 3)-dimensional complete stable minimal hypersurface in Rn+1 must have only one end. Later, Li and Wang [17,18] generalized this topological result to minimal hypersurfaces with finite index in Euclidean space and stable minimal hypersurfaces in a nonnegatively curved manifold. In this paper, motivated by the relationship between the space of harmonic functions and the geometry of submanifolds, we study the geometric structure of submanifolds by using the vanishing properties of p-harmonic functions and p-harmonic 1-forms with finite L q energy for some p > 1 and q > 0. We recall that the p Laplacian operator on a Riemannian manifold M is defined by  p u := div(|∇u| p−2 ∇u) 1, p

for any function u ∈ Wloc (M) and p > 1, which arises as the Euler–Lagrange operator associated with the p-energy functional  |∇u| p . E p (u) := M

Let E ⊂ M be an end of M. In other words, E is an unbounded connected component of M \  for a sufficiently large compact subset  ⊂ M with smooth boundary. As in usual harmonic function theory, we define the p-parabolicity and p-nonparabolicity of E as follows (see also [1,3,25]): Definition 1.1 An end E of the Riemannian manifold M is called p-parabolic if for every compact subset K ⊂ E  cap p (K , E) := inf |∇ f | p = 0, E

where the infimum is taken among all f ∈ end E is called p-nonparabolic.

Cc∞ (E)

such that f ≥ 1 on K . Otherwise, the

In [3], Buckuley and Koskela gave a volume estimate of p-parabolic ends and pnonparabolic ends in terms of the first eigenvalue of the p Laplacian. Recently, Batista et al. [1] proved that if E is an end of a complete Riemannian manifold and satisfies a Sobolev-type inequality, then E must either have finite volume or to be p-nonparabolic (see also [25]). Let M an n-dimensional complete orientable noncompact (not necessarily minimal) hypersurface in a complete manifold N of nonnegative sectional curvature. Assume further that M satisfies the δ-stability inequality (1.1) for some 0 < δ ≤ 1. We note that whenever M is a δ-stable minimal hypersurface in N , the δ-stability inequality holds on M. Kim and Yun

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[14] proved that if δ = 1 and 2 ≤ n ≤ 4, then there is no nontrivial L 2 harmonic 1-form on M. Recently, the authors [11] generalized their result to hypersurfaces with 0 < δ ≤ 1 and 2 ≤ n ≤ 6. In Sect. 2, we prove that if a low-dimensional complete orientable noncompact hypersurface in a complete manifold N with nonnegative sectional curvature supports the δ-stability inequality (1.1) , then M must have only one p-nonparabolic end as follows (see Corollary 3.6): Theorem Let M n be an n-dimensional complete orientable noncompact hypersurface in a complete manifold N of nonnegative sectional curvature for 2 ≤ n ≤ 6. Assume that the √ δ-stability inequality holds on M for p 8n−1 < δ ≤ 1. Then M has only one p-nonparabolic √ end for p ≥ 1 + n − 1. The similar vanishing results can be obtained by imposing that the minimal hypersurface M has sufficiently small L n norm of the second fundamental form instead of assuming that M is stable. If M ⊂ Rn+1 is a complete minimal hypersurface with sufficiently small L n norm of the second fundamental form, then a vanishing theorem for L 2 harmonic 1-forms holds on M [23,27,37]. It turned out that these vanishing results are still valid for more general cases [10,28,29,31]. Moreover, Cavalcante et al. [5] extended the previous results to n-dimensional complete noncompact submanifolds in a complete simply connected manifold N with sectional curvature K N satisfying −k 2 ≤ K N ≤ 0 for some constant k in terms of the traceless second fundamental form φ. They proved that if such submanifold M has a sufficiently small L n norm φ n and if the first eigenvalue λ1 (M) of the Laplacian is bigger than some constant depending on n, k, and the infimum of the mean curvature, then there is no nontrivial L 2 harmonic 1-form on M. Recently, the authors [11] obtained a similar nonexistence theorem for L 2 harmonic 1-forms on complete noncompact submanifolds under the same assumptions as in [5] except the condition on the lower bound of the fundamental tone λ1 (M) depending on φ n . In Sect. 3, we obtain a rigidity theorem of the p-nonparabolic ends of complete noncompact submanifolds in terms of φ n and the p-fundamental tone λ1, p (M) for p Laplacian on M, which gives a generalization of the previous rigidity theorems for usual harmonic functions and forms. If M is an n-dimensional complete noncompact submanifold in a complete simply connected manifold N with sectional curvature K N satisfying that −k 2 ≤ K N ≤ 0 for some constant k and if L n norm of the traceless second fundamental form φ of M is greater than some constant depending only on k, n, φ n , then M has only one p-nonparabolic end (see Theorem 4.2). Moreover, we also obtain a lower bound of p-fundamental tone of the p Laplace operator on complete submanifolds, which is a generalization of the results obtained by Bessa and Montenegro [2], Cheung and Leung [8], and the second author [30].

2 Vanishing property of p-harmonic functions Let M be a complete Riemannian manifold and  ⊂ M be an open subset of M. We recall 1, p that a function u ∈ Wloc () is said to be (weakly) p-harmonic if  p u := div(|∇u| p−2 ∇u) = 0 in the weak sense, i.e.,

 

|∇u| p−2 ∇u, ∇ψ = 0

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for all ψ ∈ W0 (). In general, it is known that the regularity of (weakly) p-harmonic 1,α (see [19,33,36] and the references therein). Moreover, function u is not better than Cloc 2, p 2,2 it is also known that u ∈ Wloc if p ≥ 2; u ∈ Wloc if 1 < p < 2 by Tolksdorf [33]. In fact, any nontrivial (weakly) p-harmonic function u on M is smooth away from the set S := {x ∈ M : ∇u(x) = 0} (see [21,36] for example). Theorem 2.1 Let M n (n ≥ 2) be an n-dimensional complete noncompact Riemannian manifold. Assume that the Ricci curvature of M satisfies that for some constant a ∈ R Ric M (x) ≥ −aτ (x), x ∈ M where τ (x) satisfies the weighted Poincaré inequality   δ τ ϕ 2 ≤ |∇ϕ|2 , ∀ϕ ∈ C0∞ (M)

(Pτ,δ )

