The blow-up analysis of an affine Toda system corresponding to superconformal minimal surfaces in ${\mathbb{S}}^4$

Abstract

In this paper, we study the blow-up analysis of an affine Toda system corresponding to minimal surfaces into ${\mathbb{S}}^4$. This system is an integrable system which is a natural generalization of sinh-Gordon equation. By exploring a refined blow-up analysis in the bubble domain, we prove that the blow-up values are multiple of 8π, which generalizes the previous results proved in [39], [37], [27], [22] for the sinh-Gordon equation. More precisely, let $(u_k^1,u_k^2,u_k^3)$ be a sequence of solutions of

$$-\mathrm{\Delta }{u}^1=e^{u^1}-e^{u^3},-\mathrm{\Delta }{u}^2=e^{u^2}-e^{u^3},-\mathrm{\Delta }{u}^3=-\frac{1}{2}{e}^{u^1}-\frac{1}{2}{e}^{u^2}+e^{u^3},u^1+u^2+2u^3=0,$$

in $B_1(0)$, which has a uniformly bounded energy in $B_1(0)$, a uniformly bounded oscillation on $\partial B_1(0)$ and blows up at an isolated blow-up point {0}, then the local masses $({\sigma }_1,{\sigma }_2,{\sigma }_3)\ne 0$ satisfy

$$\begin{gathered} \begin{array}{ccc}\hfill {\sigma }_1& \hfill =\hfill & m_1(m_1+3)+m_2(m_2-1)\hfill \\ \hfill {\sigma }_2& \hfill =\hfill & m_1(m_1-1)+m_2(m_2+3)\hfill \\ \hfill {\sigma }_3& \hfill =\hfill & m_1(m_1-1)+m_2(m_2-1)\hfill \end{array} \\ \text{for some}\begin{array}{c}(m_1,m_2)\in \mathbb{Z}\text{ either}\hfill \\ m_1,m_2=0,1\text{ mod }4,\text{ or }\hfill \\ m_1,m_2=2,3\text{ mod }4.\hfill \end{array} \end{gathered}$$

Here ${\sigma }_i:=\frac{1}{2\pi }{\lim }_{\delta \to 0}{\lim }_{k\to \infty }{\int }_{B_{\delta }(0)}{e}^{u_k^i}dx$.

Introduction

The compactness of a solution space of a nonlinear equation or a system plays an important role in the study of the existence of solutions. For most of interesting geometric partial differential equations it is a challenging problem to understand their solution space. One of important methods is the so-called blow-up analysis, which gives us better information, when the solution space is lack of the compactness. This method goes back at least to the work of Sacks-Uhlenbeck [38] on harmonic maps. Since then, there has been a lot of work on the blow-up analysis for harmonic maps, and also for minimal surfaces, the equation of the constant mean curvature surface, pseudo-holomorphic curves, Yang-Miles fields and the Yamabe equation [1], [4], [40], which play an important role in geometric analysis. In this paper we are interested in the blow-up analysis for the following Liouville type system,

$$-\mathrm{\Delta }{u}_i=\sum _{j=1}^N{a}_{ij}{e}^{u_j},i=1,2,\cdots ,N,$$

where $A=(a_{ij})$ is an $N\times N$ matrix, which has usually a geometric meaning, for example the Cartan matrix of $SU(N+1)$. The Toda system is an important object in integrable system [29], [18], appears in many physical models [15] and relates to many geometric objects, holomorphic curve, minimal surfaces, harmonic maps and flat connections. See for example [2], [3], [14], [16], [25].

When $N=1$ and $a_{11}=1$, it is the Liouville equation

$$-\mathrm{\Delta }u=2e^u,$$

which plays a fundamental role in many two dimensional physical models and also in the problem of prescribed Gaussian curvature [7], [6], [11]. Its blow-up analysis has been carried out by Brezis-Merle [5], Li [30] and Li-Shafrir [31]. When $N\ge 2$ and A is the Cartan matrix for $SU(N+1)$, the blow-up analysis for (1.1) was initiated by Jost-Wang [24] and continued by Jost-Lin-Wang [26], Lin-Wei-Ye [32] and many mathematicians. There also have been a lot of work on the blow-up analysis for the Liouville system with conical singularities. Here we just mention the work of Chen-Li [10], Lin-Wei-Ye [32] and the recent work of Lin-Wei-Yang [33], Lin-Wei-Yang-Zhang [34]. When A is the Cartan matrix for a simple group, for example $S(N+1)$, (1.1) is called a Toda system, or a 2-dimensional open (or finite) Toda system. There are also work on the Toda system for other simple groups. See for example Karmakar-Lin-Nie [28]. Here we remark that such a system could be embedded in the Toda system for $SU(N+1)$. The solutions of an (open) Toda system are closely related to holomorphic curves, harmonic maps and flat connections, see [2], [3], [14], [16].

