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Creators/Authors contains: "Tam, Luen-Fai"

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  1. ; (, Canadian Journal of Mathematics)
    null (Ed.)
    Abstract In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $$\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$$ for $t>0$ for some constant $${\alpha }>0$$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature . By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ . 
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  2. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown–York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them. 
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