An Example of a Nontrivial Bubble Tree in the Harmonic Map Heat Flow ∗
An Example of a Nontrivial Bubble Tree in the Harmonic Map Heat Flow ∗
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谐波映射热流中非平凡气泡树的示例 *
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发表时间：
2000
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通讯作者：
P. Topping
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作者：
P. Topping
我们提出的第一个例子中的调和映射热流的非平凡泡树的形成。换句话说，我们给出了一个在同一点上产生多个气泡的流动。气泡在无限时间内发生，并以不同的尺度发展。设（M，γ）是紧致黎曼曲面，（N，g）是紧致无边界黎曼流形.当调和映射能量E（v）= λ M1 2时，调和映射热流是L2梯度下降的|DV| 1964年由Eells和Sampson提出[4]。说明该流动是抛物方程<$u <$u <$xα <$uj <$xβ ; u（·，0）= u 0; u（·，t）的解u：M× [0，∞）→ N| M = u0|其中，R是M上的Laplace-Beltrami算子，并且Γij表示目标N的Christoffel符号。我们把这个方程称为“热方程”，把地图u 0称为“初始地图”，把地图u 0称为“初始地图”。|以Boundary Value为界。流u不随时间变化的映射u 0称为调和映射。有关谐波映射流程的介绍，读者应参考[10，第1章]。然而，我们简要地调查这项工作所需的主要成果。我们将使用的基本存在结果是由于Struwe [8]。定理1给定一个正则的初始映射和边值，热方程（1）存在解u ∈ W1，2 loc（M× [0，∞），N），它在M ×（0，∞）中是光滑的，且至多只离开有限个奇点. Struwe的工作也给了我们第一个信息的渐近流在无限的时间。定理2设u为定理1中引入的热方程（1）的解。则存在一个时间序列ti → ∞，一个调和映射u∞：M → N和一个有限点集{x1，. . . xm} M使得出现在1997年7月在法国布雷斯特举行的关于“调和态射、调和映射和相关主题”的会议记录中。这项工作构成了作者博士论文的一部分[10]。
We present the first example of the formation of a nontrivial bubble tree in the harmonic map heat flow. In other words, we give a flow in which more than one bubble develops at the same point. The bubbles occur at infinite time and develop at different scales. Let (M, γ) be a compact Riemannian surface, and (N , g) a compact Riemannian manifold without boundary. The harmonic map heat flow is L2-gradient descent for the harmonic map energy E(v) = ∫ M 1 2 |dv|, and was introduced in 1964 by Eells and Sampson [4]. Explicitly, the flow is a solution u : M× [0,∞)→ N to the parabolic equation ∂ul ∂t = ∆u + γΓij(u) ∂ui ∂xα ∂uj ∂xβ ; u(·, 0) = u0; u(·, t)|∂M = u0|∂M, (1) where ∆ is the Laplace-Beltrami operator on M and Γij denote the Christoffel symbols of the target N . We refer to this equation as the ‘heat equation,’ to the map u0 as the ‘initial map,’ and to the map u0|∂M as the boundary values. Maps u0 whose flows u do not vary in time are known as harmonic maps. For an introduction to the harmonic map flow, the reader should refer to [10, Chapter 1]. However, we briefly survey the main results required for this work. The basic existence result we will use is due to Struwe [8]. Theorem 1 Given a regular initial map and boundary values, there exists a solution u ∈W 1,2 loc (M× [0,∞),N ) of the heat equation (1) which is smooth inM×(0,∞) away from at most a finite number of singular points. The work of Struwe also gave us the first information on the asymptotics of the flow at infinite time. Theorem 2 Let u be the solution of the heat equation (1) introduced in Theorem 1. Then there exist a sequence of times ti → ∞, a harmonic map u∞ : M → N and a finite set of points {x1, . . . xm} ⊂ M such that ∗To appear in the proceedings of the conference on “Harmonic Morphisms, Harmonic Maps and Related Topics” held in Brest, France, in July 1997. This work constituted part of the author’s PhD thesis [10].