Zero modes on product Riemannian manifolds

TL;DR

This paper derives sharp bounds on the vector field norm for zero mode solutions of the Dirac equation on product Riemannian manifolds, linking geometric invariants like the Yamabe constant.

Contribution

It establishes new sharp estimates for zero modes on product manifolds, connecting Dirac solutions with Yamabe invariants and extending understanding of geometric analysis.

Findings

Established a lower bound for the vector field norm in zero mode equations.

Proved the estimate is sharp in even dimensions.

Extended results to solutions of Dirac equations with scalar functions.

Abstract

This paper is concerned with the zero mode equation $D_g\varphi=iA\cdot\varphi$ on product of closed spin manifolds $(M_1^{n_1}\times M_2^{n_2},g_1+g_2,\sigma)$ 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 $|\varphi|$ we prove that $$\parallel A\parallel_n^2\geq\frac{n_2}{4(n_2-1)}Y(M^n,[g]),$$ 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 $D_g\varphi=f\varphi$, where $f$ is a scalar function.