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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 Salón de seminarios "Graciela Salicrup" vCal iCal

Abstract.

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).

References.

[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.