Paper introduces a new convergence for Lorentzian spaces using causal diamonds.
problem No specific problem stated; focuses on a new geometric convergence.
method Uses causal diamonds to define a new convergence for Lorentzian spaces.
result Proves a pre-compactness theorem for Lorentzian spaces.
Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.
problem Projective compactness for torsion-free linear connections on a manifold.
method Introduce and study a weakening of projective compactness for torsion-free linear connections on a manifold.
result Induces projective structure on the boundary and relates to asymptotic forms in GR.
The article introduces a new convergence concept for Lorentzian spaces and applies it to generalized cones.
problem Stability of curvature bounds in generalized Lorentzian cones.
method Introduces ℓ-convergence for Lorentzian pre-length spaces, applies it to generalized cones, and proves stability of curvature bounds. result Sharp timelike curvature and curvature-dimension bounds for generalized cones are established.
Given a class of closed Riemannian manifolds with prescribed geometric conditions, we introduce an embedding of the manifolds into ℓ2 based on the heat kernel of the Connection Laplacian associated with the Levi-Civita connection on the tangent bundle. As a result, we can construct a distance in this class which …
It is proved that the category of simplicial complete bornological spaces over R carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particu…
Compact metrics found near Kähler manifold's canonical class.
problem Finding unique cscK metrics near Kähler manifold's canonical class.
method Proved existence and uniqueness of cscK metrics for Kähler classes near canonical class.
result Metric spaces are pre-compact in Gromov-Hausdorff sense.
The purpose of this note is to study the complex structures orthogonal to a given Riemannian metric. For another paper on this topic, we highly recommend the work of Salamon. His work describes in great detail the role that curvature plays in this question. We instead focus on torsion, which lends itself to somewhat di…
We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension n, assuming uniform volume bounds and Ln/2 bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are pre-compact in the Gr…
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
The paper proves convergence of normalized Ricci flow on compact manifolds.
problem Convergence of normalized Ricci flow on compact manifolds.
method Gradient inequality of Łojasiewicz type to show convergence to steady-states.
result Convergence of normalized Ricci flow to steady-states on compact manifolds.
We characterize lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy on the L2-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of \cite{giglinonsmooth}. This ex…
The abstract discusses a new type of space and its properties.
problem The abstract tackles the concept of non-Hilbertian (Lorentzian) length spaces.
method The abstract introduces a new type of space and analyzes its properties.
result The abstract finds that normed spaces without inner products have no sectional curvature bounds.
Complete left-invariant metrics on Lie groups with specific properties.
problem Completeness of left-invariant semi-Riemannian metrics on Lie groups.
method Introducing bi-Lipschitz Riemannian Clairaut metrics and proving completeness conditions.
result All left-invariant metrics are complete for certain Lie groups.
Study compact sequences of warped product circles over spheres with nonnegative scalar curvature.
problem Compactness of sequences of warped product circles over spheres with nonnegative scalar curvature.
method Proved subsequence convergence to a W1,p Riemannian metric for all p<2 with nonnegative scalar curvature in the distributional sense. result Proved compactness of sequences of warped product circles over spheres with nonnegative scalar curvature.
The study extends classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.
problem Extending classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.
method Investigated the restricted mean-value property on Riemannian manifolds, focusing on non-tangential boundary behavior.
result Extended a classical result of Fenton to non-positively curved Harmonic manifolds of purely exponential volume growth.
Defines new metrics for Lorentzian spaces and their convergence.
problem Defining metrics for Lorentzian spaces and their convergence.
method Abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, defining bounded Lorentzian-metric spaces, and proving stability under GH limits.
result GH limits of Lorentzian-metric spaces are isometric and homeomorphic.