Inequalities between volume, center of mass, circumscribed radius, order, and mean curvature (Q1346376)

The paper is devoted to the search for sharp geometric inequalities for submanifolds of the Euclidean space. The main tools are the spectral decomposition of the position vector of a Euclidean submanifold together with the notions of order and finite type of a Euclidean submanifold introduced by the first author in the late seventies. In the first part, they obtain relations between volume, total mean curvature and circumscribed radius for compact submanifolds of \(E^n\), improving some well-known results. The equalities characterize minimal submanifolds of the sphere. They also obtain bounds for the first eigenvalue of the Laplacian of a compact Euclidean submanifold in terms of the circumscribed and incribed radius and applications to special cases such as ellipsoids and tubes in \(E^3\). In the last part, they give lower bounds for the squared mean curvature of special kinds of isometric immersions (immersions of non-negative kind, pointwise orthogonal,\ \dots) and other geometric inequalities.