The Second Variation for Null-Torsion Holomorphic Curves in the 6-Sphere

In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces. This class surfaces is quite rich: By a theorem Bryant, extended by Rowland, every closed Riemann surface may be conformally embedded in 6-sphere as curve. this work, we study second variation area for compact $$\Sigma $$ genus g and $$4\pi d$$ , focusing on spectrum Jacobi operator. We show that if $$g \le 6$$ then multiplicity lowest eigenvalue $$\lambda _1 = -2$$ equal to 4d. Moreover, any genus, nullity at least $$2d + 2 - 2g$$ . These results likely have implications deformation theory asymptotically conical associative 3-folds $${\mathbb {R}}^7$$ studied Lotay.