1,α 2,2 (M) ∩ Wloc (M) and u is smooth outside S = {x ∈ for some δ > 0. For p ≥ 2, let u ∈ Cloc M : ∇u(x) = 0}. Suppose that on M \ S we have   |du|( + aτ )(|du| p−1 ) ≥ b|du| p−2 |∇|du||2 − d ∗ d(|du| p−2 du), du , (2.2)

where b is a constant and d ∗ denotes the adjoint operator of d. If the constants a and b satisfy that 1+

a 4δ(b + p − 1) (b − 1) > 0 and a < δ p2

and if |du| ∈ L 2β (M) for p δ ≤β< 2 a



 1+

1+

a (b − 1) , δ

then the function u is constant. Proof Choose any number q ≥ 0 and a smooth nonnegative function ϕ with a compact support in M+ := M \ S. Multiplying both sides of the inequality (2.2) by |du|q ϕ 2 and integrating over M gives   q+1 2 p−1 |du| ϕ |du| +a τ |du| p+q ϕ 2 M+ M+  ≥b |du|q+ p−2 |∇|du||2 ϕ 2 M  +  ∗  d d(|du| p−2 du), |du|q du ϕ 2 , − M+

where ϕ ∈ C0∞ (M+ ) ⊂ C0∞ (M). Therefore,     τ |du| p+q ϕ 2 ∇(|du|q+1 ϕ 2 ), ∇|du| p−1 − a M+ M+     ≤−b |du|q+ p−2 |∇|du||2 ϕ 2 + d(|∇| p−2 du), d(|du|q ϕ 2 du) . M+

123

M+

(2.3)

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On the other hand,     q+1 2 p−1 ∇(|du| ϕ ), ∇|du| = (q + 1)( p − 1) |du|q+ p−2 |∇|du||2 ϕ 2 M+ M+  + 2( p − 1) ϕ|du| p+q−1 ∇ϕ, ∇|du| .

(2.4)

M+

Since |d(ϕω)| = |dϕ ∧ ω| ≤ |dϕ||ω| for any smooth function ϕ : M → R and any closed 1-form ω (see also Lemma 13 in [24]), we have    d(|du| p−2 du), d(|du|q ϕ 2 du) M+  ≤ |∇(|du| p−2 )| · |du| · |∇(|du|q ϕ 2 )| · |du| M+   = ( p − 2)q |du| p+q−2 |∇|du||2 ϕ 2 + 2( p − 2) |du| p+q−1 ϕ ∇ϕ, ∇|du| . M+

M+

(2.5) By our assumption, we have the following weighted Poincaré inequality (Pτ,δ ) on M+ :   |∇ϕ|2 ≥ δ τ ϕ2 M+

M+

for any ϕ ∈ C0∞ (M+ ). Replacing ϕ by |du| p+q ϕ in the above inequality, we obtain     2 p+q



δ τ |du| p+q ϕ 2 ≤

∇ |du| 2 ϕ

M+

M+

 p+q 2 ≤ (1 + ε) |du| p+q−2 |∇|du||2 ϕ 2 2 M+   1 + 1+ |du| p+q |∇ϕ|2 . ε M+ 

(2.6)

Combining the inequalities (2.3), (2.4), (2.5), and (2.6), we see that for any ε > 0   a p+q 2 b − 1 + ( p + q) − (1 + ε) |du| p+q−2 |∇|du||2 ϕ 2 δ 2 M+    1 a 1+ ≤ |du| p+q |∇ϕ|2 + 2(2 p − 3) ϕ|du| p+q−1 |∇ϕ||∇|du||. δ ε M+ M+

(2.7)

Since 1 2ϕ|du| p+q−1 |∇ϕ||∇|du|| ≤ ε|du| p+q−2 |∇|du||2 + |du| p+q |∇ϕ|2 ε by (2.7), we have Cε



 M+

|du| p+q−2 |∇|du||2 ϕ 2 ≤ Dε

|du| p+q |∇ϕ|2 M+

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for any ϕ ∈ C0∞ (M+ ), where Cε = b − 1 + ( p + q) − (1 + ε) and a Dε = δ



1 1+ ε

a δ



+

p+q 2

2 − (2 p − 3)ε

2p − 3 . ε

Choose a sufficiently small ε > 0 in the above. Then there exists a positive constant C = C(ε, n, δ, p, q) such that for any ϕ ∈ C0∞ (M+ )   |du| p+q−2 |∇|du||2 ϕ 2 ≤ C |du| p+q |∇ϕ|2 (2.8) M+

M+

provided b − 1 + ( p + q) −

a δ



p+q 2

2 > 0.

(2.9)

Applying a variation of the Duzaar–Fuchs cutoff method (see also [9,22,34]), we shall show that (2.8) holds for every ψ ∈ C0∞ (M). We define   |du| η ε = min ,1

ε for ε˜ > 0. Let ϕε˜ = ψ 2 ηε˜ . It is easy to see that ϕε˜ is a compactly supported continuous function and ϕε˜ = 0 on M \ M+ . By [22], we know that ϕε˜ ∈ W01,2 (M+ ). As ε˜ → 0, ηε˜ → 1 pointwisely in M+ . Using the similar argument as in [34], we can replace ϕ by ϕε˜ in (2.8) and obtain  ψ 4 (ηε˜ )2 |du| p+q−2 |∇|du||2 M+   ≤ 6C |du| p+q |∇ψ|2 ψ 2 (ηε˜ )2 + 3C |du| p+q |∇ηε˜ |2 ψ 4 . (2.10) M+

Observe that

M+



 M+

|du| p+q |∇ηε˜ |2 ψ 4 ≤ ε˜ p+q−2

M+

|∇|du||2 ψ 4 χ{|du|≤˜ε}

(2.11)

and the right-hand side vanishes by dominated convergence as ε˜ → 0, because |∇|du|| ∈ 2 (M). Letting ε˜ → 0 and applying Fatou lemma to the integral on the left-hand side and L loc dominated convergence to the first integral in the right-hand side of (2.10), we obtain   ψ 4 |du| p+q−2 |∇|du||2 ≤ 6C |du| p+q |∇ψ|2 ψ 2 , (2.12) M+

M+

where ψ ∈ C0∞ (M). Let β = following condition:

p+q 2

β2 −

123



p 2.

