Proves curvature comparison for Riemannian bands in low dimensions.
problem Curvature comparison in Riemannian bands with lower bounds.
method Uses warped products over scalar-flat manifolds with log-concave warping.
result Scalar and mean curvature comparison results proven.
Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
problem Proving equivalence between Brunn-Minkowski inequality and curvature dimension condition.
method Analyzes weighted Riemannian manifolds, proving equivalence without optimal transport or differential structure.
result Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
New sub-Riemannian spaces with boundary meet curvature-dimension condition.
problem Finding sub-Riemannian manifolds with boundary satisfying curvature-dimension condition.
method Constructing specific sub-Riemannian structures on half-spaces and hemispheres.
result Provided new examples of sub-Riemannian manifolds with boundary that meet RCD(K,N) condition. The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n-volumic scalar curvature. method Using the curvature-dimension condition mCD(κ,n) and smGH-convergence. result The stability of n-volumic scalar curvature ≥κ under smGH-convergence. This paper generalizes biharmonic Riemannian submersions to higher dimensions.
problem Classifying biharmonic Riemannian submersions from manifolds with constant sectional curvature.
method Constructing an adapted orthonormal frame to simplify the biharmonic equation and analyzing curvature properties.
result A Riemannian submersion is biharmonic if and only if it is harmonic from an (n+1)-dimensional manifold with constant sectional curvature to an n-dimensional manifold. Odd co-dimension Riemannian foliations don't work on curved surfaces.
problem Proving Riemannian foliations can't exist on positively curved manifolds.
method Reasonable conditions on co-dimension and curvature.
result Odd co-dimension Riemannian foliations cannot occur in positively curved manifolds.
Study shows failure of curvature-dimension conditions on sub-Riemannian manifolds.
problem Failure of curvature-dimension conditions on sub-Riemannian manifolds.
method Proves failure of curvature-dimension conditions using tangent isometries and Killing vector fields.
result Proves failure of curvature-dimension conditions on sub-Riemannian manifolds.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. Proves Penrose inequality in all dimensions for specific manifolds.
problem Proving Penrose inequality in arbitrary dimensions for certain manifolds.
method Extends Bray's conformal-flow method to higher dimensions, dealing with singular outer-minimizing enclosures.
result Proves the Riemannian Penrose inequality in arbitrary dimensions.
Paper proves curvature conditions are preserved in metric spaces.
problem Preserving curvature conditions in metric spaces.
method Doubling and gluing constructions to preserve RCD(K,N) condition. result Proves RCD(K,N) condition is preserved. The study proves conditions for complete Riemannian manifolds to be Einstein.
problem Conditions for complete Riemannian manifolds to be Einstein.
method Proving conditions using harmonic curvature and curvature operator of the second kind.
result Complete Riemannian manifolds with specific curvature conditions are Einstein.
Lower Ricci curvature bound prevents first Betti number from dropping more than dimension in collapsing manifolds.
problem Understanding how the first Betti number behaves under manifold collapse with Ricci curvature bounds.
method Analyzing sequences of Riemannian manifolds with lower Ricci curvature bounds.
result The first Betti number cannot drop more than the dimension in collapsing manifolds.
New results on tori restrict sectional curvature when Ricci curvature is negative and bounded.
problem Restricting sectional curvature on tori with mixed Ricci bounds.
method Using Lohkamp's theorem and explicit constants.
result Explicit constants show sectional curvature is positive in some directions.
We prove a family of new Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension…
The study shows a 3D manifold's macroscopic dimension is 1 under specific curvature constraints.
problem Understanding the macroscopic dimension of 3D Riemannian manifolds with curvature restrictions.
method Analyzing the volume and homology of balls in the manifold.
result A 3D manifold with the specified curvature constraints has macroscopic dimension 1.
SubRiemannian structures fail to meet Riemannian Brunn--Minkowski inequalities.
problem SubRiemannian structures do not satisfy Riemannian Brunn--Minkowski inequalities.
method The proof relies on the method used for the Heisenberg group and new investigations by Agrachev, Barillari, and Rizzi on subRiemannian structures.
result No Brunn--Minkowski inequality can be satisfied by strictly subRiemannian structures.
Study confirms equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.
problem Equivalence between curvature-dimension conditions and strong Brunn-Minkowski inequalities in Heisenberg groups.
method Optimal transport and approximation techniques in sub-Riemannian Heisenberg group Hn, combined with previous works.
result Confirms the equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.
New curvature-dimension condition for Lagrangians on manifolds.
problem Establishing a curvature-dimension condition for autonomous Lagrangians.
method Generalizing Klartag's needle decomposition technique to Lagrangian setting.
result Equivalence of curvature-dimension condition to displacement convexity of entropy.
The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.
problem Curvature-dimension conditions and related inequalities on Riemannian manifolds.
method Information-theoretic approach to study curvature-dimension condition, rigidity theorems, and entropy differential inequalities.
result Equivalence of curvature-dimension condition and entropy differential inequalities on Riemannian manifolds.
Almost-Riemannian manifolds fail to meet a synthetic curvature condition.
problem Proving almost-Riemannian manifolds do not satisfy the CD condition. method Developed a new strategy to contradict the 1-dimensional CD condition. result 2D and strongly regular almost-Riemannian manifolds do not satisfy CD(K,N) for any K and N. 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.
