We show that one-dimensional circle is the only case for closed smooth metricmeasure spaces with nonnegative Bakry-\'{E}mery Ricci curvature whose spectrumof the weighted Laplacian has an optimal positive upper bound. This resultextends the work of Hang-Wang in the manifold case (Int. Math. Res. Not. 18(2007), Art. ID rnm064, 9pp).
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机译：我们表明，一维圆是具有非负Bakry-\'{E} mery Ricci曲率的闭式光滑度量空间的唯一情况，其加权Laplacian谱具有最佳正上限。该结果扩展了Hang-Wang在流形情况下的工作（Int。Math。Res。Not。18（2007），Art ID ID rnm064，9pp）。
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