Finite time blowup of the \(n\)-harmonic flow on \(n\)-manifolds. (English) Zbl 1390.58010
Let \(M^n\) be an \(n\)-dimensional Riemannian manifold without boundary, and let \(N^m\) be another \(m\)-dimensional compact Riemannian manifold without boundary (isometrically embedded into \({\mathbb R}^L\)). For a map \(u : M \to N \hookrightarrow {\mathbb R}^L\), the \(n\)-energy functional of \(u\) is defined by
\[
E_n(u) = \frac{1}{n}\int_M |\nabla u|^2\, dv.
\]
A \(C^1\)-map \(u\) from \(M\) to \(N\) is said to be an \(n\)-harmonic map if \(u\) is a critical point of the \(n\)-energy functional, i.e., it satisfies
\[
\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial x^i}\left[|\nabla u|^{n-2}g^{ij}\sqrt{|g|} \frac{\partial}{\partial x^j}u\right] + |\nabla u|^{n-2}A(u)(\nabla u, \nabla u) =0,
\]
where \(A\) is the second fundamental form of \(N\).
In this paper, the authors consider the \(n\)-harmonic map flow in the following setting: \[ \frac{\partial u}{\partial t} = \frac{1}{\sqrt{|g|}}\frac{\partial}{\partial x^i}\left[|\nabla u|^{n-2}g^{ij}\sqrt{|g|} \frac{\partial}{\partial x^j}u\right] + |\nabla u|^{n-2}A(u)(\nabla u, \nabla u). \] N. Hungerbühler [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 4, 593–631 (1997; Zbl 0911.58011)] proved that there exists a global weak solution \(u : M\times [0, \infty) \to N\) of the \(n\)-harmonic map flow with initial value \(u_0\) such that \(u_0\in C^{1, \alpha}(M \times (0, \infty)\setminus \{\Sigma_k \times T_k\}_{k=1}^m)\)for a finite number of times \(\{T_k\}\) and a finite number of singular closed sets \(\Sigma_k \subset M\). Based on the fundamental result of K.-C. Chang et al. [J. Differ. Geom. 36, No. 2, 507–515 (1992; Zbl 0765.53026)] for \(n=2\) and supported by some numerical evidences, Hungerbühler also conjectured the phenomenon of finite time blowup of the \(n\)-harmonic flow for \(n \geq 3\).
In this paper, the authors generalize the no-neck result of J. Qing and G. Tian [Commun. Pure Appl. Math. 50, No. 4, 295–310 (1997; Zbl 0879.58017)] to the \(n\)-harmonic map flow, and as an application of this generalized no-neck result, they settle the conjecture proposed by Hungerbühler by constructing an example to show that the \(n\)-harmonic map flow on an \(n\)-dimensional Riemannian manifold blows up in finite time for \(n \geq 3\).
In this paper, the authors consider the \(n\)-harmonic map flow in the following setting: \[ \frac{\partial u}{\partial t} = \frac{1}{\sqrt{|g|}}\frac{\partial}{\partial x^i}\left[|\nabla u|^{n-2}g^{ij}\sqrt{|g|} \frac{\partial}{\partial x^j}u\right] + |\nabla u|^{n-2}A(u)(\nabla u, \nabla u). \] N. Hungerbühler [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 4, 593–631 (1997; Zbl 0911.58011)] proved that there exists a global weak solution \(u : M\times [0, \infty) \to N\) of the \(n\)-harmonic map flow with initial value \(u_0\) such that \(u_0\in C^{1, \alpha}(M \times (0, \infty)\setminus \{\Sigma_k \times T_k\}_{k=1}^m)\)for a finite number of times \(\{T_k\}\) and a finite number of singular closed sets \(\Sigma_k \subset M\). Based on the fundamental result of K.-C. Chang et al. [J. Differ. Geom. 36, No. 2, 507–515 (1992; Zbl 0765.53026)] for \(n=2\) and supported by some numerical evidences, Hungerbühler also conjectured the phenomenon of finite time blowup of the \(n\)-harmonic flow for \(n \geq 3\).
In this paper, the authors generalize the no-neck result of J. Qing and G. Tian [Commun. Pure Appl. Math. 50, No. 4, 295–310 (1997; Zbl 0879.58017)] to the \(n\)-harmonic map flow, and as an application of this generalized no-neck result, they settle the conjecture proposed by Hungerbühler by constructing an example to show that the \(n\)-harmonic map flow on an \(n\)-dimensional Riemannian manifold blows up in finite time for \(n \geq 3\).
Reviewer: Gabjin Yun (Yongin)
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