2026/07/21 by Erjuan Fu, Chang-Shou Lin
Mathematics · #math.AP
Classifying the spherical metrics on a torus Eτ with 4π conic angle at each half period point ωk/2, k=0,1,2,3 is equivalent to classify solutions of the following curvature equation Δu+eu=4π∑k=03δ(ωk)/(2) on Eτ where τ∈ ℍ:=\z∈ ℂ| Im z>0\ and δp is the Dirac measure at p∈ Eτ. By constructing a multiple Green function G2(z1, z2;τ):=G(z1-z2;τ)-(1)/(2)∑j=03(G(z1-(ωj)/(2);τ)+G(z2-(ωj)/(2);τ)), in terms of the Green function G(z;τ) on Eτ, we classify the solutions of (\refeq0731093154) into two types: special and non-special. Furthermore, we obtain the following conclusion about the solutions of (\refeq0731093154): \beginenumerate \item any special solution is an even function and the set of special solutions is isomorphic to SL(2,ℂ)/SU(2) for all τ∈ ℍ. \item a non-special solution exists if and only if τ∈ E. Moreover, if τ∈ E, then there are six one-parameter families of nonspecial solutions. \endenumerate where E:=\τ∈ ℍ| G(z;τ) has exactly 5 critical points.\. The set E is completely determined in \citeCLW2018, Lin, which is a union of countable many open triangular domains. As a byproduct, we completely determine and classify the critical points of G2 and then obtain the degeneracy criterion of critical points for G2, which may be of independent interest.