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.
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. 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.
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 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.
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.
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.
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 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.
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 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.
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.
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.
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.
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 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 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 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.
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 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 …
Paper explores entropic curvature in Markov chains, comparing it to other curvatures.
problem Comparing entropic curvature to other curvatures in Markov chains.
method Adapted Γ-calculus for θ-curvatures, explicit lower bounds, curvature perturbation.
result Entropic curvature differs significantly from other curvature notions.
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.
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
The paper reformulates Bakry-Émery curvature on graphs using eigenvalues.
problem Analyzing curvature on weighted graphs.
method Reformulating curvature as the smallest eigenvalue of a rank one perturbation of the curvature matrix.
result The curvature function is analytic, strictly monotone increasing, and concave until a threshold, after which it is constant.
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 study some function-theoretic properties on a complete smooth metric measure space (M,g,e−fdv) 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^…
Study compares sub-Riemannian curvature to optimal control variational problems.
problem Comparing sub-Riemannian curvature to optimal control variational problems.
method Introducing sub-Riemannian Bakry-Émery curvature and proving sub-Laplacian comparison theorems.
result Established sharp measure contraction property for 3-Sasakian manifolds.
The study introduces a new curvature concept for weighted graphs and applies it to warped products.
problem Establishing curvature bounds for doubly warped product graphs.
method Developed a new notion of curvature for weighted graphs and applied it to warped products, establishing bounds in terms of constituent graph curvatures.
result Established curvature bounds for $\left(R_1,R_2
ight)$-doubly warped products of smooth measure spaces.
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
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 Lp spaces. 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…
Graphs satisfy Li-Yau inequality under CD(0,n) curvature condition.
problem Proving Li-Yau inequality for graphs under CD(0,n) condition. method Introduced modified heat equation and used Bakry Emery curvature condition.
result Proved Li-Yau inequality −Δut≤2tn under CD(0,n) condition. 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.
Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.
problem Spectral rigidity of hypercube graphs
method Interplay between global spectral embedding and local curvature analysis
result Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.
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 …
The paper explores curvatures on graphs and their implications for Ricci flatness.
problem Comparing and understanding different curvature notions on graphs and their implications for Ricci flatness.
method Analyzing Ollivier Ricci curvature and Bakry-Émery curvature on combinatorial graphs, investigating graph products, and proving curvature properties.
result Non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under specific conditions.
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.
We study the Bakry-Émery curvature function KG,x:(0,∞]→R of a vertex x in a locally finite graph G systematically. Here KG,x(N) is defined as the optimal curvature lower bound K in the Bakry-Émery curvature-dimension inequality $CD(\mathcal{K},\ma…
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.
New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the…
For Riemannian manifolds with a measure (M,g,e−fdvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to ext…
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.
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.
We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result i…
Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.
problem Proving properties of graphs with specific curvature conditions.
method Graph-theoretic modified nonlinear heat-flow method, including point-mass consequences and diffusive exit-time control.
result Volume doubling and Poincaré inequalities for graphs with nonnegative Bakry-Émery curvature.