Bi-harmonic maps and bi-Yang-Mills fields. (English) Zbl 1201.58012
“This is an expository and research paper.” The authors recall the energy and the bi-energy functional \(E_2(\varphi)=(1/2)\int_M|\tau(\phi)|^2v_g\) for maps \(\varphi:M\to N\) between Riemannian manifolds \((M,g)\) and \((N,h)\), and compute the first and second variations at critical points, obtaining the corresponding Jacobi equations. Harmonic maps are trivially bi-harmonic maps. Results on bi-harmonic maps into unit spheres and bi-harmonic isoparametric hypersurfaces are revised. The authors characterize the bi-harmonic immersed compact (real) hypersurfaces of constant mean curvature, bi-harmonic homogeneous real hypersurfaces into complex and quaternionic projective spaces. In section 7, some answers to some conjectures on biharmonic submanifolds are given. Namely, the authors prove that, if \(M\) has bounded sectional curvature and \(N\) has nonpositive sectional curvature and the tension field \(\tau(\varphi)\) and its first covariant derivative are in \(L^2\), then \(\varphi:(M,g) \to (N,h)\) is a harmonic map.
In section 8, the Yang-Mills functional \({\mathcal Y}M(\nabla)\) and the bi-Yang-Mills functional \({\mathcal Y}M_2(\nabla)=(1/2)\int_M\| \delta^{\nabla}R^{\nabla}\|^2v_g\) and their first variation formulas are recalled. The critical points correspond to connections that are, respectively, Yang-Mills and bi-Yang-Mills fields (the first one is trivially a second one and it is an absolute minimum for \({\mathcal Y}M_2\)). In this paper the second variation for the bi-Yang-Mills is computed, and specialized for co-closed vector variation. This defines a fourth order selfadjoint elliptic differential operator \({\mathcal S}^{\nabla}_2\) acting on the space of 1-forms of a compact Riemannian manifold \(M\) with values on the bundle of skew symmetric endomorphisms of a Riemannian vector bundle \(E\). Yang-Mills fields have zero index for the corresponding forth-order operator \({\mathcal S}^{\nabla}_2\), and the same nullity when comparing with the second-order operator \({\mathcal S}^{\nabla}\) corresponding to the Yang-Mills functional. The authors show a “bounded isolation phenomena” for a bi-Yang Mills field with \(\|R^{\nabla}\|<1/2\) and \(M\) with strictly positive Ricci operator, by concluding it must be a Yang-Mills field. Under another \(L^2\)-condition on \(\|R^{\nabla}\|\), involving an isoperimetric constant of \((M,g)\), they obtain the same conclusion. In an appendix, the Euler-Lagrange equation for the \(k\)-energy functional is computed.
In section 8, the Yang-Mills functional \({\mathcal Y}M(\nabla)\) and the bi-Yang-Mills functional \({\mathcal Y}M_2(\nabla)=(1/2)\int_M\| \delta^{\nabla}R^{\nabla}\|^2v_g\) and their first variation formulas are recalled. The critical points correspond to connections that are, respectively, Yang-Mills and bi-Yang-Mills fields (the first one is trivially a second one and it is an absolute minimum for \({\mathcal Y}M_2\)). In this paper the second variation for the bi-Yang-Mills is computed, and specialized for co-closed vector variation. This defines a fourth order selfadjoint elliptic differential operator \({\mathcal S}^{\nabla}_2\) acting on the space of 1-forms of a compact Riemannian manifold \(M\) with values on the bundle of skew symmetric endomorphisms of a Riemannian vector bundle \(E\). Yang-Mills fields have zero index for the corresponding forth-order operator \({\mathcal S}^{\nabla}_2\), and the same nullity when comparing with the second-order operator \({\mathcal S}^{\nabla}\) corresponding to the Yang-Mills functional. The authors show a “bounded isolation phenomena” for a bi-Yang Mills field with \(\|R^{\nabla}\|<1/2\) and \(M\) with strictly positive Ricci operator, by concluding it must be a Yang-Mills field. Under another \(L^2\)-condition on \(\|R^{\nabla}\|\), involving an isoperimetric constant of \((M,g)\), they obtain the same conclusion. In an appendix, the Euler-Lagrange equation for the \(k\)-energy functional is computed.
Reviewer: Isabel Salavessa (Lisboa)
