arXiv research

The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.

We show that near-horizon geometries in the presence of a positive cosmological constant cannot exist with ring topology. In particular, de Sitter black rings with vanishing surface gravity do not exist. Our result relies on a known mathematical theorem which is a straightforward consequence of a type of energy conditi…

We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einst…

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-Émery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m…

We call a metric $m$-quasi-Einstein if $Ric_X^m$ (a modification of the $m$-Bakry-Emery Ricci tensor in terms of a suitable vector field $X$) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and…

The paper proves inequalities under Bakry-Émery-Ricci curvature bounds.

We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the…

The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.

Scalar-tensor gravitation theories, such as the Brans-Dicke family of theories, are commonly partly described by a modified Einstein equation in which the Ricci tensor is replaced by the Bakry-Émery-Ricci tensor of a Lorentzian metric and scalar field. In physics this formulation is sometimes referred to as the "Jordan…

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure spac…

We call a metric $m$-quasi-Einstein if $Ric_X^m$, which replaces a gradient of a smooth function $f$ by a vector field $X$ in $m$-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant met…

Study on 4D solitons with curvature constraints.

The study shows black hole horizons at low temperatures have limited topology.

New proof confirms noncompact locally conformally flat manifolds are compact.

The paper characterizes rigidity in harmonic-Ricci solitons.

Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.

We introduce a flow of Kähler structures over Fano manifolds with formal limit at infinite time a Kähler-Ricci soliton. This flow correspond to a Perelman's modified backward Kähler-Ricci type flow that we call Soliton-Kähler-Ricci flow. It can be generated by the Soliton-Ricci flow. We assume that the Soliton-Ricci fl…

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by c…

We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhof…

The paper studies modified Einstein tensors and their positivity properties on compact manifolds.

Sharp eigenvalue bounds and splitting for modified Ricci flow.

Study new Einstein-like metrics and their properties.

Sharp gradient estimates for a weighted p-Laplacian equation on metric measure spaces.

We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.

Study on type-D Ricci-flat metrics with Killing spinors and Killing vectors.

Study vanishing and splitting results on metric measure spaces with specific curvature bounds.

Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.

Enhances understanding of Kähler-Ricci flow singularities.

Modified Gibbs-Helmholtz equation geometric models for thermodynamics.

The study characterizes spacetime and modified gravity models using projective curvature tensor.

In this note, a modified Kähler-Ricci flow is introduced and studied. The main point is to show the flexibility of Kähler-Ricci flow and summarize some useful techniques.

We investigate the Kähler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a Kähler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.

New proof shows gradient Ricci solitons with harmonic Weyl tensor have at most three eigenvalues.

Let $(N,g)$ be an $n$-dimensional complete Riemannian manifold with nonempty boundary $\pt N$. Assume that the Ricci curvature of $N$ has a negative lower bound $Ric\geq -(n-1)c^2$ for some $c>0$, and the mean curvature of the boundary $\pt N$ satisfies $H\geq (n-1)c_0>(n-1)c$ for some $c_0>c>0$. Then a known result (s…

Study classifies 3D Einstein manifolds with cyclic Ricci tensor.

In this paper, we propose a method of studying the modified Kahler-Ricci flow on projective bundles and give the explicit equation from the view point of symplectic geometry.

Defines Ricci tensor for graded geometry manifolds.

The paper explores symmetries in Kähler manifolds using Ricci tensor properties.

The paper introduces and analyzes pseudo generalized Ricci-recurrent spacetimes in modified gravity.

The paper studies geometric constants under modified Ricci flows with variable parameters.

We show a quite simple second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow over Fano manifolds.

Among other results, a compact almost Kähler manifold is proved to be Kähler if the Ricci tensor is semi-negative and its length coincides with that of the star Ricci tensor or if the Ricci tensor is semi-positive and its first order covariant derivatives are Hermitian. Moreover, it is shown that there are no compact a…

New classification of gradient steady Ricci solitons with vanishing D-tensor.

Study compares spectral properties of a specific tensor in geometry.

The Bakry-Emery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Emery tensor. We show that the Bakry-Emery tensor is nondecreasing under a Riemannia…

The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.

Study of $η$-Ricci solitons on Kenmotsu 3-manifolds.