Smooth scalar curvature decrease of big scale on a sphere

초록

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 < pi, we obtain a C-infinity-continuous path of Riemannian metrics g(t), 0 <= t <= 1, on the 4-dimensional sphere S-4, with go being the round metric of constant curvature 1, such that the scalar curvatures s(g(t)) are strictly decreasing in t on the open ball B-b(g0) (p) of g(0)-radius b centered at a point p, s(g(1)) < -N on B-a(g0) (p) and g(t) = g(0) on the complement of the ball B-b(g0) (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. (C) 2014 Elsevier B.V. All rights reserved.

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