A note on the p-Parabolicity of Submanifolds

We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for p bigger or equal than 2.

manifolds, namely, to decide when a Riemannian manifold is p-parabolic of p-hyperbolic.
In particular, in the Corollary 5.2 of [T1], (see too Proposition 4 in [GT]), it is presented a characterization of p-parabolicity for Riemannian manifolds with a warped cilindrical end, and in Corollary 5.4 of [T1], (see too Proposition 3 in [GT]), we have a sufficient codition for p-parabolcity in terms of the volume growth of the manifold.
While these two criteria are intrinsic, we are going to present in this paper a geometric criterion to decide if a submanifold S m properly immersed in an ambient manifold N n with a pole is p-parabolic, which involves (lower) bounds for the mean curvature and the second fundamental form of S.
This criterion is based, (as in [MP1] and, specially, [HMP] from which this paper can be considered a spin-off), in the Hessian-Index analysis of the (restricted to S) extrinsic distance function from the pole, (see [GreW]).
1.1. Outline of the paper. We shall present the basic definitions concerning the p-Laplacian in Section 2. Section 3 is devoted to the study of the curvature setting where our results hold, together with the Hessian and Laplacian analysis needed. Main results are stated and proved in Sections 4, 5 and 6.
1.2. Acknowledgements. We would like to acknowledge professors Ilkka Holopainen and Steen Markvorsen their useful comments concerning these results.

The p-Laplacian
Let M be a non-compact Riemannian manifold, with the Riemannian metric ·, · and the Riemannian volume form dµ.
The p-Laplacian of a C 2 function u is defined as ∆ p u = div( ∇u p−2 ∇u).
When p = 2, we have the usual Laplacian, and the classical potential theory developed from the study of the solutions of the Laplace equation However, when p = 2, equation is nonlinear and degenerates at the zeroes of the gradient of u. Then, the solutions of (2.1), known usually as p-harmonic functions, need not be smooth, nor even C 2 and equation (2.1) must be interpreted in a weak sense.
In this way, and given 1 < p < ∞, we say that a function u ∈ W 1,p loc (M) is a (weak) solution to the p-Laplace equation . Furthermore, L 1 (M) denotes the space of measurable functions f : M −→ R with finite norm f 1 < ∞, and L 1 loc (M) is its corresponding local space defined through the open sets in M with compact closure, (see [HKM] p. 13). In its turn, the space W 1,p (M), 1 ≤ p < ∞ is the Sobolev space of all functions u ∈ L p (M) whose distributional gradient ∇u belongs to L p (M), equipped with the norm u 1,p = u p + ∇u p . The corresponding local space W 1,p loc (M) is defined as in [HKM]. Continuous solutions of (2.2) are called p-harmonic. Here the continuity assumption makes no restriction since every solution of (2.2) has a continuous representative. The extension of regularity results of this kind, (see [E] and [Li]), from the Euclidean setting to the Riemannian setting is detailed in [HMP], Remark 9.2.
Similarly, a function v ∈ W 1,p loc (M) is called a p-subsolution in M if −v is a p-supersolution. If, moreover, v is lower semicontinuous, then v is psubharmonic, (∆ p v ≥ 0).
A fundamental feature of solutions of (2.2) is the following well-known maximum (or comparison) principle which will be instrumental for the comparison technique presented below in Sections 5 and 6: If We refer to [HKM,3.18] for a short proof of the comparison principle.

Comparison Constellations
We assume throughout the paper that S m is a non-compact, properly immersed, and connected Riemannian submanifold of a complete Riemannian manifold N n . Furthermore, we assume that N n possesses at least one pole. Recall that a pole is a point o such that the exponential map exp o : T o N n → N n is a diffeomorphism. For every x ∈ N n \{o} we define r(x) = dist N (o, x), and this distance is realized by the length of a unique geodesic from o to x, which is the radial geodesic from o. We also denote by r the restriction r| S : S → R + ∪ {0}. This restriction is called the extrinsic distance function from o in S m . The gradients of r in N and S are denoted by ∇ N r and ∇ S r, respectively. Let us remark that ∇ S r(x) is just the tangential component in S of ∇ N r(x), for all x ∈ S. Then we have the following basic relation: With the extrinsic distance at hand, we define the following domains: Definition 3.1. Given a connected and complete m-dimensional submanifold S m in a complete Riemannian manifold N n with a pole o, we denote the extrinsic metric balls of (sufficiently large) radius R and center o by D R (o). They are defined as any connected component of the intersection denotes the open geodesic ball of radius R centered at the pole o in N n . Using these extrinsic balls we define the o-centered extrinsic annuli Remark 3.2. We must to point out that these extrinsic domains are precompact, (because the submanifold S is properly immersed), and that the radii R that produce smooth boundaries ∂D R (o) are dense in R by Sard's theorem and the Regular Level Set Theorem, because the distance function r is smooth in N n \ {o}, and hence, its restriction to S, r| S .
Definition 3.4. The o-radial mean convexity C(x) of S in N, is defined as follows: Moreover, for p > 2 we define: is the unit tangent vector to S in the direction of ∇ S r(x) (resp. tacitly assumed to be 0 in case ∇ S r(x) = 0).