Other than these open Toda systems there are another type of Liouville systems closely related to constant mean curvature surfaces, superconformal minimal surfaces and harmonic maps rather than holomorphic curves, which are usually called affine Toda systems or periodic Toda systems. The corresponding Liouville systems (or Toda systems) have also different features, though they are very close. The simplest equation is the well-known equation, especially in mathematical physics, the sinh-Gordon equation

$$-\mathrm{\Delta }u=e^u-e^{-u}.$$

The blow-up analysis of (1.3) was initiated by Spruck in [39] and was continued by Ohtsuka-Suzuki in [37], where the existence of solutions were also studied. A more precise blow-up analysis was obtained by Jost-Wang-Ye-Zhou [27], where they showed that the local masses of blow-up must be multiples of 8π. The proof uses the connection of the solutions of (1.3) with constant mean curvature surfaces and harmonic maps. A possible drawback of this geometric proof is that it may work only for (1.3), not for the system with variable coefficients, namely,

$$-\mathrm{\Delta }u=h_1{e}^u-h_2{e}^{-u},$$

where $h_1$ and $h_2$ are two positive functions.¹ The precise blow up analysis for (1.4) has been carried out recently by Jevnikar-Wei-Yang [22] by using a powerful analytic method. A similar blow-up analysis for the Tzitzeica equation was carried out in [21]. For the topological degree of the sinh-Gordon equation, see Jevnikar-Wei-Yang [23]. In this paper we want to generalize their analysis to the following system

$$-\mathrm{\Delta }w=e^w-\frac{1}{2}{e}^{-w+\eta }-\frac{1}{2}{e}^{-w-\eta },$$

$$-\mathrm{\Delta }\eta =\frac{1}{2}{e}^{-w+\eta }-\frac{1}{2}{e}^{-w-\eta }.$$

It is clear that the sinh-Gordon equation (1.3) is a special case of (1.5)-(1.6) with $\eta =0$. This system is related to minimal surfaces into ${\mathbb{S}}^4$ without superminimal points [19]. It was appeared already in the work on integrable system by Fordy-Gibbons in [18]. For the geometric formulation, we refer to the classical paper by Ferus-Pinkall-Pedit-Sterling [19]. For System (1.5)-(1.6) it is convenient to consider the following equivalent system: Let

$$u^1=-w+\eta ,u^2=-w-\eta ,u^3=w.$$

System (1.5)-(1.6) is now equivalent to the following system

$$-\mathrm{\Delta }{u}^1=e^{u^1}-e^{u^3},$$

$$-\mathrm{\Delta }{u}^2=e^{u^2}-e^{u^3},$$

$$-\mathrm{\Delta }{u}^3=-\frac{1}{2}{e}^{u^1}-\frac{1}{2}{e}^{u^2}+e^{u^3},$$

$$u^1+u^2+2u^3=0,$$

in $B_1(0)$.

Our main result in this paper is the following

Theorem 1.1

Let $u^k=(u_k^1,u_k^2,u_k^3)$ be a sequence of solutions of System (1.7)-(1.10) satisfying

$$\sum _{i=1}^3\underset{B_1(0)}{\int }{e}^{u_i^k}(x)dx\le C,\sum _{i=1}^{3}os{c}_{\partial B_1(0)}{u}_k^i\le C$$

and with 0 being its only blow-up point in $B_1(0)$, i.e.

$$\underset{K\subset \subset \overline{B_1(0)}\setminus \{0\}}{\max }\{u_k^1,u_k^2,u_k^3\}\le C(K),\underset{\overline{B_1(0)}}{\max }\{u_k^1,u_k^2,u_k^3\}\to +\infty .$$

Denote

$${\sigma }_i:=\frac{1}{2\pi }\underset{\delta \to 0}{\lim }\underset{k\to \infty }{\lim }\underset{B_{\delta }(0)}{\int }{e}^{u_k^i}dx.$$

Then

$$({\sigma }_1,{\sigma }_2,{\sigma }_3)\in \mathbf{V},$$

where V is defined

$$\mathbf{V}:=\{({\sigma }_1,{\sigma }_2,{\sigma }_3)|{\sigma }_i=4n_i,n_i\in \mathbb{N}\cup \{0\},i=1,2,3;{({\sigma }_1-{\sigma }_3)}^2+{({\sigma }_2-{\sigma }_3)}^2=4({\sigma }_1+{\sigma }_2+2{\sigma }_3)\}﹨\{0,0,0\}.$$

