Study finds open manifolds without complete metrics with positive scalar curvature.
problem Topological obstruction to positive scalar curvature on open manifolds.
method Defined Schoen-Yau-Schick and weak Schoen-Yau-Schick manifolds to prove the absence of complete metrics with positive scalar curvature.
result Proved no complete metric with positive scalar curvature on open Schoen-Yau-Schick manifolds.
The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.
problem Conditions for positive scalar curvature on manifolds with boundaries and their doubles.
method Analyzes the relationship between boundary conditions and positive scalar curvature metrics on manifolds and their doubles.
result Provides conditions for positive scalar curvature metrics on manifolds with boundaries and their doubles.
Extending results about positive scalar curvature to non-negative curvature.
problem Extending results about positive scalar curvature to non-negative curvature.
method Using index theory to generalize results.
result Explicit generalizations of classical results about positive scalar curvature metrics to non-negative curvature metrics.
Research examines metrics with positive scalar curvature in domains with corners.
problem Positive scalar curvature in domains with corners.
method Study of metrics with positive scalar curvatures in domains with corners.
result Possible extensions of positive scalar curvature to singular spaces.
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. Study torsion obstructions to positive scalar curvature on manifolds.
problem Obstructions to positive scalar curvature metrics on manifolds.
method Analysis of torsion classes in integral homology.
result Construction of manifolds without positive scalar curvature metrics but whose Cartesian products do.
Study shows no complete positive scalar curvature for certain 3-manifolds.
problem Existence of complete metrics with positive scalar curvature on specific 3-manifolds.
method Analyzing Whitehead manifold and genus one 3-manifolds.
result No contractible genus one 3-manifold admits a complete metric of positive scalar curvature.
Constructs uniformly positive scalar curvature metrics on open manifolds
problem Finding uniformly positive scalar curvature metrics on open manifolds
method Using Morse functions and exhaustion
result Proving the existence of uniformly positive scalar curvature metrics
Paper finds conditions for non-Einstein relative Yamabe metrics.
problem Finding relative Yamabe metrics with positive scalar curvature.
method Sufficient condition for positive constant scalar curvature metrics on manifolds with boundary.
result Examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
The study proves that certain manifolds can have metrics with specific volume growth.
problem Determining if manifolds with positive scalar curvature can have metrics with a given volume growth.
method Using Gromov-Lawson and Grimaldi-Pansu constructions, the study proves the existence of metrics with the desired volume growth on specific manifolds.
result The study positively answers the question for manifolds that are infinite connected sums of closed manifolds with positive scalar curvature.
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.
The paper constructs positive scalar curvature metrics via immersions.
problem Existence of positive scalar curvature metrics on manifolds.
method Extrinsic Gromov-Lawson surgery procedure.
result Positive scalar curvature metrics can be realized by immersions.
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.
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 sets new bounds on metrics with positive scalar curvature.
problem Classifying metrics with positive scalar curvature.
method Lower bounds on the rank of the group of psc metrics over M.
result Lower bounds on the rank of psc metrics up to bordism.
Researchers prove spaces of positive scalar curvature metrics are contractible with symmetry.
problem Contractibility of spaces of invariant positive scalar curvature metrics.
method Combining equivariant Morse theory with conformal deformations and local flexibility properties.
result Spaces of invariant positive scalar curvature metrics are contractible.
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
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.
Constructs metrics with positive scalar curvature on odd-order abelian fundamental group manifolds.
problem Creating metrics with positive scalar curvature on manifolds with odd-order abelian fundamental groups.
method Introduced Riemannian metrics, proved homology invariance, constructed metrics using admissible products.
result Solved the Gromov-Lawson-Rosenberg conjecture for manifolds with odd-order abelian fundamental groups.
Study on positive scalar curvature metrics with fundamental group constraints.
problem Nontriviality of space of metrics with positive scalar curvature on manifolds with torsionfree fundamental groups.
method Introduce stable metrics and round corners of manifolds, prove existence theorem.
result Existence of compact Spin 6-manifolds with infinite-dimensional rational homotopy groups for positive scalar curvature metrics.
Obstructs complete metrics with positive scalar curvature on non-compact manifolds.
problem Obstructing complete metrics with positive scalar curvature on non-compact manifolds.
method Using minimal hypersurfaces and MOTS, the study provides topological obstructions and proves the Liouville theorem.
result The Liouville theorem for locally conformally flat n-manifolds of non-negative scalar curvature follows from the impossibility of positive scalar curvature metrics.
Study on moduli space of metrics with positive scalar curvature.
problem Topology of moduli space of metrics with positive scalar curvature.
method Analyzes the moduli space of concordances of metrics with positive scalar curvature.
result Shows non-trivial homotopy groups in the moduli space.
New examples show positive scalar curvature metrics on manifolds with boundary that cannot be extended.
problem Positive scalar curvature metrics on manifolds with boundary that cannot be extended.
method Analytic techniques related to the prescribed scalar curvature problem in conformal geometry.
result Obstruction to positivity of conformal Laplacians given by a real-valued ξ-invariant.
