Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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MELTING OF THE EUCLIDEAN METRIC TO NEGATIVE SCALAR CURVATURE

  • Received : 2012.03.12
  • Published : 2013.07.31
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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.

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Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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