Spiro Karigiannis, Pure Mathematics Department, University of Waterloo
“The first and second variation formulas for the volume functional in Riemannian geometry”
Let L be an immersed oriented submanifold of a Riemannian manifold (M, g) endowed with the induced metric. We derive the first variation formula of the volume functional of this immersion, and identity the critical points as those submanifolds with vanishing mean curvature vector field. Then, at such a critical submanifold, we compute the second variation formula to determine the stability of the critical immersion. [This is the analogue of the “second derivative test” in calculus.]