arXiv research

Paper proves static triples with specific curvature are standard hemispheres.

We prove that a four-dimensional gradient shrinking Ricci soliton with $δW^{\pm}=0$ is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$ or $\mathbb{R}^4$. We also prove that a four-dimensional cscK gradient Ricci soliton is either Kähler-Einstein, or a finite quotient of $M\times\…

We first provide an alternative proof of the classical Weitzneböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of canonical metrics on four-manifolds (or a Weitzenböck formula for "Einstein metr…

Study classifies 4D Ricci solitons with specific curvature conditions.

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

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

Study proves rigidity of certain gradient steady Ricci solitons with harmonic Weyl curvature.

New findings on compact manifolds with specific curvature properties.

The paper classifies quasi-Einstein manifolds with harmonic Weyl curvature.

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

In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature

Study shows Sasaki solitons with harmonic Weyl tensor are spheres.

Defines half-volume spectrum for manifolds and proves Weyl law holds.

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor,…

Classifies gradient Ricci solitons with harmonic Weyl curvature in dimensions 5 and above.

New conformal geometry method solves Einstein-Weyl equations.

Let $(M,g)$ be a noncompact complete $n$-manifold with harmonic curvature and positive Sobolev constant. Assume that $L_2$ norms of Weyl curvature and traceless Ricci curvature are finite. We prove that $(M,g)$ is Einstein if $n \ge 5$ and $L_{n/2}$ norms of Weyl curvature and traceless Ricci curvature are small enough…

In this paper we study 4-dimensional $(m,ρ)$-quasi-Einstein manifolds with harmonic Weyl curvature when $m\notin\{0,\pm1,-2,\pm\infty\}$ and $ρ\notin\{\frac{1}{4},\frac{1}{6}\}$. We prove that a non-trivial $(m,ρ)$-quasi-Einstein metric $g$ (not necessarily complete) is locally isometric to one of the followings: (i) $…

The paper studies hypersurfaces in 5D space forms with topological and rigidity results.

We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $σ_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positi…

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

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…

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…

In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of…

Study rigidifies geometry of electrostatic systems with specific tensor properties.

The study shows rigidity of Kähler-Ricci solitons on a specific 4D manifold.

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n, g)$ be a compact gradient shrinking Ricci soliton satisfying ${\rm Ric}_g + Ddf = ρg$ with $ρ>0$ constant. We show that if $(M,g)$ satisfies $δ\mathcal W (\cdot, \cdot, \nabla f) = 0$, t…

We construct infinitely many Einstein-Weyl structures on $S^2 \times R$ of signature (-++) which is sufficiently close to the model case of constant curvature, and whose space-like geodesics are all closed. Such structures are obtained from small perturbations of the diagonal of $CP^1 \times CP^1$ using the method of L…

Study pseudo-harmonic maps on Weyl manifolds.

We establish a $C^{1,α}$ compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the $C^{1,α}$ harmonic radius, where we use the blow up analysis as in \cite{Anderson90}. The result is motivated by, and may be applied to the Calabi flow on complex surfa…

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. …

We show that Weyl spaces provide a natural context for harmonic morphisms.

Study heat flow for half-harmonic maps and harmonic maps with free boundary.

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons $(M, g, f)$ which have harmonic Weyl curvature, i.e. $δW=0$. Roughly speaking, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein metric, the product $ \mathbb{R}^2 \tim…

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harm…

Simply connected 4-manifolds with specific Weyl tensor are geodesic balls in space forms.

New approach to heat flow for half-harmonic maps, related to minimal surfaces.

Study classifies gradient almost Ricci solitons with harmonic Weyl tensor.

New proof shows gradient Ricci solitons with harmonic Weyl tensor have at most three eigenvalues.

Study on stability of half-harmonic maps from R to S, proving non-degeneracy and quantitative stability.

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…

We define, on smooth manifolds, the notions of almost twistorial structure and twistorial map, thus providing a unified framework for all known examples of twistor spaces. The condition of being harmonic morphisms naturally appears among the geometric properties of submersive twistorial maps between low-dimensional Wey…

New rigidity results for tensors on non-compact manifolds with curvature conditions.

The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic …

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

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We obtain an extension of Siu's rigidity to Kahler-Weyl geometry and apply the latter to Vaisman's conjecture. Other applications include topological obstructions to the existence of Kahler-Weyl structures. Fo…

Paper extends Weyl's lemma to RCD(K,N) spaces.