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.

Cosmological Density Perturbations with Modified Gravity

Abstract: We determine how the usual formula for density perturbations from inflationary cosmology should be modified when the inflaton is coupled to the scalar curvature at the level of the Lagrangian. By applying a conformal transformation to a single gauge invariant quantity, we rederive a previously proposed formula for the fully coupled system in a simple and unambiguous way

Special Lagrangians, stable bundles and mean curvature flow

Abstract: We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-Holder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.

Avoiding Dark Energy with 1/R Modifications of Gravity

Abstract: Scalar quintessence seems epicyclic because one can choose the potential to reproduce any cosmology (I review the construction) and because the properties of this scalar seem to raise more questions than they answer. This is why there has been so much recent interest in modified gravity. I review the powerful theorem of Ostrogradski which demonstrates that the only potentially stable, local modification of general relativity is to make the Lagrangian an arbitrary function of the Ricci scalar. Such a theory can certainly reproduce the current phase of cosmic acceleration without Dark Energy. However, this explanation again seems epicyclic in that one can construct a function of the Ricci scalar to support any cosmology (I give the technique). Models of this form are also liable to problems in the way they couple to matter, both in terms of matter's impact upon them and in terms of the long range gravitational force they predict. Because of these problems my own preference for avoiding Dark Energy is to bypass Ostrogradski's theorem by considering the fully nonlocal effective action built up by quantum gravitational processes during the epoch of primordial inflation.

Spectra of Compact Locally Symmetric Manifolds of Negative Curvature.

Abstract: Let S be a Riemannian symmetric space of noncompact type, and let G be the group of motions of S. Then the algebra L-~ of G-invariant differential operators on S is commutative, and its spectrum A(S) can be canonically identified with ~/w where ~ is a complex vector space with dimension equal to the rank of S, and to is a finite subgroup of G L ( ~ ) generated by reflexions. Let P be a discrete subgroup of G that acts freely on S and let X = E \\ S . Then the members of 5~ may be regarded as differential operators on X. Let us now assume that X is compact and define the spectrum A of X as the set of those elements of A(S) for which one can find a nonzero eigenfunction defined on X. In this paper we study the relationship of A to the geometry of X and determine the asymptotic growth of A as a subset of A(S). In subsequent papers we plan to study the asymptotic behaviour of the eigenfunctions and to examine the problem of obtaining improvements on the error estimates. It is well-known that G, which is transitive on S, is a connected real semisimple Lie group with trivial center, and that the stabilizers in G of the points of S are the maximal compact subgroups of G. So we can take S = G/K, X =F\\G/K, where K is a fixed maximal compact subgroup of G, and F is a discrete subgroup of G containing no elliptic elements (= elements conjugate to an element of K) other than e, such that F\\G is compact. Let G = K A N be an Iwasawa decomposit ion of G; let o be the Lie algebra of A; and let to be the Weyl group of (G, A). If we take ,~to be the dual of the complexification a c of a, then A ( S ) ~ / w canonically. In what follows we shall commit an abuse of notation and identify A(S) with ,~, but with the proviso that points of ~ in the same w-orbit represent the same element of A(S).

Noncollapsing in mean-convex mean curvature flow

Abstract: We provide a direct proof of a noncollapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial hypersurface admits an interior sphere with radius inversely proportional to the mean curvature at that point, then this remains true for all positive times in the interval of existence. 53C44; 58J35, 35K93