Speaker: Yunqing Wu（Tsinghua University）
Time: 10:30-11:30, Nov 23, 2023
Room: USTC Shanghai Institute for Advanced Studies, Room 414
We prove that given a closed Riemann surface (\Sigma,g_0) and a real divisor \mathscr{D}=\sum^{m}_{i=1}\beta_{i}p_i with \beta_i\geq-1, p_i\in\Sigma, i=1,\cdots,m, then for any function K\in L^{\infty}(\Sigma) with K\leq-1, there exists u\in\cap_{p\in[1,2)}W^{1,p}(\Sigma) such that g=e^{2u}g_0 represents \mathscr{D} with curvature K in the sense of distributions. If \Sigma is the Riemann sphere, Dey gave a condition on \beta=(\beta_1,\cdots,\beta_m) with \beta_i>-1 such that \beta is not conformally admissible. We show that by adding certain conditions on \beta, there exists a sequence \beta^{k}\rightarrow\beta such that \beta^{k} is still not conformally admissible. This is a joint with Prof.Jingyi Chen and Prof.Yuxiang Li.