One can easily show that

$$\mathbf{V}=\{\left(\begin{array}{c}\hfill {\sigma }_1\hfill \\ \hfill {\sigma }_2\hfill \\ \hfill {\sigma }_3\hfill \end{array}\right)\in {\mathbb{N}}_0^3|\begin{array}{ccc}\hfill {\sigma }_1& \hfill =\hfill & m_1(m_1+3)+m_2(m_2-1)\hfill \\ \hfill {\sigma }_2& \hfill =\hfill & m_1(m_1-1)+m_2(m_2+3)\hfill \\ \hfill {\sigma }_3& \hfill =\hfill & m_1(m_1-1)+m_2(m_2-1)\hfill \end{array}\mathrm{\&}\begin{array}{c}(m_1,m_2)\in \mathbb{Z}\hfill \\ m_1,m_2=0\text{ or }1\text{ mod }4,\text{ or }\hfill \\ m_1,m_2=2\text{ or }3\text{ mod }4\hfill \end{array}\}﹨\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right).$$

Here ${\mathbb{N}}_0=\mathbb{N}\cup \{0\}$. The first 5 possibilities of $({\sigma }_1,{\sigma }_2,{\sigma }_3)$ are

$$(0,0,4),(4,0,0),(4,4,0),(4,4,12),(12,12,4).$$

Since the roles of ${\sigma }_1$ and ${\sigma }_2$ are symmetric, we have not listed the obvious possibilities by changing ${\sigma }_1$ and ${\sigma }_2$. When $u_1=u_2$, it is easy to see that System (1.7)–(1.10) is reduced to the sinh-Gordon equation. The above result generalizes the previous results given in [39], [37], [27], [22] for sinh-Gordon equation.

The result holds also for System (1.7)–(1.9) with variable coefficients. We remark that like the sinh-Gordon equation one may expect all cases listed in V could occur. For the related work we refer to the work of Esposito-Wei [17] and Grossi-Pistoia [20] and reference therein. We leave this problem to the interested reader. With Theorem 1.1 one can obtain the existence of solutions for system (1.5)-(1.6) under suitable conditions. See for example [12], [26] and the work of Malchiodi for an approach to the existence [36].

This paper follows closely the arguments given in [22]. We do blow-up analysis for our system and obtain blow-up limits which are solution of either the Liouville equation (1.2) or the following system

$$-\mathrm{\Delta }{v}_1=e^{v_1}-e^{v_2}-\mathrm{\Delta }{v}_2=-\frac{1}{2}{e}^{v_1}+e^{v_2}.$$

Except this complexity, there is one big difference. In order to describe this difference, let us first recall the key ideas in [22] for the sinh-Gordon equation. By using a selection process for describing the blow-up situations in the framework of prescribed curvature problems (see for example [8], [30]), the key point in [22] is to show that in each bubbling disk at least local mass (${\sigma }_1$ or ${\sigma }_2$) is a multiple of 4. More precisely, let $B(0,{\lambda }_k)$ be a bubbling disk. By using a standard blow-up analysis, one can easily get that there exists a sequence of numbers $r_k^1\to 0$ with $r_k^1=o(1){\lambda }_k$, such that

$$({\sigma }_1(r_k^1),{\sigma }_2(r_k^1))=(4,0)+o(1),\text{ or }(0,4)+o(1),$$

which roughly means that in $B_{r_k^1}(0)$ one component of $u_k$ converges to a solution of the Liouville equation while the other converges to −∞. Here ${\sigma }_i(r)=\frac{1}{2\pi }{\int }_{B_r}{e}^{u_i}$ is the so-called local mass (or local energy) of $u^i$. As r increases from $r_k^1$ to ${\lambda }_k$, there are two possibilities: either

(1) $({\sigma }_1({\lambda }_k),{\sigma }_2({\lambda }_k))=(4,0)+o(1)$ or $(0,4)+o(1)$, or

(2) there exists $s_k\to 0$ with $s_k/r_k^1\to +\infty$ and $s_k=o(1){\lambda }_k$, such that one component of the solutions has slow decay (for the definition, see Definition 2.3 below). One only needs to care about case (2). In this case, the authors in [22] managed to show that as r increases across $s_k$, one of ${\sigma }_i$ ($i=1,2$) almost does not change while the other increases at least a positive quantity. More precisely they proved that there exists $r_k^2\to 0$ with $r_k^2=o(1){\lambda }_k$ and $r_k^2/s_k\to +\infty$ such that

$${\sigma }_1(r_k^2)-{\sigma }_1(r_k^1)=o(1),{\sigma }_2(r_k^2)-{\sigma }_2(r_k^1)\ge \delta >0,\text{ for some small }\delta >0.$$

