1,720,983 research outputs found
Bochner-type Formulas for the Weyl Tensor on Four-dimensional Einstein Manifolds
Full text linkThe very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner-type formulas for the Weyl tensor on a four-dimensional Einstein manifold. In particular, we prove a 2nd Bochner-type formula that, formally, extends to the covariant derivative level the classical one for the Weyl tensor obtained by Derdziński in 1983. As a consequence, we deduce new integral identities involving the Weyl tensor and its derivatives on a compact four-dimensional Einstein manifold and we derive a new rigidity result
A potential generalization of some canonical Riemannian metrics
No full textThe aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases, we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper, we also describe the nongradient version of this construction
Rigidity of critical metrics for quadratic curvature functionals
Full text linkIn this paper we prove new rigidity results for complete, possibly non-compact, t f R2 critical metrics of the quadratic curvature functionals a2t = f | Ricg |2dVg + gdVg, t is an element of R, and 672 = f R2gdVg. We show that (i) flat surfaces are the only critical points of 672, (ii) flat three-dimensional manifolds are the only critical points of a2t for every t > -13, (iii) three-dimensional scalar flat manifolds are the only critical points of 672 with finite energy and (iv) n-dimensional, n > 4, scalar flat manifolds are the only critical points of 672 with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M.T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates
Two Rigidity Results for Stable Minimal Hypersurfaces
Full text linkThe aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in R^4, while they do not exist in positively curved closed Riemannian (n+1)-manifold when n≤5; in particular, there are no stable minimal hypersurfaces in S^(n+1) when n≤5. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical work of Fischer-Colbrie
Some canonical metrics via Aubin's local deformations
Full text linkEnglish: In this paper, using special metric deformations introduced by Aubin, we construct Riemannian metrics satisfying non-vanishing conditions concerning the Weyl tensor, on every compact manifold. In particular, in dimension four, we show that there are no topological obstructions for the existence of metrics with non-vanishing Bach tensor
On the relation between conformally invariant operators and some geometric tensors
No full textIn this note we introduce and study some new tensors on general Riemannian manifolds which provide a link between the geometry of the underlying manifold and conformally invariant operators (up to order four). We study some of their properties and their relations with well-known geometric objects, such as the scalar curvature, the Q-curvature, the Paneitz operator and the Schouten tensor, and with the elementary conformal tensors recently constructed on the Euclidean space
On Riemannian 4‐manifolds and their twistor spaces: A moving frame approach
Full text linkIn this paper, we study the twistor space (Formula presented.) of an oriented Riemannian 4-manifold (Formula presented.) using the moving frame approach, focusing, in particular, on the Einstein, non-self-dual setting. We prove that any general first-order linear condition on the almost complex structures of (Formula presented.) forces the underlying manifold (Formula presented.) to be self-dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first-order quadratic conditions, showing that the Atiyah–Hitchin–Singer almost Hermitian twistor space of an Einstein 4-manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold
Some triviality results for quasi-Einstein manifolds and Einstein warped products
No full textIn this paper we prove a number of triviality results for Einstein warped products and quasi-Einstein manifolds using different techniques and under assumptions of various nature. In particular we obtain and exploit gradient estimates for solutions of weighted Poisson-type equations and adaptations to the weighted setting of some Liouville-type theorems
On conformally invariant equations on Rn
No full textAbstract. In this paper we provide a complete characterization of fully nonlinear conformally invariant differential operators of any integer order on Rn, which extends the result proved for operators of the second order by A. Li and the first named author in [38]. In particular we prove existence and uniqueness of a family of tensors (suitably invariant under Möbius transformations) which are the basic building blocks that appear in the definition of all conformally invariant differential operators on Rn. We also explicitly compute the tensors that are related to operators of order up to four. 1. Introduction an
A variational characterization of flat spaces in dimension three
No full textWe prove that, in dimension three, flat metrics are the only complete metrics with nonnegative scalar curvature which are critical for the σ2-curvature functional
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