Study lower weighted Ricci curvature bounds on manifolds with boundary.
problem Comparing geometric properties of manifolds with boundary under curvature constraints.
method Lower weighted Ricci curvature bound and boundary conditions.
result Various comparison geometric results under the curvature condition.
Study on stable minimal hypersurfaces under Ricci curvature constraints.
problem Stability of weighted minimal hypersurfaces under Ricci curvature bounds.
method Derive geometric consequences and prove a Schoen-Yau type criterion.
result Structure theorem for three-dimensional weighted manifolds of non-negative Ricci curvature.
Proves equivalence of two types of Ricci curvature bounds.
problem Equivalence of distributional and synthetic Ricci curvature bounds.
method Analyzes weighted Riemannian manifolds with specific smoothness conditions.
result Proves equivalence of Ricci curvature bounds under given conditions.
Unified proof of inequality for metric measure spaces with lower Ricci curvature bounds.
problem Establishing a Fenchel-Willmore-Chen inequality for metric measure spaces.
method Using a lower bound on the weighted intermediate Ricci curvature, extending previous results.
result Unified proof of the Fenchel-Willmore-Chen inequality.
Establishes lower bounds for weighted Ricci curvature and applies to heat flow inequalities.
problem Understanding weighted Ricci curvature and its implications.
method Introduces weighted intermediate Ricci curvature and establishes lower bounds with equivalent characterizations.
result Derives intrinsic-dimensional evolution variational inequalities and Wasserstein contraction estimates for the heat flow.
Paper extends Aronson-Bénilan estimates for porous medium equations on manifolds with negative curvature.
problem Estimating gradients for porous medium equations on manifolds with negative curvature.
method Develops Aronson-Bénilan gradient estimates for porous medium equations under lower bounds of N-weighted Ricci curvature with N<0. result Generalizes gradient estimates for porous medium equations to manifolds with negative curvature.
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.
The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.
problem Analyzing geometric properties of weighted manifolds under Ricci curvature bounds.
method Develops geometric analysis techniques on weighted Riemannian manifolds with lower 0-weighted Ricci curvature bounds. result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
Characterizes a new curvature bound with convexity of entropies.
problem Lowering Ricci curvature bounds with ε-range.
method Characterization through convexity of entropies over Wasserstein space.
result Derives various interpolation and functional inequalities.
We study Riemannian manifolds with boundary under a lower N-weighted Ricci curvature bound for N at most 1, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of …
Paper proves Liouville theorems for harmonic functions under specific curvature bounds.
problem Analyzing harmonic functions on manifolds with lower bounds of N-weighted Ricci curvature. method Uses Moser's iteration procedure to prove Liouville theorems.
result Establishes Liouville theorems for harmonic functions with sublinear growth and under weaker bounds of N-weighted Ricci curvature. The paper studies heat kernels on weighted Riemannian manifolds with curvature bounds.
problem Estimating heat kernels on weighted Riemannian manifolds with lower Ricci curvature bounds.
method Establishing parabolic Harnack inequalities, proving Gaussian bounds for heat kernels, and constructing Li-Yau-type gradient estimates.
result Gaussian upper and lower bounds for the heat kernel, Liouville theorem, uniqueness property, and eigenvalue bounds.
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…
Study compares manifolds with boundary under weighted Ricci curvature bounds.
problem Understand geometric properties of manifolds with boundary under lower weighted Ricci curvature bounds.
method Use lower N-weighted Ricci curvature bounds with ε-range to study comparison geometry. result Conclude splitting theorems and comparison geometric results for inscribed radius, volume, and eigenvalues.
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.
problem Investigating inequalities on Finsler manifolds with weighted Ricci curvature.
method Volume comparison, Bonnet-Myers theorem, Poincaré-Lichnerowicz inequality.
result Sharp lower bound for the first eigenvalue on Finsler manifolds.
The paper generalizes a Steklov eigenvalue inequality for substatic triples under non-negative Ricci curvature.
problem Estimating Steklov eigenvalues for substatic triples under non-negative Ricci curvature.
method Generalization of Fraser-Li type inequality for substatic triples under non-negative Ricci curvature associated with an affine connection.
result The paper provides a new inequality for Steklov eigenvalues of substatic triples.
The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
problem Characterizing constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
method Analyzing properties of hypersurfaces in specific ambient spaces (shrinking Ricci solitons).
result Conditions for a constant weighted mean curvature hypersurface to be a level set of the potential function.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
The paper connects Ricci curvature to entropy convexity in one dimension.
problem Understanding curvature bounds in one-dimensional spaces.
method Proving equivalence between 1-weighted Ricci curvature and entropy convexity.
result Established equivalence between curvature bounds and entropy convexity.
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.
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…
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…
Researchers estimate gradients of solutions to a Finslerian Allen-Cahn equation.
problem Estimating gradients of solutions to a specific type of partial differential equation.
method Using the Finslerian Allen-Cahn equation as an Euler-Lagrange equation to a Liapunov entropy functional, proving gradient estimates on compact and noncompact Finsler metric measure spaces.
result Global and local gradient estimates of positive solutions to the Finslerian Allen-Cahn equation.
