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DOI QR Code

MELTING OF THE EUCLIDEAN METRIC TO NEGATIVE SCALAR CURVATURE

  • Received : 2012.03.12
  • Published : 2013.07.31

Abstract

We find a $C^{\infty}$-continuous path of Riemannian metrics $g_t$ on $\mathbb{R}^k$, $k{\geq}3$, for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ > 0 with the following property: $g_0$ is the Euclidean metric on $\mathbb{R}^k$, the scalar curvatures of $g_t$ are strictly decreasing in $t$ in the open unit ball and $g_t$ 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.

Keywords

scalar curvature

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Cited by

  1. Smooth scalar curvature decrease of big scale on a sphere vol.37, 2014, https://doi.org/10.1016/j.difgeo.2014.10.001
  2. A NOTE ON DECREASING SCALAR CURVATURE FROM FLAT METRICS vol.35, pp.4, 2013, https://doi.org/10.5831/HMJ.2013.35.4.647
  3. SCALAR CURVATURE DECREASE FROM A HYPERBOLIC METRIC vol.20, pp.4, 2013, https://doi.org/10.7468/jksmeb.2013.20.4.269

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)