Local and global conformal invariants of submanifolds

Abstract

We develop methods for constructing and computing conformal invariants of submanifolds, with a particular emphasis on conformal submanifold scalars and conformally invariant integrals of natural submanifold scalars. These methods include a direct construction of the extrinsic ambient space, a construction of global invariants of conformally compact minimal submanifolds of conformally compact Einstein manifolds via renormalized extrinsic curvature integrals, and the introduction of a large class of conformal submanifold scalars that are easily computed at minimal submanifolds of Einstein manifolds. As an application, we derive an explicit Gauss--Bonnet--Chern-type formula relating the renormalized area of a conformally compact $k$-dimensional minimal submanifold of a conformally compact Einstein manifold to its Euler characteristic and the integral of a conformal submanifold scalar of weight $-k$. As another application, we prove a rigidity result for conformally compact minimal submanifolds of conformally compact hyperbolic manifolds.