On spin conformal invariant on manifolds with boundary

Let M be an n-dimensional connected compact manifold with non- empty boundary equipped with a Riemannian metric g, a spin structure sigma and a chirality operator Gamma. We define and study some properties of a spin conformal invariant given by: lambda(min)(M, partial derivative M) := inf((g) over bar epsilon[g])vertical bar lambda(+/-)(1)((g) over bar)vertical bar Vol(M, (g) over bar)(1/n), where lambda(+/-)(1) ((g) over bar) is the smallest eigenvalue of the Dirac operator under the chiral bag boundary condition B((g) over bar)(+/-). More precisely, we show that if n >= 2 then: lambda(min)(M, partial derivative M) <= lambda(min)(S(+)(n), partial derivative S(+)(n))