Then the inequality (2.9) is equivalent to the

2δ δ β − (b − 1) < 0. a a

(2.13)

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Moreover, it is easy to see that the inequality (2.13) is satisfied if and only if the assumption on a, b, and β in Theorem 2.1 is satisfied, that is,   p a a ≤β< 1 + 1 + (b − 1) , 2 δ δ a 4δ(b + p − 1) . 1 + (b − 1) > 0, and a < δ p2 Choose a nonnegative smooth function ψ such that 1 on B(R) ψ= 0 on M \ B(2R) and |∇ψ| ≤

2 R.

Then the inequality (2.12) implies   4C |du| p+q−2 |∇|du||2 ≤ 2 |du|2β . R M+ M+

Letting R → ∞, we see that |du| is constant on each connected component of M+ , since 1,α |du| ∈ L 2β (M). Note that u ∈ Cloc (M) and du = 0 on ∂ M+ . Thus, du = 0 on each connected component of M+ provided ∂ M+  = ∅, which is a contradiction. It follows that M+ = M, and hence, |du| is a nonzero constant on M. Since M satisfies the weighted Poincaré inequality, M must have infinity volume. Therefore, we obtain du = 0 since |du| ∈ L 2β (M). This shows that u is constant, which completes the proof.   In order to study p-harmonic functions, we introduce the following Kato-type inequality, which can be regarded as a refinement of Kato-type inequality in [12]. Lemma 2.2 Let u be a p-harmonic function on an n-dimensional Riemannian manifold M for p ≥ 2. Outside the singular set S = {x ∈ M : du(x) = 0}, we have  κp |∇(|du| p−2 du)| ≥ 1 + (2.14) |∇|du| p−1 |2 , ( p − 1)2 where  ( p − 1)2 κ p := min 1, . n−1 

Proof It is well known that (2.14) holds for p = 2. Thus, we may assume p > 2. For any fixed point x ∈ M \ S, choose a local orthonormal frame {e1 , . . . , en } and its dual frame {θ1 , . . . , θn } on a neighborhood of x such that ∇ei e j (x) = 0, u 1 (x) = du(e1 )(x) = |∇u|(x) = |du|(x)  = 0 and du(ei )(x) = 0 for i ≥ 2. At the fixed point x ∈ M, we have ∇e j |∇u| = ∇ j |∇u| =

n  ui ui j i=1

|∇u|

= u1 j

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for 1 ≤ i, j ≤ n. Therefore, |∇(|du| p−2 du)|2 − |∇|du| p−1 |2 =

n 

 2 |du|2( p−2) ( p − 2)∇i (ln |∇u|)u j + u i j

i, j=1



n 

|du|2( p−2) {( p − 2)∇i (ln |∇u|)|∇u| + ∇i |∇u|}2

i=1

=

n 

 2 |du|2( p−2) ( p − 2)∇i (ln |∇u|)u j + u i j

i, j=1



n 

|du|2( p−2) {( p − 2)∇i (ln |∇u|)u 1 + u 1i }2

i=1



n 

|du|2( p−2) {( p − 2)∇1 (ln |∇u|)u i + u 1i }2

i≥2

+

n 

|du|2( p−2) {( p − 2)∇i (ln |∇u|)u i + u ii }2

i≥2

≥|du|2( p−2)



u 21i

i≥2

⎛ ⎞2 n |du|2( p−2) ⎝ + [( p − 2)∇i (ln |∇u|)u i + u ii ]⎠ . n−1 i≥2

(2.15) Since u is a p-harmonic function, we see that 0 = d ∗ (|du| p−2 du) = −|du| p−2

n 

[( p − 2)∇i (ln(|∇u|))u i + u ii ] .

(2.16)

i=1

Combining (2.15) with (2.16), we have |∇(|du| p−2 du)|2 − |∇|du| p−1 |2 ≥ |du|2( p−2)



u 21i

i≥2

|du|2( p−2) + (( p − 2)∇1 (ln |∇u|)u 1 + u 11 )2 n−1  ( p − 1)2 ≥ |du|2( p−2) u 21i + |du|2( p−2) u 211 n−1 i≥2   n  ( p − 1)2 ≥ min 1, u 21i |du|2( p−2) n−1 κp = |∇|du| p−1 |2 . ( p − 1)2

i=1

 

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Remark 2.1 As p ≥ 2, we have ( p − 1)2 1 ≥ , n−1 n−1 1 . Therefore, we recovered the following Kato-type inequality which implies that κ p ≥ n−1 which was obtained by Han and Pan [12]:  1 |∇(|du| p−2 du)| ≥ 1 + |∇|du| p−1 |2 . (n − 1)( p − 1)2

In particular, if we put δ = 1 in Theorem 2.1, that is, the stability inequality holds on M, we obtain the following result applying Theorem 2.1 and Kato-type inequality (2.14). Theorem 2.3 Let M n (n ≥ 2) be a complete noncompact Riemannian manifold. Assume that the Ricci curvature of M satisfies that for some constant a ∈ R Ric M (x) ≥ −aτ (x), ∀x ∈ M and assume that τ (x) satisfies the weighted Poincaré inequality (Pτ,1 ), where (Pτ,δ ) is the same as in Theorem 2.1. If one of the following conditions holds: (i) κ p = 1 and a < 4p or   4( p−1+κ p ) 1 , (ii) κ p < 1 and a < min 1−κ p , p2 p 2 ≤ β < 4( p−1+κ p ) , then p2

then every p-harmonic function u with finite L 2β energy is constant for √ 1+ 1+a(b−1) . a

Here κ p is the same as in Lemma 2.2. In particular, if a < every p-harmonic function u with finite L p energy is constant.

Proof Consider a p-harmonic L 2β function u on M. Recall that p-harmonic function u is 1,α 2,2 smooth away from the set S := {x ∈ M : ∇u(x) = 0}. Moreover, u ∈ Cloc (M) ∩ Wloc (M). p−2 Applying the Bochner–Weitzenböck formula for |du| du, we obtain 1 (||du| p−2 du|2 ) = |∇(|du| p−2 du)|2 2   − (d ∗ d + dd ∗ )|du| p−2 du, |du| p−2 du + Ric M (|du| p−2 du, |du| p−2 du). Thus, by assumption on Ricci curvature and the Kato-type inequality (2.14), we have κp |du| p−1 (|du| p−1 ) ≥ |∇(|du| p−1 )|2 ( p − 1)2   − d ∗ d(|du| p−2 du), |du| p−2 du − aτ |du|2( p−1) . Here we used the fact that d ∗ (|du| p−2 du) = 0, since u is p-harmonic. Therefore,   |du|( + aτ )(|du| p−1 ) ≥ b|du| p−2 |∇|du||2 − d ∗ d(|du| p−2 du), du ,

(2.17)

where b = κ p . The assumption on a and p guarantees that the hypothesis of Theorem 2.1 is satisfied. Hence, applying Theorem 2.1, we obtain that every p-harmonic function u with finite L 2β energy is constant. Furthermore, we note that the condition a < 4(b+pp−1) is 2  p 2 p b−1 equivalent to 2 − a − a < 0. Therefore, the inequality (2.13) in the proof of Theorem 2.1 is satisfied. Using the argument in the proof of Theorem 2.1, we obtain that every pharmonic function with finite p energy is constant, which completes the proof.  

