arXiv research

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundar…

According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation $e^Φ = \det D^2 Φ$ on proper convex cones. We…

Gradient estimates for special harmonic functions on manifolds.

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

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…

Lower bounds for eigenvalues on Bakry-Emery manifolds proven.

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

Paper generalizes Bakry-Émery calculus for curvature and applies to Markov chains.

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

The article explores the fundamental gap in Bakry-Emery geometry.

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

Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.

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

In this article, we prove gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians which satisfy an ellipticity assumption. As applications, we study completeness and finiteness of stochastically complete graphs under Bakry-Emery curvature bounds.

Proves Sobolev inequality on manifolds with specific curvature properties.

Extends a result on manifolds with specific curvature properties.

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

Study compares spectral properties of a specific tensor in geometry.

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

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 …

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

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…

Study preserves metrics with positive Bakry-Émry Ricci curvature via surgery.

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…

The study generalizes curvature bounds for manifolds with boundary.

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

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…

Eigenvalue estimates for Beltrami-Laplacian under specific curvature conditions.

New curvature tensor and matrices for connection graphs derived from Bakry-Émery curvature.

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

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…

In this paper, we prove two generalized versions of the Cheeger-Gromoll splitting theorem via the non-negativity of the Bakry-Émery Ricci curavture on complete Riemannian manifolds.

Comparison theorems in centro-affine differential geometry

Motivated and inspired by the recent work of Colding [5] and Colding-Minicozzi [6] we derive several families of monotonicity formulas for manifolds with nonnegative Bakry-Emery Ricci curvature, extending the formulas in [5, 6].

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 prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the …

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.

In this note, by extending the arguments of Ling (Illinois J. Math. 51, 853-860, 2007) to Bakry-Emery geometry, we shall give lower bounds for the first nonzero eigenvalue of the Witten-Laplacian on compact Bakry-Emery manifolds in the case that the Bakry-Emery Ricci curvature has some negative lower bounds and the man…

In this paper, we prove the extensions of Bonnet--Myers' type theorems obtained by Calabi and Cheeger--Gromov--Taylor via Bakry--Emery Ricci curvature, which generalize the results of \cite{FG, Lim1, Wan, Wang, WW, Wu}.

We prove a comparison theorem on the first Neumann eigenvalue on Bakry-Emery manifolds. Examples are constructed to illustrate the sharpness of the result. A linear explicit lower bound is also proved. We also discuss the asymptotic sharpness of such a result.

We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^…

Analyticity of heat equation extended to Bakry-Émery Ricci curvature manifolds.

Flow on weighted graphs sharpens Bakry-Émery curvature.

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…

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…

We develop a general framework on Dirichlet spaces to prove a weak form of the Bakry-Émery estimate and study its consequences. This estimate may be satisfied in situations, like metric graphs, where generalized notions of Ricci curvature lower bounds are not available.

Study spectral properties on manifolds with conical singularities, proving new inequalities.