Scalar Curvature, Inequality and Submanifold

Hardy’s Inequality and Curvature

A Hardy inequality of the form ∫ Ω |∇f(x)|dx ≥ ( p− 1 p )p ∫ Ω {1 + a(δ, ∂Ω)(x)} |f(x)| p δ(x)p dx, for all f ∈ C∞ 0 (Ω \ R(Ω)), is considered for p ∈ (1,∞), where Ω is a domain in R, n ≥ 2, R(Ω) is the ridge of Ω, and δ(x) is the distance from x ∈ Ω to the boundary ∂Ω. The main emphasis is on determining the dependance of a(δ, ∂Ω) on the geometric properties of ∂Ω. A Hardy inequality is also e...

Positive Scalar Curvature and Surgery

Constant mean curvature hypersurfaces condensing along a submanifold

Constant mean curvature (CMC) hypersurfaces in a compact Riemannian manifold (Mm+1, g) constitute an important class of submanifolds and have been studied extensively. In this paper we are interested in degenerating families of such submanifolds which ‘condense’ to a submanifold Kk ⊂ Mm+1 of codimension greater than 1. It is not hard to see that the closer a CMC hypersurface is (e.g. in the Hau...

Positive Scalar Curvature

One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [12, 11] derived from the Dirac operator. If a manifo...

An Application of the Isoperimetric Inequality to the Conformal Scalar Curvature Equation

We study the radial symmetry and asymptotic behavior of positive solutions of a certain class of nonlinear elliptic equations, a typical example of which is u = e jxj u2; x 2 IRn and jxj large ( ) where n 2. In particular we show each positive solution u(x) of ( ) satis es lim jxj!1[u(x) u(jxj)] = 0 where u(r) is the average of u on the sphere jxj = r. We also determine the asymptotic behavior ...