Finally,
Definition 3.6. The o-radial tangency T (x) of S in N is defined as follows: for all x ∈ S.
Upper and lower bounds of C(x), B(x) and T (x) together with a suitable control on the o-radial sectional curvatures of the ambient space will eventually control the p-Laplacian of restricted radial functions on S.
Proposition 3.9 (See [GreW] and [Gri]). Let M m w be a w−model with warping function w(r) and center o w . The distance sphere of radius r and center o w in M m w is the fiber π −1 (r). This distance sphere has the constant mean curvature η w (r) = w ′ (r) w(r) . On the other hand, the o w -radial sectional curvatures of M m w at every x ∈ π −1 (r) (for r > 0) are all identical and determined by 3.3. Comparison constellations. We now collect the previous ingredients and formulate the general framework for our p-parabolicity comparison result, which results a dual setting with respect the curvature assumptions stated in [HMP] to obtain p-hyperbolicity. (3.1) Then the triple {N n , S m , M m w } is called a comparison constellation with lower tangency on the interval [0, R] if the radial tangency T and the radial convexity functions B and C of the submanifold S m are all bounded from below by smooth radial functions g(r), λ(r), and h(r), respectively: Note that the radial tangency is, in a natural way, bounded from above by 1. This fact motivates the following Definition 3.11. We assume the same general hypothesis on S and N than in the definition above. The triple {N n , S m , M m w } is called a comparison constellation with upper tangency on the interval [0, R] when the radial convexity functions B and C of the submanifold S m are all bounded from below by smooth radial functions λ(r), and h(r), respectively: B(x) ≥ λ(r(x)), and 3.4. Hessian and Laplacian comparison analysis. The 2nd order analysis of the restricted distance function r | P defined on manifolds with a pole is firstly and foremost governed by the Hessian comparison Theorem A in [GreW]: Theorem 3.12 (See [GreW], Theorem A). Let N = N n be a manifold with a pole o, let M = M m w denote a w−model with center o w , and m ≤ n. Suppose that every o-radial sectional curvature at x ∈ N \ {o} is bounded from below by the o w -radial sectional curvatures in M m w as follows: for every radial two-plane σ x ∈ T x N at distance r = r(x) = dist N (o, x) from o in N. Then the Hessian of the distance function in N satisfies for every unit vector X in T x N and for every unit vector Y in T y M with r(y) = r(x) = r and ∇ M r(y), Y M = ∇ N r(x), X N .
As a consequence of this result, we have the following Laplacian inequality: Proposition 3.13. Let N n be a manifold with a pole o, let M m w denote a w−model with center o w . Suppose that every p-radial sectional curvature at x ∈ N − {o} is bounded from below by the o w -radial sectional curvatures in M m w as follows: Then we have for every smooth function f (r) with f ′ (r) ≤ 0 for all r, (respectively f ′ (r) ≥ 0 for all r): where H S denotes the mean curvature vector of S in N.

