## 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:

 5.8e+16 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 5.8e+21 Harmonic maps, etc.

### Keywords:

exponential Yang-Mills; energy gap; Simons-type inequality
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### References:

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