English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The problem of prescribing Gaussian curvature on Riemann surface with conicalsingularity is considered. Let $(\Sigma,\beta)$ be a closed Riemann surfacewith a divisor $\beta$, and $K_\lambda=K+\lambda$, where$K:\Sigma\rightarrow\mathbb{R}$ is a H\"older continuous function satisfying$\max_\Sigma K= 0$, $K\not\equiv 0$, and $\lambda\in\mathbb{R}$. If the Eulercharacteristic $\chi(\Sigma,\beta)$ is negative, then by a variational method,it is proved that there exists a constant $\lambda^\ast>0$ such that for any$\lambda\leq 0$, there is a unique conformal metric with the Gaussian curvature$K_\lambda$; for any $\lambda$, $0<\lambda<\lambda^\ast$, there are at leasttwo conformal metrics having $K_\lambda$ its Gaussian curvature; for$\lambda=\lambda^\ast$, there is at least one conformal metric with theGaussian curvature $K_{\lambda^\ast}$; for any $\lambda>\lambda^\ast$, there isno certain conformal metric having $K_{\lambda}$ its Gaussian curvature. Thisresult is an analog of that of Ding and Liu \cite{Ding-Liu}, partly resemblesthat of Borer, Galimberti and Struwe \cite{B-G-Stru}, and generalizes that ofTroyanov \cite{Troyanov} in the negative case.

Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case