# An area bound for surfaces in Riemannian manifolds

## TL;DR

This paper establishes an upper bound on the area of complete surfaces immersed in certain Riemannian manifolds, linking it to the integral of their extrinsic curvature, specifically the squared norm of the second fundamental form.

## Contribution

It provides a new area bound for surfaces in Riemannian manifolds based on extrinsic curvature energy, under the condition that the manifold contains no totally geodesic surfaces.

## Key findings

- Area of immersed surfaces is bounded by extrinsic curvature energy.
- Bound applies to surfaces in manifolds without totally geodesic surfaces.
- Results connect geometric energy to surface area in Riemannian geometry.

## Abstract

Let $M$ be a compact Riemannian manifold not containing any totally geodesic surface. Our main result shows that then the area of any complete surface immersed into $M$ is bounded by a multiple of its extrinsic curvature energy, i.e. by a multiple of the integral of the squared norm of its second fundamental form.