Min-max theory and existence of H-spheres with arbitrary codimensions

We demonstrate the existence of branched immersed 2-spheres with prescribed mean curvature, with controlled Morse index and with arbitrary codimensions in closed Riemannian manifold N admitting finite fundamental group, where πₖ(N) ≠ 0 and k ≥ 2, for certain generic choice of prescribed mean curvature vector. Moreover, we enhance this existence result to encompass all possible choices of prescribed mean curvatures under certain Ricci curvature condition on N when N = 3. When N ≥ 4, we establish a Morse index lower bound while N satisfies some isotropic curvature condition. As a consequence, we can leverage latter strengthened result to construct 2-spheres with parallel mean curvature when N has positive isotropic curvature and N ≥ 4. At last, we partially resolve the homotopy problem concerning the existence of a representative surface with prescribed mean curvature type vector field in some given homotopy classes.