Let M be a complete Riemannian manifold with Riemannian volume volg and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator 􏰂 f = 􏰂 − ∇ f · ∇ was given by Wu (J Math Anal Appl 361:10–18, 2010) and Wang (Ann Glob Anal Geom 37:393–402, 2010) under a condition that finite dimensional Bakry–Émery Ricci curvature is bounded below, independently. They propounded an open problem is whether there is some rigidity on the estimate. In this note, we will solve this problem to obtain a splitting type theorem, which generalizes Li–Wang’s result in (J Differ Geom 58:501–534, 2001, J Differ Geom 62:143–162, 2002). For the case that infinite dimensional Bakry–Emery Ricci curvature of M is bounded below, we donotexpectanyupperboundestimateonthefirsteigenvalueof􏰂f withoutanyadditional assumption (see the example in Sect. 2). In this case, we will give a sharp upper bound estimate on the first eigenvalue of􏰂f under the additional assuption that |∇f| is bounded.We also obtain the rigidity result on this estimate, as another Li–Wang type splitting theorem.

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We study integral and pointwise bounds on the curvature of gradient shrinking Ricci solitons. As applications, we discuss gap and compactness results for gradient shrinkers.

In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M × R2 has nonnegative isotropic curvature.

This note discusses the recent work of Huang-Schoen-Wang [13] on specifying total conserved quantities of a vacuum initial data set.