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Remark 2.2 Pigola et al. obtained a Liouville-type theorem for p-harmonic functions with finite p energy in Theorem 5.1 of [25]. We note that Theorem 2.3 improves their result even in case 2β = p. Remark 2.3 In Theorem 1.1 of [6], Chang, Chen, and Wei obtained a vanishing theorem for p-harmonic functions with finite p energy ( p ≥ 2) on complete noncompact Riemannian manifold supporting the weighted Poincaré inequality (Pτ,1 ) and satisfying

where a <

4( p−1+κ) p2

Ric M (x) ≥ −aτ (x), x ∈ M    2 1 . It follows that and κ = max n−1 , min 1, ( p−1) n  κ ≤ max

  1 ( p − 1)2 = κp. , min 1, n−1 n−1

Here we used the condition that p ≥ 2 in the last equality. Therefore, the range of value of a we obtained is better than that in [6]. It is also worth noting that when p is sufficiently large, that is, ( p − 1)2 ≥ n, the range of a in Theorem 2.3 is the same that in [6]. Note that, in [6], the authors assumed that u is p-harmonic and u ∈ C 3 (M) and a < a1 :=

˜ 4(Q − 1 + κ p + b) , Q2

where b˜ := min{0, ( p − 2)(Q − p)}. Under an additional assumption on the value of Q, Chang et al. obtained their vanishing result (Theorem 1.2 in [6]). We would like to mention that Theorem 1.2 in [6] is a consequence of Theorem 2.1 when Q > 2 or p ≥ 4. Indeed, we recall the conditions (2) and (3) in Theorem 1.2 in [6].   (2) p = 4, Q > max 1, 1 − κ − b˜ , (3) p > 2, p  = 4 and either     ( p − 4)2 n κ < q ≤ min 2, p − ; max 1, p − 1 − p−1 4( p − 2)   or q > max 2, 1 − κ − b˜ . We note that, due to the proof of Theorem 2.1, if p + q ≥ 2, the right-hand side of (2.11) tends to zero when ˜ goes to zero. Hence, we can require β ≥ 1 in Theorem 2.1. Observe that since b˜ ≤ 0, we have a1 ≤

4(Q − 1 + κ p ) . Q2

This implies β = Q/2 satisfies the condition (2.13). Therefore, by the proof of Theorem 2.1, we infer that if u is weakly p-harmonic function with finite L Q energy then u is constant. On the other hand, if p ≥ 4, we see that   Q > max 1, 1 − κ − b˜ ≥ 1 − κ − min {0, ( p − 2)(Q − p)} .

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If Q > p then Q > 2. By the above argument if u is weakly p-harmonic function with finite Q energy, then u is constant. Assume that Q ≤ p then Q > 1 − κ − ( p − 2)(Q − p) ≥ 1 − κ − 2(Q − 4) ≥ 8 − 2Q κ Hence, Q > 8/3. Moreover, if p ≥ 4, then p − 1 − p−1 > 2. This implies that there is no Q such that     κ ( p − 4)2 m max 1, p − 1 − < q ≤ min 2, p − . p−1 4( p − 2)

Consequently, when p ≥ 4, the conditions (2) and (3) show that Q > 2. In conclusion, we have proved the following result which is a refined version of Theorem 1.2 in [6]. Corollary 2.4 ([6]) Let M n (n ≥ 2) be an n-dimensional complete noncompact Riemannian manifold satisfying the weighted Poincaré inequality (Pτ,1 ) and Ric M (x) ≥ −aτ (x) for all x ∈ M, where a is a constant such that for Q ≥ 2 a < a1 :=

4(Q − 1 + κ p ) . Q2

If u is a weakly p-harmonic function ( p ≥ 2) with finite L Q energy then u is constant. We remark that Theorem 1.2 in [6] requires that u ∈ C 3 (M) and Q is bounded by some constant. Corollary 2.4 is stronger than Theorem 1.2 in [6], when p ≥ 4 or Q > 2.

3 Applications to complete submanifolds in a Riemannian manifold In this section, we obtain some geometric applications to complete submanifolds in a Riemannian manifold using Theorems 2.1 and 2.3. First we need the following useful Ricci curvature estimate for submanifolds, which was obtained by Leung [15]. Lemma 3.1 [15] Let M be an n-dimensional submanifold in a Riemannian manifold N with sectional curvature K N satisfying that K ≤ K N where K is a constant. Then the Ricci curvature Ric M of M satisfies  n−1  √ 1  Ric M ≥ (n − 1)K + 2 2(n − 1)|H |2 −(n − 2) n − 1|H | n|A|2 − |H |2 − |A|2 . n n In particular, if the sectional curvature of the ambient space is nonnegative, we see that K = 0 in Lemma 3.1. Moreover, a straightforward computation (see [11] for more details) shows that √  √ (n − 2)2 n − 1 2(n − 1)|H |2 − (n − 2) n − 1|H | n|A|2 − |H |2 ≥ −n 2 |A|2 √ . 2n( n − 1 + 1)2 Using this inequality and Lemma 3.1, we obtain the following estimate. Proposition 3.2 [11] Let M n be a complete orientable noncompact submanifold in a Riemannian manifold N of nonnegative sectional curvature. Then √ n−1 2 |A| . Ric M ≥ − (3.1) 2

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Let M be an n-dimensional complete orientable noncompact submanifold in a Riemannian √ manifold N of nonnegative sectional curvature. If we put a = n−1 and τ (x) = |A|2 (x), 2 then Proposition 3.2 tells us that the Ricci curvature condition on M in Theorem 2.1 is satisfied. Moreover, if we assume that δ = 1, that is, the stability inequality holds on M, then the weighted Poincaré inequality (Pτ,δ ) holds on M. If we further assume that b = κ p and β = 2p (i.e., q = 0), then the condition (2.13) is satisfied, which is equivalent to the following √ n−1 2 κp − 1 + p − (3.2) p > 0, 8   2 . We see that these assumptions on a, b, and β meet the requirewhere κ p = min 1, ( p−1) n−1 ment in Theorem 2.1. From this observation, we obtain the following vanishing result of p-harmonic L p function on complete noncompact hypersurfaces. Theorem 3.3 Let M n be an n-dimensional complete orientable noncompact hypersurface in a complete manifold N of nonnegative sectional curvature. Assume that the stability 8 inequality holds on M. If 2 ≤ n ≤ 6 and 2 ≤ p < √n−1 , then there is no nontrivial p p-harmonic L function on M . Proof It suffices to show that the condition (3.2) is satisfied under our assumption. To see 2 this, we divide into two cases: (i) κ p = 1 and (ii) κ p = ( p−1) n−1 . √ √ 2 (i) If κ p = 1, then p ≥ 1 n − 1. The inequality (3.2) becomes p − n−1 8 p > 0. Thus, 1+

