An obstruction for the mean curvature of a conformal immersion 𝑆ⁿ→ℝⁿ⁺¹

Abstract

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H H of a conformal immersion S n → R n + 1 S^n\to \mathbb {R}^{n+1} satisfies ∫ ∂ X H = 0 \int \partial _X H=0 where X X is a conformal vector field on S n S^n and where the integration is carried out with respect to the Euclidean volume measure of the image. This identity is analogous to the Kazdan-Warner obstruction that appears in the problem of prescribing the scalar curvature on S n S^n inside the standard conformal class.