Elliptic KdV potentials and conical metrics of positive constant curvature, I

In this paper, we consider curvature equations Δu+eu=16πδ0onEτ,and Δu+eu=8π(δ0+δp+δq)onEτ,p≠q,where eitherp≠-qand℘′(p)+℘′(q)=0∀τ, orq=-p,℘″(p)=0andg2(τ)≠0.Here E_τ= C/ Λ _τ, Λ _τ is the lattice generated by ω₁= 1 and ω₂= τ, τ∈ H, the upper half plane. We prove, among other things that (i) If (0.1) has a solution u then there is a solution u_p of (0.2) with p= (p, q) satisfying (0.3). Moreover, u_p(z) is continuous with respect to p and uniformly converges to u(z) in any compact subset of E_τ\ { 0 } as p→ 0, however, u_p blows up at z= 0. This provides an example for describing blowing-up phenomena without concentration;(ii) If (0.2) is invariant under the change z→ - z, i.e., (p, q) is either (ωi2,ωj2)i≠ j for any τ∈ H, or q= - p and ℘^″(p) = 0 if g₂(τ) ≠ 0 , then (0.1) has the same number of even solutions as (0.2). The converse of (i) remains open. In this paper, we establish a connection between curvature equations and the elliptic KdV theory. The results (i) and (ii) are proved by using this connection.