Deformation of scalar curvature and volume

Existence of Metrics with Prescribed Scalar Curvature on the Volume Element Preserving Deformation

In this paper,we obtain two results on closed Reimainnian manifold M × [0, T ].When T is small enough,to any prescribed scalar curvature, the existence and uniqueness of metrics are obtained on the volume element preserving deformation.When T is large and the given scalar curvature is small enough,the same result holds.

Conformal Deformation of a Riemannian Metric to Constant Scalar Curvature

A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 tha...

The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds

Abstract. We study the equation ∆gu− n−2 4(n−1)R(g)u+Ku p = 0 (1+ ζ ≤ p ≤ n+2 n−2 ) on locally conformally flat compact manifolds (M, g). We prove the following: (i) When the scalar curvature R(g) > 0 and the dimension n ≥ 4, under suitable conditions on K, all positive solutions u have uniform upper and lower bounds; (ii) When the scalar curvature R(g) ≡ 0 and n ≥ 5, under suitable conditions ...

Some Compactness Results Related to Scalar Curvature Deformation

Motivated by the prescribing scalar curvature problem, we study the equation ∆gu+Ku p = 0 (1 + ζ ≤ p ≤ n+2 n−2 ) on locally conformally flat manifolds (M, g) with R(g) = 0. We prove that when K satisfies certain conditions and the dimension of M is 3 or 4, any solution u of this equation with bounded energy has uniform upper and lower bounds. Similar techniques can also be applied to prove that...

Positive Scalar Curvature and Surgery