Large Conformal metrics with prescribed sign-changing Gauss curvature

Abstract

Let (M, g) be a two dimensional compact Riemannian manifold of genus g(M) > . Let f be a smooth function on M such that

f >= 0, f not equivalent to 0, min(M) f = 0.

Let be any set of points at which f (P-i) = 0 and D2 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 (1/root 2 log 1/lambda) + O(1)

and

lambda(2)e(u lambda) -> 8 pi Sigma(n)(i=1) delta(pi), as lambda --> 0,

where delta(p) designates Dirac mass at the point p.