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

PY - 2024/1

Y1 - 2024/1

N2 - In this paper, we consider curvature equations (Formula presented.) and (Formula presented.) where (Formula presented.) Here Eτ=C/Λτ, Λτ 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 Eτ\{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 (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 (Formula presented.) and (Formula presented.) where (Formula presented.) Here Eτ=C/Λτ, Λτ 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 Eτ\{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 (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

VL - 388

SP - 2241

EP - 2274

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3

ER -