Study angles between curves in metric measure spaces.
problem Define and analyze the angle between curves in metric measure spaces.
method Introduce a new notion of angle and prove the cosine formula on RCD∗(K,N) spaces. result The new notion of angle is compatible with classical notions in Riemannian manifolds and Alexandrov spaces.
Paper proves almost Schur Lemma on smooth metric measure spaces.
problem Proving a specific lemma on metric measure spaces.
method Proves almost Schur Lemma using closed smooth metric measure spaces.
result Implications of the lemma for X. Cheng's and De Lellis-Topping's results.
Study on conformal transformations in metric measure spaces.
problem Preserving curvature bounds under conformal transformations.
method Analysis of Sobolev space, differential structure, and curvature-dimension condition.
result First result showing preservation of lower curvature bounds under perturbation.
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 inequality in metric measure spaces with non-branching structure.
problem Proving Heintze-Karcher inequality in metric measure spaces.
method Used needle decomposition technique for metric measure spaces.
result Characterizes equality case in spaces with positive curvature.
Study Ricci tensor on spaces with boundary, offering new curvature conditions.
problem Curvature conditions on smooth metric measure spaces with boundaries.
method Generalization of Bakry-Emery's Ricci tensor to spaces with boundaries.
result New approach to curvature-dimension conditions on spaces with boundaries.
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.
Proves Weyl's law for metric spaces with Ricci curvature.
problem Proving Weyl's law for metric measure spaces with bounded Ricci curvature.
method Analyzes RCD∗(K,N) spaces to prove asymptotic eigenvalue formula. result Establishes Weyl's law for Dirichlet eigenvalues in metric measure spaces.
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…
Defines weighted σ_k-curvature for smooth metric measure spaces.
problem Prescribing weighted σ_k-curvature in smooth metric measure spaces.
method Proposes a definition and justifies it through variational and stability properties.
result Quasi-Einstein metrics are stable with respect to the total weighted σ_k-curvature functional in variational cases.
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 paper establishes estimates for heat equations and harmonic functions on metric spaces with curvature-dimension condition.
problem Analyzing geometric properties of metric measure spaces with curvature-dimension condition.
method Establishing local Li-Yau estimates and proving sharp Yau's gradient estimates for heat equations and harmonic functions.
result Sharp Li-Yau and gradient estimates for weak solutions of heat equations and harmonic functions on RCD∗(K,N) spaces. The paper proves metric measure spaces with specific inequalities have n-dimensional volume growth.
problem Proving geometric and topological properties of spaces with Caffarelli-Kohn-Nirenberg inequalities.
method Volume doubling condition and Caffarelli-Kohn-Nirenberg inequality with same exponent n.
result Metric measure spaces with these inequalities have exactly n-dimensional volume growth.
A new method calculates a barycenter for probability measures using Wasserstein distance.
problem Finding a central measure for a set of probability distributions.
method Regularizing the pushforward measure of a set of probability distributions into the Wasserstein space and then finding the barycenter.
result The method yields a uniquely defined barycenter measure supported on the barycentric points of the input measures.
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.
New approach to convexity and monotonicity on metric spaces.
problem Characterizing convexity and monotonicity in non-smooth metric spaces.
method Characterization of convexity and monotonicity using Riemannian Ricci curvature.
result Offers new rigidity theorems like splitting theorem and volume cone implies metric cone theorem.
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.
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.
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. Sharp gradient estimate for heat kernels on metric measure spaces.
problem Establishing gradient estimates for heat kernels on metric measure spaces.
method Elliptic local Li-Yau gradient estimate for weak solutions of the heat equation.
result Sharp gradient estimate for the logarithm of heat kernels.
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. Statistical hyperbolicity proven for Teichmüller space.
problem Harmonic measures from random walks on mapping class groups.
method Proving statistical hyperbolicity using Teichmüller metric.
result Teichmüller space is statistically hyperbolic for certain harmonic measures.
Study shows Sasakian manifolds with non-negative Ricci curvature have synthetic lower bounds.
problem Synthetic Ricci curvature bounds on Sasakian manifolds.
method Used measure contraction property and synthetic Ricci curvature lower bounds.
result Sasakian manifolds with non-negative Ricci curvature satisfy measure contraction property.
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.
Study bounds expansion coefficient from observable diameter in metric measure spaces.
problem Bound the expansion coefficient from below in terms of the observable diameter.
method Considered concentration of measure phenomenon, connected observable diameter and expansion coefficient, derived upper bound, combined with lower bound to obtain upper bound for observable diameter.
result Obtained upper bound for observable diameter in terms of expansion coefficient.
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.
Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.
problem Finding a function that attains the Escobar weighted constant.
method Introducing Escobar quotient, infimum, and resolving the problem when the weighted constant is negative.
result Obtained an Aubin type inequality connecting weighted Escobar constant and optimal constant for trace inequality.
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.
The paper proves conditions for metric measure spaces to have specific volume growth and curvature properties.
problem Conditions for metric measure spaces to have specific volume growth and curvature properties.
method Proves conditions using volume doubling and Gagliardo-Nirenberg inequalities.
result Metric measure spaces with specific conditions have exactly the n-dimensional volume growth and zero flag curvature. 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…
Extends heat flow estimates to non-smooth spaces.
problem Heat flow estimates on non-smooth metric measure spaces.
method Extends Hamilton's gradient estimates and monotonicity formula to metric measure spaces.
result Establishes heat flow estimates for metric measure spaces.
We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces.