TY - JOUR

T1 - Elliptic KdV potentials and conical metrics of positive constant curvature, I

AU - Kuo, Ting Jung

AU - Lin, Chang Shou

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023

Y1 - 2023

N2 - 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 Eτ= 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 up of (0.2) with p= (p, q) satisfying (0.3). Moreover, up(z) is continuous with respect to p and uniformly converges to u(z) in any compact subset of Eτ\ { 0 } as p→ 0, however, up 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.

AB - 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 Eτ= 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 up of (0.2) with p= (p, q) satisfying (0.3). Moreover, up(z) is continuous with respect to p and uniformly converges to u(z) in any compact subset of Eτ\ { 0 } as p→ 0, however, up 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.

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U2 - 10.1007/s00208-023-02580-3

DO - 10.1007/s00208-023-02580-3

M3 - Article

AN - SCOPUS:85147911284

SN - 0025-5831

JO - Mathematische Annalen

JF - Mathematische Annalen

ER -