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  1. Abstract

    The notions of function growth and sharp global integral estimates inp-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 arep-subharmonic andp-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 forp-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 underp-parabolicity condition (without any volume growth condition, cf. Theorem 5.1) or underp-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 withp-moderate volume growth (2.4) must bep-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 ap-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 top-harmonic maps and stablep-harmonic maps. Notably, these outcomes address an intriguingquestion 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 tobiharmonic maps, isometric immersions, holomorphic functionson Kähler manifolds,generalized harmonic forms, andYang–Mills fieldson 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 underp-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).

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  2. Reversible lithium metal anodes (LMAs) are the holy grail for future rechargeable lithium metal batteries. Threedimensional (3-D) conductive hosts have been extensively explored as an effective approach to suppressing dendrite formation and enabling reversible Li plating/stripping. However, the microscopic morphologies of Li plating and their correlation with the cell performance are not clear. Herein we unravel these issues using the vertically aligned carbon nanofiber (VACNF) array as a model 3-D conductive carbon host which has a welldefined vertical low-tortuosity structure allowing observation of the intrinsic Li morphologies infiltrated into the 3-D host. The VACNF array indeed provides much higher stability and reversibility for Li plating/stripping due to its high surface area and lithiophilic properties. We found that Li plating on both VACNF array and planar Cu electrodes follows the classical nucleation and growth model. Though the low plating current density (≤0.10 mA/cm2) provides better cycling stability consistent with the Sand’s equation, it forms sparse irregular grains stacked with dendrite-like long Li fibers. In contrast, the moderate to high plating current densities (1.0 - 5.0 mA/cm2) produce more uniform Li morphologies consisting of smaller micro-columns or micro-spheres. By decoupling the plating and stripping current densities, we unravel that the more uniform micro-columnar Li infiltrated in the VACNF array obtained at the moderate plating current density (~1.0 mA/cm2) indeed exhibits the highest cycling performance. This provides new insights into the relationship between macroscopic electrochemical tests and microscopic Li morphologies, aiding in optimizing the performance of LMAs based on 3-D conductive hosts. 
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  4. Amorphous molybdenum sulfide (a-MoS3) is a promising non-precious electrocatalyst for hydrogen evolution reaction owing to the abundant defective active sites. Here in, we show a rapid microwave-assisted synthesis method to produce a-MoS3 catalysts on reduced graphene oxide (rGO) substrates. The a-MoS3 reported in this study comprise of two possible 1D chain-like structures, i.e., with molybdenum (IV) in Weber’s model and molybdenum (V) in Hibble’s model, unlike the polymeric cluster type a-MoS3 structures reported in literature. Thermal annealing of the microwave-prepared a-MoS3 produced a family of defect-engineered MoSx/rGO hybrids, from a-MoS3 to crystalline MoS2, which showed tunable HER activities. XPS analysis provided in-depth understanding of the compositional changes in MoSx/rGO with thermal annealing. The a-MoS3/rGO 250 (annealed at 250 ◦C) exhibited the highest HER catalytic activity among all the MoSx/rGO hybrids, with an overpotential of 208 mV at 10 mA/cm2, a low Tafel slope of 52 mV/decade, a high double layer capacitance of 3.7 mF/cm2 and a high TOF value of 0.43 H2/s per site at the HER overpotential of 208 mV. The excellent HER activity is attributed to both MoV and sulfur active sites. This study provides a controllable, scalable and rapid synthesis method to produce 1D chain-like a-MoS3 structures for HER electrocatalysis. 
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