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.
Abstract: Overview of scalar curvature constraints in Riemannian spaces.
problem Geometric constraints on Riemannian spaces with lower scalar curvature bounds.
method Exposition of known and new constraints.
result Presentation of geometric constraints implied by scalar curvature bounds.
Study adds scalar curvatures of mapped manifolds to Riemannian products.
problem Additivity of scalar curvatures in Riemannian products.
method Stabilized scalar curvatures of mapped manifolds.
result Proves additivity in some cases.
Sharp bounds on scalar curvature spectrum and rigidity theorems.
problem Understanding scalar curvature bounds and rigidity on manifolds.
method Sharp upper bounds for the bottom spectrum of the Beltrami Laplacian, scalar curvature rigidity theorem.
result Sharp upper bound for the bottom spectrum of the Beltrami Laplacian and scalar curvature rigidity theorem.
Formula for scalar curvature under metric collapse.
problem Finding positive scalar curvature metrics.
method Formula involving scalar curvature and adapted orthonormal frame.
result Effect of metric collapse on scalar curvature.
We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space …
Study shows limits of metrics with positive scalar curvature on spheres.
problem Non-negativity of scalar curvature is not preserved under certain limits.
method Examined metrics conformal to the round metric on Sn for n≥4. result Any conformal metric to the round metric on Sn for n≥4 can be a limit of metrics with positive scalar curvature. Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for C0 metrics. method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.
problem Integral of scalar curvature on manifolds with a pole.
method Asymptotic scaling invariant integral of scalar curvature equals a term determined by asymptotic volume ratio.
result Integral of scalar curvature equals a volume ratio term.
The paper studies metrics on manifolds with scalar curvature properties.
problem Finding metrics with specific scalar curvature properties.
method Analyzing the squared L2-norm of the scalar curvature over constant volume metrics. result Critical points of the functional correspond to Einstein or scalar flat metrics.
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. Study shows no large mean curvature fill-ins for nonnegative scalar curvature.
problem Existence of fill-ins with nonnegative scalar curvature and large mean curvature.
method Analyzes closed Riemannian manifolds and scalar curvature properties.
result No closed Riemannian manifold admits a fill-in with nonnegative scalar curvature and large mean curvature.
The study examines special properties of compact Riemannian manifolds with harmonic Weyl curvature.
problem Characterizing compact Riemannian manifolds with specific curvature properties.
method Rigidity theorems and geometric analysis on manifolds with positive scalar curvature and constant σ2. result Theorems proving the isometry of certain manifolds under specific curvature conditions.
The study examines conditions for scalar curvature and Dirac operators on singular spaces.
problem Existence of scalar curvature measures and Dirac operators on singular spaces.
method Investigation of smooth manifolds with singular Riemannian metrics.
result Sufficient conditions for the existence of scalar curvature measures and Dirac operators.
Study gives conditions for noncompact manifolds to have positive scalar curvature.
problem Conditions for noncompact manifolds to have positive scalar curvature.
method Analyzes finite and infinite volume cases to find obstructions.
result Provides conditions for noncompact manifolds to admit metrics with positive scalar curvature.
New manifolds with negative curvature limit to one with negative curvature.
problem Preserving nonnegative scalar curvature under intrinsic flat convergence.
method Constructing sequences of manifolds with positive scalar curvature.
result Intrinsic flat limit of manifolds with negative scalar curvature.
New examples of manifolds with lower scalar curvature bounds and submanifold collapse.
problem Stability of scalar curvature rigidity phenomena.
method Constructing Riemannian manifolds with specific curvature and collapse properties.
result Examples demonstrating stability and rigidity of scalar curvature.
We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation theory to prove the existence of a infinite number of conformal classes with at leas…
Proves a cube cannot have positive scalar curvature with specific conditions.
problem Proving impossibility of a cube with positive scalar curvature under certain conditions.
method Produced a metric on the n-torus from the cube's metric, contradicting Geroch conjecture.
result Cube cannot have positive scalar curvature with specified conditions.
New Einstein metrics found from para-Sasaki-like Riemannian manifolds.
problem Finding new Einstein metrics in Riemannian geometry.
method Cone construction and hyperbolic extension of paracontact paracomplex Riemannian manifolds.
result New complete Einstein para-Sasaki-like Riemannian manifolds with negative scalar curvature.
The study shows that certain manifolds cannot have metrics with positive scalar curvature.
problem The existence of metrics with positive scalar curvature on manifolds.
method Definition and analysis of enlargeable length-structures and their properties.
result Closed manifolds with certain properties are obstructions to the existence of metrics with positive scalar curvature.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.
problem Integral and variation formulas for mixed scalar curvature in multi-product manifolds.
method Generalizes results from pseudo-Riemannian almost product manifolds to multi-product structures.
result Generalizes formulas for mixed scalar curvature in multi-product manifolds.
Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results involving CVIs, generalizing many such results for the scalar curvature.
