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Energy gaps for exponential Yang-Mills fields. (English) Zbl 1463.58008

The exponential Yang-Mills functional for connections \(\nabla\) on a vector bundle over \(M\) is the integral \(\int_M\exp(|R^\nabla|^2/2)\). The critical points of that functional are called exponential Yang-Mills fields.
The paper is concerned with energy gaps for such fields. Their occurrence is based on a Simons-type inequality estimating the integrals of \(\exp(|R^\nabla|^2/2)|R^\nabla|^2|\nabla|R^\nabla||^2\) and of \(\exp(|R^\nabla|^2/2)|\nabla R^\nabla|^2\) in terms of the integral of \(\exp(|R^\nabla|^2/2)(c_m|R^\nabla|^3-\lambda|R^\nabla|^2)\) with \(c_m\) depending only on the manifold dimension and \(\lambda\) being the lowest eigenvalue of some curvature operator derived from the Riemannian curvature of \(M\).
The consequence of such a Simons-type inequality is a gap theorem which states that, under suitable conditions on the integrability of \(R^\nabla\) and on the constants in the corresponding Sobolev embeddings, the smallness of the \(r\)-norm of \(R^\nabla\exp(|R^\nabla|^2/2)\) for some \(r\ge m/2\) implies that \(\nabla\) is flat. For an exponential Yang-Mills field over the round sphere \(S^m\), \(m\ge3\), and \(r\ge m/2\), the condition for flatness reads \[\Big\|R^\nabla\exp\Big(\frac{|R^\nabla|}2\Big)\Big\|_r<\frac{\sqrt{m(m-1)}}{4(m-2)}\,\omega_m^{1/r}\,\min\Big\{\frac{m(r-1)}2\,,2(m-2)\Big\}.\]

MSC:

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
58E20 Harmonic maps, etc.
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