Dezhong Chen

Bio: Dezhong Chen is an academic researcher from McMaster University. The author has contributed to research in topics: Ricci curvature & Scalar curvature. The author has an hindex of 1, co-authored 1 publications receiving 13 citations.

Remarks on complete non-compact gradient Ricci expanding solitons

Abstract: In this paper, we study gradient Ricci expanding solitons (X,g) satisfying Rc = cg + D2f, where Rc is the Ricci curvature, c 0 on X unless (X,g) is Ricci flat. We also show that there is exponentially decay for scalar curvature on a complete non-compact expanding soliton with its Ricci curvature being e-pinched.

On the conditions to control curvature tensors of Ricci flow

Abstract: An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M n , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that $${L^\frac{n+2}{2}}$$ norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C 0 bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M n × [0, T), the curvature tensor stays uniformly bounded on M n × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.

Properties of complete non-compact Yamabe solitons

Abstract: In this article, we first study the local volume estimate of the complete non-compact Yamabe soliton. Then we study the behavior of the potential function of the steady Yamabe soliton with positive Ricci curvature. We also study the scalar curvature decay of steady and expanding Yamabe solitons with Ricci pinching condition.

Expanding Ricci solitons with pinched Ricci curvature

Abstract: In this paper, we prove that there is no expanding gradient Ricci soliton with (positively) pinched Ricci curvature in dimension three. The same result is true for higher dimensions with the extra decay condition about the full curvature.

Eigenvalues of the drifting Laplacian on complete noncompact Riemannian manifolds

Abstract: In this paper, we investigate eigenvalues of the eigenvalue problem with Dirichlet boundary condition of the drifting Laplacian on an n -dimensional, complete noncompact Riemannian manifold. Some estimates for eigenvalues are obtained. By utilizing Cheng and Yang recursion formula, we give a sharp upper bound of the k th eigenvalue. As we know, product Riemannian manifolds, Ricci solitons and self-shrinkers are some important Riemannian manifolds. Therefore, we investigate the eigenvalues of the drifting Laplacian on those Riemannian manifolds. In particular, by some theorems of classification for Ricci solitons, we can obtain some eigenvalue inequalities of drifting Laplacian on the Ricci solitons with certain conditions.