We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman’s entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

中文翻译：

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锐利的上限直径，可紧凑地缩小Ricci孤子

就其压缩的Ricci孤子的标量曲率积分和Perelman熵函数而言，我们给出了一个锐利的上限。尖锐的情况可能发生在圆球上。证明主要依靠梯度收缩的Ricci孤子的对数Sobolev不等式和Vitali型覆盖论点。