Study on Yamabe flow for negative scalar curvature.
problem Prescribed scalar curvature problem on compact manifolds.
method Yamabe flow with conditions on scalar curvature function.
result Long time existence and convergence of the flow.
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The paper examines complete Yamabe solitons with finite total scalar curvature.
problem Characterizing complete Yamabe solitons with specific curvature properties.
method Analyzing steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature.
result Steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.
The paper studies special Finsler spaces with -scalar curvature.
problem Characterizing and investigating Finsler spaces with specific scalar curvatures.
method Intrinsic investigation and various conditions for transformations between Finsler spaces.
result Conditions for transforming Finsler spaces of scalar curvature to those of -scalar curvature.
The paper explores time consistency for scalar multivariate risk measures in markets with transaction costs.
problem Time consistency of scalar multivariate risk measures in markets with transaction costs.
method Presented dual representations and derived an equivalent recursive formulation for multivariate scalar risk measures.
result Developed a direct notion of a 'moving scalarization' for scalar time consistency.
The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the -volumic scalar curvature.
method Using the curvature-dimension condition and smGH-convergence.
result The stability of -volumic scalar curvature under smGH-convergence.
Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.
problem Stability and rigidity of scalar curvature in quaternion-Kähler manifolds.
method Analysis of stability and rigidity conditions using Einstein manifold properties.
result Quaternion-Kähler manifolds of negative scalar curvature are stable and scalar curvature rigid.
Study on scalar curvature deformations in pseudohermitian manifolds.
problem Deformation of scalar curvature in pseudohermitian manifolds.
method Analogy with Riemannian manifolds, introduction of -singular spaces, stability conditions, partial infinitesimal rigidity.
result Partial infinitesimal rigidity result for scalar curvature of compact pseudohermitian manifolds.
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.
The paper studies metrics on manifolds with scalar curvature properties.
problem Finding metrics with specific scalar curvature properties.
method Analyzing the squared -norm of the scalar curvature over constant volume metrics.
result Critical points of the functional correspond to Einstein or scalar flat metrics.
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.
Small Weyl infimum on 4-manifolds with positive scalar curvature.
problem Understanding scalar curvature on 4-manifolds.
method Analyzing the Weyl functional and comparing scalar and self-dual Weyl curvatures.
result The infimum of the Weyl functional is small on many 4-manifolds with positive scalar curvature.
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 -curvature and are either Minkowskian or Riemannian.
The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.
Minimal splitting factors help study scalar curvature constraints.
problem Scalar curvature constraints in geometry.
method Introducing minimal splitting factors with positive scalar curvature.
result Minimal splitting factors have properties similar to area minimizing hypersurfaces.
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.
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.
In this paper we investigate complete critical metrics of the -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.
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.
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.
New CVIs generalize scalar curvature stability results.
problem Stability and rigidity of Riemannian invariants.
method Establishing stability and rigidity results for CVIs.
result Generalizing scalar curvature stability results.
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.
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 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.
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
Classifies radial scalar flat metrics with a constant third coefficient.
problem Classifying radial scalar flat metrics with a specific coefficient.
method Analyzing the TYZ expansion and Simanca's scalar flat metric.
result Characterization of Simanca's scalar flat metric as a byproduct.
The paper classifies Randers metrics with scalar flag curvature.
problem Classifying Finsler metrics of scalar flag curvature.
method Investigating Randers metrics under the condition that β is a Killing 1-form.
result Obtained necessary conditions for Randers metrics to be of scalar flag curvature.
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.
The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.
problem Classifying flag manifolds with specific isotropy components and finding conditions for Kähler-like scalar curvature.
method Investigating invariant almost Hermitian structures on generalized flag manifolds with two or three irreducible components.
result Classification of flag manifolds admitting Kähler-like scalar curvature and conditions for such structures.
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 generalizes scalar curvature theorem to weighted manifolds.
problem Generalizing scalar curvature rigidity theorem to weighted manifolds.
method Proves a refinement of Llarull's theorem for P-scalar curvature.
result Establishes a Llarull type theorem for .
