arXiv research

Formula for scalar curvature under metric collapse.

The paper establishes bounds on scalar curvature on asymptotically flat manifolds.

Positive mass theorem for asymptotically flat manifolds with non-negative distributional scalar curvature

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

Study compact sequences of warped product circles over spheres with nonnegative scalar curvature.

We give a complete classification of 1-dimensional exponential families $\mathcal{E}$ defined over a finite space $Ω=\{x_{0}, ...,x_{n}\}$ whose Hessian scalar curvature is constant. We observe an interesting phenomenon: if $\mathcal{E}$ has constant Hessian scalar curvature, say $λ$, then $λ=\tfrac{2}{k}$ for some pos…

Paper proves convergence of warped product manifolds to a nonnegative scalar curvature limit.

The study classifies Riemannian manifolds with curvature nullity.

The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.

The Freund family of distributions becomes a Riemannian 4-manifold with Fisher information as metric; we derive the induced $α$-geometry, i.e., the $α$-curvature, $α$-Ricci curvature with its eigenvales and eigenvectors, the $α$-scalar curvature etc. We show that the Freund manifold has a positive constant 0-scalar cur…

The paper discusses a flow for almost continuous metrics with bounded curvature, leading to smooth metrics with bounded scalar curvature.

The paper is devoted to differential geometry of singular distributions (i.e., of varying dimension) on a Riemannian manifold. Such distributions are defined as images of the tangent bundle under smooth endomorphisms. We prove the novel divergence theorem with the divergence type operator and deduce the Codazzi equatio…

Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prov…

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold $(M^n, g)$ endowed with the complementary orthogonal distributions $D_1$ and $D_2$. We provide conditions for symmetric $(0,2)$-tensors $T$ of a simple form (defined on $M$) to admit metrics $\tilde g$, conformal to …

We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case …

The paper derives inequalities for Riemannian submersions and their applications.

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a dis…

The paper proves compactness of warped product metrics on S²×S¹ with varying base metrics.

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ 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…

The curvature-dimension condition implies a new weighted scalar curvature.

Paper establishes a relation between Berwald scalar curvature and S-curvature.

The paper examines Randers metrics with isotropic scalar curvature properties.

The paper studies Kropina metrics with a specific curvature property.

The paper studies Berwald scalar curvature properties in Finsler geometry.

Sharp volume growth ratio for 3D manifolds with positive scalar curvature.

Small Weyl infimum on 4-manifolds with positive scalar curvature.

New scalar curvature defined from Ollivier-Ricci curvature for graphs.

The aim of the present paper is to provide an \emph{intrinsic} investigation of special Finsler spaces of $H_{p}$-scalar curvature and of $H_{p}\,$-constant curvature. Characterizations of such spaces are shown. Sufficient condition for Finsler space of $H_{p}$-scalar curvature to be of perpendicular scalar curvature i…

Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.

Proves curvature comparison for Riemannian bands in low dimensions.

In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold $(M,g)$ equipped with a vector field $X$. We define several functions ($q$th Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The n…

Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.

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…

We examine the total mixed scalar curvature of a fixed distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also v…

The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.

Minimal splitting factors help study scalar curvature constraints.

Study curvature invariants in sub-Riemannian manifolds.

Sharp bounds on scalar curvature spectrum and rigidity theorems.

Study finds open manifolds without complete metrics with positive scalar curvature.

In this paper we investigate complete critical metrics of the $L^{2}$-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.

Study on scalar curvature deformations in pseudohermitian manifolds.

Paper investigates prescribing Chern scalar curvatures on specific manifolds.

The paper explores geometry and positive scalar curvature on non-compact manifolds.

Study on the convergence rate of prescribed scalar curvature flow.

The study restricts scalar curvature on certain noncompact manifolds.

Study local properties of Chern-scalar curvature through linearization stability.

We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with co…

Proves positive mass theorem for non-spin manifolds with distributional curvature.