A vanishing theorem for elliptic genera under a Ricci curvature bound

We show that given $n \in \mathbb{N}$ and positive numbers $p, \lambda_1, \lambda_2$ with $p>2n$, there exists some $\epsilon=\epsilon(p, n, \lambda_1, \lambda_2)>0$ such that if a compact $4n$-dimensional spin Rimannian manifold $(M, g)$ satisfies $-\lambda_1 \leq Ric(g) \leq \epsilon, diam (g)\leq 1$, $\frac{1}{V(g)}\int_M |\Re_g|^p dV \leq \lambda_2$ and has infinite isometry group, then the elliptic genera of $M$ vanish. This extends our simple observation that a compact $4n$-dimensional spin Rimannian manifold with nonpositive Ricci curvature and infinite isometry group has vanishing elliptic genera. To the author's best knowledge, it is the first time in the literature that the elliptic genera are shown to vanish based on curvature assumptions and analytic methods.