# MELTING OF THE EUCLIDEAN METRIC TO NEGATIVE SCALAR CURVATURE

• Published : 2013.07.31
• 35 5

#### 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.

scalar curvature

#### References

1. M. Abreu, Kahler geometry of toric varieties and extremal metrics, Internat. J. Math. 9 (1998), no. 6, 641-651. https://doi.org/10.1142/S0129167X98000282
2. V. Apostolov and T. Draghici, The curvature and the integrability of almost Kahler manifolds: A survey, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), 25-53, Fields Inst. Commun., 35, Amer. Math. Soc., Providence, RI, 2003.
3. R. Beig, P. T. Chrusciel, and R. Schoen, KIDs are non-generic, Ann. Henri Poincare 6 (2005), no. 1, 155-194. https://doi.org/10.1007/s00023-005-0202-3
4. A. L. Besse, Einstein Manifolds, Ergebnisse der Mathematik, 3. Folge, Band 10, Springer-Verlag, 1987.
5. M. J. Calderbank, L. David, and P. Gauduchon, The Guillemin formula and Kahler metrics on toric symplectic manifolds, J. Symplectic Geometry 1 (2002), no. 4, 767- 784.
6. J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137-189. https://doi.org/10.1007/PL00005533
7. Y. Kang and J. Kim, Almost Kahler metrics with non-positive scalar curvature which are Euclidean away from a compact set, J. Korean. Math. Soc. 41 (2004), no. 5, 809-820. https://doi.org/10.4134/JKMS.2004.41.5.809
8. Y. Kang, J. Kim, and S. Kwak, Melting of the Euclidean metric to negative scalar curvature in 3 dimension, Bull. Korean Math. Soc. 49 (2012), no. 3, 581-588. https://doi.org/10.4134/BKMS.2012.49.3.581
9. M. Lejmi, Extremal almost-Kahler metrics, Internat. J. Math. 21 (2010), no. 12, 1639-1662. https://doi.org/10.1142/S0129167X10006690
10. J. Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140 (1994), no. 3, 655-683. https://doi.org/10.2307/2118620
11. J. Lohkamp, Curvature h-principles, Ann. of Math. (2) 142 (1995), no. 3, 457-498. https://doi.org/10.2307/2118552

#### 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)