The Ricci flow on surfaces with boundary

Abstract

We study a boundary value problem for the Ricci flow on a surface with boundary, where the geodesic curvature of the boundary is prescribed.

The Ricci flow on a cylinder

“…We show the previous result for the normalised flow for initial data of the form g 0 = dr 2 + f (r) 2 dθ 2 in Section 3, and in Section 4 we show that whenever the normalised flow exists for all time so does the unnormalised flow (no symmetry assumptions are required to prove this claim). This result should be compared with Proposition 2.3 in [3], where it is shown, in the case of a disk, that under the conditions R g0 > 0 and k g0 ≤ 0, the unnormalised flow becomes singular in finite time.…”

“…The following evolution equation for the normalised flow is well known (a similar formula for the unnormalised flow holds, see Proposition 2.1 in [3]). Lemma 2.1.…”

“…The equation in the interior of M is well known (see [4]). The formula for the normal derivative requires some clarification: it comes from the formula for the normal derivative in the case of the unnormalised flow (Proposition 2.1 in [3]) by noticing that this identity is scaling invariant.…”

“…From now on, all quantities referring to the unnormalised flow will have a tilde, whereas those referring to the normalised flow will not. The existence theory for the unnormalised flow, and hence that of the normalised flow, is well known and we refer to [3] for a short discussion on the matter. Hence, in this work, we shall be concerned with the asymptotic behavior of both versions of the flow, under certain geometric hypotheses, and what prompted us to write this note is how curious this behavior seems to be.…”

Cohomogeneity-one quasi-Einstein metrics

“…In this case, we search for a quasi-Einstein metric (g, u) on G/H × (0, 1) which can be smoothly extended to G/H ×[0, 1] so that for i = 0, 1, (g, u) coincides with (ĝ i , u(i)) when restricted to G/H × {i}, whereĝ i is a fixed G-invariant Riemannian metric on G/H, and u(i) is a fixed real number. The Dirichlet problem for Einstein metrics has been studied in [1,6], but various other boundary-value problems for equations involving the Ricci curvature have also been studied by a number of authors; for example, see [1,6,28,26,25,24,4,5,17,10,16,11].…”