In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.

Integral formulas in riemannian geometry

From the Ricci flow evolution equation to vanishing theorems for Ricci solitons

On generalized special pseudo Ricci symmetric manifolds

We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 n\mu^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the relation between $\mu$ and the upper or lower bound of the normal curvatures of the evolving hypersurface at time 0.

Gaussian Mean curvature flow

In this paper we give an explicit bound of $\Delta_{g(t)}u(t)$ and the local curvature estimates for the Ricci-harmonic flow under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author \cite{LY1}, whose stable points give Ricci-flat metrics on a complete manifold, and which is very close to the $(K, N)$-super Ricci flow recently defined by Xiangdong Li and Songzi Li \cite{LL2014}. Next we propose a conjecture for Einstein's scalar field equations motivated by a result in the first part and the bounded $L^{2}$-curvature conjecture recently solved by Klainerman, Rodnianski and Szeftel \cite{KRS2015}. In the last two parts of this paper, we discuss two notions of "Riemann curvature tensor" in the sense of Wylie-Yeroshkin \cite{KW2017, KWY2017, Wylie2015, WY2016}, respectively, and Li \cite{LY3}, whose "Ricci curvature" both give the standard Bakey-\'Emery Ricci curvature \cite{BE1985}, and the forward and backward uniqueness for the Ricci-harmonic flow.

Local curvature estimates for the Ricci-harmonic flow

A curvature obstruction to covering a deformation of con- nections of a Levi-Civita connection of a Riemannian metric by a metric de- formation is derived and used to study deformations of connections related to the deformation of connections arising from a conformal deformation of a Riemannian metric. In this note we assume familiarity with the difference tensor of two con- nections, as explained for instance in (5, §5.4). Everything will, for conven- ience, be taken to be C. Given a manifold M", n>2, and a Riemannian metric g for M, a conformal deformation git) = e ''g of g results in the following classical formula for the Levi-Civita connection D of git):

https://www.ams.org/journals/proc/1975-052-01/S0002-9939-1975-0383293-4/S0002-9939-1975-0383293-4.pdf

Integral formulas in riemannian geometry

Citations

From the Ricci flow evolution equation to vanishing theorems for Ricci solitons

On generalized special pseudo Ricci symmetric manifolds

Gaussian Mean curvature flow

Local curvature estimates for the Ricci-harmonic flow

A curvature obstruction to covering deformations of connections by metric deformations

Related Papers (5)

Riemannian Geometry of Contact and Symplectic Manifolds

The entropy formula for the Ricci flow and its geometric applications

Certain conditions for a Riemannian manifold to be isometric with a sphere

A class of almost contact riemannian manifolds

Curvature and Betti numbers