Abstract
Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = e h dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian \({\triangle_{\mu}}\) has a positive lower bound λ1(M) > 0 and the m(m > n)-dimensional Bakry-Émery curvature is bounded from below by \({-\frac{m-1}{m-2}\lambda_1(M)}\), then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.
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Wang, L.F. A splitting theorem for the weighted measure. Ann Glob Anal Geom 42, 79–89 (2012). https://doi.org/10.1007/s10455-011-9302-0
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DOI: https://doi.org/10.1007/s10455-011-9302-0