On the curvature ODE associated to the Ricci flow
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- 作者：Atreyee Bhattacharya (1)
1. Department of Mathematics ; Indian Institute of Science ; Bangalore ; 560012 ; India - 关键词：The curvature ODE ; Ricci flow ; Curvature operator ; Stability of curvature ; Primary 53C21 ; Secondary 53C20
- 刊名：Geometriae Dedicata
- 出版年：2015
- 出版时间：April 2015
- 年：2015
- 卷：175
- 期：1
- 页码：189-209
- 全文大小：282 KB
- 参考文献：1. Arthur, L (1987) Besse, Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3)[Results in Mathematics and Related Areas (3)]. Springer, Berlin
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4. Hamilton, R (1982) Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17: pp. 255-306
5. Hamilton, R (1986) Four-manifolds with positive curvature operator. J. Differ. Geom. 24: pp. 153-179
6. Hamilton, R (1995) The formation of singularities in the Ricci flow. Surv. Differ. Geom. II: pp. 7-136
7. Huisken, G (1985) Ricci deformation on the metric on a Riemannian manifold. J. Differ. Geom. 21: pp. 47-62
8. Topping, P (2006) Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, No. 325. Cambridge University Press, Cambridge - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Geometry - 出版者：Springer Netherlands
- ISSN：1572-9168
文摘
In the vector space of algebraic curvature operators we study the reaction ODE $$\begin{aligned} \frac{dR}{dt} = R^2+R^{\#}= Q(R) \end{aligned}$$ which is associated to the evolution equation of the Riemann curvature operator along the Ricci flow. More precisely, we give a partial classification of the zeros of this ODE up to suitable normalization and analyze the stability of a special class of zeros of the same. In particular, we show that the ODE is unstable near the curvature operators of the Riemannian product spaces \(M \times \mathbb {R}^k, \ k \ge 0\) where \(M\) is an Einstein (locally) symmetric space of compact type and not a spherical space form when \(k=0\) .