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.
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 S k i m e s T n − k \mathbb{S}^k imes\mathbb{T}^{n-k} S k im es T n − k . The paper explores Kähler-like metrics on generalized flag manifolds.
problem Finding invariant almost Hermitian structures with specific scalar curvature properties.
method Investigating invariant almost Hermitian geometry on generalized flag manifolds, focusing on Kähler-like metrics.
result Examples of Kähler-like metrics satisfying s = 2 s m C s=2s_{
m C} s = 2 s m C are provided. 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 extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidi…
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.
Localized deformation of scalar curvature and mean curvature on manifolds.
problem Deforming scalar curvature and mean curvature on compact manifolds with boundary.
method Proving localized surjection of scalar curvature and mean curvature map, handling non-variational linearized problem.
result Localized deformations of scalar curvature and mean curvature on compact manifolds are possible.
Generalizes rigidity of scalar curvature for convex domains.
problem Rigidity of scalar curvature for convex domains.
method Harmonic spinors on convex domains with boundary conditions constructed by Brendle.
result Rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain.
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.
We show that results about spaces or moduli spaces of positive scalar curvature metrics proved using index theory can typically be extended to non-negative scalar curvature metrics. We illustrate this by providing explicit generalizations of some classical results concerning moduli spaces of positive scalar curvature m…
The study proves manifold properties related to positive scalar curvature.
problem Proving the non-existence of metrics with positive scalar curvature on certain manifolds.
method Use of generalized soap bubbles and prescribed-mean-curvature functionals.
result Proves non-existence of metrics with positive scalar curvature on specific manifolds.
New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below, in the spirit of the classical bound on the distances between conjugates points in surfaces with positive sectional curvature.
Study of prescribing scalar curvature and mean curvature on compact manifolds with boundary.
problem Prescribing scalar curvature and mean curvature on compact manifolds with boundary.
method Introducing singular metrics inspired by previous work on closed manifolds, proving rigidity results for flat manifolds with totally geodesic boundary.
result Generic scalar-flat manifolds with minimal boundary can have scalar curvature and mean curvature prescribed simultaneously.
Surveying Ricci flow for weak lower scalar curvature bounds.
problem Creating local definitions for weak lower scalar curvature bounds for C 0 C^0 C 0 metrics. method Using Ricci flow to define and analyze weak lower scalar curvature bounds.
result Properties and applications of Ricci flow in defining weak lower scalar curvature bounds.
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
Study symplectic scalar curvature on supermanifolds.
problem Define and analyze symplectic scalar curvature on supermanifolds.
method Introduced two families of odd super-Fedosov structures using graded symmetric and non-symmetric connections.
result Found non-trivial odd symplectic scalar curvature for the second family.
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ μ μ --bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
Paper explores prescribing Chern scalar curvatures on noncompact Hermitian manifolds.
problem Prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds.
method Generalizes Aviles-McOwen's existence results to higher-dimensional Hermitian manifolds.
result Existence results for Chern scalar curvatures on Hermitian manifolds.
New rigidity results for scalar curvature with stabilized conditions.
problem Establishing rigidity for scalar curvature with stabilized conditions.
method Construction of foliations and development of a monotone quantity using Ricci flow and heat equation.
result Generalized classical scalar curvature rigidity results to the \(T^{
times}\)-stabilized setting.
We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed \emph{coarse scalar curvature}, generalizes a recent definition of Ricci curvature for…
The paper studies 3D manifolds with positive scalar curvature and volume growth.
problem Understanding the geometry of 3D manifolds with positive scalar curvature.
method Analyzes volume and geometric properties of 3D complete manifolds with positive scalar curvature, considering different curvature conditions.
result Volume growth estimates for 3D manifolds with positive scalar curvature, answering Gromov's question affirmatively.
Study on scalar curvature bounds and manifold topological complexity.
problem Understanding the topological complexity of manifolds with scalar curvature constraints.
method Introduced a small scale index theorem to establish bounds for Gromov's simplicial norm.
result Upper bound for Gromov's simplicial norm established in terms of scalar curvature, volume, and injectivity radius.
