arXiv research

In this letter we provide an invariant characterization for all spacetimes with all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives vanishing except those zeroth order curvature invariants expressed as polynomials in $Λ$, the cosmological constant. Using this invariant des…

$\mathcal{I}$-non-degenerate spaces are spacetimes that can be characterized uniquely by their scalar curvature invariants. The ultimate goal of the current work is to construct a basis for the scalar polynomial curvature invariants in three dimensional Lorentzian spacetimes. In particular, we seek a minimal set of alg…

A pseudo-Riemannian manifold is called CSI if all scalar polynomial invariants constructed from the curvature tensor and its covariant derivatives are constant. In the Lorentzian case, the CSI spacetimes have been studied extensively due to their application to gravity theories. It is conjectured that a CSI spacetime i…

A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or none independent Killing vectors. We present an explicit algorithm for computing dimension of the infinitesimal isometry algebra. It branches according to the values of curvature invariants. These are relative differential invariants computed via curv…

Study on scalar curvature decay on non-compact manifolds linked at infinity.

The paper explores Kähler-like metrics on generalized flag manifolds.

We will prove that \emph{there are no stable complete hypersurfaces of $\mathbb{R}^4$ with zero scalar curvature, polynomial volume growth and such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere, for some constant $c>0$}, where $K$ denotes the Gauss-Kronecker curvature and $H$ denotes the mean curvature of the immersion. …

We examine the existence of one parameter groups of diffeomorphisms whose infinitesimal generators annihilate all scalar polynomial curvature invariants through the application of the Lie derivative, known as $\mathcal{I}$-preserving diffeomorphisms. Such mappings are a generalization of isometries and appear to be rel…

The paper explores constant holomorphic d-scalar curvature on specific manifolds.

Study adds scalar curvatures of mapped manifolds to Riemannian products.

In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac …

Holomorphic functions grow polynomially on Kähler-Ricci shrinkers, proving ring finitely generated.

The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.

Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.

Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results involving CVIs, generalizing many such results for the scalar curvature.

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions g…

Finite element method approximates scalar curvature in arbitrary dimensions.

Paper proves a Liouville theorem for solitons with constant curvature.

Explains peculiarities of 4D scalar curvature via Yamabe invariant.

Study on metrics with positive scalar curvature on manifolds with singularities.

The paper proves inequalities for scalar curvature on various manifolds.

We prove that every quasitoric manifold admits an invariant metric of positive scalar curvature.

The paper connects curvature data to polynomial coefficients in gluing formulas.

New scalars measure failure of CC metrics to solve singular Yamabe problem.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

The paper studies invariant metrics with positive scalar curvature on 3-manifolds.

New approach linking CR Yamabe invariant to Sasaki structures.

Study determines scalar curvature invariants for 3-spheres embedded in 4-manifolds.

Conformal invariants of manifolds of non-positive scalar curvature are studied in association with growth in volume and fundamental group.

Survey on scalar curvature stability and related questions.

Researchers prove spaces of positive scalar curvature metrics are contractible with symmetry.

The paper proves scalar curvature lower bounds along Ricci flow on compact manifolds.

Abstract: Characterizes spaces with positive scalar curvature.

We compute renormalized curvature integrals on Poincaré-Einstein manifolds.

Study shows incompatibility of certain scalar curvatures on manifolds.

Study negative scalar curvature metrics with positive boundary mean curvature.

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 construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metr…

We show that the higher homotopy groups of the moduli space of torus-invariant positive scalar curvature metrics on certain quasitoric manifolds are non-trivial.

Proves the Kundt conjecture in arbitrary dimensions, confirming its validity.

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bo…

New Kazdan-Warner problem for equivariant metrics on manifolds.

Study eta invariant on non-compact manifolds with positive scalar curvature.

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant …

Researchers solve the negative Yamabe case for scalar curvature prescription.

Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.

The paper provides obstructions to positive scalar curvature for certain manifolds with group actions.

On a pre-quantized symplectic manifold, we show that the symplectic Futaki invariant, which is an obstruction to the existence of constant Hermitian scalar curvature almost-Kähler metrics, is actually an asymptotic invariant. This allows us to deduce a lower bound for the L^2-norm of the Hermitian scalar curvature as o…