Geometric evolution equations in critical dimensions. (English) Zbl 1127.58003
There is a difference in the behaviours of two geometric evolution equations that otherwise show a lot of similarities: the harmonic map heat flow and the Yang-Mills heat flow. Equivariant solutions in the critical dimension can blow up for the former flow [K.-C. Chang, W.-Y. Ding and R. Ye, J. Differ.Geom.36, 507–515 (1992; Zbl 0765.53026)], but they do not for the latter [A. E. Schlatter, M. Struwe and A. S. Tahvildar-Zadeh, Am.J. Math.120, 117–128 (1998; Zbl 0938.58007)].
The current paper discusses what could be the reason for that different behaviour. Basically, the answer is that the calculations for equivariant Yang-Mills solutions behave like they had degree 2 singularities, while equivariant singularities to the harmonic map heat flow want to have degree 1. To support this statement, the authors prove two things: (1) The equivariant harmonic map heat flow does not blow up if the group action forces any possible singularity to have degree \(>1\). (2) The Yang-Mills heat equation, reduced by symmetry and modified in such a way that singularities would correspond to degree 1 singularities of the harmonic map flow, does blow up in finite time for suitable initial data.
The current paper discusses what could be the reason for that different behaviour. Basically, the answer is that the calculations for equivariant Yang-Mills solutions behave like they had degree 2 singularities, while equivariant singularities to the harmonic map heat flow want to have degree 1. To support this statement, the authors prove two things: (1) The equivariant harmonic map heat flow does not blow up if the group action forces any possible singularity to have degree \(>1\). (2) The Yang-Mills heat equation, reduced by symmetry and modified in such a way that singularities would correspond to degree 1 singularities of the harmonic map flow, does blow up in finite time for suitable initial data.
Reviewer: Andreas Gastel (Erlangen)
References:
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