Compactness theorems of gradient Ricci solitons

Details Compactness theorems of gradient Ricci solitons Compactness theorems of gradient Ricci solitons

2005-07-30

arxiv.org/abs/math/0508009v1

Mathematics

Differential Geometry

21pages

Scientific paper

10.1016/j.geomphys.2006.01.004

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_{\alpha}, g_{\alpha})$ be a sequence of compact gradient Ricci solitons of dimension $n\geq 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$ norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded volume and with uniformly upper bounded diameter, then there must exists a subsequence $(M_{\alpha}, g_{\alpha})$ converging to a compact orbifold $(M_{\infty}, g_{\infty})$ with finitly many isolated singularities, where $g_{\infty}$ is a gradient Ricci soliton metric in an orbifold sense.

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