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.
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.
Study on scalar curvature deformations in pseudohermitian manifolds.
problem Deformation of scalar curvature in pseudohermitian manifolds.
method Analogy with Riemannian manifolds, introduction of R-singular spaces, stability conditions, partial infinitesimal rigidity. result Partial infinitesimal rigidity result for scalar curvature of compact pseudohermitian manifolds.
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 S-curvature and are either Minkowskian or Riemannian. In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive form…
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.
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.
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.
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.
The aim of the present paper is to provide an \emph{intrinsic} investigation of special Finsler spaces of Hp-scalar curvature and of Hp-constant curvature. Characterizations of such spaces are shown. Sufficient condition for Finsler space of Hp-scalar curvature to be of perpendicular scalar curvature i…
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.
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.
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 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.
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 SkimesTn−k. 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.
In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scala…
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.
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.
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.
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.
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 L2-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.
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.
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.
Let M be a simply-connected closed manifold of dimension ≥5 which does not admit a metric with positive scalar curvature. We give necessary conditions for M to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of M. As a consequence any simply-connected …
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 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.
New bounds on scalar curvature for metric sequences.
problem Bounding scalar curvature in metric sequences.
method Integral convergence of scalar curvature; point-wise scalar curvature lower bound.
result Limiting metric has scalar curvature lower bound.
Study on the convergence rate of prescribed scalar curvature flow.
problem Prescribing scalar curvature on manifolds.
method Inspired by Yamabe flow convergence rate study, analyze the prescribed scalar curvature flow convergence rate.
result Determine the convergence rate of the prescribed scalar curvature flow.
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…
Study on deformation of weighted scalar curvature, proving geometric results and stability.
problem Deformation of weighted scalar curvature and related geometric properties.
method Linearization of weighted scalar curvature, studying kernel of formal adjoint.
result Definition and study of weighted vacuum static spaces, stability results on flat spaces.
New scalars measure failure of CC metrics to solve singular Yamabe problem.
problem Measuring failure of CC metrics to solve singular Yamabe problem.
method Introducing conformally invariant scalar curvature quantities along conformal infinity.
result CC boundary curvature scalars compute canonical expansion coefficients for singular Yamabe metrics.
In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.
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.
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global ε-regularity for 4D Ricci flow with integral scalar curvature bound. Upper diameter bound for manifolds with positive scalar curvature.
problem Estimating the maximum size of manifolds with positive scalar curvature.
method Proving an upper diameter bound using scalar curvature integral, Yamabe constant, and manifold dimension.
result The power of scalar curvature integral in diameter estimates is sharp and occurs at round spheres with canonical metric.
This note relaxes conditions for Kähler metrics with bounded entropy and scalar curvature.
problem Boundedness conditions for Kähler metrics with bounded entropy and scalar curvature.
method Slightly relaxes the boundedness condition on the scalar curvature.
result Apriori estimates and C3,α estimate for the potential of the Kähler metrics under relaxed conditions. 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.
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.
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
Sharp bound on scalar curvature integral in 3-manifolds.
problem Bounding the integral of scalar curvature on 3-manifolds.
method Geodesic ball analysis with nonnegative Ricci curvature.
result Integral of scalar curvature is bounded by 8πR for large radii.