| 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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