Prescribing Curvatures on Surfaces with Conical Singularities and Corners

This article is concerned with the problem of prescribing Gaussian curvature K and geodesic curvature h in a compact surface with boundary with conical singularities {p1, ... , pn} and corners {q1, ... , qn}. This is equivalent to solving the Liouville-type equation: { - Delta u+2k(0) = 2ke(u) - 4 pi Sigma (n)( i=1)alpha( i) (delta( pi) - 1 /|Sigma|) -2 pi Sigma (m )(j=1) beta(j) (delta(qi) -1 /|Sigma|) in Sigma partial derivative(v)u+2h(0) = 2he (u /2) on partial derivative Sigma where K0, h0 are the pre-existing Gaussian curvature and the geodesic curvature, respectively, and alpha(i), beta(j )> -1 are given. Solutions are obtained using a new variational formulation, first introduced in Thierry (1998) in [1] for the regular counterpart of the problem and extended here to the singular case. As far as we know, this is the first result for the problem of prescribed curvatures in surfaces with the two types of singularities. Key ingredients are a blow-up analysis around a sequence of points different from local maxima and Morse index estimates.