On the reduced \(L^2\) cohomology on complete hypersurfaces in Euclidean spaces
A \(p\)-form \(\omega\) is called \(L^{2}\) harmonic if \(\Delta \omega=0\) and \(\int_{M} \omega \wedge \ast \omega < \infty\) for the Laplacian operator \(\Delta\). The space of all \(L^{2}\) harmonic \(p\)-forms on \(M\) is denoted by \(H^{p}(L^{2}(M))\) and let \(Z_{2}^{p}(M)=\{\alpha \in L^{2}(\wedge^{p}(T^{\ast}M)):d\alpha=0\}\) and \(D^{p}(d)=\{\alpha \in L^{2}(\wedge^{p}(T^{\ast} M))\}\). The \(p\)-space of reduced \(L^{2}\) cohomology is denoted as \(H^{p}_{2}(M)=\frac{Z^{p}_{2}(M)}{dD^{p-1}(d)}\). The main result of this paper is that if \(M^{n}(n\geq 3)\) is an \(n\)-dimensional complete noncompact hypersurface isometrically immersed in \(\mathbb{R}^{n+1}\) with finite total curvature, then the \(p\)th space of reduced \(L^{2}\) cohomology on \(M\) has finite dimension for \(0\leq p\leq n\). Several results were found, for example, \textit{M. P. Cavalcante} et al. [J. Geom. Anal. 24, No. 1, 205--222 (2014; Zbl 1308.53056)] showed that if a complete noncompact submanifold \(M^{n}\) (\(n\geq 3\)) isometric immersed in \(\mathbb{R}^{n+p}\) has finite total curvature, then the space of the \(L^{2}\) harmonic 1-forms on \(M\) has finite dimension. \textit{G. Carron} [Math. Ann. 314, No. 4, 613--639 (1999; Zbl 0933.35054)] also obtained the same result if \(M^{n}\) (\(n\geq 3\)) is a complete noncompact submanifold of \(\mathbb{R}^{n+p}\) with finite total curvature and finite mean curvature. So, this paper generalizes these two papers.
scientific article; zbMATH DE number 6477367
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