√ 8 n−1≤ p < √ , n−1

which holds for 2 ≤ n ≤ 6. √ 2 (ii) If κ p = ( p−1) n−1 , then p < 1 + n − 1. The inequality (3.2) gives √ ( p − 1)2 n−1 2 −1+ p− p > 0, n−1 8  √ √ 4 3+

 25−10 5 √ 5 5−8

 5.816 if n = 6. which holds for all p ≥ 2 if 2 ≤ p ≤ 5 and holds for p < √ Thus, in this case, we see that (3.2) holds for p < 1 + n − 1 and 2 ≤ n ≤ 6. Therefore, from the case (i) and (ii), it follows that (3.2) is satisfied for 2 ≤ n ≤ 6 and 8 2 ≤ p < √n−1 , which gives the conclusion.  

Furthermore, when p = 2 and β = 1 in Theorem 2.1, one immediately obtains the following consequence, which was proved by the authors [11]. Corollary 3.4 [11] For 2 ≤ n ≤ 6, let M n be a complete orientable noncompact hypersurface in a complete manifold√N with nonnegative sectional curvature. If δ-stability inequality holds on M for some (n−1)2n n−1 < δ ≤ 1, then there is no nontrivial L 2 harmonic 1-form on M. In order to give another geometric application of Theorem 2.1, we recall the following result about the existence of p-harmonic function on a Riemannian manifold. Theorem 3.5 [6,25] Let M be a Riemannian manifold with at least two p-nonparabolic ends. Then, there exists a nonconstant, bounded p-harmonic function u ∈ C 1,α (M) for some α > 0 such that |∇u| ∈ L p (M).

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As an immediate consequence, we obtain the following: in Theorem 3.6 Let M n be an n-dimensional complete orientable noncompact hypersurface √ p n−1 a complete manifold N of nonnegative sectional curvature for 2 ≤ n ≤ 6. For 8 < δ ≤ 1, assume that the √ δ-stability inequality holds on M. Then M has only one p-nonparabolic end for p ≥ 1 + n − 1. Proof Suppose that M has at least two p-nonparabolic ends. From Theorem 3.5, it follows that M admits a nonconstant, bounded√p-harmonic function u ∈ C 1,α (M) for some α > 0 p such that |∇u| ∈ L p (M). Put a = n−1 2 , b = κ p , and β = 2 in Theorem 2.1. Since √ √ p n−1 < δ ≤ 1 and p ≥ 1 + n − 1, the constants a, b, p, n, β, δ satisfy the assumption in 8 Theorem 2.1. Thus, we see that u is constant by Theorem 2.1, which gives a contradiction. Therefore, we obtain the conclusion.   For higher-codimensional cases, Q. Wang [35] introduced the concept of super-stability of minimal submanifolds. Motivated by this, we define δ super-stability of submanifolds as follows: Definition 3.7 Let M n be an n-dimensional orientable submanifold in the (n + k)dimensional Euclidean space Rn+k . It is called that the δ super-stability inequality holds on M for 0 < δ ≤ 1 if   δ |A|2 ϕ 2 ≤ |∇ϕ|2 M

M

for any compactly supported Lipschitz function ϕ on M. In particular, when k = 1 and δ = 1, the concept of super-stability is the same as the usual definition of stability. Using the same argument as in the proof of Theorem 2.1, we have the following: Theorem 3.8 Let M n be an n-dimensional complete orientable noncompact submanifold in Rn+k for 2 ≤ n ≤ 6. If the δ super-stability inequality holds on M for some 0 < δ ≤ 1, then there is no nontrivial p-harmonic function with finite L 2β energy on M provided    4δβ 2δ ( p − 1)2 2 − 1 < 0. β −√ −√ min 1, n−1 n−1 n−1 Consequently, M has only one p-nonparabolic end if the stability inequality holds on M and 8 if 2 ≤ n ≤ 6 and 2 ≤ p < √n−1 .

4 Uniqueness of p-nonparabolic ends In this section, we prove the rigidity of p-nonparabolic ends of complete noncompact submanifolds in a complete simply connected manifold with nonpositive sectional curvature. We begin with the following Sobolev inequality. Lemma 4.1 [13] Let M n (n ≥ 3) be an n-dimensional complete submanifold in a complete simply connected manifold with nonpositive sectional curvature. Then for any f ∈ W01,2 (M) we have    n−2   2n n | f | n−2 dv ≤ CS |∇ f |2 + |H |2 f 2 dv, (4.3) M

M

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where C S is the Sobolev constant which depends only on n and H denotes the mean curvature vector of M. Let M n be an n-dimensional submanifold in an (n +k)-dimensional Riemannian manifold N n+k . For x ∈ M, let {e1 , . . . , en+k } be a local orthonormal frame such that {e1 , . . . , en } is an orthonormal basis of the tangent space Tx M and {en+1 , . . . , en+k } is a orthonormal basis of the normal space N x M. For each α ∈ {n +1, . . . , n +k}, a shape operator Aα : Tx M → Tx M is defined by

Aα X, Y = ∇¯ X Y, eα , where X, Y are tangent vector fields and ∇¯ denotes the Levi-Civita connection on N . Then the mean curvature vector H is defined by H=

n+k 

(trace Aα )eα .

α=n+1

and a linear map φα : Tx M → Tx M is defined by

φα X, Y = Aα X, Y − X, Y H, eα . The traceless second fundamental form φ : Tx M × Tx M → N x M is defined by n+k 

φ(X, Y ) =

φα X, Y eα .