The paper studies manifolds with positive weighted Ricci curvature of negative effective dimension.
problem Investigating properties of Riemannian manifolds with specific curvature conditions.
method Analyzing complete Riemannian manifolds with lower weighted Ricci curvature bound and discussing eigenvalues.
result If the minimum first nonzero eigenvalue is attained, the manifold splits off the real line as a warped product of hyperbolic nature.
Study compares nonsmooth spaces with integrable Ricci bounds.
problem Comparing geometric and functional inequalities on nonsmooth spaces.
method Localization method and one-dimensional comparison estimates.
result Extension of comparison principles to nonsmooth settings.
The study preserves lower bounds of total scalar curvature under specific metric convergence.
problem Preserving lower bounds of total scalar curvature on smooth manifolds.
method Used stability of Ricci flow and heat flow with Ricci flow background.
result Lower bound of weighted total scalar curvature is preserved under specified convergence conditions.
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.
In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorem on Finsler manifolds with nonnegative Ricci curvature.
The paper analyzes graph Laplacians on manifolds with curvature bounds and applies to non-collapsed spaces.
problem Analyzing spectral properties of graph Laplacians on manifolds with curvature constraints.
method Quantitative bounds on eigenvalues and eigenfunctions of graph Laplacians constructed from random variables on manifolds with uniform lower Ricci curvature bounds.
result Spectral convergence of graph Laplacians on manifolds with curvature bounds and in non-collapsed spaces.
In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.
The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
method Assumes a Sobolev inequality and integral Ricci bounds, proving local gradient estimates and Liouville type results.
result Proves local gradient estimates and Liouville type results on manifolds with lower bounds of Ricci curvature.
Estimates for Poisson equation on manifolds with weighted Poincare inequality.
problem Existence and estimates of Poisson equation solutions on manifolds.
method Develops Green's function estimate using weighted Poincare inequality and Ricci curvature.
result Proves Liouville property for finite energy holomorphic functions on Kähler manifolds.
The paper studies how to glue spaces with lower Ricci curvature bounds.
problem Lower Ricci curvature bounds in glued spaces.
method Analyzes conditions for a glued space to satisfy curvature-dimension conditions.
result Conditions (*) and (**) are necessary and sufficient for a glued space to satisfy CD(K,N). The paper studies weighted Ricci curvatures and characterizes Randers metrics.
problem Characterizing Randers metrics with weighted Ricci curvatures.
method General weighted Ricci curvatures and characterization of Randers metrics.
result Characterization of Randers metrics with almost isotropic weighted Ricci curvatures.
Optimizes transport in Finsler spacetimes with lower Ricci bounds.
problem Optimizing transport in Finsler spacetimes with lower Ricci bounds.
method Using optimal transport and weighted Ricci curvature bounds.
result Proves timelike curvature-dimension condition for Finsler spacetimes.
The paper explores inequalities for strongly-convex sets in weighted Riemannian manifolds.
problem Investigating dilation type inequalities on weighted Riemannian manifolds.
method Introducing dilation profile and comparing it with model space under lower weighted Ricci curvature bounds.
result Showed several functional inequalities related to various entropies.
The paper extends Ricci curvature in Finsler geometry and finds conditions for specific metrics.
problem Characterizing and finding conditions for specific metrics in Finsler geometry.
method Introducing weighted projective Ricci curvature and analyzing Randers and Kropina metrics.
result Conditions for metrics to have weighted projective Ricci flat curvature.
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.
Sharp bounds found for bi-drifting Laplacian eigenvalues.
problem Eigenvalue problems for bi-drifting Laplacian on manifolds.
method Sharp lower bounds derived for the first eigenvalue.
result Found sharp lower bounds for the first eigenvalue.
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…
We study both function theoretic and spectral properties of the weighted Laplacian Δf on complete smooth metric measure space (M,g,e−fdv) with its Bakry-Émery curvature Ricf bounded from below by a constant. In particular, we establish a gradient estimate for positive f−harmonic functions and a sharp upper…
We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension co…
We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields …
New invariants help solve existence of weighted cscK metrics.
problem Existence of weighted cscK metrics in K-stability.
method Introduced weighted analytic delta invariant and beta invariant.
result Sufficient condition for existence of weighted cscK metrics.
Introduces new curvature concept for Kähler manifolds.
problem Optimizing curvature constraints for projective Kähler manifolds.
method Introduces weighted orthogonal Ricci curvature and proves vanishing theorems.
result Proves optimal curvature constraints for projective Kähler manifolds.
New method calculates Ricci curvature from distances between weighted volumes.
problem Calculating Ricci curvature for weighted Riemannian manifolds.
method Asymptotic retrieval of generalized Ricci tensor from scaled metric derivatives of Wasserstein 1-distances.
result Limiting coarse curvature of random graphs converges to generalized Ricci tensor.
Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.
problem Extending Ricci flow theory under Kato-type curvature bounds.
method Extends non-collapsed Ricci flow existence theory to Kato-type lower bounds.
result Compact three-dimensional non-collapsed strong Kato limit spaces are homeomorphic to smooth manifolds.