Conference Agenda

Stolz conjectured that the Witten genus of a closed string manifold with positive Ricci curvature vanishes. We will discuss a recent result that confirms the conjecture for string complete intersections X in Fano manifolds M such that M has second Betti number one and admits a smooth effective action of a two-dimensional torus.

In mathematics, there has long been an intrinsic motivation to formulate static problems in a flow context.
Typically, elliptic theories are regarded as the static case of the corresponding parabolic problem; in that sense, many times the better-understood elliptic theory has been a source of intuition to generalise the corresponding results in the parabolic case.
Examples for this duality are minimal surfaces and the mean curvature flow, harmonic maps and the solutions of the heat equation, as well as the Uniformisation Theorem and the two-dimensional normalised Ricci flow.

In this talk, we give a glimpse into several results regarding the prescribed Gauss curvature problem---a problem raised by Kazdan and Warner dealing with the question which smooth functions $f:M\to\R$ arise as the Gauss curvature $K_g$ of a conformal metric $g(x)=\e^{2u(x)}\bar g(x)$ on a closed Riemannian manifold $(M,\bar g)$---and its corresponding flows.

We consider Riemannian manifolds $(M^n,g_0)$, $(M^n,h)$, where $(M^n,h)$ is smooth, complete, with curvature bounded in absolute value by $K_0 < \infty$, and $(1-\varepsilon_0(n)) h \leq g_0 \leq (1+\varepsilon_0(n)) h$ for some small $\varepsilon_0(n)>0$.
It was shown by Simon (2002) that a Ricci–DeTurck flow solution $g(t)_{t \in (0,T)}$ related to $g_0$ exists for some $T=T(n,K_0)>0$.
If $g_0 \in L^2_{\mathrm{loc}}$ or $g_0 \in W^{1,2+2\sigma}_{\mathrm{loc}}$, $\sigma \in (0,\frac{1}{4})$, respectively, we show that $g(t) \to g_0$ in the $L^2_{\mathrm{loc}}$- or $W^{1,2+\sigma}_{\mathrm{loc}}$-sense, respectively.
If $M$ is closed, $g_0 \in W^{1,2+\sigma}(M)$ for some $\sigma>0$, and the distributional scalar curvature of Lee--LeFloch (2015) is not less than $b \in \mathbb{R}$, then we show that $g(t)$ has scalar curvature not less than $b$ in the smooth sense for all $t>0$.

This is joint work with Miles Simon.