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A new result for the porous medium equation derived from the Ricci flow. (English) Zbl 0780.58009

On the space of all Riemannian metrics on a manifold the Ricci curvature operator can be interpreted as a vector field. In this paper, on \(\mathbb{R}^ 2\), with a “good” complete metric as initial point, it is shown that the Ricci flow exists up to time infinity. It may converge to 0, but after reparametrizing with a suitable 1-parameter family of diffeomorphisms, cluster points for \(t\to\infty\) of the Ricci flow are only soliton like metrics, namely either a flat metric or a “cigar- soliton” \({dx^ 2+dy^ 2\over 1+x^ 2+y^ 2}\).
On \(\mathbb{R}^ 2\), the porous medium equation in the limiting case \(m\to 0\) is equivalent to the Ricci flow. So the results about the Ricci flow can be interpreted as sufficient conditions on the initial data which guarantee that the solution for the porous medium equation on the whole plane asymptotically behave like a “soliton-solution”.
Reviewer: P.Michor (Wien)

MSC:

58D25 Equations in function spaces; evolution equations
58D17 Manifolds of metrics (especially Riemannian)
35K05 Heat equation
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References:

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