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Geometric evolution equations in critical dimensions

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Abstract

We make a qualitative comparison of phenomena occurring in two different geometric flows: the harmonic map heat flow in two space dimensions and the Yang–Mills heat flow in four space dimensions. Our results are a regularity result for the degree-2 equivariant harmonic map flow, and a blow-up result for an equivariant Yang–Mills-like flow. The results show that qualitatively differing behaviours observed in the two flows can be attributed to the degree of the equivariance.

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Authors and Affiliations

  1. Department of Mathematics, The University of Queensland, St Lucia, QLD, 4072, Australia

    Joseph F. Grotowski

  2. Courant Institute of Mathemtical Sciences, New York University, New York, NY, 10012, USA

    Jalal Shatah

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  1. Joseph F. Grotowski
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  2. Jalal Shatah
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Correspondence to Jalal Shatah.

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Grotowski, J.F., Shatah, J. Geometric evolution equations in critical dimensions. Calc. Var. 30, 499–512 (2007). https://doi.org/10.1007/s00526-007-0100-2

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