Melting of the Euclidean Metric to Negative Scalar Curvature in 3 Dimension

Abstract

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.

Scalar Curvature Decrease From a Hyperbolic Metric

“…Our argument in this section follows those in [4,Section 4] and [5,Section 4] with just a few differences in estimation.…”

“…In the previous works we have studied explicit scalar curvature meltings of Euclidean metrics and one positive Einstein metric [4,5]. In this article we study the hyperbolic metric g h , i.e.…”

Melting of the Euclidean Metric to Negative Scalar Curvature

“…We easily get ds(e 2φ tg t ) dt | t=0 = 0. Using the conformal deformation formula s(e 2φt g t ) = e −2φt (s gt + 4n∆ gt φ t − 2n(2n− 1)|∇ gt φ t | 2 ), we calculate as in [8,Section 4] to show that d 2 s(e 2φ tg t ) dt 2 | t=0 < 0 on B 9+ǫ1 (p) for small m > 0. Note that e 2φtg t = g 0 on R 2n+1 \B 9+ǫ1 (p).…”

“…In a recent paper [8], we explained the scalar-curvature melting of Euclidean metric in 3 dimension. The purpose of this article is to complete the scalar-curvature melting of Euclidean metrics in any dimension ≥ 3 and then extend the discussion to the Fubini-Study metric in a similar way.…”