Abstract
Locally, a screen integrable globally null manifold $M$ splits through a Riemannian leaf $M'$ of its screen distribution and a null curve $\mathcal{C}$ tangent to its radical distribution. The leaf $M'$ carries a lot of geometric information about $M$ and, in fact, forms a basis for the study of expanding and non-expanding horizons in black hole theory. In the present paper, we introduce a degenerate Ricci-type flow in $M'$ via the intrinsic Ricci tensor of $M$. Several new gradient estimates regarding the flow are proved.
Highlights
Let M be a compact m-dimensional Riemannian manifold on which a one parameter family of Riemannian metrics g(t), t ∈ [0, T ], T < T, where T is the time where there is a blow-up of the curvature, is defined
The evolution equation for the metric tensor implies the evolution equation for the curvature tensor R in the form ∂tR = ∆R + Q, where ∆ denotes the Laplacian operator on M and Q is a quadratic expression of the curvatures
The scalar curvature R satisfies ∂tR = ∆R + 2|Ric|2, so by the maximum principle its minimum is non-decreasing along the flow
Summary
Let M be a compact m-dimensional Riemannian manifold on which a one parameter family of Riemannian metrics g(t), t ∈ [0, T ], T < T , where T is the time where there is (possibly) a blow-up of the curvature, is defined. When the underlying manifold M is null (sometimes called degenerate or lightlike), one may not define, in the usual way, the Ricci flow associated to the degenerate metric g on M. It is well known (see [7]) that, in general, there is no Ricci tensor on M via the null metric g.
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