The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar-curvature Riemannian metrics g on M . (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g. affect the sign of the answer.) In this article, it is shown that many 4-manifolds ...

We study the Cauchy problem for the semilinear parabolic equations ∆u−Ru+ up − ut = 0 on Mn × (0,∞) with initial value u0 ≥ 0, where Mn is a Riemannian manifold including the ones with nonnegative Ricci curvature. In the Euclidean case and when R = 0, it is well known that 1 + 2 n is the critical exponent, i.e., if p > 1+ 2 n and u0 is smaller than a small Gaussian, then the Cauchy problem has ...

One of the most interesting questions in Riemannian geometry is that of deciding whether a manifold admits curvatures of certain kinds. More specifically, one might want to know whether some given manifold admits a canonical metric, i.e. one with constant curvature of some form (sectional curvature, scalar curvature, etc.). (This will in fact have many topological implications.). One such probl...

Recall that a conformal 4-manifold is called self-dual if its Weyl curvature, considered as a bundle valued 2-form, is in the +1 eigenspace of the Hodge star-operator [1]. Due to Schoen’s proof [19] of the Yamabe conjecture it is known that within any conformal class on a compact manifold is a metric whose scalar curvature is constant and the sign of this constant is a conformal invariant. The ...

A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived. A classical result of I. M. Singer [7] states that a Riemannian manifold is locally homogeneous if and only if its Riemannian curvature tensor together with its ...

The Yamabe problem, solved by Trudinger [14], Aubin [1], and Schoen [12], asserts that any Riemannian metric on a closed manifold is conformal to a metric with constant scalar curvature. Escobar [8], [9] has studied analogous questions on manifolds with boundary. To fix notation, let (M,g) be a compact Riemannian manifold of dimension n ≥ 3 with boundary ∂M . We denote by Rg the scalar curvatur...

Abstract We analyze the stress-energy tensor, and the resulting energy conditions, for a scalar field with general curvature coupling, outside a perfectly reflecting sphere with Dirichlet boundary conditions. For conformal coupling we find that the null energy condition is always obeyed, and therefore the averaged null energy condition (ANEC) is also obeyed. Since the ANEC is independent of cur...

The possibility of realization of a curvaton scenario is studied in a theory in which two dilatons are introduced along with coupling to the scalar curvature. It is shown that when two dilatons have an approximate O(2) symmetric coupling, a scalar field playing the role of the curvaton may exist in the framework of this theory without introducing any other scalar field for the curvaton. Thus th...