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

In this paper, we consider curvature equations (Formula presented.) and (Formula presented.) where (Formula presented.) 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 (Formula presented.) of (0.2) with p=(p,q) satisfying (0.3). Moreover, (Formula presented.) is continuous with respect to p and uniformly converges to u(z) in any compact subset of E_τ\{0} as p→ 0, however, (Formula presented.) 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 (ω_i2,ω_j2)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.