A Steklov eigenvalue estimate for affine connections and its application to substatic triples
Yasuaki Fujitani
TL;DR
The paper develops a framework for affine-connection geometry on weighted manifolds and uses it to generalize Fraser–Li type inequalities and Steklov eigenvalue estimates to the $N=1$ setting and to substatic triples. By establishing a $D$-Reilly formula and a second variation formula, it relates the affine Ricci curvature $Ric^D$ to the $1$-weighted Ricci curvature $Ric_f^1$ and to substatic conditions, enabling isoperimetric and Frankel-type results for $D$-minimal hypersurfaces with free boundary. The main results include a Fraser–Li type isoperimetric inequality, a Frankel-type property, and a sharp lower bound $ rac{k}{2}$ for the first Steklov eigenvalue on $D$-minimal hypersurfaces, with compactness and topological rigidity consequences in three dimensions. These findings extend previous weighted and unweighted Choi–Wang/Fraser–Li theory to accommodate affine connections, providing new tools for stability, compactness, and rigidity in substatic geometric contexts.
Abstract
Choi-Wang obtained a lower bound of the first eigenvalue of the Laplacian on closed minimal hypersurfaces. On minimal hypersurfaces with boundary, Fraser-Li established an inequality giving a lower bound of the first Steklov eigenvalue as a counterpart of the Choi-Wang type inequality. These inequalities were shown under lower bounds of the Ricci curvature. In this paper, under non-negative Ricci curvature associated with an affine connection introduced by Wylie-Yeroshkin, we give a generalization of Fraser-Li type inequality. Our results hold not only for weighted manifolds under non-negative $1$-weighted Ricci curvature but also for substatic triples.