"Bubbling" of the prescribed curvature flow on the torus

Abstract

For given functions f and j on the disc B and its boundary ∂B = S 1 , we study the existence of conformal metrics g = e 2u g Ê 2 with prescribed Gauss curvature Kg = f and boundary geodesic curvature kg = j. Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz [9], we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work [19] on the prescribed curvature problem on the sphere, we are able to exhibit a 2-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz [17] by degree-theory.

A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic

“…Proof. We follow the proof of [19,Lemma 2.5]. By using the fact that u(t) is a volume preserving solution of (5.11) with u(0) = u 0 ∈ C p,A and therefore M e 2u(t) dµ ḡ ≡ A, we get with (4.3) and the fact that K < 0 that…”

“…Let f 0 ≤ 0 be a smooth, nonconstant function withmax x∈M f 0 (x) = 0. Following here the argumentation of [19], and using (5.31), we know that for a suitable sequence t l → ∞, l → ∞, with associated metrics g l = g(t l ) we obtain convergence…”

“…(5. 19) In the following, the letter C denotes different positive constants. Multiplying (5.19) with 2u and integrating over M gives…”

“…In the present paper we focus on the case χ(M ) < 0, so M is a surface of genus greater than one and K < 0. The complementary cases χ(M ) ≥ 0-i.e., the cases where M = S 2 or M = T , the 2-torus-will be discussed briefly at the end of this introduction, and we also refer the reader to [18,19,2,8] and the references therein. Multiplying equation (1.2) with the factor e −2u and integrating over M with respect to the measure µ ḡ , we get the following necessary condition-already mentioned by Kazdan and Warner in [11]-for the average f := 1 volḡ M f (x)dµ ḡ (x), with vol ḡ := M dµ ḡ (x): x) dµ ḡ (x) < 0.…”

“…At this point, it should be noted that the normalisation factor α(t) in the prescribed Gauss curvature flow given by (1.8) is also not the appropriate choice in the case of the torus, where, as noted before, f has to change sign or be identically zero in order to arise as the Gauss curvature of a conformal metric. The case of the torus was considered by Struwe in [19], where, in particular, he used to a flow approach to reprove and partially improve a result by Galimberti [8] on the static problem. In this approach, the normalisation in (1.8) is replaced by .11) With this choice, Struwe shows that for any smooth…”