Area growth rate of the level surface of the potential function on the 3-dimensional steady gradient Ricci soliton. (English) Zbl 1167.53056
Summary: We show that on a three-dimensional steady gradient Ricci soliton with positive curvature and which is \( \kappa\)-noncollapsed on all scales, the scalar curvature and the mean curvature of the level surface of the potential function both decay linearly. Consequently we prove that the area of the level surface grows linearly.
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
Keywords:
self-similar solutions; rigidity; gradient solitons; strictly positive sectional curvature; \(\kappa\)-noncollapsedReferences:
| [1] | Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, vol. 135, American Mathematical Society, Providence, RI, 2007. Geometric aspects. · Zbl 1157.53034 |
| [2] | Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006. · Zbl 1118.53001 |
| [3] | Sun-Chin Chu, Geometry of 3-dimensional gradient Ricci solitons with positive curvature, Comm. Anal. Geom. 13 (2005), no. 1, 129 – 150. · Zbl 1094.53062 |
| [4] | Chu, Sun-Chin. Personal communications. |
| [5] | Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7 – 136. · Zbl 0867.53030 |
| [6] | Lott, John. Dimensional reduction and the long-time behavior of Ricci flow. arXiv: 0711.4063. · Zbl 1195.53094 |
| [7] | Naber, Aaron. Noncompact shrinking \( 4\)-solitons with nonnegative curvature. arXiv: 0710.5579. · Zbl 1196.53041 |
| [8] | Ni, Lei; Wallach, Nolan. On a classification of the gradient shrinking solitons. Math. Res. Lett. 15 (2008), no. 5, 941-955. · Zbl 1158.53052 |
| [9] | Ni, Lei; Wallach, Nolan. On \( 4\)-dimensional gradient shrinking solitons. Int. Math. Res. Notices (2008), vol. 2008, article ID rnm152. · Zbl 1146.53039 |
| [10] | Perelman, Grisha. The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. · Zbl 1130.53001 |
| [11] | Perelman, Grisha. Ricci flow with surgery on three-manifolds. arXiv:math.DG/ 0303109. · Zbl 1130.53002 |
| [12] | Petersen, Peter; Wylie, William. Rigidity of gradient Ricci solitons. arXiv:0710.3174. · Zbl 1176.53048 |
| [13] | Petersen, Peter; Wylie, William. On the classification of gradient Ricci solitons. arXiv:0712.1298. · Zbl 1202.53049 |
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