Variational problems for integral invariants of the second fundamental form of a map between pseudo-Riemannian manifolds

TL;DR

This paper investigates variational problems involving integral invariants derived from the second fundamental form of maps between pseudo-Riemannian manifolds, deriving variational formulas and analyzing specific energy functionals.

Contribution

It derives the first variational formulas for integral invariants from invariant polynomials and reduces the Euler-Lagrange equation of the Chern-Federer energy to a second order PDE.

Findings

Euler-Lagrange equation reduces to a second order PDE

Examples of Chern-Federer submanifolds in Riemannian space forms

Variational formulas for invariant integral functionals

Abstract

We study variational problems for integral invariants, which are defined as integrations of invariant functions of the second fundamental form, of a smooth map between pseudo-Riemannian manifolds. We derive the first variational formulae for integral invariants defined from invariant homogeneous polynomials of degree two. Among these integral invariants, we show that the Euler-Lagrange equation of the Chern-Federer energy functional is reduced to a second order PDE. Then we give some examples of Chern-Federer submanifolds in Riemannian space forms.