Solving Liouville-type Problems on Manifolds with Poincaré-Sobolev Inequality by Broadening q-Energy from Finite to Infinite

The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincaré-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincaré-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincaré-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of q-energy from finite to infinite. At this point, a technique of p-balanced growth has been introduced to overcome difficulties of broadening from finite q-energy in Lq spaces to infinite q-energy in non-Lq spaces. An innovative computational method and new estimation techniques are illustrated. At the end of this article, Liouville-type results including vanishing properties for differential forms and constancy properties for differential maps have been verified on manifolds with Poincaré-Sobolev Inequality approaching to infinite q-energy growth.


Introduction
The study of Liouville-type problems is to obtain constancy properties for differential forms and differential maps on manifolds equipped with a wide variety of metric structures.It has been one of the most valuable and challenging research topics in the mathematical society.Mathematicians have obtained Liouville-type results as follows: 1. Liouville-type results for differential forms: The first result goes back to Greene and Wu (Greene & Wu, 1981).
In 1981, Greene and Wu solved Liouville-type problems for harmonic 1-forms and obtained vanishing property for harmonic 1-forms with finite q-energy in L q (1 < q < ∞) spaces on non-negatively curved manifolds M with Ricci M ≥ 0. In 2001, Zhang (Zhang, 2001) studied Liouville-type problems for closed and p-co-closed differential 1-forms (p > 1) in L q (0 < q < ∞) spaces on positively curved manifolds M.
2. Liouville-type results for differential maps: In 1976, Schoen and Yau (Schoen & Yau, 1976) solved Liouville-type problems for harmonic maps and established constancy property for harmonic maps on M with Ricci M ≥ 0. In 1995 Cheung and Leung (Cheung & Leung, 1995) showed Liouville Theorems for p-harmonic maps (p ≥ 2) in L q (q = p − 1) spaces on the target spaces of Cartan-Hadamand Manifolds.In 1999, Kawai (Kawai, 1999) derived Liouville Theorems for p-harmonic maps (p ≥ 2) from p-parabolic manifolds to non-positively curved targets.
The aim of this article is to investigate Liouville-type problems for both differential forms and differential maps on complete non-compact manifolds M in the presence of Poincaré-Sobolev Inequality.It is well-known that any non-flat manifold can be determined by the sign of its curvatures.A curved non-flat manifold can be classified into either a manifold with only one sign of its curvatures (such as a globally positive curved manifold or a globally negative curved manifold) or a manifold with the mixed signs of its curvatures (a combination of locally positive curvatures with locally negative curvatures).The main study of Liouville-type problems on manifolds is to study various manifolds M equipped with all possible different metric structures (M, g) such that there exist constancy properties or vanishing properties for differential forms and differential maps on (M, g).
Liouville-type problems on either globally positive curved manifolds or globally negative curved manifolds have been studied by mathematicians for decades.Most research findings have been obtained in L q spaces with finite q-energy on curved manifolds with only one sign of curvatures.Many questions are still open.For example, how to solve Liouvilletype problems on curved manifolds with the mixed curvature signs and how to solve Liouville-type problems with infinite q-energy in non-L q spaces.
Liouville-type results have been generalized from finite q-energy in L q spaces to infinite q-energy in non-L q spaces as well as the extended scope of q.For example, Liouville-type results such as vanishing properties for harmonic 1-forms with infinite q-energy on manifolds with positive semi-definite Ricci curvatures were discovered in 2015 by S.W. Wei and Wu (Wei & Wu, 2015).Liouville-type results for closed and p-pseudo-coclosed differential 1-forms with infinite q-energy on manifolds with non-negative Ricci curvatures were obtained in 2016 by Y. Li and Wu (Wu & Li, 2016).However, the generalization of q-energy in Liouville-type theorems in the past research work had been proven on manifolds with only one sign of curvatures.
The goal of this article is to solve Liouville-type problems on curved manifolds with the mixed curvature signs.Research findings of Liouville-type results on various manifold structures determined by the mixed signs of curvatures are obtained.
In particular, vanishing properties for differential forms and constancy properties for differential maps are verified in both finite q-energy in L q spaces and infinite q-energy in non-L q spaces.Two significant technical breakthroughs are demonstrated.The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures at any values, which vary among negative values, zero, and positive values.Poincaré-Sobolev Inequality has been applied to overcome difficulties of manifold structure extensions.Poincaré-Sobolev Inequality has offered a special structure on curved manifolds for Ricci curvatures with the mixed signs.It turns out that Poincaré-Sobolev Inequality survives on various manifolds with quite different structures in the large scope of metric changes.The second breakthrough is an extension from finite q-energy L q spaces to broader spaces, which include both finite q-energy L q spaces and infinite q-energy non-L q spaces.At this point, the technique of p-balanced growth has been studied to overcome difficulties of the q-energy generalization.
As compared with traditional calculation methods used to solve Liouville-type problems in the context of finite q-energy, an innovative computational method leading to infinite q-energy is demonstrated in this article.Weitzenböck Bochner Formula, Poincaré-Sobolev Inequality, Hölder Inequality, Cauchy-Schwarz Inequality, and Calculus skills as estimation techniques are presented.Weitzenböck Bochner Formula is used at the beginning as the foundation to establish the first inequality regarding integrals of differential forms or differential maps on curved manifolds.After that, an appropriate test function with its power varying in a certain range is carefully selected.Changes of power in this selected test function play an important role to reveal the way how changes of manifold structures have an impact on changes of energy growth rates for differential maps or differential maps.The maximum range of power in this test function determines the maximum scope of energy growth rates.Energy growth rates being too fast or too slow as indicated by too high or too low power in the test function lead to contradictions with the maximum scope of manifolds whose structures are compatible with Poincaré-Sobolev Inequality.At the end, a balance between reasonable energy growth rates and appropriate manifolds supported by Poincaré-Sobolev Inequality is made to obtain Liouville-type results.

