论文标题
Hodge Laplacian的散射理论
Scattering theory for the Hodge Laplacian
论文作者
论文摘要
我们证明，使用一个整体标准，波操作员的存在和完整性$ w _ {\ pm}（Δ_h^{（k）}，Δ_g^{（k）}，i_ {g，h}^{（k）}^{（k）} $ socatians $ k $ k $ k $ k $ k） $ν\ in \ {g，h \} $，由两个准iS-Imotem iSmoterricric Riemannian指标$ g $和$ h $在一个完整的打开光滑歧管$ m $上。特别是，该结果提供了绝对连续光谱的标准$σ_{\ Mathrm {ac}}}（Δ_g^{（k）}）=σ_{\ Mathrm {ac}}}（Δ_H^{（k）}）该证明是基于概率的bismut型公式获得的梯度估计，用于光谱微积分定义的热半群。通过这些局部公式，如果Weitzenböck曲率内态呈现在Kato类中，则积分标准需要局部曲率界限，并且对功能的热内核的某些上部局部控制作用，但对注射性放射线没有控制。结果是在RICCI流下绝对连续光谱的稳定性结果。作为应用程序，我们集中于保形扰动的重要情况。
We prove using an integral criterion the existence and completeness of the wave operators $W_{\pm}(Δ_h^{(k)}, Δ_g^{(k)}, I_{g,h}^{(k)})$ corresponding to the Hodge Laplacians $Δ_ν^{(k)}$ acting on differential $k$-forms, for $ν\in\{g,h\}$, induced by two quasi-isometric Riemannian metrics $g$ and $h$ on a complete open smooth manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra $σ_{\mathrm{ac}}(Δ_g^{(k)}) = σ_{\mathrm{ac}}(Δ_h^{(k)})$ of $Δ_ν^{(k)}$ to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion requires local curvature bounds and some upper local control on the heat kernel acting on functions provided the Weitzenböck curvature endomorphism is in the Kato class, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.
