arXiv research

The paper examines properties of generalized τ-quasi Ricci-harmonic metrics and proves rigidity results.

In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In add…

Paper studies equations for a type of soliton with applications.

Study eigenvalue variation in $(p,q)$-Laplacian on Ricci-harmonic flow.

The paper studies gradient Ricci-Harmonic solitons on warped product manifolds.

Static spacetimes are stable attractors in a flow equation.

The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.

In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of…

In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fernández-López and García-Río (Q. J. Math. 61, 319--327, 2010), Futaki and Sano (Asian J. Math. 17, 17-…

We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times $t$ \in[0, T), then the curvature tensor has to be uniformly bounded as well.

New flow for G2-structures helps find torsion-free structures.

Paper bounds local curvature for Ricci-harmonic flow under Ricci bounded condition.

The paper examines triviality of Ricci-Bourguignon harmonic solitons.

In the present paper, by using estimates for the generalized Ricci curvature, we shall give some gap theorems for Ricci-harmonic solitons showing some necessary and sufficient conditions for the solitons to be harmonic-Einstein. Our results may be regarded as a generalization of recent works by H. Li, and M. Fernandez-…

The paper estimates curvature for a specific flow on manifolds.

Study new Einstein-like metrics and their properties.

Study solutions and singularities of G2-structures flows on specific manifolds.

Advances geometric structure flows, proving short-time existence and uniqueness for various flows.

We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In provin…

Proves uniqueness of Ricci flow with scaling invariant estimates.

Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.

We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures.

The paper considers the Ricci flow, coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analog of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy fu…

The paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold $M$ evolving under the Ricci flow, coupled with the harmonic map flow between $M$ and a second manifold $N$. We prove Li-Yau type Harnack inequalities and we consider the cases when $M$ is a complete manifold …

We estimate the heat kernel on a closed Riemannian manifold $M$, with $dim(M)\geq 3$, evolving under the Ricci-harmonic map flow and the result depends on some constants arising from a Sobolev imbedding theorem. In a special case, when the scalar curvature satisfies a certain natural inequality, we obtain, as a corolla…

In this article we derive Harnack estimates for conjugate heat kernel in an abstract geometric flow. Our calculation involves a correction term D. When D is nonnegative, we are able to obtain a Harnack inequality. Our abstract formulation provides a unified framework for some known results, in particular including corr…

Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.

We characterize $η$-Ricci solitons $(g,ξ,λ,μ)$ in some special cases when the $1$-form $η$, which is the $g$-dual of $ξ$, is a harmonic or a Schrödinger-Ricci harmonic form. We also provide necessary and sufficient conditions for $η$ to be a solution of the Schrödinger-Ricci equation and point out the relation between …

We study the Ricci flow of initial metrics which are C^0-perturbations of the hyperbolic metric on H^n. If the perturbation is bounded in the L^2-sense, and small enough in the C^0-sense, then we show the following: In dimensions four and higher, the scaled Ricci harmonic map heat flow of such a metric converges smooth…

Constructs explicit solutions to Spin(7)-structures gradient flow.

The paper classifies flows of SU(2)-structures on 4-manifolds.

B List has proposed a geometric flow whose fixed points correspond to solutions of the static Einstein equations of general relativity. This flow is now known to be a certain Hamilton-DeTurck flow (the pullback of a Ricci flow by an evolving diffeomorphism) on RxM^n. We study the SO(n) rotationally symmetric case of Li…

Smooth Ricci flows from metric spaces can be extended to smooth solutions.