New metrics found without topological restrictions.
problem Finding metrics with constant negative scalar-Weyl curvature.
method Extended Aubin's construction to prove existence.
result Every manifold admits a metric with constant negative scalar-Weyl 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.
Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.
problem Creating metrics with negative scalar curvature.
method Constructed two types of Eguchi-Hanson metrics.
result Found metrics with negative scalar curvature.
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.
Study Kähler metrics on complex tori with almost non-negative scalar curvature.
problem Stability of Kähler metrics on complex tori.
method Proved convergence of non-collapsing subsequence of Kähler metrics to flat torus.
result Kähler metrics with almost non-negative scalar curvature on complex tori converge to flat torus.
New metrics with non-negative scalar curvature are always Ricci-flat on certain surgeries.
problem Understanding metrics with non-negative scalar curvature on surgeries of manifolds.
method Analyzing spin surgeries and their impact on metrics with non-negative scalar curvature.
result Complete metrics with non-negative scalar curvature are Ricci-flat on certain surgeries.
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…
Researchers solve the negative Yamabe case for scalar curvature prescription.
problem Prescribing scalar curvature on manifolds with negative Yamabe invariant.
method A new variational approach to the problem.
result Existence of solutions for sign-changing scalar curvature functions, but not uniqueness.
New rigidity theorems for spin fill-ins with non-negative scalar curvature.
problem Mean curvature rigidity for spin fill-ins with non-negative scalar curvature.
method Two spinorial techniques: extending boundary spinors and comparison using index theory.
result New Witten-type integral inequality for the mass of asymptotically Schwarzschild manifolds.
Study finds infinitely many non-radial solutions for negative scalar curvature in higher dimensions.
problem Prescribing scalar and boundary mean curvature in a ball with negative scalar curvature.
method Analyzes the existence of infinitely many non-radial positive solutions in dimensions 5 and above.
result First existence result for negative scalar curvature in higher dimensions.
Unified flow approach to curvature problem on specific manifolds.
problem Prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree.
method Unified flow approach with conditions on curvature function f. result Flow converges to a conformal Hermitian metric with specified curvature.
Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.
problem Bounding Laplacian eigenvalues on manifolds with non-negative scalar curvature.
method Investigation of invariant spectrum on compact Riemannian manifolds with large isometry groups.
result Upper bounds for eigenvalues of the invariant spectrum assuming non-negative 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.
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.
Study on prescribing curvature on specific manifolds with negative Gauduchon degree.
problem Prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree.
method Analysis of geometric flow convergence to obtain existence results.
result Existence results for curvature functions that are nonzero and nonpositive, and sign-changing cases.
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…
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.
We show that every closed symplectic four-dimensional manifold admits compatible almost Kaehler metrics of negative scalar curvature.
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space Wloc2,n/2 for manifolds of dimension less than or equal to 7 or spin-manifolds of any dimension. More generally, we give a (negative) lower bound on the ADM mass of metrics for which the scalar curvat…
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.
We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex bound…
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.
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.
Kähler blowups can have scalar curvature arbitrarily close to any given metric.
problem Understanding scalar curvature on Kähler blowups.
method Analyzing scalar curvature on blowups of Kähler manifolds.
result The scalar curvature of Kähler blowups can be made arbitrarily close to any given metric.
In this article we show that any finite cover of the moduli space of closed Riemann surfaces of g genus with g≥2 does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which i…
Study rigidifies Einstein manifolds with symmetry, proving conjecture.
problem Einstein manifolds with negative scalar curvature and Lie group action.
method Rigidity result for nilradical action and minimal Einstein submanifolds.
result Alekseevskii conjecture proven for negative scalar curvature homogeneous manifolds.
Study shows convergence of certain metrics to flat torus.
problem Stability of metrics on three-torus with negative scalar curvature.
method Defined metrics and used Stern's inequality to show convergence.
result Subsequence of metrics converges to flat metric.
Proves mass theorem for manifolds with arbitrary ends.
problem Proving the positive mass theorem for manifolds with various ends.
method Quantitative analysis of scalar curvature on manifolds with arbitrary ends.
result Proves the positive mass theorem for a wide class of manifolds.
The paper solves a problem related to curvature in complex geometry.
problem Resolving the prescribed Chern scalar curvature problem.
method Divided into three cases based on the sign of the Gauduchon degree, analyzed separately.
result Proves that certain functions are Chern scalar curvatures of conformal metrics.
Study of singular metrics with negative scalar curvature on compact manifolds.
problem Understanding metrics with negative scalar curvature on compact manifolds with singularities.
method Analyzes metrics with edge singularities and isolated point singularities, showing they are Einstein.
result Uniformly Euclidean metrics with negative scalar curvature are Einstein on compact manifolds.
Paper investigates prescribing Chern scalar curvatures on specific manifolds.
problem Prescribing Chern scalar curvatures on noncompact Hermitian manifolds with nonpositive curvatures.
method Establishes existence results and sufficient conditions for negative curvature metrics.
result Obtains sufficient conditions for the existence of a constant negative Chern scalar curvature metric.
We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geode…
Positive mass theorem for asymptotically flat manifolds with non-negative distributional scalar curvature
problem Positive mass theorem
method Ricci flow smoothing
result Asymptotically flat manifolds with non-negative ADM mass
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.
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.
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.
The paper shows that certain 3-manifolds are essentially Euclidean space.
problem Understanding the relationship between topological rigidity and positive scalar curvature.
method Analyzing complete contractible 3-manifolds with non-negative scalar curvature.
result Any complete contractible 3-manifold with non-negative scalar curvature is homeomorphic to \(\mathbf{R}^3\).
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.
Study shows tori metrics converging to flat under specific conditions.
problem Understanding convergence of metrics on tori with non-negative scalar curvature.
method Uniformly conformal metrics and controlled geometry sequences.
result Sequence of metrics converges to flat metric in multiple senses.
We extend the vanishing theorem for the Seiberg-Witten invariants of a manifold with positive scalar curvature to the case when the curvature is allowed to be negative on a set of small volume. (The precise curvature bounds are described in the paper.) The idea is to combine the method of `semigroup domination' with th…
We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacel…
We classify manifolds of small dimension that admit both, a Riemannian metric of non-negative scalar curvature, and a -- a priori different -- metric for which all wedge products of harmonic forms are harmonic. For manifolds whose first Betti numbers are sufficiently large, this classification extends to higher dimensi…
The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using spinors and prove that if a complete, asymptotically flat manifold with non-negative…
Constructs Kahler-Einstein metrics near isolated log canonical singularities.
problem Constructing metrics near singularities in complex geometry.
method Constructs Kahler-Einstein metrics with negative scalar curvature near isolated log canonical singularities.
result Metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable.
Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem.
problem Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. method Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. result Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. Study on scalar curvature minimizability loss and saddle point solutions.
problem Loss of minimizability in prescribing scalar curvature.
method Variational approach, focusing on saddle point solutions.
result Existence of saddle point solutions despite minimizability loss.
The paper extends Gray's result to quaternion-Kähler manifolds.
problem Understanding quaternion-Kähler manifolds with non-negative quaternionic sectional curvature.
method Introducing quaternionic sectional curvature, proving Wolf spaces have non-negative curvature, and using nearly Kähler twistor spaces.
result Every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space.