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Bubbling and non-bubbling behaviour of solutions for different prescribed curvature problems

Ponente: Franziska Monika Borer
Institución: Universidad de Frankfurt
Tipo de Evento: Investigación
Cuándo 14/08/2018
de 11:00 a 12:00
Dónde Salón de seminarios "Graciela Salicrup"
Agregar evento al calendario vCal


Let \((M, g_0)\) be a closed, connected Riemann surface endowed with a smooth metric \(g_0\). A classical problem in differential geometry is the question if there exists a constant Gauss curvature metric \(g = e^{2u}g_0\) on \(M\). By the Uniformisation Theorem we know that this question has a positive answer. Therefore we can expand the problem in two directions: one option is to consider a family of metrics \(g(t)\) such that over time the curvature \(K_{g(t)}\) will be uniformly distributed over the manifold and so \(g(t)\) converges (in the best case) to \(\bar{g}\) (the so-called normalized Ricci flow). Motivated by other geometric flows we can observe if in this case non-uniqueness through reverse bubbling may occur, see [1]. The other option we can investigate is which smooth functions \(f : M \to \mathbb{R}\) arise as the Gauss curvature \(K_g\) of a conformal metric \(g = e^{2u}g_0\) on \(M\) and to characterize the set of all such metrics with \(K_g = f\). For the case where \((M, g_0)\) is a closed Riemann surface of \(genus (M) > 1\) and \(f_0\) is a smooth, non-constant function with \(\max_{p\in M} f_0(p) = 0\) such that all of whose maximum points are non-degenerate, Ding-Liu [3] showed that for sufficiently small \(\lambda > 0\) there exist at least two distinct conformal metrics \(g_{\lambda} = e^{2u_\lambda}g_0\), \(g^{\lambda} = e^{2u^{\lambda}}g_0\) of Gauss curvature \(K_{g_{\lambda}} = K_{g^{\lambda}} = f_0 + \lambda\), where \(u_{\lambda}\) is a relative minimiser of the associated variational integral and where \(u^{\lambda} \neq u_{\lambda}\) is a further critical point not of minimum type. In [2], by means of a more refined mountain-pass technique we obtain additional estimates for the "large" solutions \(u^{\lambda}\) that allow to characterize their "bubbling behavior" as \(\lambda \downarrow 0\).

The second part is a joint work with Luca Galimberti (University of Oslo, Norway) and Michael Struwe (ETH Zurich, Switzerland).


[1]  F. Borer. Uniqueness of Weak Solutions for the Normalised Ricci Flow in Two Dimensions. Calculus of Variations and Partial Differential Equations, 55:4:1-14, 2016.

[2]  F. Borer and L. Galimberti and M. Struwe, "Large" conformal Metrics of prescribed Gauss Curvature on Surfaces of higher Genus. Commentarii Mathematici Helvetici, 90:2:407-428, 2015.

[3]  W.-Y. Ding and J.-Q. Liu. A Note on the Problem of Prescribing Gaussian Curvature on Surfaces. Transactions of the American Mathematical Society, 347:3:1059-1066, 1995.