Sharp Estimate for the Critical Parameters of S U (3) Toda System with Arbitrary Singularities, I
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Sharp Estimate for the Critical Parameters of SU (3) Toda System with Arbitrary Singularities, I Chang-Shou Lin1 · Wen Yang2,3 Received: 18 April 2020 / Accepted: 20 June 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract To obtain the a priori estimate of Toda system, the first step is to determine all the possible local masses of blow up solutions. In this paper we study this problem and improve the main results in (Anal. PDE 8, 807–837, 2015). Our method is based on a recent work by Eremenko–Gabrielov–Tarasov (Illinois J. Math. 58, 739–745, 2014). Keywords SU (3)-Toda system · Conical singularity · Critical parameter · A priori estimate · Blowup solutions Mathematics Subject Classification (2010) 35J60 · 35J55
1 Introduction Let (M, g) be a Riemann compact surface without boundary and |M| = 1. We consider the solution (u1 , . . . , un ) of the following system defined on M: n hj e u j aij ρj − 1 = 4π αti (δpt − 1), i = 1, . . . , n, (1.1) g ui + u j h e dV j g M j =1
pt ∈A
where g is the Laplace–Beltrami operator, A is a finite set on M, h1 , . . . , hn are positive and smooth functions on M, αt1 , . . . , αtn > −1 are the strength of the Dirac measure δpt , Dedicated to Professor J¨urgen Jost on the occasion of his 65th birthday. Wen Yang
[email protected] Chang-Shou Lin [email protected] 1
Department of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan
2
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, People’s Republic of China
3
Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China
C.-S. Lin, W. Yang
ρ = (ρ1 , . . . , ρn ) are nonnegative constants and A = (aij )n×n is the Cartan matrix of the Lie algebra sl(n + 1), i.e., ⎞ ⎛ 2 −1 0 · · · 0 ⎜ −1 2 −1 · · · 0 ⎟ ⎟ ⎜ ⎟ ⎜ (aij )n×n = ⎜ ... ... ... . . . ... ⎟ . ⎟ ⎜ ⎝ 0 · · · −1 2 −1 ⎠ 0 · · · 0 −1 2 It is well known that equation (1.1) is closely related to geometry [11, 14] and the Chern– Simons–Higgs system in physics model. We refer the readers to [1–4, 12, 15, 17, 20, 21, 25, 27, 28, 30–33] and the references therein for the details about its background in physics and recent developments. In complex geometry, the solutions of the Toda systems are connected to the holomorphic curves in projective spaces. Particularly, the classical Pl¨ucker formula can be written as a local version of the SU (n + 1) Toda system and the branch points of these curves correspond to the singularities of the solutions. It is also remarkable to mention that a local version of (1.1) is an integrable system [22] and the single equation (n = 1) has deep connections with algebraic geometry, modular forms and the Painlev´e VI equation [5, 10]. So it is natural to study equation (1.1) from the perspectives of integrable system. While from the analytic side, one of the most interesting and important question is t
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