Study on metrics with positive scalar curvature on manifolds with singularities.
problem Understanding metrics with positive scalar curvature on manifolds with singularities.
method Proving homotopy invariance of the space of metrics with positive scalar curvature on manifolds with fibred singularities.
result Proved that the space of metrics with positive scalar curvature is homotopy invariant under certain surgeries.
The study sets limits on scalar curvature in positive curvature manifolds.
problem Bounding scalar curvature in positive curvature manifolds.
method Establishing inequalities for manifolds with positive scalar curvature and scalar curvature bounded from below.
result Established inequalities for manifolds with positive scalar curvature.
Study negative scalar curvature metrics with positive boundary mean curvature.
problem Bounding conformal metrics with specific curvature properties.
method Analyzing Riemannian manifolds with boundary conditions.
result A priori boundedness of metrics in specific cases.
Study on metrics with positive scalar curvature on spherical space forms.
problem Determining metrics with positive scalar curvature on spherical space forms.
method Analysis of moduli spaces of metrics with positive scalar curvature on topological spherical space forms.
result Determination of the number of path components of moduli spaces for dimensions at least 5 and non-simply connected forms.
Strengthened surgery theorem for positive scalar curvature metrics.
problem Metrics of positive scalar curvature on manifolds.
method Complete account of Chernysh's strengthening of the Gromov-Lawson surgery theorem.
result Complete account of the strengthened theorem.
Constructs metrics for surfaces to show uniform positive scalar curvature.
problem Proving positive scalar curvature on surfaces and their bundles.
method Constructs complete Riemannian metrics on total spaces of vector bundles.
result Shows total space of tangent bundles on non-torus surfaces admit uniform positive scalar curvature.
Study non-positive scalar curvature on hyperbolic manifolds, finding necessary and sufficient conditions.
problem Finding metrics with non-positive scalar curvature on asymptotically hyperbolic manifolds.
method Analyzing the prescribed scalar curvature problem for non-positive scalar curvature on asymptotically hyperbolic manifolds.
result Obtained a necessary and sufficient condition for the existence of solutions.
Paper bounds the rank of positive scalar curvature metrics on manifolds.
problem Measuring the space of positive scalar curvature metrics.
method Analyzing the relative higher index map and higher rho-invariant.
result Rational containment of Baum-Connes assembly map image.
Study shows non-trivial higher homotopy groups in moduli space of metrics.
problem Understanding invariant metrics with positive scalar curvature.
method Analysis of higher homotopy groups on quasitoric manifolds.
result Non-trivial higher homotopy groups found in moduli space.
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 study explores positive scalar curvature metrics on non-orientable manifolds and their covers.
problem Existence of positive scalar curvature metrics on non-orientable manifolds and their covers.
method Extends Schoen-Yau inductive descent approach to non-orientable manifolds.
result Examples of non-orientable manifolds with positive scalar curvature metrics on their orientation double covers but not on homotopy equivalent manifolds.
Abstract: Characterizes spaces with positive scalar curvature.
problem Spaces with positive scalar curvature and invariant metrics.
method Cohomogeneity one manifolds and homogeneous spaces with compact Lie group actions.
result Characterization of spaces with positive scalar curvature.
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
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.
Extends metric properties over surgeries to higher codimensions.
problem Extending metrics of positive scalar curvature over surgeries in higher codimensions.
method Generalizes Gromov-Lawson, Schoen-Yau, and Walsh's construction to (p,n)-intermediate scalar curvature. result Shows infinitely many path components for certain metrics on (4n−1)-manifolds. 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.
Study positive scalar curvature metrics using cobordism category.
problem Homotopy type of positive scalar curvature metrics on spin manifolds.
method Cobordism category and recent theorems of Botvinnik et al.
result Alternative proof and extension of positive scalar curvature metrics.
Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group of positive scalar curvature metrics is…
Study positive scalar curvature metrics on even-dimensional compact spin manifolds.
problem Obstructions and path components of positive scalar curvature metrics.
method End-periodic eta invariants and K-homology/bordism theory.
result Obtained obstructions and refined path component results.
We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold…
In this paper we investigate complete critical metrics of the L2-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize critical metrics with nonnegative scalar curvature in dimension three and four.
Study shows inequalities for 4-manifolds with periodic ends and constructs new non-SC manifolds.
problem Finding inequalities for 4-manifolds with positive scalar curvature metrics on ends.
method Analyzes end-periodic 4-manifolds and constructs new examples.
result Constructs new 4-manifolds that do not admit positive scalar curvature metrics.
We prove that every quasitoric manifold admits an invariant metric of positive scalar curvature.
Proves conditions for positive scalar curvature on certain manifolds with conical singularities.
problem Conditions for positive scalar curvature on manifolds with isolated conical singularities.
method Analyzes isolated conical singularities and uses Geroch type results.
result No metric with positive scalar curvature on X#Tn with isolated conical singularity. New invariant for spaces of positive scalar curvature metrics.
problem No invariant defined for certain product manifolds.
method Constructing a new path-component invariant for specific manifolds.
result Analogous invariant defined for product manifolds.