Local and global conformal invariants of submanifolds

TL;DR

This paper introduces new methods for constructing and computing conformal invariants of submanifolds, including global invariants and scalars, with applications to formulas relating area, Euler characteristic, and rigidity results.

Contribution

It develops direct constructions of conformal invariants, introduces computable scalars at minimal submanifolds, and derives a Gauss-Bonnet-Chern-type formula and rigidity results.

Findings

Derived an explicit Gauss-Bonnet-Chern-type formula for minimal submanifolds.

Introduced a class of conformal scalars that are easily computed at minimal submanifolds.

Proved a rigidity theorem for conformally compact minimal submanifolds in hyperbolic manifolds.

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.