Main results
Applying the notion of a comparison constellation as defined in the previous section, we now formulate our main p-parabolicity results. The proofs are developed through the following sections.
Let Λ g,p (r) denote the function dt .
Suppose finally that p ≥ 2 and that Then S m is p-parabolic.
Theorem 4.2. Consider a comparison constellation with upper tangency {N n , S m , M m w } on the interval [ 0, ∞[ . Assume further that the functions h(r) and λ(r) are balanced with respect to the warping function w(r) by the following inequality: Suppose finally that p ≥ 2 and that Then S m is p-parabolic. With this consideration at hand, we can find a version of Theorem 4.1 for p = 2 in the paper [MP1], where it is used a more restrictive balance condition which implies condition (4.1). On the other hand, we have, based on the same consideration, a version of Theorem 4.2 for p = 2 in the paper [EP], where we can find a direct proof and some consequences in connection with [MP1]. Proof. To obtain the result it suffices to apply Theorem 4.2, taking into account that, under the hypothesis, Λ p (r) ≥ w(r) for all r > 0. Assume furthermore that Then S m is p-parabolic, for all p ≥ q.
Proof. It is straightforward to see that, if p ≥ q, since η w (r) − h(r) < 0 for all r > 0. Then, it is easy to check that, under the hypothesis, Applying Theorem 4.2, the results follows.

Proof of Theorem 4.1
We define on model spaces M m w , the modified Laplacian for smooth functions ψ on M m w . If ψ = ψ(r) only depends on the radial distance r, then Consider now the following Dirichlet-Poisson problem associated to L: The explicit solution to the Dirichlet problem (5.2) is given in the following Proposition which is straightforward, Proposition 5.1. The solution to the Dirichlet problem (5.2) only depends on r and is given explicitly -via the function Λ g,p (r) introduced in Theorem 4.1, by: The corresponding 'drifted' 2-capacity is It is easy to see, using equation (5.3) and the balance condition (4.1) that Now, we need the following result, which relates the p-Laplacian of a radial function f (r) with the operator L.
Lemma 5.2. Let {N n , S m , M m w } be a comparison constellation with lower tangency on [0, R] for R > 0. Let f • r be a smooth real-valued function with f ′ ≥ 0, and suppose now that f (r) satisfies the following condition: Then, for all x ∈ S, ∆ S p f (r(x)) ≤ (p − 1)F p−2 (x)g 2 (r(x)) L(f (r(x))), where L is the second order differential operator defined by equation (5.1) and F is given by equation Proof. Computing as in [HMP], we have This partial 'isolation' of the factor (p − 2) is the reason behind the general assumption p ≥ 2 in this work. Once we have equation (5.8), we argue as follows: First, it is easy to see that (5.9) This quantity is bounded from above using Theorem 3.12 and the lower bound of B(x). On the other hand, since the o-radial mean convexity of S, C(x) is bounded from below by the function h(r(x)), we obtain the following estimate using Proposition 3.13, (recall that f ′ (r) ≥ 0) So, using the fact that f (r) satisfies inequality (5.6) and that ∇ S (r) ≥ g(r), we have (5.11) ∆ S p (f (r(x))) ≤ (p − 1)F p−2 (x)g 2 (r) L(f (r)), as claimed in the lemma.
As S is properly immersed, D ρ (o) and D R (o) are precompact and with regular boundary, so there exists a unique function u ∈ C(D R (o)) which is p-harmonic in D R (o) \D ρ (o) such that u = 0 inD ρ (o), u = 1 in ∂D R (o), and that (see [T1] and [HKM,).
Furthermore, let Ψ ρ,R be the transplanted p-supersolution in D R (o). By the comparison principle, we have now for all x ∈ D R (o). Hence, as u(x) = Ψ ρ,R (x) = 0 for all x ∈D ρ (o), we obtain that for all x ∈ ∂D ρ (o). With same arguments than in [HMP], but inverting all inequalities, we obtain As, on the other hand, D ρ (o) is precompact with a smooth boundary thence, ∂Dρ ∇ S r p−1 dH m−1 > 0.
So finally we have (5.13) since lim R→∞ Cap L (A w ρ,R ) = 0 by hypothesis (4.2) and equality (5.4) . Thus D ρ (o) is a compact subset with zero p-capacity in S m , and p-parabolicity of that submanifold follows.
Consider now the smooth radial solution φ ρ,R (r) of the Dirichlet-Poisson problem associated to L and defined on the annulus A w ρ,R = B w R − B w ρ . Now we transplant the model space solutions φ ρ,R (r) of this problem into the extrinsic annulus A ρ,R = D R (o) \D ρ (o) in S as in the proof of Theorem 4.1, so we have Φ ρ,R : A ρ,R → R, Φ ρ,R (x) = φ ρ,R (r(x)).