The non-parabolicity of infinite volume ends

Let $M^m$, with $m\geq 3$, be an $m$-dimensional complete noncompact manifold isometrically immersed in a Hadamard manifold $\bar M$. Assume that the mean curvature vector has finite $L^p$-norm, for some $2\leq p\leq m$. We prove that each end of $M$ must either have finite volume or be non-parabolic.


Introduction
Let (M m , , ) be a complete noncompact Riemannian manifold without boundary. We recall that M is parabolic if it does not admit a non-constant positive superharmonic function. Otherwise, it is said to be non-parabolic. There exist equivalent definitions for parabolic manifolds (see for instance Theorem 5.1 of [8]). Let E ⊂ M be an end of M , that is an unbounded connected component of M − Ω, for some compact subset Ω ⊂ M . The property of parabolicity can be localized on each end of M . Namely, we say that an end E is parabolic (see Definition 2.4 of [10]) if it does not admit a harmonic function f : E → R satisfying: (1) f | ∂E = 1; (2) lim inf y→∞ y∈E f (y) < 1.
Otherwise, we say that E is a non-parabolic end of M . It is well known that M is non-parabolic if and only if it admits a non-parabolic end. Furthermore, ends with finite volume are parabolic (see for instance Section 14.4 of [8]). In this direction we recall the following result due to Li and Wang: Theorem A (Corollary 4 of [12] and Corollary 2.9 of [10]). Let E be an end of a complete manifold. Suppose that, for some constants ν ≥ 1 and C > 0, E satisfies a Sobolev-type inequality of the form for all compactly supported Sobolev function u ∈ W 1,2 c (E). Then E must either have finite volume or be non-parabolic. Moreover, in the case ν > 1, E must be non-parabolic.
Note that if a complete manifold M that satisfies a Sobolev inequality as in Theorem A with ν = 1 (that is just the Dirichlet Poincaré inequality) then the first eigenvalue λ 1 (M ) of the Laplace-Beltrami operator is positive, hence M must be non-parabolic (see Proposition 10.1 of [8]). Example 4.1 below exhibits a complete manifold that contains a finite volume end and that also satisfies a Sobolev inequality as in Theorem A with ν = 1.
Cao, Shen and Zhu [2] showed that if M m , with m ≥ 3, is a complete manifold then each end of M is non-parabolic provided that M can be realized as a minimal submanifold in a Euclidean space R n . The same conclusion also was obtained by Fu and Xu [7] provided that there exists an isometric immersion of M in a Hadamard manifoldM with finite total mean curvature, that is, the mean curvature vector field H of the immersion satisfies H L m (M ) < ∞. In the both cases, they observed that M admits a Sobolev-type inequality as in Theorem A with ν > 1.
Our main result states the following: Example 4.3 below exhibits an example of a complete non-compact hypersurface M m in R m+1 , with m ≥ 3, of finite volume and mean curvature vector with finite L p -norm, for all 2 ≤ p < m − 1. This example shows that Theorem 1.1 is not a consequence of Theorem A (except when p = m). Note also that the catenoids in R 3 are parabolic minimal surfaces whose ends have infinite area, which shows that the hypothesis m ≥ 3 is essential.
In the present paper we also give a unified proof of the following fact: Theorem B. Let x : M →M be an isometric immersion of a complete noncompact manifold M in a manifoldM with bounded geometry (i.e.,M has sectional curvature bounded from above and injectivity radius bounded from below by a positive constant). Let E be an end of M and assume that the mean curvature vector of x satisfies H L p (E) < ∞, for some m ≤ p ≤ ∞. Then E must have infinite volume.
The fact above was proved by Frensel [4] and by do Carmo, Wang and Xia [3] for the case that the mean curvature vector field is bounded in norm (the case p = ∞), by Fu and Xu [7] for the case that the total mean curvature is finite (the case p = m) and by Cheung and Leung [1] for the case that the mean curvature vector has finite L p -norm for some p > m. Since the cylinders of the form M m = S m−1 ×R, where S m−1 is the unit Euclidean (m− 1)-dimensional sphere, are examples of complete parabolic hypersurfaces in R m+1 we conclude that boundedness of the mean curvature vector does not imply that M admits a Sobolev-type inequality. Furthermore, for all m ≥ 3, we exhibit an example of a parabolic complete noncompact hypersurface M m in R m+1 such that the mean curvature vector has finite L p -norm, for all p > 2(m−1). These examples show that Theorem B is not a consequence of Theorem A.
Two questions arise in this paper: is there an example of a complete noncompact submanifold M m , with m ≥ 3, in a Euclidean space satisfying one of the conditions below?

