Melting of the Euclidean Metric to Negative Scalar Curvature

Abstract

Abstract. We find a C ∞ -continuous path of Riemannian metrics gt on R k , k ≥ 3, for 0 ≤ t ≤ ε for some number ε > 0 with the following property: g 0 is the Euclidean metric on R k , 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. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.

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.…”

Smooth scalar curvature decrease of big scale on a sphere

“…In a pursuit for the scalar curvature melting problem, we demonstrated such meltings for Euclidean metrics in a previous work [10]. In this article we focus on the round metric on S 4 and prove; Theorem 1.…”

A Note on Decreasing Scalar Curvature From Flat Metrics

“…There exists a C ∞ -continuous one-parameter family of Riemannian metrics g t on R 4 for 0 ≤ t < ε for some number ε with the following property: g 0 is the Euclidean metric on R 4 , g t is isometric to g 0 in the complement of the polydisc D × D ⊂ R 4 , the derivative ds(gt) dt | t=0 of scalar curvatures of g t is identically zero and dt 2 | t=0 is zero somewhere in the polydisc. As explained in Section 2, one can apply the "modification" process of the section 4 of [6]. In fact, we diffuse the negativity of s(g t ) onto a ball by conformally deforming the metric tog t = e 2φt g t , where φ t is a family of functions with φ 0 = 0.…”