Extrinsic curvature flows provide powerful analytic tools to deform hypersurfaces toward canonical shapes. Among them, mean curvature flow evolves a hypersurface in the direction of its mean curvature, acting as the gradient flow of the area functional and typically smoothing the geometry until singularities occur. In many geometric and physical problems, however, additional quantities such as enclosed volume or other integral invariants must be preserved, leading to con- strained curvature flows. These introduce nonlocal effects that profoundly influence long-term behavior, as the reliance on classical maximum principles is more tricky or even fails dramatically. After reviewing classical results on constrained mean-curvature-type flows in Euclidean, spherical, and hyperbolic geometries, we highlight recent joint work with Sara Albert-Niclòs on constrained curvature flows of closed curves on pinched Hadamard surfaces. We will also give some hints about a new notion of convexity that simplifies the analysis of this type of flows on ambient spheres.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006