The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.
problem Understanding the combinatorial types of 3D polytopes in specific Riemannian manifolds.
method Analysis of mean curvature convex Riemannian polyhedra with non-obtuse dihedral angles in positive scalar curvature 3-manifolds.
result Determination of combinatorial types of 3D simple convex polytopes.
Proves conjecture for symmetric manifolds with positive scalar curvature.
problem Compactness theorem for symmetric Riemannian manifolds with positive scalar curvature.
method Proves conjecture for sequences of rotationally symmetric warped product manifolds.
result Limit spaces have nonnegative scalar curvature and Euclidean tangent cones almost everywhere.
Bound on integral of scalar curvature on non-parabolic manifolds.
problem Bounding integral of scalar curvature on non-parabolic manifolds.
method Using monotonicity formulas of Colding and Minicozzi.
result Explicit bound on asymptotic weighted scaling invariant integral of scalar curvature.
We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied…
The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.
problem Investigating the geometric properties of Riemannian manifolds influenced by average scalar curvature.
method Analyzing the conjugate radius, average area of geodesic spheres, average volume of metric balls, and total volume of closed manifolds.
result Improves the Bishop-Gromov estimate on the average volume of metric balls and proves monotone decreasing properties of certain geometric integrals.
We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…
The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Ri…
Non-compact flow lines for scalar curvature prescription on manifolds.
problem Non-compactness in scalar curvature prescription on manifolds.
method Gradient flow analysis of scalar curvature on Riemannian manifolds.
result A modification of the gradient flow leads to compact flow lines.
The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.
problem Bounding simplicial volume and L2-Betti numbers with scalar curvature constraints. method Using upper bounds on volumes of 1-balls in universal covers.
result Macroscopic versions of conjectures about scalar curvature and volume bounds are proven.
The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.
problem Disproving the macroscopic version of Gromov's Urysohn width conjecture for scalar curvature.
method Novel estimate on Urysohn width of circle bundles and a new notion of ruling for Riemannian manifolds.
result The macroscopic version of Gromov's Urysohn width conjecture for scalar curvature is false in dimensions four and above.
The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.
problem Proving the geometric stability conjecture for scalar curvature on manifolds conformal to tori.
method Reduction via the Yamabe problem and handling sequences of manifolds conformal to flat tori or constant negative scalar curvature.
result Proves the geometric stability conjecture for certain sequences of manifolds conformal to flat tori.
The paper solves scalar curvature problems under conformal deformation for Riemannian manifolds.
problem Solving scalar curvature problems under conformal deformation for Riemannian manifolds.
method Pointwise conformal deformation, Yamabe equation with Dirichlet boundary conditions.
result Positive, smooth solutions to the Yamabe equation with Dirichlet boundary conditions.
Study shows topological constraints on manifolds with non-negative scalar curvature and mean convex boundary.
problem Topological constraints on manifolds with non-negative scalar curvature and mean convex boundary.
method Constructing examples of compact manifolds that do not admit such metrics.
result Many compact manifolds with boundary do not admit a metric of non-negative scalar curvature and mean convex boundary.
Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.
problem Finding a conformal metric with zero scalar curvature and prescribed boundary mean curvature.
method Construction of local test functions to resolve open cases and establish new solvability conditions.
result Established new solvability conditions for the problem.
Study on 4-manifolds with positive scalar curvature.
problem Characterizing 4-manifolds with positive scalar curvature.
method Geography problem approach focusing on four-dimensional case.
result Survey of mathematical work and strengthening of Carr's result.
The paper provides obstructions to positive scalar curvature for certain manifolds with group actions.
problem Obstructions to the existence of complete invariant metrics with positive scalar curvature.
method Callias-type index theorem applied to proper actions by locally compact groups.
result Obstructions to positive scalar curvature vanish for certain Lie group actions.
Develops a method to deform metrics on manifolds with non-compact boundaries.
problem Creating metrics with positive scalar curvature on manifolds with boundary.
method General deformation principle for Riemannian metrics on manifolds with non-compact boundaries.
result Non-existence of metrics with positive scalar curvature and mean convex boundary.
The study preserves upper bounds of total scalar curvature in conformal classes.
problem Preserving upper bounds of total scalar curvature in conformal classes.
method Analyzing Yamabe constant and scalar curvature conditions.
result The upper bound condition of total scalar curvature is preserved in a conformal class.
Survey on scalar curvature stability and related questions.
problem Understanding scalar curvature stability and rigidity phenomena.
method Survey and discussion of existing tools and questions.
result Survey of known results and open questions in scalar curvature stability.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic S-curvature and are either Minkowskian or Riemannian. The study restricts scalar curvature on certain noncompact manifolds.
problem Ruling out complete metrics with nonnegative scalar curvature.
method Conditions and examples of positive scalar curvature metrics.
result Examples of manifolds with positive scalar curvature.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-curvature.