On the general Toda system with multiple singular points
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On the general Toda system with multiple singular points Ali Hyder1 · Jun‑cheng Wei2 · Wen Yang3,4 Received: 23 January 2020 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we consider the following elliptic Toda system associated to any general simple Lie algebra with multiple singular sources � ∑n ∑m − Δwi = j=1 ai,j e2wj + 2𝜋 𝓁=1 𝛽i,𝓁 𝛿p𝓁 in ℝ2 , wi (x) = − 2 log �x� + O(1) as �x� → ∞, i = 1, … , n,
where 𝛽i,𝓁 ∈ [0, 1) . Under some suitable assumption on 𝛽i,𝓁 we establish the existence and non-existence results. This paper generalizes Luo and Tian’s (Proc Am Math Soc 116(4):1119–1129, 1992) and Hyder et al. (Pac J Math 305(2):645–666, 2020) results to the general Toda system. Mathematics Subject Classification 35J47 · 35B44 · 35J61
1 Introduction In this paper, we shall consider the following singular Toda system with multiple singular sources
Communicated by M. Del Pino. * Wen Yang [email protected] Ali Hyder [email protected] Jun‑cheng Wei [email protected] 1
Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA
2
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
3
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, People’s Republic of China
4
Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China
13
Vol.:(0123456789)
A. Hyder et al.
− Δwi =
n ∑
ai,j e2wj + 2𝜋
j=1
m ∑
𝛽i,𝓁 𝛿p𝓁
in
ℝ2 ,
𝓁=1
(1.1)
where (ai,j )n×n is the Cartan matrix associated to a simple Lie algebra, 𝛽i,𝓁 ∈ [0, 1) , p1 , … , pm are distinct points in ℝ2 and 𝛿𝓁 denotes the Dirac measure at p𝓁 , 𝓁 = 1, … , m. When the Lie algebra is 𝐀1 = 𝔰𝔩2 , (1.1) becomes the Liouville equation
Δu + e2u = − 2𝜋
m ∑
𝛾𝓁 𝛿 p 𝓁
in ℝ2 .
(1.2)
𝓁=1
The Toda system (1.1) and the Liouville equation (1.2) arise in many physical and geometric problem. On geometric side, Liouville equation is related to the Nirenberg problem of finding a conformal metric with prescribed Gaussian curvature if {p1 , … , pm } = � , and the existence of the same curvature metric of problem (1.2) with conical singularities at {p1 , … , pm } . When the Lie algebra is 𝐀n , the Toda system (1.1) is closely related to holomorphic curves in projective spaces [5] and the Plücker formulas [9], while the periodic Toda systems are related to harmonic maps [10]. In physics, the Toda system is a wellknown integrable system and closely related to the W -algebra in conformal field theory, see [2, 8] and references therein. Liouville equation and Toda system also play an important role in Chern–Simons gauge theory. For example, 𝐀n ( n = 2 ) Toda system governs the limit equations as physical parameters tend to 0 and is used to explain the physics of high temperature, we refer the readers to [6, 32, 33] for more background on it, and [13–19, 21, 29, 30] for the recent developments on (1.1) and (1.2). For the Liouvi
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