Then a local Pohozaev identity

$${({\sigma }_1(r_k^2)-{\sigma }_2(r_k^2))}^2=4({\sigma }_1(r_k^2)+{\sigma }_2(r_k^2))+o(1)$$

for the sinh-Gordon equation implies that

$${\sigma }_1(r_k^2)-{\sigma }_1(r_k^1)=o(1),{\sigma }_2(r_k^2)-{\sigma }_2(r_k^1)=4n+o(1),n\in \mathbb{N}.$$

This means the local energies increase always a multiple of 4, whenever r increases across such an $s_k$. Then the results follow from a careful iteration argument.

For our system (1.7)–(1.9) we also have a similar local Pohozaev identity

$${({\sigma }_k^1(r_k)-{\sigma }_k^3(r_k))}^2+{({\sigma }_k^2(r_k)-{\sigma }_k^3(r_k))}^2=4({\sigma }_k^1(r_k)+{\sigma }_k^2(r_k)+2{\sigma }_k^3(r_k))+o(1).$$

See Section 2 below. This identity plays also a crucial role in our proof. However, alone with this identity is not enough for our system, since we have 3 local energies. It could happen that when r increases, one of local energies keeps almost no change, other two local energies increase at least a positive quantity. One could not use the local Pohozaev identity to conclude that these two energies must increase a multiple of 4. In order to deal with this problem, we manage to do another blow-up near $s_k$ and obtain a “singular bubble”, which is either a solution of the Liouville equation (1.2) or the system (1.15), but with a singular source at the origin. See (3.22) and (3.23). The classification result in the recent work of Lin-Yang-Zhong [35] tells us that the corresponding local energies are a multiple of 4. Moreover, we show that between the previous blow-up and the singular bubble there is no energy loss. In order to show this we crucially use an oscillation estimate Lemma 2.2 and the local Pohozaev identity. This is the main difference to the paper of Jevnikar-Wei-Yang [22]. This proof's ideas come from the study of the harmonic maps, where one proves the so-called energy identity, see for example the work of Ding-Tian [13].

Our methods work at least also for the affine Toda system for $SU(4)$, which includes the Tzitzeica equation as a special case. See Section 5 below. The blow-up analysis of the Tzitzeica equation [41] was carried out by Jevnikar-Yang [21] recently.

We want to emphasize that in the above blow-up analysis the classification of all entire solution of the blow-up limits plays a crucial role. The classification for the Liouville equation (1.2) was given by Chen-Li [9], for (1.1) for $SU(N+1)$ by Jost-Wang [24]. As mentioned above we need also the classification of entire solutions to the Liouville equation (1.2) and system (1.15) with a singular source. For such a result see the work of Lin-Wei-Ye [32] and Lin-Yang-Zhong [35] or Appendix below.

Recently there have been many interesting further results on the study of local mass of a blow-up sequence of open Toda systems (with or without singular sources) with the number of components more than two, for example the Toda systems with Lie algebras of $A_n,B_n,C_n,G_2,D_4,F_4$. See the work of Lin-Yang-Zhong [35] and Karmakar-Lin-Nie [28] where there are also complicated (or even more complicated) blow-up phenomena. The main difference between our results, or other results for affine Toda systems in [39], [37], [27], [22], and the results in [35] and [28], or previous results for open Toda systems, is that affine Toda systems have infinitely many local blow-up possibilities, while open Toda systems have only finitely many local blow-up possibilities. Nevertheless, the blow-up analysis for affine Toda systems is established on the blow-up analysis for the corresponding Toda systems without or even with singular sources. This is showed in this paper. We believe that it is a general phenomenon.

The rest of the paper is organized as follows. In Section 2 we establish a selection process for finite number of bubbling areas, the oscillation estimate outside the blow-up set and the local Pohozaev identities corresponding to the blow-up analysis of system (2.1)-(2.4). In Section 3, we prove a local blow-up behavior Theorem 3.1 where we need to explore more careful blow-up analysis in the bubbling areas. With the help of local Theorem 3.1, by using a standard argument of combining the blow-up areas and a global Pohozaev identity, we give the proof of our main Theorem 1.1 in Section 4. In Section 5, we discuss another systems, affine Toda system for $SU(4)$. The blow-up analysis holds for this system. In Section 6, we recall two classification theorems which are used in our proof. In Section 7 we review the relationship between the system (1.5)-(1.6) and the minimal surface into ${\mathbb{S}}^4$.