The paper explores geometry and positive scalar curvature on non-compact manifolds.
problem Understanding the relationship between geometry and positive scalar curvature on non-compact manifolds.
method Analysis of volume growth, scalar curvature integral, and width in different dimensions.
result Proves minimal volume growth and integral of scalar curvature in three dimensions, and volume growth with stronger conditions in higher dimensions.
Develops scalar curvature in generalized Kahler geometry and shows constant scalar curvature on compact Lie groups.
problem Defines scalar curvature in generalized Kahler geometry.
method Introduces scalar curvature in terms of pure spinors formalism and develops a moment map framework.
result Scalar curvature is given by the moment map, generalizing results from ordinary Kahler geometry.
Survey of nonnegative scalar curvature sequences and their limits.
problem Understanding sequences of manifolds with nonnegative scalar curvature.
method Analyzing sequences of manifolds with nonnegative scalar curvature and proving convergence.
result Proved the GH and SWIF convergence of an extreme example.
Study local properties of Chern-scalar curvature through linearization stability.
problem Local properties of Chern-scalar curvature function.
method Linearization analysis of the Chern-scalar curvature function.
result Stability of linearization and structure of metrics with prescribed curvature.
Estimates for scalar curvature equations on Kähler manifolds with singularities.
problem Developing estimates for scalar curvature equations with singular metrics.
method Estimates and Laplacian estimates for scalar curvature equations of degenerate Kähler metrics.
result Derivation of estimates for singular constant scalar curvature Kähler metrics and singular Kähler-Einstein metrics.
Researchers confirm scalar-flatness for critical metrics in 5-9 dimensions.
problem Verifying scalar-flatness for critical metrics in specific dimensions.
method Analyzing complete Riemannian manifolds with critical metrics of the -scalar curvature functional.
result The conjecture that all complete noncompact critical metrics with finite energy are scalar-flat is confirmed for dimensions 5 to 9.
Summarizes geometric formulation of Einstein-Scalar-Maxwell theories.
problem Geometric formulation of Einstein-Scalar-Maxwell theories.
method Global geometric formulation using flat symplectic vector bundles.
result Describes scalar-electromagnetic symmetry group and Dirac quantization condition.
The paper studies Berwald scalar curvature properties in Finsler geometry.
problem Characterizing Finsler manifolds based on Berwald scalar curvature.
method Analyzes properties of Berwald scalar curvature and its implications for Finsler manifolds.
result Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.
Scalar Curvature of Manifolds with Boundaries: Natural Questions and Artificial Constructionsmath.DG
The study explores scalar curvatures on manifolds with boundary properties.
problem Understanding scalar curvatures on manifolds with boundary constraints.
method Presentation of problems and results related to scalar curvatures and mean curvatures of boundaries.
result Exploration of natural and artificial constructions in scalar curvature studies.
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
Study on prescribing scalar curvature on spheres using flow method.
problem Prescribing scalar curvature on n-sphere with sign-changing function.
method Scalar curvature flow approach.
result Flow converges to a metric with prescribed sign-changing function.
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.
The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.
problem Bounding simplicial volume and -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.
Let be a simply-connected closed manifold of dimension which does not admit a metric with positive scalar curvature. We give necessary conditions for to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of . As a consequence any simply-connected …
Einstein discovered a unique metric in conformal structures, leading to scalar invariants.
problem Exploring unique metrics in conformal structures.
method Geometrical exploration and calculation of scalar conformal invariants.
result A family of scalar conformal invariants of weight 0 for pseudo-Riemannian conformal structures were derived.
The study classifies specific types of solitons with bounded scalar curvature.
problem Classifying quasi-Yamabe gradient solitons with bounded scalar curvature.
method Analyzing complete, nontrivial solitons with scalar curvature bounded above or below.
result Classification of specific types of solitons with bounded 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.