Stability and integrability of horizontally conformal maps and harmonic morphisms (Q2862560)

The authors investigate stability of minimal fibers of horizontally conformal maps and harmonic morphisms. For every regular point \(x\in M\) let \(H_x\) denote the orthogonal complement of \(\ker d\varphi \subset T_xM\). A map \(\varphi:(M,g) \to (N,h)\) is horizonally conformal if \(d\varphi: H\to TN\) preserves angles. Using classical tools (second variation of area, O'Neill's formulas, techniques of conformal geometry) the authors obtain several facts concerning horizontally conformal maps. Next they apply them to horizontally conformal maps with stable fibres (a submanifold is stable if for any normal variation with compact support, the second derivative of the volume functional is non-negative). They show that there is a relation between the integrability of \(H\) and the stability of the fibres of \(\varphi\). Namely, (Theorem 4.6) if all the fibers of \(\varphi\) are minimal submanifolds of \(M\) and \(H\) is integrable then the fibers are volume stable. Since harmonic morphisms are horizontally conformal, the authors obtain several facts concerning harmonic morphisms. In particular, (Corollary 4.8) if \(\varphi\) is a submersion, all fibres of \(\varphi\) are totally geodesic submanifolds of \(M\) and \(H\) is integrable, then the fibers are volume stable.