α=n+1

Theorem 4.2 Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in a complete simply connected manifold N with sectional curvature K N satisfying that −k 2 ≤ K N ≤ 0 for a real number k. Assume that the traceless second fundamental form φ satisfies for any p ≥ 2 φ n <  := min{1 , 2 }, where the constants 1 and 2 are defined by 1 , n(n − 1)C S ⎛ ⎞ 1 ⎝ 4n(κ p + p − 1) (n − 2)2 (n − 2)2 ⎠ 2 = 2 (n, p) := √ + − , p 2 (n − 1) 16n(n − 1) 16n(n − 1) CS 1 = 1 (n) := √

  2 . In case k  = 0, assume and C S is the Sobolev constant in (4.3) and κ p = min 1, ( p−1) n−1 further that the fundamental tone λ1 (M) on M satisfies λ1 (M) >

k 2 (n − 1) , g( φ n )

where the function g(t) is defined by g(t) :=

4(κ p + p − 1) n − 2  n−1 − n(n − 1)C S t − CS t 2. 2 2 p 2n n

Then M has only one p-nonparabolic end.

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Proof Suppose that M has at least two p-nonparabolic ends. Then it follows from Theorem 3.5 that there exists a nonconstant bounded p-harmonic function u ∈ C 1,α (M) with |∇u|∈ L p (M). The differential ω := du is obviously p-harmonic 1-form on M satisfying that M |ω| p < ∞. Let us denote p-harmonic 1-form and its dual p-harmonic vector field by ω. As before, we consider the set M+ := {x ∈ M : |∇u(x)| > 0}. On M+ , using (2.14), we have the following Kato-type inequality for a p-harmonic function:  κp |∇(|du| p−2 du)|2 ≥ 1 + |∇|du| p−1 |2 . ( p − 1)2 Applying Bochner formula for |du| p−2 du and the above Kato-type inequality, we see that on M+ |ω| p−1 |ω| p−1 ≥ − (dd ∗ + d ∗ d)(|ω| p−2 ω), |ω| p−2 ω + |ω|2 p−4 Ric M (ω, ω) κp |∇|ω| p−1 |2 + ( p − 1)2 = − d ∗ d(|ω| p−2 ω), |ω| p−2 ω + |ω|2 p−4 Ric M (ω, ω) κp |∇|ω| p−1 |2 , + ( p − 1)2 since δ(|ω| p−2 ω) = 0. The Ricci curvature estimate (Lemma 3.1) shows that on M+  |H |2 p−1 p−1 ∗ p−2 p−2 2 |ω| |ω| ≥ − d d(|ω| ω), |ω| ω + (n − 1) − k |ω|2 p−2 n2 n − 2 n − 1 2 2 p−2 |φ| |ω| − n(n − 1)|φ||H ||ω|2 p−2 − n2 n κp + |∇|ω| p−1 |2 . (4.4) ( p − 1)2 Dividing both sides of the inequality (4.4) by |ω| p−2 on M+ ,  |H |2 2 |ω||ω| p−1 ≥ − d ∗ d(|ω| p−2 ω), ω + (n − 1) − k |ω| p n2 n − 2 n−1 2 p |φ| |ω| − n(n − 1)|φ||H ||ω| p − n2 n + κ p |ω| p−2 |∇|ω||2 . Fix a point x ∈ M and choose a geodesic ball Bx (R) of radius R centered at the point x. Choose a test function f ∈ C0∞ (M) satisfying that 0 ≤ f ≤ 1 on M, f ≡ 1 on Bx (R/2), f ≡ 0 on M \ Bx (R) and |∇ f | ≤ R1 . Multiplying both sides by f p and integrating over Bx (R) gives   f p |ω||ω| p−1 ≥ −

d(|ω| p−2 ω), d( f p ω) Bx (R)

Bx (R)

|H |2 2 |ω| p f p − k 2 n Bx (R)  n − 2 n(n − 1) − |φ||H ||ω| p f p n2 Bx (R) 



+ (n − 1)

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 n−1 − |φ|2 |ω| p f p n Bx (R)  + κp |ω| p−2 |∇|ω||2 f p Bx (R)





≥ −

d(|ω| p−2 ω), d( f p ω)

Bx (R)

|H |2 2 − k |ω| p f p n2 Bx (R)  n − 2 − n(n − 1) |φ||H ||ω| p f p n2 Bx (R)  n−1 − |φ|2 |ω| p f p n Bx (R)  + κp |ω| p−2 |∇|ω||2 f p . 



+ (n − 1)

Bx (R)

(4.5)

Since ω is a p-harmonic 1-form, we have dω = 0. Thus, we have |d(|ω| p−2 ω)| = |d|ω| p−2 ∧ ω| ≤ |∇|ω| p−2 ||ω| and |d( f p ω)| = |d f p ∧ ω| ≤ |∇ f p ||ω|, where we used the fact that |d(ϕω)| = |dϕ ∧ ω| ≤ |dϕ||ω|. Therefore, we obtain







p−2 p



d(|ω| ω), d( f ω) |d(|ω| p−2 ω)||d( f p ω)|



Bx (R) Bx (R)  ≤ |∇|ω| p−2 ||∇ f p ||ω|2 Bx (R)  = p( p − 2) f p−1 |∇ f ||ω| p−1 |∇|ω||. Bx (R)

(4.6)

Moreover, the left-hand side of the inequality (4.5) is given by   p p−1 f |ω||ω| =−

∇( f p |ω|), ∇|ω| p−1 Bx (R) Bx (R)   f p−1 |ω| ∇ f, ∇|ω| p−1 − f p ∇|ω|, ∇|ω| p−1 =− p Bx (R) Bx (R)  = − p( p − 1) f p−1 |ω| p−1 ∇ f, ∇|ω| Bx (R)  f p |ω| p−2 |∇|ω||2 − ( p − 1) Bx (R)  ≤ p( p − 1) f p−1 |ω| p−1 |∇ f ||∇|ω|| Bx (R)  f p |ω| p−2 |∇|ω||2 , (4.7) − ( p − 1) Bx (R)

