arXiv research

Lower bounds for eigenvalues on Bakry-Emery manifolds proven.

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

Proves Sobolev inequality on manifolds with specific curvature properties.

Extends a result on manifolds with specific curvature properties.

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 study generalizes curvature bounds for manifolds with boundary.

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

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

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

Eigenvalue estimates for Beltrami-Laplacian under specific curvature conditions.

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…

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…

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…

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…

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

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

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.

Comparison theorems in centro-affine differential geometry

Paper proves new theorems about curvature in weighted manifolds.

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

The study connects Kato bounds to finite-dimensional RCD spaces.

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.

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…

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].

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 paper we prove that on a complete smooth metric measure space with non-negative Bakry-Émery-Ricci curvature if the space of weighted L^2 harmonic one-forms is non-trivial then the weighted volume of the manifold is finite and universal cover of the manifold splits isometrically as the product of the real line w…

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

We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…

In this paper, we develop a new approach to prove the $W$-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds and give a natural geometric interpretation of a quantity appeared in the $W$-entropy formula. Then we prove the $W$-entropy formula for the Witten Laplacian on compact Riemannia…

Optimizes maps and eigenvalues on manifolds.

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

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…

The paper classifies holonomy groups of special connections on manifolds.

The paper reformulates Bakry-Émery curvature on graphs using eigenvalues.

In this paper, we will present some characterizations for the upper bound of the Bakry-Emery curvature on a Riemannian manifold by using functional inequalities on path space. Moreover, some characterizations for general lower and upper bounds of Ricci curvature are also given, which extends the recent results derived …

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

We prove a mixed curvature analogue of Gromov's almost flat manifolds theorem for upper sectional and lower Bakry-Emery Ricci curvature bounds.

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 …

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

The paper proves conditions for Einstein solitons to split into line and manifold.

We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-Émery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold. Specializing to the Bakry-Émery Ricci curvature condition,…

Study semilinear equations on weighted manifolds to prove rigidity.

Improved Liouville theorems for ancient solutions to V-harmonic map heat flows.

Let $L=Δ-\nabla\varphi\cdot\nabla$ be a symmetric diffusion operator with an invariant measure $dμ=e^{-\varphi}dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $|\nabla \varphi|\leqθ$ and $\infty$-dimensional Bakry-Émer…

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

We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-Émery curvature. The model…

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