Spherical metrics with one singularity and odd integer angle on flat tori, I
Journal
Journal of Differential Geometry
Journal Volume
132
Journal Issue
2
Start Page
321
End Page
382
ISSN
0022-040X
Date Issued
2026-02-01
Author(s)
Abstract
Let Eτbe the torus with periods 1, τ and δ0be the Dirac measure at 0. Consider the following curvature equation (0.1) ∆u + eu= 8πnδ0on Eτ, and the integral Lame equation (0.2) y′′= (n(n + 1)℘(z | τ ) + B)y,where ℘(z | τ ) is the Weierstrass elliptic function. Motivated by our previous works, we conjecture that solvability of (0.1) depends on the geometry of Eτdetermined by the multiple Green function Gnof Eτ. The main purpose of this paper is to confirm our conjecture. Let F0be a fundamental domain of Γ2(see (1.4) for the definition). Define LWn= the continuous part of {τ ∈ F0| Gn(z1, . . . , zn; τ ) has a degenerate trivial critical point}. Among other things, we prove the following results. (i) A necessary and sufficient condition for the existence of solutions of (0.1) with τ ∈ F0, that is, there are [formula present here] connected open domains Λ(k)in F0such that (0.1) has an even solution iff [formula present here] and ∂Λ(k)⊆ LWn. (ii) If the monodromy matrices Si, i = 1, 2, of (0.2) are unitarizable, and λ is an eigenvalue of S1or S2, then λ ∈/ {±1}. The is the best possible result for the integral Lame equation (0.2). (iii) For n = 2, 3, 4, we prove the asymptotic behavior of Re τ for those τ where (0.1) has a solution, as τ → ∞. We remark that the paper is a culmination of our previous works over more than ten years.
Publisher
International Press of Boston
Type
journal article