arXiv research

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 compares spectral properties of a specific tensor in geometry.

The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.

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

We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-Émery Ricci tensor bounds, extending Wei-Wylie's comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates and volume growth estimates on smoo…

In this paper, we first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded from below and $|f|$ is bounded. Based on this, we define a Myers-type compactness theorem by generalizing the results of Cheeger, Gromov, and Taylor and of Wan for the Bakry-Eme…

For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.

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…

For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\partial_r f$ is bounded from below, generalizing the classical ones (i.e. when $f$ is constant). This leads to ext…

In this paper, we study stable weighted minimal hypersurfaces in manifolds with nonnegative Bakry-Emery Ricci curvature. We will give some geometric and topological applications. In particular, we give some partial classification of complete 3-manifolds with nonnegative Bakry-Emery Ricci curvature assuming that $f$ is …

For smooth metric measure spaces $(M, g, e^{-f} dvol)$ we prove a Liuoville-type theorem when the Bakry-Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case $f$ is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spac…

In this paper we prove a new Myers' type diameter estimate on a complete connected Reimannian manifold which admits a bounded vector field such that the Bakry-Émery Ricci tensor has a positive lower bound. The result is sharper than previous Myers' type results. The proof uses the generalized mean curvature comparison …

Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.

Study on Kähler-Ricci solitons on toric manifolds, proving rigidity.

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

The study bounds heat kernel for manifolds with specific curvature conditions.

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 consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature $Ric_f^m$ in timelike directions. We prove that they still hold when $m$ is finite, and when $m$ is infinite, they hold under the additional assumptio…

The Bakry-Emery generalized Ricci tensor arises in scalar-tensor gravitation theories in the conformal gauge known as the Jordan frame. Recent results from the mathematics literature show that standard singularity and splitting theorems that hold when an energy condition is applied in general relativity also hold when …

Gradient estimate for harmonic functions with boundary condition proved.

We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry-Émery-Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry-Émery-Ricci tensor and obtain singularity theorems of a cosmol…

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike spli…

The Bakry-Emery Ricci tensor of a metric-measure space (M,g,e^{-f}dv_{g}) plays an important role in both geometric measure theory and the study of Hamilton's Ricci flow. Under a uniform positivity condition on this tensor and with bounded Ricci curvature we show the underlying space has finite f-volume. As a consequen…

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…

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth …

We establish a lower bound for the real eigenvalues of a Laplace-Beltrami operator with an $L^\infty$-drift term. We make no assumptions that the operator is self-adjoint or that the drift has any additional regularity. In the case where the operator is self-adjoint, this establishes a lower bound on the spectrum witho…

Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.

New decay estimates for scalar curvature of steady gradient Ricci solitons.

The paper extends topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-Émery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvat…

The paper extends a splitting theorem for a specific type of tensor in Riemannian geometry.

In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometri…

Proves volume comparison and monotonicity for Bakry-Émery Ricci curvature.

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…

Derives gradient bounds for f-heat equations on manifolds with Bakry-Emery Ricci curvature.

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

We prove the Bochner-Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li-Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry-Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.

The paper sets limits on the number of ends of certain geometric structures.

In this paper we give some results on the topology of manifolds with $\infty$-Bakry-Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifold…

Proves Sobolev inequality on manifolds with specific curvature properties.

Extends a result on manifolds with specific curvature properties.

In this paper we derive Cheng-Yau, Li-Yau, Hamilton estimates for Riemannian manifolds with Bakry-Emery Ricci curvature bounded from below, and also global and local upper bounds, in terms of Bakry-Emery Ricci curvature, for the Hessian of positive and bounded solutions of the weighted heat equation on a closed Riemann…

In this paper we study some splitting properties on complete noncompact manifolds with smooth measures when $\infty$-dimensional Bakry-Émery Ricci curvature is bounded from below by some negative constant and spectrum of the weighted Laplacian has a positive lower bound. These results extend the cases of Ricci curvatur…

Eigenvalue estimates for Beltrami-Laplacian under specific curvature conditions.

Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.

We characterize lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy on the $L^2$-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable $1$-form in the sense of \cite{giglinonsmooth}. This ex…