In this paper, we give some C 0 or C 1 limit theorems for total scalar curvatures. More precisely, we show that the lower bound of the total scalar curvatures on a closed manifold is preserved under the C 0 or C 1 convergence of the Riemannain metrics under some conditions. Moreover, we give some counterexamples to the above theorem on an open manifold.

No abstract

In this paper, for a compact manifold M with non-empty boundary ∂M , we give a Koiso-type decomposition theorem, as well as an Ebin-type slice theorem, for the space of all Riemannian metrics on M endowed with a fixed conformal class on ∂M . As a corollary, we give a characterization of relative Einstein metrics.

In this paper, we give a sufficient condition for a positive constant scalar curvature metric on a manifold with boundary to be a relative Yamabe metric, which is a natural relative version of the classical Yamabe metric. We also give examples of non-Einstein relative Yamabe metrics with positive scalar curvature.

In this paper, we study the Ricci flow on a closed manifold of dimension n ≥ 4 and finite time interval [0, T ) (T < ∞) on which the scalar curvature are uniformly bounded. We prove that if such flow of dimension 4 ≤ n ≤ 7 has finite time singularities, then every blow-up sequence of a locally Type I singularity has certain property. Here, locally Type I singularity is what Buzano and Di-Matteo defined.