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.
Develops framework for Bakry-Emery estimate on Dirichlet spaces.
problem Lack of Ricci curvature lower bounds in metric graphs.
method General framework on Dirichlet spaces for weak Bakry-Emery estimate.
result Establishes estimate in metric graphs where traditional curvature bounds are not applicable.
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.
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 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.
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.
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…
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.
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.
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.
Eigenvalue estimates without Bakry-Emery-Ricci bounds established.
problem Eigenvalue estimates for Laplace-Beltrami operators with drift.
method Establishes a lower bound for real eigenvalues without assuming self-adjointness or additional regularity of the drift.
result Establishes a lower bound on the spectrum of Laplace-Beltrami operators without assuming a lower bound for the Bakry-Emery Ricci tensor.
Establishes new lower bounds for the first eigenvalue of the weighted p-Laplacian.
problem Finding lower bounds for the first eigenvalue of the weighted p-Laplacian on metric measure spaces.
method Uses weighted p-Bochner and p-Reilly formulas to derive estimates under curvature conditions.
result Proves new lower bounds for the first eigenvalue with specific curvature assumptions.
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…
Study spectral properties on manifolds with conical singularities, proving new inequalities.
problem Analyzing spectral properties and geometric inequalities on manifolds with conical singularities.
method Develops new inequalities for manifolds with conical singularities, not covered by existing methods.
result Proves a Bakry-Émery inequality, Hardy inequality, and spectral gap estimate.
Gradient estimate for harmonic functions with boundary condition proved.
problem Proving gradient estimates for harmonic functions with boundary conditions.
method Using weighted f-harmonic functions and infinite dimensional Bakry-Emery Ricci tensor. result Gradient estimates for positive f-harmonic functions with Dirichlet boundary condition. New curvature measure on graphs improves diameter and eigenvalue bounds.
problem Improving curvature bounds on graph structures.
method Hybrid curvature definition on variable neighborhoods.
result Gradient estimates and curvature bounds proven.
For smooth metric measure spaces (M,g,e−fdvol) 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…
The paper extends curvature bounds for nonsymmetric diffusion operators.
problem Characterizing lower bounds for nonsymmetric diffusion operators.
method Using convexity of entropy on the L2-Wasserstein space and curvature-dimension condition. result New curvature bounds and estimates for nonsymmetric diffusion operators.
Let Ω be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed embedded f-minimal hypersurfaces contained in Ω. Using this estimat…
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…
We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: ut=Δφ(up) associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the m-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which…
The study shows black hole horizons at low temperatures have limited topology.
problem Topology of black hole horizons at low temperatures.
method Almost nonnegative generalized m-Bakry-Émery Ricci curvature, diameter upper bound, volume lower bound. result Low temperature black hole horizons have limited topology.
Improved Liouville theorems for ancient solutions to V-harmonic map heat flows.
problem Establishing Liouville theorems for ancient solutions to V-harmonic map heat flows.
method Refined gradient estimates and exponential growth conditions.
result Better Liouville theorems for ancient solutions to V-harmonic map heat flows.
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 Mn is a complete, noncompact Riemannian manifold with nonnegative N-Bakry Émery Ricci curvature where N>n, then Hn−1(M,Z) is 0. Sharp lower bounds for eigenvalues on specific manifolds proved.
problem Proving sharp lower bounds for eigenvalues on compact Bakry-Emery manifolds.
method Using weighted p-Laplacian and Bakry-Emery manifolds with Ricci curvature lower bound. result Sharp lower bound estimates for the first nonzero eigenvalue proved.
The paper studies gradient estimates and Liouville theorems for a nonlinear elliptic equation on metric measure spaces.
problem Gradient estimates and Liouville theorems for positive solutions to a specific nonlinear elliptic equation.
method Analyzes the nonlinear elliptic equation \( \Delta_{V}u^{m} + \mu(x)u + p(x)u^{\alpha} = 0 \) on smooth metric measure spaces with bounded Bakry-Émery curvature.
result Establishes gradient estimates and related Liouville theorems and Harnack inequalities.
We study Bakry-Emery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the Γ2 operator may not be bounded from below but the horizontal Bakry-Emery curvature is. As we prove it, under a bracket generating condition, this weaker …
The paper provides new gradient estimates and Liouville theorems for specific Poisson equations.
problem Addressing gradient estimates and Liouville theorems for specific Poisson equations on smooth metric measure spaces.
method Analyzing the parabolic equation \(u_t = \Delta_f u + F(u)\) on smooth metric measure spaces with Bakry-Émery curvature bounded from below.
result New gradient estimates and Liouville theorems for positive or bounded solutions to the equation when \(F\) is specific functions.
The paper proves a compactness theorem for spaces with Bakry-Emery Ricci tensor.
problem Compactness in spaces with Bakry-Emery Ricci tensor.
method Proving f-mean curvature comparison and defining a Myers-type compactness theorem. result Improves a result from Soylu by using a weaker condition on f′(t). 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 Γ2 criterion inequalities and a modified inequality. result Yields a family of sharp Sobolev inequalities and monotonicity of Tsallis entropy.
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.
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 proves inequalities under Bakry-Émery-Ricci curvature bounds.
problem Proving functional inequalities under lower Bakry-Émery-Ricci curvature bounds.
method Lower m-Bakry-Émery-Ricci curvature bounds with ε-range. result Proves Cheng type inequality and local Sobolev inequality.
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.
In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry-Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (J. Differential Geom…
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.
Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.
problem Eigenvalue problems for Bi-drifted Laplacian on compact manifolds with boundary conditions.
method Obtained lower bounds using specific curvature conditions.
result Lower bounds for the first eigenvalue of Bi-drifted Laplacian.
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 N-Bakry-Emery Ricci tensor. result Established diameter and volume comparisons for tensors with positive lower bounds.
New method bounds graph curvature with exceptions.
problem Global curvature bounds with no exceptions.
method Perpetual cutoff method.
result Sharp upper bounds on distance to exception sets.
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 LP-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 f is …