Proof of Theorem 1.1
Choose r 0 > 0 so that the geodesic ball B r 0 ⊂ M of radius r 0 and center at some point ξ 0 ∈ M satisfies ∂E ⊂ B r 0 . For each r > r 0 , consider E r = E ∩ B r and let f r : E r → R be a solution of the Dirichlet Problem: It follows from the maximum principle that 0 < f r ≤ f s < 1 in E r , for all s ≥ r. Hence, by standard gradient estimates it follows that {f r } is an equicontinous family which converges uniformly on compact subsets, when r goes to infinity, to a function f : If f ≡ 1 then it follows from the maximum principle that lim inf x→E(∞) f (x) < 1, which shows that E is nonparabolic. Furthermore, it is well known that an end of finite volume is parabolic (see section 14.4 of [8]). Hence, to prove Theorem 1.1, it is sufficient to show the following: Suppose, by contradiction, that f ≡ 1 and vol(E) is infinite. This implies that, given any L > 1, there exists r 1 > r 0 such that vol(E r 1 − E r 0 ) > 2L. Since f r → 1 uniformly on compact subsets, there exists r 2 > r 1 such that f , with r > r 0 , we obtain for all r > r 2 . In particular, we have that lim r→∞ h(r) = ∞. Now, for each r > r 0 , let ϕ = ϕ r ∈ C ∞ 0 (E) be a cut-off function satisfying: By Hoffmann-Spruck Inequality [9] we have where S is a positive constant.
Note that since f r is harmonic. Using that f r ϕ vanishes on ∂E r we obtain Thus, since 0 ≤ ϕ ≤ 1 in E and ϕ ≡ 1 in E r − E r 0 , we obtain (2.2) First, assume that H L 2 (E) is finite. Then, since 0 ≤ f r ≤ 1, we have Thus, lim r→∞ h(r) < ∞, which is a contradiction. Now, assume that H L p (E) is finite, for some 2 < p ≤ m. Note that m m−2 ≤ p p−2 . Since 0 ≤ f r ≤ 1 and h(r) > 1, for all r > r 2 , we have: m , for all r > r 2 . Thus, using Hölder Inequality, we have for all r > r 2 .

Proof of Theorem B
SinceM has bounded geometry, the sectional curvatureK and the injectivity radius i(M ) ofM satisfy: provided that the volume vol(N ) < Λ. Take R 0 > 0 sufficiently large so that ∂E ⊂ B R 0 and vol(E − B R 0 ) < min{Λ, 1}. Since H L p (E) is finite, for some m ≤ p ≤ +∞, we can take R 0 sufficiently large to satisfy further: Thus, by Hölder inequality, we obtain Since the distance function of M from ξ 0 is a Lipschitz function, by using By the coarea formula, we have that vol(∂B R (q)) = d dR vol(B R (q)). Thus using (3.6) we obtain d dR (vol(B R (q))) 1 m ≥ 1 2Sm . This implies that for all q ∈ E − B R 1 and 0 < R < R 1 .
Since M is complete and E ⊂ M is connected and unbounded, there exists a sequence p 2 , p 3 , . . . in E such that it follows from (3.7) that vol(E) is infinite, which is a contradiction. Theorem B is proved. Fix k ∈ R and let h κ : M → R be the function defined by h κ (t, x) = κt. The gradient vector field of h κ satisfies
Consider M the product S m−1 × R endowed with the metric induced by x. The metric of M is given by where , v denotes the metric of S m−1 . Note that M is a complete manifold with two ends. We claim that M is parabolic. To do this, it is sufficient to prove that the following ends of M : are parabolic (see Proposition 14.1 of [8]). In fact, we define: Using (4.3) and that f (t) = t 1 m−1 , for all |t| ≥ 1, we obtain that for some constant D > 0 and for all s ≥ 1. In particular, This implies that M is parabolic (see section 14.4 of [8]). We claim that the mean curvature vector H of the isometric immersion x has finite L p -norm, for all p > m. In fact, a simple computation shows that (4.4) mH(x(v, t)) = (m − 1) Using that f (t) = t 1 m−1 , for all |t| ≥ 1, we obtain that |H(x(v, t))| ≤ Ct − 1 m−1 , for some C > 0 and for all x(v, t) ∈ E + ∪ E − . Thus, we obtain for some D > 0. This implies that H L p (M ) is finite when p > 2(m − 1). where ω m−1 is the volume of S m−1 . This implies that vol(M ) is finite, since the integral +∞ −∞ e −(m−1)t 2 dt is finite and the function t ∈ R → 1 + 4t 2 e −2t 2 is bounded. In particular, M is parabolic since it has finite volume (see Theorem 7.3 of [8]).
The mean curvature vector H of the isometric immersion x is given by .