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…
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48 results for “Bakry-Émery-Ricci”
Gradient estimates, Bakry-Emery Ricci curvature and ellipticity for unbounded graph Laplaciansmath.DG
Gradient estimates for unbounded graph Laplacians under Bakry-Emery curvature.
problem Gradient estimates for unbounded graph Laplacians.
method Proving gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians with ellipticity assumption.
result Gradient estimates and applications to completeness and finiteness of stochastically complete graphs.
Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.
problem Analyzing the behavior of Ricci flow on finite graphs.
method Local existence and uniqueness proof for solutions of the Bakry-Émery Ricci flow.
result Local existence and uniqueness of solutions to the Ricci flow 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.
problem Eigenvalue estimates on Bakry-Emery manifolds.
method Generalised maximum principle and heat kernel estimates.
result Lower bounds for all eigenvalues proven.
The paper extends Bonnet-Myers theorems using Bakry-Emery Ricci curvature.
problem Proving extensions of Bonnet-Myers theorems with a new curvature measure.
method Using Bakry-Emery Ricci curvature to extend Bonnet-Myers theorems.
result Proves extensions of Bonnet-Myers theorems.
The paper extends topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.
problem Generalizing topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.
method Study of the Splitting Theorem and geodesic loops to infinity property in relation to spaces with nonnegative N-Bakry Émery Ricci curvature.
result If is a complete, noncompact Riemannian manifold with nonnegative N-Bakry Émery Ricci curvature where , then is 0.
Paper generalizes Bakry-Émery calculus for curvature and applies to Markov chains.
problem Formulating both Bakry-Émery and entropic curvature simultaneously.
method Generalization of Bakry-Émery calculus, new measure optimality criterion, dimension parameter in entropic curvature.
result Diameter estimates for Markov chains with strictly positive entropic curvature and spectral gap.
Study improves comparison geometry for spaces with Bakry-Émery Ricci tensor bounds.
problem Improving comparison geometry for spaces with specific Ricci tensor bounds.
method Proved mean curvature and volume comparison estimates on smooth metric measure spaces.
result Generalized diameter, eigenvalue, and volume growth estimates.
The paper proves a compactness theorem for spaces with Bakry-Emery Ricci tensor.
problem Compactness in spaces with Bakry-Emery Ricci tensor.
method Proving -mean curvature comparison and defining a Myers-type compactness theorem.
result Improves a result from Soylu by using a weaker condition on .
The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.
problem Extending the Bakry-Émery-Ricci tensor and proving comparison theorems.
method Generalizations of the drifted Laplacian and Bakry-Émery-Ricci tensor, mean curvature comparison theorem, Myers-type theorem, Cheeger-Gromoll splitting theorem.
result Proved a version of the mean curvature comparison theorem and its consequences.
The article explores the fundamental gap in Bakry-Emery geometry.
problem The fundamental gap in Bakry-Emery geometry.
method Recalled Bakry-Emery geometry and connected eigenvalues with boundary conditions. Showed a connection between fundamental gap and Bakry-Emery geometry.
result Presented key ideas in Andrews's and Clutterbuck's proof of the fundamental gap conjecture.
Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.
problem Establishing improved logarithmic Sobolev inequalities and monotonicity of Tsallis entropy.
method Proving a one-parameter family of weighted Bakry-Émery criterion inequalities and a modified inequality.
result Yields a family of sharp Sobolev inequalities and monotonicity of Tsallis entropy.
Derives gradient bounds for f-heat equations on manifolds with Bakry-Emery Ricci curvature.
problem Gradient estimates for positive solutions of f-heat equations on manifolds with specific curvature conditions.
method Applies Li-Yau gradient estimates to positive solutions of the f-heat equation on closed manifolds with Bakry-Emery Ricci curvature bounded below.
result Derives Li-Yau gradient bounds for positive solutions of the f-heat equation.
Graphs with some negative Bakry-Émery curvature have explicit diameter bounds.
problem Understanding graphs with non-constant Bakry-Émery curvature.
method Proving distance bounds for graphs with positive Bakry-Émery curvature except for a finite or infinite set of non-positively curved vertices.
result Explicit upper bounds for the diameter of graphs with non-constant Bakry-Émery curvature.
The paper proves inequalities under Bakry-Émery-Ricci curvature bounds.
problem Proving functional inequalities under lower Bakry-Émery-Ricci curvature bounds.
method Lower -Bakry-Émery-Ricci curvature bounds with -range.
result Proves Cheng type inequality and local Sobolev inequality.
The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.
problem Geometric comparisons on metric measure spaces with specific tensor bounds.
method Integral radial Bakry-Émery Ricci tensor bounds and potential function/gradient bounds.
result Diameter and eigenvalue estimates on smooth metric measure spaces.
Proves Sobolev inequality on manifolds with specific curvature properties.
problem Proving Sobolev inequality on manifolds with asymptotically nonnegative Bakry-Émery Ricci curvature.
method Density and Bakry-Émery Ricci curvature.
result Proves Sobolev inequality on manifolds with asymptotically nonnegative Bakry-Émery Ricci curvature.
Extends a result on manifolds with specific curvature properties.
problem Extending a result on manifolds with nonnegative Bakry-Émery Ricci curvature.
method Extends a recent result by S. Brendle to manifolds with densities and nonnegative Bakry-Émery Ricci curvature.
result Extends a result on manifolds with specific curvature properties.
The paper characterizes upper bounds of Bakry-Emery curvature.
problem Characterizing upper bounds of Bakry-Emery curvature on Riemannian manifolds.
method Using functional inequalities on path space.
result Characterizations for upper bounds of Bakry-Emery curvature are provided.
Study compares spectral properties of a specific tensor in geometry.
problem Comparing spectral properties of a specific tensor in geometry.
method Diameter and global weighted volume comparison with a positive lower bound on the -Bakry-Emery Ricci tensor.
result Established diameter and volume comparisons for tensors with positive lower bounds.
Proves volume comparison and monotonicity for Bakry-Émery Ricci curvature.
problem Volume comparison and monotonicity for Bakry-Émery Ricci curvature.
method Relative volume comparison theorem for -bound of Bakry-Émery Ricci curvature and gradient of potential function.
result Modified proof for volume comparison and monotonicity of Kähler-Ricci flow.
The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
problem Analyzing modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
method Proving Laplacian comparison theorems using modified m-Bakry-Emery Ricci tensors under m≤1.
result Optimal conditions for modified m-Bakry-Emery Ricci tensors under m≤1 are derived.
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 is …
Graphs with Bakry-Emery curvature have bounded diameter.
problem Bounding the diameter of graphs with positive Ricci curvature.
method Proving diameter bounds using Bakry-Emery curvature and vertex degree.
result First Bonnet-Myers type theorem for unbounded graph Laplacians.
New diameter estimate on manifolds with positive Bakry-Émery Ricci tensor.
problem Estimating the diameter of manifolds with specific curvature conditions.
method Using generalized mean curvature comparison on excess function.
result Sharper diameter estimate than previous results.
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.
problem Preserving metrics with positive Bakry-Émry Ricci curvature after surgery.
method Established theorems for connected sums and surgeries along higher-dimensional spheres.
result Local surgery results for positive Bakry-Émry Ricci curvature.
In this paper we study some splitting properties on complete noncompact manifolds with smooth measures when -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 proves a compactness theorem for manifolds with specific curvature conditions.
problem Proving compactness theorems for manifolds with Bakry-Emery Ricci tensor.
method Using Bakry-Emery Ricci tensor and smooth measure.
result Generalized Myers compactness theorem proved.
The study generalizes curvature bounds for manifolds with boundary.
problem Proving curvature bounds for manifolds with boundary.
method Bakry-Émery curvature bounds and splitting theorems.
result Proves curvature bounds for manifolds with boundary.
Diameter theorems on Kähler and quaternionic Kähler manifolds under a positive lower curvature boundmath.DG
Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.
problem Determine diameter bounds for Kähler and quaternionic Kähler manifolds under curvature positivity.
method Define orthogonal Bakry-Émery tensor, study diameter theorems, and derive Bonnet-Myers type bounds.
result Sharper diameter bounds than in Riemannian case under specific curvature assumptions.
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.
problem Estimating eigenvalues of the Beltrami-Laplacian on manifolds with Bakry-Émery Ricci curvature.
method Using Bakry-Émery Ricci curvature conditions, the paper derives lower bounds for eigenvalues of the Beltrami-Laplacian.
result Lower bounds for eigenvalues of the Beltrami-Laplacian depend on curvature, gradient bounds, dimension, and diameter of the manifold.
New curvature tensor and matrices for connection graphs derived from Bakry-Émery curvature.
problem Deriving Buser-type bounds on eigenvalues of connection Laplacians.
method Reformulation of Bakry-Émery curvature through curvature matrices and tensor representations.
result Extension of curvature matrices to connection graphs, addressing eigenfunction challenges.
The study bounds heat kernel for manifolds with specific curvature conditions.
problem Estimating heat kernel for manifolds with Bakry-Émery Ricci curvature.
method Gaussian upper bound for heat kernel, proving L^1-Liouville property, deriving eigenvalue bounds.
result Established Gaussian upper bound for heat kernel, derived eigenvalue bounds.
The Bakry-Émery condition is satisfied for glued spaces of Riemannian manifolds.
problem Conditions for metric measure spaces to satisfy the Bakry-Émery condition.
method Sufficient and necessary conditions for the Bakry-Émery condition on glued spaces of Riemannian manifolds.
result The Bakry-Émery condition is strictly weaker than the RCD condition and the local dimension is not constant.
Classifies graphs with Bonnet-Myers sharp curvature and connects it to Bakry-Émery curvature.
problem Classifying graphs with specific curvature properties.
method Introducing Bonnet-Myers and Lichnerowicz sharpness in Ollivier Ricci curvature, combinatorial reformulation, and relating to Bakry-Émery curvature.
result Classification of graphs including hypercubes, cocktail party graphs, and Johnson graphs .
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
problem rigidity phenomena of comparison theorems
method study of centro-affine differential geometry
result examples of rigidity phenomena
In this paper, we first prove a compactness theorem for the space of closed embedded -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 -Laplacian on compact manifold with positive $m…
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 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…
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 with Bakry-Émery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the -heat equation, which leads to upper and lower Gaussian bounds on the -heat kernel. We also prove $L^…
Analyticity of heat equation extended to Bakry-Émery Ricci curvature manifolds.
problem Analyticity of solutions to heat equation under specific curvature conditions.
method Analyzes analyticity in time for smooth solutions on Riemannian manifolds with Bakry-Émery Ricci curvature.
result Analyticity extended to all gradient Ricci solitons and certain spaces.
Flow on weighted graphs sharpens Bakry-Émery curvature.
problem Sharp curvature in weighted graphs.
method Bakry-Émery curvature flow on mixed weighted graphs.
result Limits of curvature flow are curvature sharp.
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…