The paper extends results on minimal hypersurfaces in Riemannian manifolds to higher dimensions.
problem Characterizing properties of minimal hypersurfaces in higher-dimensional Riemannian manifolds.
method Maximum principle at infinity for two-sided, parabolic, properly embedded minimal hypersurfaces.
result Two disjoint properly embedded minimal hypersurfaces bound a slab in specific conditions.
Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.
problem Finding metrics with specific curvature properties on Brieskorn quotients.
method Analyzing moduli spaces of metrics with nonnegative sectional or positive Ricci curvature.
result Moduli spaces have infinitely many path components for both nonnegative sectional and positive Ricci curvature.
Classifies 4D spaces with circle symmetry and positive curvature.
problem Classifying spaces with specific geometric properties.
method Equivariant homeomorphism classification, infinitesimal geometry analysis.
result Classification of 4D Alexandrov spaces and orbifolds with circle symmetry.
Eigenfunction maps embed manifolds with a dimension bound.
problem Bounding the embedding dimension of Riemannian manifolds.
method Analyzing Laplacian eigenfunction maps and their embedding properties.
result The maximal embedding dimension is bounded by geometric properties.
We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in t…
Compact manifolds with positive scalar curvature have negative Kodaira dimension.
problem Understanding the properties of compact manifolds with positive scalar curvature.
method Analyzing the canonical bundle and complex structures on compact Riemannian manifolds.
result Compact manifolds with positive scalar curvature have negative Kodaira dimension.
New metrics with special curvature properties are shown to be parallel in certain Lie groups.
problem Characterizing metrics with harmonic curvature in Lie groups.
method Analyzing left invariant metrics on solvable and low-dimensional Lie groups.
result Left invariant metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6.
The paper extends Liouville theorems to sub-Riemannian manifolds.
problem Generalizing Liouville theorems to sub-Riemannian manifolds.
method Constructing 'good' cut-off functions and applying a nonnegative generalized curvature-dimension inequality.
result The Liouville theorems are extended to sub-Riemannian manifolds.
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
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.
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…
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
The paper studies complex Finsler metrics and curvature inequalities.
problem Analyzing holomorphic sectional curvature in complex Finsler manifolds.
method Developed an inequality relating holomorphic sectional curvature and proved Schwarz Lemma.
result Complex Finsler manifolds with semi-positive but not identically zero holomorphic sectional curvature have negative Kodaira dimension under certain conditions.
Study on stability of curvature functionals on manifolds.
problem Stability of quadratic curvature functionals on manifolds.
method Analysis of stability at constant sectional curvature metrics.
result Stability properties of quadratic curvature functionals.
The study develops inequalities for Riemannian foliations without bundle-like assumptions.
problem Developing inequalities for Riemannian foliations without restrictive conditions.
method Bochner theory and Bakry-Emery calculus for horizontal Laplacians, derived explicit Bochner formulas, generalized curvature dimension inequalities.
result Established generalized curvature dimension inequalities for Riemannian foliations.
We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension n>2, and for the Q-curvature of random Riemannian metrics.
Study curvature and symplectic properties of symmetric products of surfaces.
problem Distinguishing between macroscopic dimensions in Riemannian manifolds.
method Detailed study of curvature and symplectic properties using symmetric products of surfaces.
result Symmetric products of surfaces sharply distinguish between two macroscopic dimensions.
Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
Closed Riemannian manifolds with positive mixed sectional curvature
problem Constructing closed Riemannian manifolds with positive mixed sectional curvature
method Explicit construction using totally geodesic foliations
result Positive mixed sectional curvature alone does not imply the Ferus--Adams estimate on closed manifolds
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma z calculus and z--Bochner's formula. result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
The paper proves a foliation of a manifold with specific properties.
problem Proving a foliation with constant mean curvature and perpendicular boundary condition.
method Analyzing a Riemannian manifold with smooth boundary and proving the existence of a foliation.
result Existence of a smooth foliation around a nondegenerate critical point of the mean curvature function of the boundary.
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 explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.
problem Investigating Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds.
method Analyzing compact almost Hermitian manifolds in the Gray-Hervella class and Hermitian manifolds with nonnegative scalar curvature.
result For compact almost Hermitian manifolds with nonnegative scalar curvature, the Kodaira dimension is either -∞ or 0, with specific conditions.
We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-di…
Proves isometry group of RCD*(K,N) spaces is a Lie group with optimal dimension.
problem Understanding the isometry group of RCD*(K,N) spaces.
method Analyzes metric measure spaces satisfying Riemannian curvature condition.
result Optimal upper bound on the dimension of the isometry group and classification of spaces achieving this bound.
On Riemannian manifolds of dimension 4, for prescribed scalar curvature equation, under lipschitzian condition on the prescribed curvature, we have an uniform estimate for the solutions of the equation if we control their minimas.
Paper defines new curvature measures for foliated manifolds and proves new theorems.
problem Estimating the diameter and splitting theorems for foliated manifolds.
method Introduced weighted mixed curvatures and new conditions to update estimates.
result Updated estimates of diameter and proved new splitting theorems.