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

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let $p_1,\ldots ,p_n$ be any set of points at which $f(p_i)=0$ and $D^2f(p_i)$ 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

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where $\delta _{p}$ designates Dirac mass at the point $p$.