Smooth scalar curvature decrease of big scale on a sphere

Abstract

Motivated by Lohkamp's conjecture on curvature deformation in [13], we present a local smooth decrease of scalar curvature by big scale on a sphere as follows. Given any positive numbers N, a,b with a<b<π, we obtain a C∞-continuous path of Riemannian metrics gt, 0≤t≤1, on the 4-dimensional sphere S4, with g0 being the round metric of constant curvature 1, such that the scalar curvatures s(gt) are strictly decreasing in t on the open ball Bbg0(p) of g0-radius b centered at a point p, s(g1)<−N on Bag0(p) and gt=g0 on the complement of the ball Bbg0(p).This result goes beyond what can be done with Corvino's local first-order deformation theory of scalar curvature [5]. Albeit done on a sphere, the argument here seems generalizable to a larger class of metrics.