论文标题
具有几何应用的非恰当歧管上广义谐波形式的增长估计值
Growth Estimates for Generalized Harmonic Forms on Noncompact Manifolds with Geometric Applications
论文作者
论文摘要
我们在完整的非政策Riemannian歧管$ M $上引入条件W $ \，$（1.2），用于平滑的微分形式$ω$。我们证明，$ω$是$ m $的谐波形式，并且仅当$ω$既关闭又是$ m \，$ω$的$ω$具有$ 2 $ 2 $的增长$ q = 2 $，或者以$ q = 2 $或$ 1 <q（\ ne ne 2）<3 \，$ $ om omω<3 \，$ op omω$满意的条件W $ w $ w $（$ q Assesting W $ \，$（1.2）。特别是，每种$ l^2 $谐波表单，或每种$ l^q $谐波表格，$ 1 <q（\ ne 2）<3 \，$满足条件w $ \，$（1.2）均已关闭和共封闭（参见定理1.1）。这概括了A. Andreotti和E. vesentini [av]的每项$ l^2 $谐波表格$ω\，。 3 $ \ big）。$我们然后研究差异形式的非线性部分差分不等式$ \langleΩ，ΔΩ\ rangle \ ge 0，$ $ω$可以将其视为广义谐波形式。我们证明，在$ω\，$（如定理1.1或1.2或1.3中的相同增长假设下），以下六个陈述：（i）$ \langleΩ，ΔΩ\ rangle \ ge 0 \，，$（$（$（$），$（ii）$δΩ= 0 \ 0 \，，$（$ iii $ $ $）$ $ \ q. 0 \，，$（iv）$ \ langle \ star \，ω，δ\ star \，ω\ the \ rangle \ ge 0 \，，$（v）$（v）$Δ\ star \ star \，ω= 0 \，，$，和（vi）$ d \ d \ d \ d \ d \，\ d \，\ star \，\ star = d d^= d^= d^\ are equent \ star \ star} （参见定理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 $\langleω, ΔΩ\ rangle \ ge 0 \，$ m $上的$（参见定理8.2和定理9.3）。
We introduce Condition W $\,$(1.2) for a smooth differential form $ω$ on a complete noncompact Riemannian manifold $M$. We prove that $ω$ is a harmonic form on $M$ if and only if $ω$ is both closed and co-closed on $M\, ,$ where $ω$ has $2$-balanced growth either for $q=2$, or for $1 < q(\ne 2) < 3\, $ with $ω$ 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 $ω\, .$ In extending $ω$ 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 $ \langleω, Δω\rangle \ge 0, $ in which solutions $ω$ can be viewed as generalized harmonic forms. We prove that under the same growth assumption on $ω\, $ (as in Theorem 1.1, or 1.2, or 1.3), the following six statements: (i) $\langleω, Δω\rangle \ge 0\, ,$ (ii) $Δω= 0\, ,$ $($iii$)$$\quad d\, ω= d^{\star}ω= 0\, ,$ (iv) $\langle \star\, ω, Δ\star\, ω\rangle \ge 0\, ,$ (v) $Δ\star\, ω= 0\, ,$ and (vi) $d\, \star\, ω= d^{\star} \star\, ω= 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 $\langleω, Δω\rangle \ge 0\, $ on $M$ (cf. Theorem 8.2 and Theorem 9.3).
