Growth Estimates for Generalized Harmonic Forms on Noncompact Manifolds with Geometric Applications

TL;DR

This paper establishes conditions under which harmonic forms on noncompact manifolds are characterized by being closed and co-closed, extending classical results to broader growth conditions and exploring related nonlinear PDE inequalities with geometric applications.

Contribution

It introduces Condition W for differential forms, generalizes harmonic form characterization to $L^q$ spaces with growth conditions, and studies nonlinear PDE inequalities with geometric implications.

Findings

Characterization of harmonic forms under Condition W and growth conditions.

Equivalence of several nonlinear PDE inequalities for differential forms.

Applications to Dirichlet problems, monotonicity formulas, and vanishing theorems.

Abstract

We introduce Condition W $\,$(1.2) for a smooth differential form $\omega$ on a complete noncompact Riemannian manifold $M$. We prove that $\omega$ is a harmonic form on $M$ if and only if $\omega$ is both closed and co-closed on $M\, ,$ where $\omega$ has $2$-balanced growth either for $q=2$, or for $1 < q(\ne 2) < 3\, $ with $\omega$ satisfying Condition W $\,$(1.2). In particular, every $L^2$ harmonic form, or every $L^q$ harmonic form, $1<q(\ne 2)<3\, $ satisfying Condition W $\,$(1.2) is both closed and co-closed (cf. Theorem 1.1). This generalizes the work of A. Andreotti and E. Vesentini [AV] for every $L^2$ harmonic form $\omega\, .$ In extending $\omega$ in $L^2$ to $L^q$, for $q \ne 2$, Condition W $\,$(1.2) has to be imposed due to counter-examples of D. Alexandru-Rugina$\big($ [AR] p. 81, Remarque 3$\big).$ We then study nonlinear partial differential inequalities for differential forms $ \langle\omega, \Delta \omega\rangle \ge 0, $ in which solutions $\omega$ can be viewed as generalized harmonic forms. We prove that under the same growth assumption on $\omega\, $ (as in Theorem 1.1, or 1.2, or 1.3), the following six statements: (i) $\langle\omega, \Delta \omega\rangle \ge 0\, ,$ (ii) $\Delta \omega = 0\, ,$ $($iii$)$$\quad d\, \omega = d^{\star}\omega = 0\, ,$ (iv) $\langle \star\, \omega, \Delta \star\, \omega\rangle \ge 0\, ,$ (v) $\Delta \star\, \omega = 0\, ,$ and (vi) $d\, \star\, \omega = d^{\star} \star\, \omega = 0\, $ are equivalent (cf. Theorem 4.1). We also study As geometric applications, we employ the theory in [DW] and [W3], solve constant Dirichlet problems for generalized harmonic $1$-forms and $F$-harmomic maps (cf. Theorems 10.3 and 10.2), derive monotonicity formulas for $2$-balanced solutions, and vanishing theorems for $2$-moderate solutions of $\langle\omega, \Delta \omega\rangle \ge 0\, $ on $M$ (cf. Theorem 8.2 and Theorem 9.3).