Evolution equations of curvature tensors along the hyperbolic geometric flow

详细信息    查看全文

• 作者：Weijun Lu (1) (2)
  
• 关键词：Hyperbolic geometric flow ; Evolution equations ; Singularity ; 53C21 ; 53C44 ; 58J45
• 刊名：Chinese Annals of Mathematics - Series B
• 出版年：2014
• 出版时间：November 2014
• 年：2014
• 卷：35
• 期：6
• 页码：955-968
• 全文大小：208 KB
• 参考文献：1. Brendle, S., Convergence of the Yamabe flow for arbitrary initial energy, / J. Differential Geom., 69, 2005, 217鈥?78.
  2. Brendle, S., Convergence of the Yamabe flow in dimension 6 and higher, / Invent. Math., 170, 2007, 541鈥?76. CrossRef
  3. Brendle, S., Ricci flow and the sphere theorem, Grad. Studies in Math., 111, AMS., Providence RI, 2010.
  4. Brendle, S. and Schoen, R., Manifolds with \(\tfrac{1} {4}\) -pinched curvature are space forms, / J. Amer. Math. Sco., 200, 2009, 1鈥?3.
  5. Chow, B., The Yamabe fow on locally conformally flat manifolds with positive Ricci curvature, / Comm. Pure Appl. Math., 65, 1992, 1003鈥?014. CrossRef
  6. Chow, B., Lu, P. and Ni, L., Harmilton鈥檚 Ricci Flow, American Mathematical Society and Science Press, Lectures in Contemp. Math. 3, AMS, Providence RI, 2006.
  7. Dai, W. R., Kong, D. X. and Liu, K. F., Dissipative hyperbolic geometric flow, / Asia J. Math., 12, 2008, 345鈥?64. CrossRef
  8. Dai, W. R., Kong, D. X. and Liu, K. F., Hyperbolic geometric flow (I): short-time existence and nonlinear stability, / Pure and Applied Mathematics Quarterly, 6(2), 2010, 331鈥?59 (Special Issue: In honor of Michael Atiyah and Isadore Singer). CrossRef
  9. Hamilton, R., Three-manifolds with positive Ricci curvature, / J. Differential Geom., 17, 1982, 255鈥?06.
  10. He, C. L., Exact solutions for Einstein鈥檚 hyperbolic geometric flow, / Communications in Theoretical Physics, 50, 2008, 1331鈥?553. CrossRef
  11. Hiric菐, I. E. and Udriste, C., Basic evolution PDEs in Riemannian geometry, / Balkan Journal of Geometry and Its Applications, 17(1), 2012, 30鈥?0.
  12. Jost, J., Riemannian Geometry and Geometric Analysis, 6th ed., Springer-Verlag, Berlin, 2011. CrossRef
  13. Kong, D. X., Hyperbolic geometric flow, The Proceedings of ICCM 2007, Vol. II, Higher Educationial Press, Beijing, 2007, 95鈥?10.
  14. Kong, D. X. and Liu, K. F., Wave character of metrics and hyperbolic geometric flow, / J. Math. Phys., 48, 2007, 103508-1鈥?03508-14. CrossRef
  15. Kong, D. X., Liu, K. F. and Wang, Y. Z., Life-span of classical solutions to hyperbolic geometric flow in two space variables with slow decay initial data, / Communications in Partial Differential Equations, 36, 2011, 162鈥?84. CrossRef
  16. Kong, D. X., Liu, K. F. and Xu, D. L., The hyperbolic geometric flow on Riemann surfaces, / Communications in Partial Differntial Equations, 34, 2009, 553鈥?80. CrossRef
  17. Liu, K. F., Hyperbolic geometric flow, Lecture at International Conference of Elliptic and Parabolic Differential Equations, Hangzhou, 2007. Available at preprint webpage of Center of Mathematical Science, Zhejiang University.
  18. Milgram, A. and Rosenblum, P., Harmonic forms and heat conduction, I: Closed Riemannian manifolds, / Proc. Nat. Acad. Sci., 37, 1951, 180鈥?84. CrossRef
  19. Schoen, R. and Yau, S. T., Lectures on Differential Geometry, International Press, Boston, 1994.
  20. Schwetlick, H. and Struwe, M., Convergence of the Yamabe flow for 鈥渓arge鈥?energies, / J. Reine Angew. Math., 562, 2003, 59鈥?00.
  21. Topping, P., Lectures on the Ricci Flow, LMS Lecture Notes, 325, London Math. Society Lecture Note Series, 325, Cambridge University Press, Cambridge, 2006. CrossRef
  22. Turk, D., Deforming metrics in the derection of their Ricci tensors, / J. Diffrential Geom., 18, 1983, 157鈥?62.
  23. Udri艧te, C., Riemann flow and Riemann wave via bialternate product Riemannian metric, arXiv:1112.4279v1
  24. Yau, S. T. and Nadis, S., The shape of inner space: String theory and the geometry of the universes hidden dimensions, Basic Books, A Member of the Perseus Books Group, New York, 2010.
  25. Ye, R., Global existence and convergence of Yamabe flow, / J. Differential Geom., 39, 1994, 35鈥?0.
  26. Zhu, X. P., Lectures on mean curvature flows, Studies in Advanced Mathematics, 32, AMS/IP, Providence RI/Somerville, MA, 2002.
  
• 作者单位：Weijun Lu (1) (2)
  
  1. School of Science, Guangxi University for Nationalities, Naning, 530006, China
  2. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China
  
• ISSN：1860-6261

文摘

The author considers the hyperbolic geometric flow \(\tfrac{{\partial ^2 }} {{\partial t^2 }}g(t) = - 2Ric_{g(t)}\) introduced by Kong and Liu. Using the techniques and ideas to deal with the evolution equations along the Ricci flow by Brendle, the author derives the global forms of evolution equations for Levi-Civita connection and curvature tensors under the hyperbolic geometric flow. In addition, similarly to the Ricci flow case, it is shown that any solution to the hyperbolic geometric flow that develops a singularity in finite time has unbounded Ricci curvature.