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Special Session 105: Nonlinear Differential Problems on Flat and Curved Structures: Variational and Topological Methods

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

Franziska Borer

Technical University of Berlin
Germany

Co-Author(s): Peter Elbau, Tobias Weth

Abstract:

Let

$(M,\bar g)$ be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth background metric

$\bar g$ . A classical problem raised by Kazdan and Warner is the question which smooth functions

$f:M\to\mathbb{R}$ arise as the Gauss curvature

$K_g$ of a conformal metric

$g(x)=\mathrm{e}^{2u(x)}\bar g(x)$ on

$M$ and to characterise the set of all such metrics. In this talk, we give an overview on several results concerning prescribed Gauss curvature problems depending on the given function

$f$ as well as the Euler characteristic

$\chi(M)$ of the manifold

$M$ , and consider everything in the flow context. Finally we will see, that in the case where the characteristic

$\chi(M)$ is negative and

$f$ is sign-changing, we have to introduce a new kind of prescribed Gauss curvature flow to solve the problem. This is a joint work with Peter Elbau and Tobias Weth.

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