For the weighted Dirac eigenproblem on a compact spin manifold with the chiral boundary condition \begin{ equation*} \left\{ \begin{ array}{ ll} D\varphi = \lambda f\varphi & \text{ in } M, \\ \mathbf{ B}\varphi = 0 & \text{ on } \partial M, \end{ array} \right. \end{ equation*} we first give a lower bound of the eigenvalue using the relative Yamabe constant \begin{ equation*} \lambda^2 \geq \frac{ n}{ 4(n-1)} Y(M,\partial M,[g]), \end{ equation*} then prove that equality holds if and only if (up to a conformal transformation) $M$ is a hemisphere and $\varphi$ is a Killing spinor. More generalizations are studied.
Categories: math.DG math-ph math.AP math.MP