p-Parabolicity and a Generalized Bochner’s Method with Applications
Authors: Wei Shihshu Walter , Li Jun-Fang , Wu Lina 
Published: June 2024

The notions of function growth and sharp global integral estimates in p-harmonic geometry in Wei (Contemp Math 756:247–269, 2020) and Wei et al. (Sharp estimates on $$\mathcal {A}$$
A
-harmonic functions with applications in biharmonic maps) naturally lead to a generalized uniformization theorem and a generalized Bochner’s method. These tools enable one to explore various geometric and variational problems in complete noncompact manifolds of arbitrary dimensions. In particular, we find the first set of nontrivial geometric quantities that are p-subharmonic and p-superharmonic functions (cf. Sect. 4.2). As further applications, we establish Liouville theorems for nonnegative $${\mathcal {A}}$$
A
-superharmonic functions (cf. Theorem 2.1) and for p-harmonic morphisms (cf. Theorem 4.3), Picard type theorems (cf. Sect. 4.3), existence theorems of harmonic maps (cf. Theorem 4.6), and a solution of the generalized Bernstein problem under p-parabolicity condition (without any volume growth condition, cf. Theorem 5.1) or under p-moderate volume growth (2.4) (cf. Corollary 5.1). These findings generalize and extend the work of Schoen–Simon–Yau under volume growth condition (0.1) which is due to $$({\text {i}})$$

(
i
)

Corollary 2.1(iv), which states that a manifold with p-moderate volume growth (2.4) must be p-parabolic (generalizing a result of Cheng and Yau (Commun Pure Appl 28(3):333–354, 1975) for the case $$p=2$$

p
=
2

, $$F(r)\equiv 1$$

F
(
r
)
≡
1

), $$({\text {ii}})$$

(
ii
)

Example 2.1 of a p-parabolic manifold with exponential volume growth, and $$({\text {iii}})$$

(
iii
)

volume growth condition (0.1) $$\overset{(\text {implies})}{\Longrightarrow }$$

⟹

(
implies
)

p-moderate volume growth (2.4). Furthermore, we offer applications to p-harmonic maps and stable p-harmonic maps. Notably, these outcomes address an intriguing question posed by Kobayashi if p-subharmonic functions are valuable tools for studying p-harmonic maps or related topics (cf. in: Kobayashi, Midwest Geometry Conference, University of Oklahoma, Oklahoma, 2006. Private Communication, 1996), and answer in the affirmative. Other applications of the notions and estimates to biharmonic maps, isometric immersions, holomorphic functions on Kähler manifolds, generalized harmonic forms, and Yang–Mills fields on complete noncompact manifolds can be found in Wei et al. (Sharp estimates on $$\mathcal A$$
A
-harmonic functions with applications in biharmonic maps), Chen and Wei (Glasg Math J 51(3):579–592, 2009), Wei (Bull Transilv Univ Brasov Ser III 1(50):415–453, 2008), Wei (Growth estimates for generalized harmonic forms on noncompact manifolds with geometric applications, Contemp Math 756 American Mathematical Society, Providence, 247–269, 2020), Wei (Sci China Math 64(7):1649–1702, 2021) and Wei (Isolation phenomena for Yang–Mills fields on complete manifolds). We generate the work of Schoen–Simon–Yau under p-parabolic condition, in which the result can be used in other types of new manifolds we found, by an extrinsic average variational method we proposed (cf. Wei in Rom J Math Comput Sci 13(2):100–124, 2023; in: Wei, Connecting Poincare inequality with Sobolev inequality on Riemannian manifolds, Int Electron J Geom 17(1):290–305, 2024).