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We find a C ∞ one-parameter family of Riemannian metrics gt on R 3 for 0 ≤ t ≤ ε for some number ε with the following property: g 0 is the Euclidean metric on R 3 , the scalar curvatures of gt are strictly decreasing in t in the open unit ball and gt is isometric to the Euclidean metric in the complement of the ball.

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 g t , 0 ≤ t ≤ 1, on the 4-dimensional sphere S 4 , with g 0 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 g0 b (p) of g 0 -radius b centered at a point p, s(g 1 ) < −N on B g0 a (p) and g t = g 0 on the complement of the ball B g0 b (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.

No abstract

We discuss on the classification problem of symplectic manifolds into three families according to the scalar curvature functions of almost Kähler metrics they admit. We also present a 4-dimensional solv-manifold as an example which belongs to one of the three families.