Large Conformal metrics with prescribed sign-changing Gauss curvature

TL;DR

This paper constructs a family of conformal metrics on a genus >1 surface with prescribed sign-changing Gauss curvature, exhibiting bubbling behavior and concentration of curvature at specified points as a parameter tends to zero.

Contribution

It proves the existence of bubbling conformal metrics with sign-changing curvature on higher genus surfaces, detailing their asymptotic behavior near prescribed zero points of the curvature function.

Findings

Existence of bubbling conformal metrics with prescribed sign-changing Gauss curvature.

Asymptotic behavior of the conformal factor near bubbling points.

Curvature concentration at specified points as the parameter approaches zero.

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 $$f \ge 0, \quad f\not\equiv 0, \quad \min_M f = 0. $$ 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 $$u_\lambda(p_j) = -4\log\lambda -2\log \left (\frac 1{\sqrt{2}} \log \frac 1\lambda \right ) +O(1)$$ and $$\lambda^2e^{u_\lambda}\rightharpoonup8\pi\sum_{i=1}^{n}\delta_{p_i},\quad \mbox{as }\lambda \to 0,$$ where $\delta_{p}$ designates Dirac mass at the point $p$.