Ben Andrews, Lei Ni
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.
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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 Riemannian manifold.
Spectral comparison and splitting theorems for the infinity-Bakry-Emery Ricci curvature
本文在谱意义下下界条件成立时，证明了加权流形中的直径比较、全局加权体积比较及分裂定理，扩展了Antonelli-Xu和Antonelli-Pozzetta-Xu的谱定理至加权流形，并补充了Chu-Hao和Yeung的谱定理。
Let $L=Δ-\nablaφ\cdot\nabla$ be a symmetric diffusion operator with an invariant measure $dμ=e^{-φ}dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $|\nabla φ|\leqθ$ and $\infty$-dimensional Bakry-Émery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator $L$ on this kind manifold, and thereby generalize a Cheng's result on the Laplacian case (Math. Z., 143 (1975) 289-297).
This article is a brief presentation of results surrounding the fundamental gap. We begin by recalling Bakry-Emery geometry and demonstrate connections between eigenvalues of the Laplacian with the Dirichlet and Neumann boundary conditions. We then show a connection between the fundamental gap and Bakry-Emery geometry, concluding with a presentation of the key ideas in Andrews's and Clutterbuck's proof of the fundamental gap conjecture. We conclude with a presentation of results for the fundamental gap of triangles and simplices.
We demonstrate lower bounds for the eigenvalues of compact Bakry-Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalised maximum principle which allows gradient estimates in the Riemannian setting to be directly applied to the Bakry-Emery setting. Lower bounds for all eigenvalues are demonstrated using heat kernel estimates and a suitable Sobolev inequality.
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 spaces for comparison are variational problems coming from optimal control theory. As an application we establish the sharp measure contraction property for 3-Sasakian manifolds satisfying a suitable curvature bound.
Under the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $C^α W^{1, q}$ harmonic radius for manifolds with bounded Bakry-Émery Ricci curvature when the gradient of the potential is bounded. Under these conditions, the regularity that can be imposed on the metrics under harmonic coordinates is only $C^αW^{1,q}$, where $q>2n$ and $n$ is the dimension of the manifolds. This is almost 1 order lower than that in the classical $C^{1,α} W^{2, p}$ harmonic coordinates under bounded Ricci cur…
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 $L^p$-Liouville theorems in terms of the lower bound of Bakry-Émery Ricci curvature and the bound of function $f$, which generalize the classical Ricci curvature case and the $N$-Bakry-Émery Ricci curvature case.
Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds I: Bakry-Émery curvature bounded below
本文提出一种统一方法，将Bakry-Émery曲率有下界的黎曼流形上半线性抛物型方程的微分Harnack不等式推导转化为求解常微分方程组，并由此得到对数型方程和Yamabe型方程的精确估计及Liouville型定理。
We revisit Villani's approach to the study of hypocoercive diffusion operators by applying a variant of the Bakry-Émery machinery. The method relies on a generalized Bakry-Émery type criterion that applies to Kolmogorov type operators. Our approach includes as a special case the kinetic Fokker-Planck equation and allows, in that case, to recover hypocoercive estimates first obtained by Villani.
The goal of this paper is to give a non-local sufficient condition for generalized Poincaré inequalities, which extends the well-known Bakry-Emery condition. Such generalized Poincaré inequalities have been introduced by W. Beckner in the gaussian case and provide, along the Ornstein-Uhlenbeck flow, the exponential decay of some generalized entropies which interpolate between the $L^2$ norm and the usual entropy. Our criterion improves on results which, for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock type perturbation results. In a second step, we apply the same …
A discrete Bakry-Emery method and its application to the porous-medium equation
本文提出了一种离散Bakry-Emery方法，用于证明一维离散孔隙介质方程相对熵的指数衰减，并引入了新的非线性求和-分部公式。
We prove higher summability and regularity of Γ(f) for functions f in spaces satisfying the Bakry-Émery condition BE(K,\infty). As a byproduct, we obtain various equivalent weak formulations of BE(K,N) and we prove the Local-to-Global property of the RCD*(K,N) condition in locally compact metric measure spaces (X,d,m), without assuming a priori the non-branching condition on the metric space.
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 obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry-Emery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear grow…
Inverse problems for the Bakry-Émery Laplacian on manifolds with boundary -- uniqueness and non-uniqueness
本文研究了带边界 Riemannian 流形上 Bakry-Émery Laplacian 两个相关反问题的唯一性与非唯一性，通过 Dirichlet-to-Neumann 映射恢复度量与权重在边界处的 Taylor 系数。
We prove stability estimates for the Bakry-Emery bound on Poincaré and logarithmic Sobolev constants of uniformly log-concave measures. In particular, we improve the quantitative bound in a result of De Philippis and Figalli asserting that if a $1$-uniformly log-concave measure has almost the same Poincaré constant as the standard Gaussian measure, then it almost splits off a Gaussian factor, and establish similar new results for logarithmic Sobolev inequalities. As a consequence, we obtain dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs are bas…
Bounds for the optimal constant of the Bakry-Émery $Γ_2$ criterion inequality on $ RP^{d-1}$
本文证明了球面与实射影空间上 Bakry-Émery $Γ_2$ 准则不等式最优常数 $Λ_d$ 的上下界，从而扩大了展现 Fisher 信息单调性的交互势范围，并具体计算出 $Λ_3$ 介于 $5.5$ 和 $5.739$ 之间。
A Spectral Splitting Theorem for the $N$-Bakry Émery Ricci tensor
将 Cheeger-Gromoll 分裂定理的谱推广到光滑度量测度空间，在加权黎曼流形具有至少两个端且满足谱条件时，证明其等距分裂为 $\mathbb{R}\times X$，并恢复已知结果的锐利常数。
A Bakry-Émery approach to Lipschitz transportation on manifolds
该研究在加权黎曼流形满足特定曲率-维数条件下，通过扩散输运映射证明了权重测度与其对数Lipschitz扰动间存在全局Lipschitz输运映射，并得到Poincaré不等式保持性、Monge映射增长估计等推论。
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.
Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1< p< \infty$ on a compact Bakry-Émery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+\nabla^2 f \geq κ\, g$ for some $κ\in \mathbb{R}$.
We prove the Bochner-Weitzenboeck 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-Emery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.
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