# Variational formulas for submanifolds of fixed degree

## TL;DR

This paper develops variational formulas for submanifolds of fixed degree in graded manifolds, deriving conditions for admissible variations and computing the associated Euler-Lagrange equations, including a potentially third-order mean curvature operator.

## Contribution

It introduces a framework for analyzing area functionals on degree-fixed submanifolds, establishing PDE conditions for variations and deformability, and deriving the Euler-Lagrange equations in this setting.

## Key findings

- Derived PDE system for admissible variations.
- Provided sufficient conditions for submanifold deformability.
- Computed Euler-Lagrange equations and identified third-order mean curvature operator.

## Abstract

We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler-Lagrange equations. The resulting mean curvature operator can be of third order.