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where we used the divergence theorem in the first equality. We remark that both f and |ω| vanish along the boundary of Bx (R). Note that p2 |ω| p−2 |∇|ω||2 4

p

|∇|ω| 2 |2 = and p

p

|ω| 2 ∇|ω| 2 =

p p−1 ∇|ω|. |ω| 2

Applying the above two identities to (4.6) and (4.7), the inequality (4.5) becomes  2( p − 1)

p

Bx (R)

p

f p−1 |ω| 2 |∇ f ||∇|ω| 2 | −

4( p − 1) p2



p

Bx (R)

f p |∇|ω| 2 |2

  |H |2 2 |ω| p f p f p−1 |ω| |∇ f ||∇|ω| | + (n − 1) − k n2 Bx (R) Bx (R)  n − 2 − n(n − 1) |φ||H ||ω| p f p n2 Bx (R)   p 4κ p n−1 − |φ|2 |ω| p f p + 2 f p |∇|ω| 2 |2 . n p Bx (R) Bx (R) 

p 2

≥ − 2( p − 2)

p 2

Thus, we obtain 

 p 4( p − 1) 4κ p + f p |∇|ω| 2 |2 2 2 p p Bx (R) Bx (R)     |H |2 n − 2 ≥(n − 1) − k 2 |ω| p f p − n(n − 1) |φ||H ||ω| p f p n2 n2 Bx (R) Bx (R)  n−1 − |φ|2 |ω| p f p . (4.8) n Bx (R) 

2(2 p − 3)

p

p

f p−1 |ω| 2 |∇ f ||∇|ω| 2 | −

Using the Schwarz inequality for positive real numbers α and β which will be chosen later, we have for any p ≥ 2  2

p

Bx (R)



p

f p−1 |ω| 2 |∇ f ||∇|ω| 2 | ≤ α  ≤α

Bx (R) Bx (R)

 p 1 f 2 p−2 |∇|ω| 2 |2 + |ω| p |∇ f |2 α Bx (R)  p 1 f p |∇|ω| 2 |2 + |ω| p |∇ f |2 (4.9) α Bx (R)

and 

 2

Bx (R)

|φ||H ||ω| p f p ≤ β

Bx (R)

|H |2 |ω| p f p +

1 β

 Bx (R)

|φ|2 |ω| p f p .

(4.10)

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Therefore, it follows from the inequalities (4.8), (4.9), and (4.10) that   p 2p − 3 f p |∇|ω| 2 |2 + |ω| p |∇ f |2 α(2 p − 3) α Bx (R) Bx (R)    n−1 n−1 ≥ 2 |H |2 |ω| p f p − |φ|2 |ω| p f p − k 2 (n − 1) |ω| p f p n n Bx (R) Bx (R) B (R) x    n − 2 1 2 p p 2 p p − n(n − 1) β |H | |ω| f + |φ| |ω| f 2n 2 β Bx (R) Bx (R)   p 4( p − 1) 4κ p + + f p |∇|ω| 2 |2 2 2 p p Bx (R)    n − 1 β(n − 2)  2 p p 2 n(n − 1) = − |H | |ω| f − k (n − 1) |ω| p f p n2 2n 2 Bx (R) Bx (R)   n − 2 n−1 − n(n − 1) + |φ|2 |ω| p f p 2βn 2 n Bx (R)   p 4( p − 1) 4κ p + + 2 f p |∇|ω| 2 |2 . (4.11) p2 p Bx (R) On the other hand, applying the Hölder inequality and the Sobolev inequality (Lemma 4.1), 

 Bx (R)

|φ|2 |ω| p f p ≤ φ 2n

p

B (R)

p

2n

x 

≤ C S φ 2n

n−2 n

(|ω| 2 f 2 ) n−2 p

Bx (R)



p

|∇(|ω| 2 f 2 )|2 +

Bx (R)

|H |2 |ω| p f p ,

where C S is the Sobolev constant depending only on n. Since  

p

p p p p p 2

2 |∇(|ω| 2 f 2 )|2 =

f ∇|ω| 2 + |ω| 2 ∇ f 2

Bx (R) Bx (R)    p p 1 ≤ (1 + α) f p |∇|ω| 2 |2 + 1 + |ω| p |∇ f 2 |2 α Bx (R) Bx (R)   2  p p 1 ≤ (1 + α) f p |∇|ω| 2 |2 + |ω| p |∇ f |2 , 1+ 4 α Bx (R) Bx (R) we obtain



 Bx (R)

|φ|2 |ω| p f p ≤ C S φ 2n (1 + α) p2 + C S φ 2n 4  + C S φ 2n



p

Bx (R)

1+

Bx (R)

1 α

f p |∇|ω| 2 |2

 Bx (R)

|H |2 |ω| p f p .

|ω| p |∇ f |2 (4.12)

In case k  = 0, we need the following eigenvalue estimate for the Laplace–Beltrami operator:  2 B (R) |∇ϕ| (4.13) λ1 (M) ≤ λ1 (Bx (R))) ≤ x 2 Bx (R) ϕ

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for any function ϕ ∈ W01,2 (Bx (R)). Since u is p-harmonic, the regularity theory shows that 2,2 1,α 1,2 0,α (M) ∩ Cloc (M). Consequently, |ω| = |du| ∈ Wloc (M) ∩ Cloc (M). Using the fact u ∈ Wloc p p 1,2 ∞ 2 2 that f ∈ C0 (M) and supp( f ) ⊂ Bx (R), we see that |ω| f ∈ W0 (Bx (R)). Therefore, p p we substitute ϕ by |ω| 2 f 2 and apply the Schwarz inequality in the above inequality (4.13) to obtain   p p λ1 (M) |ω| p f p ≤ |∇(|ω| 2 f 2 )|2 Bx (R) Bx (R)    p p 1 ≤ (1 + α) f p |∇|ω| 2 |2 + 1 + |ω| p |∇ f 2 |2 α Bx (R) Bx (R)   2  p p 1 ≤ (1 + α) f p |∇|ω| 2 |2 + |ω| p |∇ f |2 1+ 4 α Bx (R) Bx (R) (4.14) for any α > 0. Combining the inequalities (4.11), (4.12), and (4.14), we obtain    p p 2 2 p p 2 A f |∇|ω| | + B |H | |ω| f ≤ C |∇ f |2 |ω| p , Bx (R)

Bx (R)

where the constants A, B, C are defined by A=

4(κ p + p − 1) − α(2 p − 3) − C S (1 + α) φ 2n p2

(4.15)

Bx (R)



n − 2 n−1 n(n − 1) + 2βn 2 n



k 2 (n − 1) (1 + α) λ1 (M)   n − 1 β(n − 2)  n−1 2 n−2 B= n(n − 1) − C S φ n n(n − 1) + − n2 2n 2 2βn 2 n   2 n − 2 1 2p − 3 n−1 p C= n(n − 1) + + C S φ 2n 1 + α 4 α 2βn 2 n  2 2 p k (n − 1) 1 + 1+ . 4λ1 (M) α −

