The $L^{\frac{3}{2}}$-Norm of the Scalar Curvature Under the Ricci Flow on a 3-Manifold

Hongnian Huang

Communications in Mathematics and Statistics ›› 2013, Vol. 1 ›› Issue (4) : 387 -392.

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Communications in Mathematics and Statistics ›› 2013, Vol. 1 ›› Issue (4) : 387 -392. DOI: 10.1007/s40304-013-0008-4
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The $L^{\frac{3}{2}}$-Norm of the Scalar Curvature Under the Ricci Flow on a 3-Manifold

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Abstract

Assume M is a closed 3-manifold whose universal covering is not S 3. We show that the obstruction to extend the Ricci flow is the boundedness $L^{\frac{3}{2}}$-norm of the scalar curvature R(t), i.e., the Ricci flow can be extended over finite time T if and only if the $\Vert R(t)\Vert_{L^{\frac{3}{2}}}$ is uniformly bounded for 0≤t<T. On the other hand, if the fundamental group of M is finite and the $\Vert R(t)\Vert_{L^{\frac{3}{2}}}$ is bounded for all time under the Ricci flow, then M is diffeomorphic to a 3-dimensional spherical space-form.

Keywords

Ricci flow / Long time existence

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Hongnian Huang. The $L^{\frac{3}{2}}$-Norm of the Scalar Curvature Under the Ricci Flow on a 3-Manifold. Communications in Mathematics and Statistics, 2013, 1(4): 387-392 DOI:10.1007/s40304-013-0008-4

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