Researchers develop weighted GJMS operators for smooth metric measure spaces.
problem Developing mathematical tools for smooth metric measure spaces.
method Constructing and proving formal self-adjointness of weighted GJMS operators.
result Formal self-adjointness of weighted GJMS operators proved.
Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
Proves metric spaces with Euclidean heat kernel are isometric to Euclidean space.
problem Characterizing metric measure spaces with specific heat kernels.
method Analyzes Dirichlet forms and heat kernels to prove rigidity.
result Metric measure spaces with Euclidean heat kernel are isometric to Euclidean space.
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.
Proves metric measure spaces with certain properties are one-dimensional.
problem Characterizing metric measure spaces as one-dimensional.
method Analyzes properties of metric measure spaces and uses optimal transport maps.
result Metric measure spaces with specified properties are one-dimensional.
The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure …
We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces defined by P.S.Urysohn in 20-th, and generic metric triple (= metric space with…
In this short note we compare the weighted Laplacians on real and complex (Kähler) metric measure spaces. In the compact case Kähler metric measure spaces are considered on Fano manifolds for the study of Kähler-Einstein metrics while real metric measure spaces are considered with Bakry-Émery Ricci tensor. There are tw…
Metric measure boundary vanishes on certain spaces without boundary.
problem Existence of infinite geodesics on Alexandrov spaces without boundary.
method Solving conjecture by showing metric measure boundary vanishes on mRCD(K,N) spaces. result Metric measure boundary vanishes on mRCD(K,N) spaces without boundary. Compact embeddings for invariant functions in metric-measure spaces.
problem Embedding functions with symmetry in metric-measure spaces.
method Analyzing H-invariant functions in compact metric-measure spaces, extending to Riemannian manifolds. result Obtained compact Sobolev embeddings for critical exponents.
Study stability of curvature-dimension condition for negative dimensions.
problem Stability of curvature-dimension condition with negative dimension parameters.
method Introduced CD(K, N)-condition for N < 0, defined distance d_{\mathsf{iKRW}}, proved convergence stability.
result Limit structure of converging metric measure spaces remains CD(K, N) for N < 0.
The study shows finite measure-preserving isometry groups for certain metric measure spaces.
problem Understanding the structure of isometry groups in metric measure spaces.
method Analyzing synthetic negative Ricci curvature and Bakry-Émery Ricci curvature.
result The measure-preserving isometry group is finite for compact metric measure spaces with specific curvature conditions.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure spac…
We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiati…
Extended Rank-One Theorem to special metric spaces.
problem Extending a theorem to new types of spaces.
method Applied to a new class of metric measure spaces.
result Rank-One Theorem proven for RCD(K,N) spaces. Study Bochner formula on metric measure spaces for vanishing Betti numbers.
problem Vanishing Betti numbers on metric measure spaces.
method Introduce weighted curvature conditions.
result Vanishing of all Betti numbers.
Develops new synthetic Ricci flow concepts for metric measure spaces.
problem No specific problem stated; focuses on new mathematical concepts.
method Formulated in terms of dynamic convexity and local concavity of entropy, and global/short-time asymptotic transport cost estimates.
result Shows these properties characterise smooth (weighted) Ricci flows.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy…
A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
problem Defining and analyzing a mass-type invariant for smooth metric measure spaces
method Defining a mass-type quantity and showing its geometric invariance properties
result The mass-type quantity has a close relation with the fractional Yamabe problem and the relevant Green's function
Corrects errors in previous studies on metric measure spaces.
problem Identifies and corrects errors in previous research on metric measure spaces.
method Corrigendum to previous publications in Acta Math, JFA.
result Corrects errors in previous studies on metric measure spaces.
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's W-entropy and Shannon entropy power to super Ricci flows. result Equivalence between volume non-local collapsing property and lower boundedness of W-entropy on RCD(0,N) spaces. Study group actions in metric spaces, proving convergence of lens spaces.
problem Understanding convergence in metric measure spaces with group actions.
method Generalized box and observable distances, applied mass-transport theory.
result Sequence of lens spaces converging to infinite-dimensional complex projective space.
Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.
problem Understanding isoperimetric constants in metric measure spaces with measure contraction property.
method Proves local isoperimetric inequalities on essentially non-branching MCP(K,N) spaces with volume constraints and geometric conditions.
result Establishes bounds on isoperimetric constants in smaller geodesic balls.
Proves sufficiency of countable test plans for BV functions on metric spaces.
problem Recovering BV functions and their measures on arbitrary metric spaces.
method Proves sufficiency of countable test plans on arbitrary metric measure spaces and geodesics on CD(K,N) spaces. result Countable test plans are sufficient for BV functions and their measures on metric spaces.
Implementing k-NN classification using Gromov--Wasserstein distances
problem Comparing metric measure spaces
method Gromov--Wasserstein and fused Gromov--Wasserstein distances
result Universal consistency of k-NN classifiers Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q-Laplacian, and q-heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.
The study examines stability of metric measure spaces with integral Ricci curvature bounds.
problem Stability and compactness of metric measure spaces with integral Ricci curvature bounds.
method Proves convergence to metric measure spaces satisfying CD(K,n) condition under certain curvature bounds. result Proves convergence of sequences of Riemannian manifolds to metric measure spaces satisfying CD(K,n) condition. The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
Locally homogeneous RCD spaces are shown to be smooth manifolds.
problem Understanding the structure of RCD spaces.
method Adapting existing results to new spaces.
result Locally homogeneous RCD spaces are isometric to smooth manifolds.
Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces.
In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(…
We introduce a weak notion of barycenter of a probability measure μ on a metric measure space (X,d,m), with the metric d and reference measure m. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter B(μ) is well defined; it is a probability measur…
The main result of this article states that the (K;N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner…
Study solves Yamabe problems on metric measure spaces with or without boundary.
problem Yamabe-type problems on compact metric measure spaces with or without boundary.
method Analyzes uniqueness, characterization, and existence of minimizers.
result Characterizes weighted Yamabe solitons and existence of positive minimizers.
In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then introduce a notion of "being asymptotically isoperimetric" for a family of finite a…
We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's ν-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the κ-noncollapsing property. Finally, we us…
This is an exposition of the theory of differentiable structures on metric measures spaces, in the sense of Cheeger and Keith.
In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Mor…
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …
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…
Paper investigates conditions for independence of weak gradients on metric spaces.
problem Dependence of weak gradients on p in arbitrary metric measure spaces. method Investigates the Bounded Interpolation Property to ensure independence of weak gradients.
result Bounded Interpolation Property guarantees independence of weak gradients.
New formulations for comparing metric measure spaces with arbitrary positive measures.
problem Comparing metric measure spaces with arbitrary positive measures.
method Two novel formulations: a divergence and a conic lifting approach.
result Efficiently solvable formulations for comparing metric spaces with arbitrary positive measures.
Sharp estimates derived for quasilinear equations on metric measure spaces.
problem Estimating solutions and eigenvalues of quasilinear equations on smooth metric measure spaces.
method Sharp estimates derived using comparisons with one-dimensional equations.
result Optimal lower bounds for the first Dirichlet eigenvalue of quasilinear operators.
New metrics avoid high-dimensional analysis challenges, proving convergence without 'curse of dimensionality'.
problem High-dimensional analysis challenges in empirical measure convergence.
method Proposed a new class of probability metrics free of the curse of dimensionality.
result Convergence of empirical measures is free of the curse of dimensionality.
The paper derives gradient estimates for solutions of certain equations on metric measure spaces.
problem Gradient estimates for solutions of specific nonlinear and elliptic equations on metric measure spaces.
method Derives Li-Yau and Hamilton's type gradient estimates for positive solutions.
result Gradient estimates for positive solutions of the equations on complete noncompact metric measure spaces.
Global gradient estimates for Fisher-KPP equation on Finsler metric measure spaces.
problem Establishing gradient estimates for the Finslerian Fisher-KPP equation.
method Global gradient estimates on compact and noncompact Finsler metric measure spaces using the traditional CD(K,N) condition and new comparison theorems. result Global gradient estimates for positive solutions of the Finslerian Fisher-KPP equation.