An Obstruction for the Mean Curvature of a Conformal

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 of a conformal immersion S → R satisfies ∫ ∂XH = 0 where X is a conformal vector field on S 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 inside the standard conformal class. MSC 2000: 53A27, 53A30, 35J60 Let (M, g) be a compact Riemannian manifold with a conformal vector field X . Given a function s onM , then it is a classical question to ask whether s is the scalar curvature of a metric g̃ conformal to g. The determination of the set of all such functions s is still open, although several partial results are known, in particular there are necessary conditions that s has to satisfy in order to be a scalar curvature. On the one hand there are topological obstructions. If for example M is spin and has non-vanishing  genus, then the scalar curvature of any metric on M has either to be negative somewhere or the Ricci curvature vanishes everywhere on M . However, if one fixes the conformal class [g] as described above, there are further obstructions that arise from conformal vector fields. For example if M is S with the standard conformal structure, Kazdan and Warner [KW75] derived a famous obstruction. A slightly stronger version of this obstruction due to Bourguignon and Ezin [BE87] is described in the following theorem. Theorem 1. Let X be a conformal vector field on the compact manifold (M, g). If s is the scalar curvature of a metric g̃ = ug, then ∫ M ∂Xs dvg̃ = 0 where dvg̃ = u 2n n−2 dvg is the volume measure associated to g̃. Tightly related to the Kazdan-Warner obstruction is the Pohozaev identity. Let Ω be a star-shaped open set of R (n ∈ N) with smooth boundary. We denote by ∆ = − ∑n i=1 ∂ii the Laplacian on R . Let u ∈ C(Ω̄) be a positive solution of ∆u = u on Ω with u|∂Ω ≡ 0. The vector field X = ∑n i=1 x ∂i is conformal. If one uses similar methods as in the proof of the Kazdan-Warner obstruction, then one obtains the Pohozaev identity ([Po65]) which asserts that: ( 1− n 2 + n p )∫