Dualities in comparison theorems and bundle-valued generalized harmonic forms on noncompact manifolds

Abstract

We observe, utilize dualities in differential equations and differential inequalities (see Theorem 2.1), dualities between comparison theorems in differential equations (see Theorems E and 2.2), and obtain dualities in ‘swapping’ comparison theorems in differential equations. These dualities generate comparison theorems on differential equations of mixed types I and II (see Theorems 2.3 and 2.4) and lead to comparison theorems in Riemannian geometry (see Theorems 2.5 and 2.8) with analytic, geometric, PDE’s and physical applications. In particular, we prove Hessian comparison theorems (see Theorems 3.1–3.5) and Laplacian comparison theorems (see Theorems 2.6, 2.7 and 3.1–3.5) under varied radial Ricci curvature, radial curvature, Ricci curvature and sectional curvature assumptions, generalizing and extending the work of Han-Li-Ren-Wei (2014) and Wei (2016). We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship (see Theorem 5.4). These provide tools in extending the notion, integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms, introducing Condition W for bundle-valued differential forms, and proving the duality theorem and the unity theorem, generalizing the work of Andreotti and Vesentini (1965) and Wei (2020). We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature, generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds, the embedding theorem for weighted Sobolev spaces of functions on manifolds, geometric differential-integral inequalities, generalized sharp Hardy type inequalities on Riemannian manifolds, monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles, such as F-Yang-Mills fields (when F is the identity map, they are Yang-Mills fields), generalized Yang-Mills-Born-Infeld fields on manifolds, Liouville type theorems for F - harmonic maps (when $F\left(t \right) = {1 \over p}{\left({2t} \right)^{{p \over 2}}},p > 1$, they become p-harmonic maps or harmonic maps if p = 2), and Dirichlet problems on starlike domains for vector bundle valued differential 1-forms and {tF}-harmonic maps (see Theorems 4.1, 7.3–7.7, 8.1, 9.1–9.3, 10.1, 11.2, 12.1 and 12.2), generalizing the work of Caffarelli et al. (1984) and Costa (2008), in which M = ℝⁿ and its radial curvature K(r) = 0, the work of Wei and Li (2009), Chen et al. (2011, 2014), Dong and Wei (2011), Wei (2020) and Karcher and Wood (1984), etc. The boundary value problem for bundle-valued differential 1-forms is in contrast to the Dirichlet problem for p-harmonic maps to which the solution is due to Hamilton (1975) for the case p = 2 and Riem^N ⩽ 0, and Wei (1998) for 1 < p < ∞.