Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

Stable constant mean curvature surfaces

Abstract: We study relationships between stability of constant mean curvature surfaces in a Riemannian three-manifold N and the geometry of leaves of laminations and foliations of N by surfaces of possibly varying constant mean curvature (the case of minimal leaves is included as well). Many of these results extend to the case of codimension one laminations and foliations in n-dimensional Riemannian manifolds by hypersurfaces of possibly varying constant mean curvature. Since this contribution is for a handbook in Dierential Geometry, we also describe some of the basic theory of CMC (constant mean curvature) laminations and some of the new techniques and results which we feel will have an impact on the subject in future years.

On isotropy irreducible Riemannian manifolds

Abstract: A connected Riemannian manifold (M, g) is said to be isotropy irreducible if for each point p E M the isotropy group lip, i.e. all isometries of g fixing p, acts irreducibly on TpM via its isotropy representation. This class of manifolds is of great interest since they have a number of geometric properties which follow immediately from the definition. By Schur's lemma the metric g is unique up to scaling among all metrics with the same isometry group. By the same argument, the Ricci tensor of g must be proportional to g, i.e. g is an Einstein metric. Furthermore, according to a theorem of Takahashi [Ta], every eigenspace of the Laplace operator of (M, g) with eigenvalue A:#0 and of dimension k+ 1 gives rise to an isometric minimal immersion into Sk(r) with r2=dimM/~, by using the eigenfunctions as coordinates (see Li [L] and w 6 of this paper for further properties of these minimal immersions). By a theorem of D. Bleecker [BI], these metrics can also be characterised as being the only metrics which are critical points for every natural functional on the space of metrics of volume 1 on a given manifold. From the definition it follows easily that the isometry group of g must act transitively on M. Hence (M, g) is also a Riemannian homogeneous space. Conversely, we can define a connected effective homogeneous space G/H to be isotropy irreducible if H is compact and Adn acts irreducibly on ~/~. Given an isotropy irreducible homogeneous space G/H, there exists a G-invariant metric g, unique up to scaling, such that (M, g) is isotropy irreducible in the first sense. But if we start with a Riemannian manifold (M, g) which is isotropy irreducible, it can give rise to several

On the Conditions to Extend Ricci Flow

Abstract: Consider {(M n , g(t)), 0 ⩽ t < T < ∞} as an unnormalized Ricci flow solution: for t ∈ [0, T). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈ [0, T) then the solution can be extended over T. Natasa Sesum proves that a uniform bound of Ricci tensor is enough to extend the flow. We show that if Ricci is bounded from below, then a scalar curvature integral bound is enough to extend flow, and this integral bound condition is optimal in some sense.

Universality of einstein equations for the ricci squared lagrangians

Abstract: It has been recently shown that, in the first order (Palatini) formalism, there is universality of Einstein equations and Komar energy– momentum complex, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets Einstein equations and Komar’s expression for the energy–momentum complex. In this paper a similar analysis (also in the framework of the first order formalism) is performed for all nonlinear Lagrangians depending on the (symmetrized) Ricci square invariant. The main result is that the universality of Einstein equations and Komar energy–momentum complex also extends to this case (modulo a conformal transformation of the metric).