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.

Aperiodic tilings, positive scalar curvature, and amenability of spaces

Abstract: The object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity. Manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers. We shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties. The most interesting result characterizes when H uf (X) vanishes as an analogue and strengthening of F0lner's amenability criterion for groups in terms of isoperimetric inequalities. (See [4].) One can view this result as producing a successful infinite Ponzi scheme on any nonamenable space. Each point, with only finite resources, gives to some of its neighbors some of these resources, yet receives more from the remaining neighbors. As one can imagine this is useful for eliminating obstructions on noncompact spaces. This has a number of applications. We present two of them. The first produces tilings that are "unbalanced" on any nonamenable polyhedron. Unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically. Unfortunately, imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of Euclidean space (Penrose tilings) are necessarily not accessible to our method. On the other hand, most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion. The second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature. We prove a converse to a theorem of Roe. We show that for any nonamenable group F one can find a spin manifold with fundamental group F, with nonzero A-genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi-isometry class.

$E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory

Abstract: We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a $d$-dimensional manifold for all $d\leq7$. The theory is based on an extended tangent space which admits a natural $E_{d(d)} \times \mathbb{R}^+$ action. The bosonic degrees of freedom are unified as a "generalised metric", as are the diffeomorphism and gauge symmetries, while the local $O(d)$ symmetry is promoted to $H_d$, the maximally compact subgroup of $E_{d(d)}$. We introduce the analogue of the Levi--Civita connection and the Ricci tensor and show that the bosonic action and equations of motion are simply given by the generalised Ricci scalar and the vanishing of the generalised Ricci tensor respectively. The formalism also gives a unified description of the bosonic NSNS and RR sectors of type II supergravity in $d-1$ dimensions. Locally the formulation also describes M theory variants of double field theory and we derive the corresponding section condition in general dimension. We comment on the relation to other approaches to M theory with $E_{d(d)}$ symmetry, as well as the connections to flux compactifications and the embedding tensor formalism.

Generalized Riemannian spaces

Abstract: CONTENTS Introduction Chapter I. Basic concepts connected with the intrinsic metric § 1. Basic definitions § 2. General propositions on upper angles Chapter II. Spaces of curvature () § 3. Basic properties of an -domain § 4. Constructions § 5. Equivalent definitions of an -domain § 6. Area and the isoperimetric inequality § 7. Plateau's problem § 8. Spaces of curvature Chapter III. The space of directions § 9. The space of directions at a point in § 10. The tangent space Chapter IV. Spaces of bounded curvature § 11. Spaces of curvature both and § 12. Introduction of the Riemannian structure Chapter V. Smoothness of the metric in spaces of bounded curvature § 13. Parallel translation § 14. Smoothness of the metric References

Mean curvature flow with surgeries of two-convex hypersurfaces

Abstract: We consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow. It is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded. The purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure. Compared with the notions of weak solutions existing in the literature, the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold. Our construction is inspired by the procedure originally introduced by Hamilton for the Ricci flow, and then employed by Perelman in the proof of Thurston's geometrization conjecture. In this paper we consider initial hypersurfaces which have dimension at least three and are two-convex, that is, such that the sum of the two smallest principal curvatures is nonnegative everywhere. Under these assumptions, we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology. As a corollary, we obtain a classification up to diffeomorphism of the hypersurfaces under consideration.

Bounding scalar curvature and diameter along the kähler ricci flow (after perelman)

Abstract: In this short note we present a result of Perelman with detailed proof. The result states that if . We learned about this result and its proof from Grigori Perelman when he was visiting MIT in the spring of 2003. This may be helpful to people studying the Kahler Ricci flow.