From the following arithmetic–geometric mean inequality  1 β + C S φ 2n ≥ 2 C S φ n , (4.16) β √ we choose β = C S φ n to attain equality in the inequality (4.16). Thus, the assumption on φ n shows that B > 0. Furthermore, we choose the number α > 0 small enough satisfying that A > 0 by making use of the assumption on λ1 (M) and φ n . Note that C > 0 is automatically satisfied. Since |ω| ∈ L p (M), letting R → 0 in the inequality (4.15) shows that the right-hand side of the inequality (4.15) goes to zero, which implies that |∇|ω|| ≡ 0 and |H ||ω| ≡ 0 on every connected component of M+ . That is, |ω| ≡ constant and |H ||ω| ≡ 0 on every connected component of M+ . It follows that M+ = M. The reason is that, if M+ is a proper subset of M, then |ω| = 0 on ∂ M+ , which shows that |ω| = 0 on each connected component of M+ . This means that ω ≡ 0 on M. However, this is a contradiction since the function u is nonconstant. Therefore, we see that M+ = M. Now we suppose that |ω| is a nonzero constant. Since |H ||ω| ≡ 0, we see that M is a minimal submanifold. However, it is well known that the volume of a complete minimal submanifold in a Riemannian manifold of

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 nonpositive sectional curvature is infinite. Thus, it follows that M |ω| p = ∞, which is a contradiction to the assumption that ω is an L p harmonic 1-form. Hence, we see that ω ≡ 0, which completes the proof.   In particular, if the ambient space N is the Euclidean space, we obtain the following result. Corollary 4.3 Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in the Euclidean space R N . Assume that the traceless second fundamental form φ satisfies φ n <  = min{1 , 2 }, where the constants 1 and 2 are the same as in Theorem 4.2. Then M has only one p-nonparabolic end. Moreover, when the ambient space N has pinched nonpositive sectional curvature, we immediately obtain an upper bound of the fundamental tone of an n-dimensional complete noncompact submanifold with at least two p-nonparabolic ends and with φ n small enough. Corollary 4.4 Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in a complete simply connected Riemannian manifold N with sectional curvature K N satisfying that −k 2 ≤ K N ≤ 0 for some constant k  = 0. Assume that the traceless second fundamental form φ satisfies φ n <  = min{1 , 2 }, where the constants 1 and 2 are the same as in Theorem 4.2. If M has at least two p-nonparabolic ends for any p ≥ 2, then λ1 (M) ≤

k 2 (n − 1) , g()

where the function g is the same as in Theorem 4.2. Proof We argue by contraction. Suppose that λ1 (M) > guarantees that λ1 (M) >

k 2 (n−1) g() .

The assumption on φ n

k 2 (n − 1) k 2 (n − 1) > . g() g( φ n )

By Theorem 4.2, M must have only one p-nonparabolic end, which is a contradiction. Therefore, we obtain the conclusion.   The above Corollary 4.3 and Corollary 4.4 can be regarded as an extension of [5] and [11] into p-nonparabolicity of complete submanifolds.

5 Lower bound for the p-fundamental tone of the p Laplacian operator on complete submanifolds Let  be a domain in a Riemannian manifold M. The p-fundamental tone λ1, p () of  for the p Laplace operator on M is defined by   |∇u| p 1, p λ1, p () := inf  p : u ∈ W0 (), u  = 0 . u

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In the case where p = 2, we denote by λ1 () the 2-fundamental tone λ1,2 (). We have the following well-known lower bound for λ1 (M) on complete submanifolds in a Riemannian manifold [2,8,30]. Theorem [2,8,30] Let M be an n-dimensional complete submanifold in a complete simply connected Riemannian manifold N with sectional curvature K N satisfying K N ≤ −a 2 < 0 for a positive constant a. Let H denote the mean curvature vector of M in N . If |H | ≤ α for some nonnegative constant α < (n − 1)a, then we have [(n − 1)a − α]2 . 4 In this section, we give an extension of the above theorem to the p Laplace operator. Before stating our theorem, we need the following definition: λ1 (M) ≥

Definition 5.1 [2] Let  ⊂ M be a domain with compact closure in a smooth Riemannian manifold M. Let X () be the set of all smooth vector fields X on  with ||X ||∞ = sup |X | < ∞ and inf div X > 0. Define the constant c() by   inf div X : X ∈ X () . c() = sup ||X ||∞ We are now ready to prove the following result: Theorem 5.2 Let M be an n-dimensional complete noncompact submanifold in a complete simply connected Riemannian manifold N with sectional curvature K N satisfying K N ≤ −a 2 < 0 for a positive constant a. Let H denote the mean curvature vector of M in N . If |H | ≤ α for some nonnegative constant α < (n − 1)a, then we have  (n − 1)a − α p λ1, p (M) ≥ . p Proof Replacing f 2 X by f p X and applying the same argument as in the proof of Lemma 2.3 in [2], we obtain  c() p λ1, p () ≥ p for any domain  ⊂ M with compact closure. Fix a point x ∈ N \ M and let f (·) := dist(·, x) be the distance function from the point x on N . For r > 0, let  be a connected component of M ∩ B N (x, r ) and let X = ∇ f on this component, where B N (x, r ) denotes the geodesic ball centered at x with radius r in N . Repeating the computation as in Theorem 4.3 in [2], we obtain div X ≥ (n − 1)a coth(ar ) − h(x, r ) > 0 and |X | ≤ 1, where h(x, r ) := sup {|H (y)|; y ∈ φ(M) ∩ B N (x, r )}. Therefore,   (n − 1)a coth(ar ) − h( p, r ) p (n − 1)a − α p ≥ . λ1, p () ≥ p p We note that the injectivity radius at x is ∞. Thus, for any exhaustion { j }∞ j=1 of M, it follows that  (n − 1)a − α p λ1, p ( j ) ≥ . p Using the fact that λ1, p (M) = lim λ1, p ( j ), we obtain the conclusion. j→∞

 

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N. T. Dung, K. Seo

Acknowledgements The authors would like to thank the referee(s) for many helpful suggestion and comments. Owing to his/her help, we are able to obtain Theorems 2.1 and 4.2 which are much general than the previous version. Part of this paper was written while the first author visited Institut Fourier at Grenoble. He would like to express his sincere thanks to Professor Besson for hospitality and support. The second author was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF-2013R1A1A1A05006277).

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p-Harmonic functions and connectedness at infinity of ...

N. T. Dung, K. Seo for all ψ ∈ W. 1,p. 0. ( ). In general, it is known that the regularity of (weakly) p-harmonic function u is not better than C. 1,α loc (see [19,33,36] ...

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