# GRADIENT RICCI SOLITON ON O(n)-INVARIANT n-DIMENSIONAL SUBMANIFOLD IN Sn(1)×Sn(1)

• Cho, Jong Taek (Department of Mathematics Chonnam National University) ;
• Kimura, Makoto (Department of Mathematics Faculty of Science Ibaraki University)
• 투고 : 2018.01.23
• 심사 : 2018.03.27
• 발행 : 2019.01.01
• 201 12

#### 초록

We construct gradient Ricci solitons as n-dimensional submanifolds in $S^n{\times}S^n$ by using solutions of some nonlinear ODE.

#### 과제정보

연구 과제 주관 기관 : National Research Foundation of Korea(NRF), JSPS

#### 참고문헌

1. R. Bryant, Ricci flow solitons in dimensions three with SO(3)-symmetries, preprint.
2. H.-D. Cao and Q. Chen, On locally conformally at gradient steady Ricci solitons, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2377-2391. https://doi.org/10.1090/S0002-9947-2011-05446-2
3. X. Cao, B. Wang, and Z. Zhang, On locally conformally at gradient shrinking Ricci solitons, Commun. Contemp. Math. 13 (2011), no. 2, 269-282. https://doi.org/10.1142/S0219199711004191
4. B.-Y. Chen, A survey on Ricci solitons on Riemannian submanifolds, in Recent advances in the geometry of submanifolds-dedicated to the memory of Franki Dillen (1963-2013), 27-39, Contemp. Math., 674, Amer. Math. Soc., Providence, RI, 2016.
5. J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. (2) 61 (2009), no. 2, 205-212. https://doi.org/10.2748/tmj/1245849443
6. J. T. Cho and M. Kimura, Ricci solitons of compact real hypersurfaces in Kahler manifolds, Math. Nachr. 284 (2011), no. 11-12, 1385-1393. https://doi.org/10.1002/mana.200910186
7. J. T. Cho and M. Kimura, Ricci solitons on locally conformally at hypersurfaces in space forms, J. Geom. Phys. 62 (2012), no. 8, 1882-1891. https://doi.org/10.1016/j.geomphys.2012.04.006
8. B. Chow and D. Knopf, The Ricci Flow: an introduction, Mathematical Surveys and Monographs, 110, American Mathematical Society, Providence, RI, 2004.
9. A. Derdzinski, Compact Ricci solitons, preprint.
10. M. Eminenti, G. La Nave, and C. Mantegazza, Ricci solitons: the equation point of view, Manuscripta Math. 127 (2008), no. 3, 345-367. https://doi.org/10.1007/s00229-008-0210-y
11. R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988.
12. T. Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), no. 4, 301-307. https://doi.org/10.1016/0926-2245(93)90008-O
13. M. Kimura and K. Suizu, Fundamental theorems of Lagrangian surfaces in $S^2{\times}S^2$, Osaka J. Math. 44 (2007), no. 4, 829-850.
14. N. Koiso, On rotationally symmetric Hamilton's equation for Kahler-Einstein metrics, Recent topics in differential and analytic geometry, 327-337, Adv. Stud. Pure Math., 18-I, Academic Press, Boston, MA, 1990.
15. G. D. Ludden and M. Okumura, Some integral formulas and their applications to hypersurfaces of $S^n{\times}S^n$, J. Differential Geometry 9 (1974), 617-631. https://doi.org/10.4310/jdg/1214432559
16. L. Ni and N. Wallach, On a classification of gradient shrinking solitons, Math. Res. Lett. 15 (2008), no. 5, 941-955. https://doi.org/10.4310/MRL.2008.v15.n5.a9
17. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/math.DG/0211159, preprint.
18. P. Petersen and W. Wylie, On gradient Ricci solitons with symmetry, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2085-2092. https://doi.org/10.1090/S0002-9939-09-09723-8
19. P. Petersen and W. Wylie, On the classification of gradient Ricci solitons, Geom. Topol. 14 (2010), no. 4, 2277-2300. https://doi.org/10.2140/gt.2010.14.2277
20. F. Urbano, On hypersurfaces of $S^2{\times}S^2$, http://arXiv.org/abs/1606.07595, preprint.
21. Z.-H. Zhang, On the completeness of gradient Ricci solitons, Pacific J. Math. 242 (2009), 189-200. https://doi.org/10.2140/pjm.2009.242.189