Large Conformal metrics with prescribed sign-changing Gauss curvature

Abstract

Let $(M,g)$ be a two dimensional compact Riemannian manifold of genus $g(M)>1$. Let $f$ be a smooth function on $M$ such that $$f\ge 0,f\not\equiv 0,\underset{M}{\min }f=0.$$ Let $p_1,\dots ,p_n$ be any set of points at which $f(p_i)=0$ and $D^{2}f(p_i)$ is non-singular. We prove that for all sufficiently small $\lambda >0$ there exists a family of "bubbling" conformal metrics $g_{\lambda }=e^{u_{\lambda }}g$ such that their Gauss curvature is given by the sign-changing function $K_{g_{\lambda }}=-f+{\lambda }^2$. Moreover, the family $u_{\lambda }$ satisfies $$u_{\lambda }(p_j)=-4\log \lambda -2\log \left(\frac{1}{\sqrt{2}}\log \frac{1}{\lambda }\right)+O(1)$$ and $$λ^2e^{u_λ}\rightharpoonup8π\sum_{i=1}^{n}δ_{p_i},\quad \mbox{as }λ\to 0,$$ where ${\delta }_p$ designates Dirac mass at the point $p$.

29 pages, 1 figure

Cited by in corpus (4)

• Unbounded mass radial solutions for the Keller-Segel equation in the disk
• Prescribed Gauss curvature problem on singular surfaces
• Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case
• Large conformal metrics with prescribed scalar curvature