arXiv research

Study complete gradient Ricci solitons with zero radial Weyl curvature.

The paper classifies compact quasi-Einstein manifolds with boundary.

Let $\mathcal{C}$ be the space of smooth metrics $g$ on a given compact manifold $M^{n}$ ($n\geq3$) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space $\mathcal{C}$ (we shall refer to this critical poi…

The aim of this paper is to study complete (noncompact) steady $m$-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on the Weyl tensor. In this case, we are able to prove that a steady $m$-quasi-Einstein manifold ($m>1$) on a simply connected $n$-dimensional manifold $(M^{n},g)$, $(n\geq4),$ with …

In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete Einstein-type manifold with fourth-order divergence-free Weyl tensor and zero radial Weyl curvature is locally a warped product with $(n-1)$-dimensional…

In this paper we introduce the notion of generalized quasi--Einstein manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi--Einstein manifolds. We prove that a complete generalized quasi--Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a war…

We provide a general Böchner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero rad…

Study examines maximal domains of radial harmonic functions across different curvature types.

Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.

The paper classifies electrovacuum spaces in higher dimensions, proving several key results.

Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.

We prove that, in a space-time of dimension n>3 with a velocity field that is shear-free, vorticity-free and acceleration-free, the covariant divergence of the Weyl tensor is zero if the contraction of the Weyl tensor with the velocity is zero. The other way, if the covariant divergence of the Weyl tensor is zero, then…

The paper classifies closed Einstein manifolds with specific curvature properties.

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinea…

Einstein 4-manifolds become conformally Kähler with positive scalar curvature.

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the $(p,q)$-curvatures. They are a generalization of the $p$-curvat…

Let (M,g) a compact Riemannian n-dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set, provi…

In 1996, Kirk Lancaster and David Siegel investigated the existence and behavior of radial limits at a corner of the boundary of the domain of solutions of capillary and other prescribed mean curvature problems with contact angle boundary data. In Theorem 3, they provide an example of a capillary surface in a unit disk…

The paper generalizes radial curvature bounds on manifolds.

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…

The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negat…

Study classifies gradient almost Ricci solitons with harmonic Weyl tensor.

The main purpose of this paper is to investigate the Schouten-Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten-Weyl tensor are studied. Moreover, the three-dimensional metric…

Let $Ω$ be a star-shaped bounded domain in $(\mathbb{S}^{n}, ds^{2})$ with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in $Ω.$ This result is the generalization of a result given by Kuttler and Sigillito for a star-shaped bounded doma…

Radial graphs with constant mean curvature found in Euclidean space.

Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.

Study pinched self-dual Weyl curvature in compact 4-manifolds.

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

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Bach tensor and Weyl tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. …

Study on Berwald-Weyl curvature with projective invariance and vanishing results.

We generalize the maximal diameter sphere theorem due to Toponogov by means of the radial curvature. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of …

The aim of this article is to establish a Toponogov type triangle comparison theorem for Finsler manifolds, in the manner of radial curvature geometry. We consider the situation that the radial flag curvature is bounded below by the radial curvature function of a non-compact surface of revolution, the edge opposite to …

The study characterizes harmonic spaces and their radial eigen-functions and vector fields.

Proves conditions for radial Kaehler metrics to be Kaehler-Einstein.

New metrics found without topological restrictions.

An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetri…

We show that the Weyl structure of an almost-Hermitian Weyl manifold of dimension at least 6 is trivial if the associated curvature operator satisfies the Kaehler identity. Similarly if the curvature of an almost para-Hermitian Weyl manifold of dimension at least 6 satisfies the para-Kaehler identity, then the Weyl str…

We show any Weyl curvature model can be geometrically realized by a Weyl manifold

Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.

Given a parabolic geometry on a smooth manifold $M$, we study a natural affine bundle $A \to M$, whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on $A$, which induces an almost bi-Lagrangian structure on $A$ a…

The study finds conditions for almost-Kähler 4-manifolds to be Kähler.

In the presented paper left-invariant pseudo-Riemannian metrics on four-dimensional Lie groups with zero Schouten-Weyl tensor are investigated. The complete classification of these metric Lie groups is obtained in terms of the structure constants of corresponding Lie algebras.

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from t…

Local classification of surfaces and hypersurfaces with radial mean curvature.

The paper proves rigidity results for self-shrinkers and surfaces with parallel weighted mean curvature.

In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with con…

We determine the space of algebraic pseudo-Hermitian Kähler-Weyl curvature tensors and the space of para-Hermitian Kähler-Weyl curvature tensors in dimension 4 and show that every algebraic possibility is geometrically realizable. We establish the Gray identity for pseudo-Hermitian Weyl manifolds and for para-Hermitian…

Study on manifolds with density using modified Hessians for curvature comparison.