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Schrödinger Operators, Integral Curvature, and the Euler Characteristic of Riemannian Manifolds

We establish new connections between integral curvature bounds and the Euler characteristic of closed Riemannian manifolds through the perspective of Schrödinger-type operators. Central to our approach is the twisted Dirac operator \(\mathcal{D}_θ\), whose index equals \(χ(M)\). Under integral smallness conditions on the negative part of a potential \(V\) and a Sobolev--Poincaré inequality, we show that a suitable scaling of \(θ\) forces the kernel of \(\mathcal{D}_{tθ}\) to vanish, thereby implying \(χ(M)=0\). Applying this framework to geometrically natural potentials yields several topological consequences. In even dimensions, sufficiently small integral bounds on partial sums of curvature operator eigenvalues force \(χ(M)\) either to vanish or to have a sign determined by the middle dimension. For four-manifolds, a small \(L^{p}\)-norm of the negative Ricci curvature relative to the diameter guarantees \(χ(M)\ge 0\). Moreover, when \(χ(M)\neq 0\) we obtain a Li--Yau type lower bound for the first eigenvalue of the rough Laplacian on \(1\)-forms in terms of the diameter and an integral curvature quantity. Subsequently, we provide an explicit lower bound for the first eigenvalue of the Laplacian on $1$-forms under almost nonnegative curvature conditions, thereby giving an affirmative answer to Yau's Problem 79.