Multiplicative functional for the heat equation on manifolds with boundary

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Harnack inequalities on manifolds with boundary and applications (2009, methodology): “Since this condition is trivial for compact M , our proof below is much shorter than that in [5].”
On the role of convexity in functional and isoperimetric inequalities (2008, background): “In the Riemannian setting, it is known [44] (see also [29, 55]) that the gradient estimate of Lemma 2.”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, background): “We comment that in the Riemannian setting, it is known [77] (see also [44, 84]) that the gradient estimate of Lemma 3.”