Paper sharpens inequality linking curvature and spectrum on manifolds.
problem Linking scalar curvature and the bottom spectrum on complete manifolds.
method Using deformed Dirac operators and relative A ^ \widehat{A} A -cowaist. result Established a sharp inequality between scalar curvature and the bottom spectrum.
Derives scalar curvature formula in generalized Kähler geometry.
problem Formalizes scalar curvature in generalized Kähler geometry.
method Uses moment map and action of generalized Hamiltonian automorphisms.
result Derives explicit formula for Goto's scalar curvature.
The study classifies spin c ^c c manifolds with positive generalized scalar curvature.
problem Classifying spin c ^c c manifolds with positive generalized scalar curvature. method Using generalized scalar curvature, index of spin c ^c c Dirac operator, and surgery techniques. result Positive generalized scalar curvature metrics exist if and only if a specific invariant vanishes.
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.
We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use…
New method uses relative capacities of geodesic balls to determine scalar curvature.
problem Determining scalar curvature from geodesic ball volumes.
method Using relative capacities of concentric small geodesic balls.
result Scalar curvature is determined by relative capacities of geodesic balls.
Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant sc…
The paper proves properties of Kähler surfaces with zero scalar curvature.
problem Characterizing Kähler surfaces with zero scalar curvature.
method Analyzing families of generalized Taub-Nut Kähler surfaces and Burn's metric.
result Proves that certain Kähler surfaces are QCH and of specific types.
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n n n -volumic scalar curvature. method Using the curvature-dimension condition m C D ( κ , n ) {
m CD}(κ,n) m C D ( κ , n ) and smGH-convergence. result The stability of n n n -volumic scalar curvature ≥ κ \geq κ ≥ κ under smGH-convergence. We introduce mu-scalar curvature for a K"ahler metric with a moment map mu and start up a study on constant mu-scalar curvature K"ahler metric as a generalization of both cscK metric and K"ahler-Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant…
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 ) (p,n) ( p , n ) -intermediate scalar curvature. result Shows infinitely many path components for certain metrics on ( 4 n − 1 ) (4n-1) ( 4 n − 1 ) -manifolds. The paper studies invariant metrics with positive scalar curvature on 3-manifolds.
problem Classifying G G G -invariant 3-manifolds with positive scalar curvature. method Analyzes the space of G G G -invariant Riemannian metrics with positive scalar curvature on closed 3-manifolds. result The space of G G G -invariant PSC metrics is either empty or contractible. Proves constant scalar curvature Kähler metrics are very general.
problem Existence of constant scalar curvature Kähler metrics on smooth polarized varieties.
method Combining uniform arc K-stability and algebraic properties in families.
result The constant scalar curvature Kähler locus is very general.
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.
In this work we prove that the Whitehead manifold has no complete metric of positive scalar curvature. This result can be generalized to the genus one case. Precisely, we show that no contractible genus one 3 3 3 -manifold admits a complete metric of 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 S S -curvature and are either Minkowskian or Riemannian. The paper studies automorphisms of generalized Kähler manifolds and their Lie algebras.
problem Understanding the existence or non-existence of generalized Kähler structures with constant scalar curvature.
method Analyzing the Lie algebra of automorphisms of generalized complex manifolds under specific conditions.
result The Lie algebra of automorphisms is reductive if a generalized Kähler structure of symplectic type with constant scalar curvature exists.
The study finds hypersurfaces with constant scalar curvature in Minkowski space.
problem Finding hypersurfaces with constant scalar curvature in Minkowski space.
method Analyzes regular domains in Minkowski space and constructs hypersurfaces with constant scalar curvature.
result Hypersurfaces with constant scalar curvature exist and provide foliations of domains.
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.
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.
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.
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.
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 aim of the present paper is to provide an \emph{intrinsic} investigation of special Finsler spaces of H p H_{p} H p -scalar curvature and of H p H_{p}\, H p -constant curvature. Characterizations of such spaces are shown. Sufficient condition for Finsler space of H p H_{p} H p -scalar curvature to be of perpendicular scalar curvature i…