Zero modes on product Riemannian manifolds

Jurgen Julio-Batalla

paper

Zero modes on product Riemannian manifolds

arXiv:2603.23242

Abstract

This paper is concerned with the zero mode equation $D_{g} φ = i A \cdot φ$ on product of closed spin manifolds $(M_{1}^{n_{1}} \times M_{2}^{n_{2}}, g_{1} + g_{2}, σ)$ of dimensions $n_{1} \leq n_{2}$ respectively. Here $A$ is a real vector field on $M^{n} = M_{1}^{n_{1}} \times M_{2}^{n_{2}}$ . Under non-increasing condition on $∣ φ ∣$ we prove that $∥ A ∥_{n}^{2} \geq \frac{n _{2}}{4 ( n _{2} - 1 )} Y (M^{n}, [g]),$ where $Y (M^{n}, [g])$ is the Yamabe constant of $(M^{n}, g)$ . This estimate is sharp in even dimensions. We also obtain a similar estimate for non trivial solutions of the zero mode type equation $D_{g} φ = f φ$ , where $f$ is a scalar function.