Preliminary
In this section, we first define the concept of p-balanced growth for p > 1, which consists of 5 cases: p-finite growth, p-mild growth, p-obtuse growth, p-moderate growth, and p-small growth.After that, we give definitions of closed differential forms, p-pseudo-coclosed differential forms, and p-harmonic maps.At the end of this section, Poincaré-Sobolev Inequality on a complete non-compact Riemannian manifold is defined.
Throughout this paper, we assume that M is a complete non-compact n-dimensional Riemannian manifold and B(x 0 ; r) (or B(r)) is the geodesic ball of radius r centered at a point x 0 ∈ M. A function or a differential form f is said to be with p-balanced growth provided f has one of the following "p-finite, p-mild, p-obtuse, p-moderate, and p-small" growth where p > 1.Otherwise, f is said to be with p-imbalanced growth (Wei, Li & Wu, 2008).
Definition 1 A function f has p-finite growth (or, simply, is p-finite) if there exists x 0 ∈ M such that and has p-infinite growth (or, simply, is p-infinite) otherwise.
Definition 2 A function f has p-mild growth (or, simply, is p-mild) if there exist x 0 ∈ M, and a strictly increasing sequence of {r j } ∞ 0 going to infinity, such that for every l 0 > 0, we have: and has p-severe growth (or, simply, is p-severe) otherwise.
Definition 3 A function f has p-obtuse growth (or, simply, is p-obtuse) if there exists x 0 ∈ M such that for every a > 0, we have: and has p-acute growth (or, simply, is p-acute) otherwise.
Definition 4 A function f has p-moderate growth (or, simply, is p-moderate) if there exist x 0 ∈ M, and F(r) ∈ F , such that lim sup and has p-immoderate growth (or, simply, is p-immoderate) otherwise, where for some a > 0. Notice that the functions in F are not necessarily monotone.
Definition 5 A function f has p-small growth (or, simply, is p-small) if there exists x 0 ∈ M, such that for every a > 0, we have: and has p-large growth (or, simply, is p-large) otherwise.
The above definitions of "p-finite, p-mild, p-obtuse, p-moderate, p-small" and their counter-parts "p-infinite, p-severe, p-acute, p-immoderate, p-large" growth depend on q, and q will be specified in the context in which the definition is used.
Let N be a complete Riemannian manifold.
Definition 6 For a map u : M → N, we define the p-energy of u by: where dv is the volume element of M and p > 1.
Let A k (ρ) = C(∧ k T * M ⊗ V) be a space of smooth k-forms on M with values in the vector bundle ρ : V → M, and let d : A k (ρ) → A k+1 (ρ) be the exterior differential operator and let d * : A k (ρ) → A k−1 (ρ) be the exterior differential operator given by d * = − ∑ n j=1 i(e j )∇ e j where {e j } is a local orthonormal frame at x ∈ M, and i(X) is the interior multiplication by X defined as Definition 7 A map u is said to be a p-harmonic map (p > 1) if it is a critical point of p-energy functional E p .Equivalently, u is a p-harmonic map if and only if the p-tension field In particular, a map u is said to be a harmonic map if u is a p-harmonic map when p = 2. Equivalently, u is a harmonic map if and only if the 2-tension field For example, the differential of a p-harmonic function is a closed and p-pseudo-coclosed 1-form.
for every compactly supported non-negative smooth function on M (that is, φ ∈ C ∞ 0 (M) and φ ≥ 0) and for some positive constant S α > 0 where The version of Poincaré-Sobolev Inequality given in this paper is obtained by squaring both sides of the standard Poincaré-Sobolev Inequality (that is, S ∥φ∥ √ S α > 0 and α = p n and p = 2 when n > 2.

Results
In this section, on complete non-compact Riemannian manifolds M with Poincaré-Sobolev Inequality, lemmas and a sequence of Liouville-type theorems are proven in the following two categories: vanishing properties for differential forms and constancy properties for differential maps.
In particular, ω ∈ L q (M) (that is, ∫ M |ω| q dv < ∞) must satisfy an assumption of lim inf r→∞ 1 r 2 ∫ B(x;r) |ω| q dv = 0 for q = 2m + 2. We can obtain the vanishing property for ω in L q space.At the end, by combining (9) and m ≥ p ≥ 2, we compute the range of q = 2m + 2 as follows: Theorem 2 Let M be a complete non-compact Riemannian manifold satisfying Poincaré-Sobolev Inequality (1) and Ricci M ≥ −k(x) where k(x) ≥ 0 is a continuous function such that Let ω be a closed and p-pseudo-coclosed differential 1-form on M for p ≥ 2. Then ω = 0 if ω has: where q is given by (15).