New constructions of volume-critical submanifolds of the sphere

Constant mean curvature hypersurfaces in S^n are critical points of the (n-1)-volume functional subject to an enclosed-volume constraint whereas contact-stationary Legendrian (CSL) submanifolds of S^{2n+1} are n-dimensional submanifolds tangent to the contact distribution whose n-volume is critical amongst all Legendrian competitors. I will present new constructions of CMC and CSL submanifolds in spheres using gluing techniques and a good understanding of the isometries of the ambient sphere. I will also highlight some similarities between the world of CSL submanifolds and the world of CMC hypersurfaces.