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

Teng Huang, Pan Zhang

TL;DR

This work develops an analytic framework tying integral curvature bounds to global topology via a twisted Dirac operator $\mathcal{D}_{\theta}$ whose index equals the Euler characteristic $\chi(M)$. By combining a Sobolev-Poincaré setup, Bochner-type identities, and Schrodinger-type operator estimates, the authors show that under precise smallness conditions on the negative part of curvature, the kernel of $\mathcal{D}_{t\theta}$ vanishes for suitable $t$, forcing $\chi(M)=0$ or constraining signs in even dimensions. They derive explicit results for curvature operators with $L^{p}$ lower bounds, establish vanishing of Morse–Novikov cohomology under these integral curvature conditions, and obtain eigenvalue lower bounds for the rough and Hodge Laplacians on 1-forms, including Li–Yau-type estimates when $\chi(M)\neq0$. These results connect integral geometric data to topological invariants and provide quantitative obstructions to negative curvature regions on manifolds, addressing Yau’s problem in the form of concrete eigenvalue and cohomology consequences. The methods advance the program of translating pointwise curvature theorems into integral curvature analogs with explicit constants, offering new tools for studying ANCO manifolds and four-manifolds with prescribed topological constraints.

Abstract

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.

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

TL;DR

This work develops an analytic framework tying integral curvature bounds to global topology via a twisted Dirac operator whose index equals the Euler characteristic . By combining a Sobolev-Poincaré setup, Bochner-type identities, and Schrodinger-type operator estimates, the authors show that under precise smallness conditions on the negative part of curvature, the kernel of vanishes for suitable , forcing or constraining signs in even dimensions. They derive explicit results for curvature operators with lower bounds, establish vanishing of Morse–Novikov cohomology under these integral curvature conditions, and obtain eigenvalue lower bounds for the rough and Hodge Laplacians on 1-forms, including Li–Yau-type estimates when . These results connect integral geometric data to topological invariants and provide quantitative obstructions to negative curvature regions on manifolds, addressing Yau’s problem in the form of concrete eigenvalue and cohomology consequences. The methods advance the program of translating pointwise curvature theorems into integral curvature analogs with explicit constants, offering new tools for studying ANCO manifolds and four-manifolds with prescribed topological constraints.

Abstract

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 , whose index equals \(χ(M)\). Under integral smallness conditions on the negative part of a potential and a Sobolev--Poincaré inequality, we show that a suitable scaling of forces the kernel of 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 -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 -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 -forms under almost nonnegative curvature conditions, thereby giving an affirmative answer to Yau's Problem 79.
Paper Structure (13 sections, 24 theorems, 238 equations)

This paper contains 13 sections, 24 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curvature Operator
  4. A Mean Value Inequality
  5. The Twisted Dirac Operator $\mathcal{D}_{\theta}$ on Riemannian Manifolds
  6. An Integral Inequality
  7. A Priori $L^{\infty}$-Estimates for Forms in $\ker\mathcal{D}_{\theta}$
  8. $L^{\infty}$-Estimates for Solutions of Schrödinger-Type Equations
  9. Key Estimates for Forms in $\ker\mathcal{D}_{\theta}$
  10. Applications to Integral Curvature Conditions
  11. Curvature Operator with $L^p$ Lower Bound
  12. Vanishing of Morse--Novikov cohomlogy
  13. Eigenvalue Estimates for the Rough Laplacian

Key Result

Theorem 1.1

Let $(M^{n},g)$ be a closed $n$-dimensional, $n\geq4$, Riemannian manifold satisfying the Sobolev--Poincaré inequality Denote by $\lambda_{1}\leq\cdots\leq\lambda_{{n\choose 2}}$ the eigenvalues of the curvature operator and set Fix $D>0$ and $1\leq l\leq\left\lfloor\frac{n}{2}\right\rfloor$. Assume that $\theta$ is a non-trivial $1$-form satisfying where $V\in C^{\infty}(\operatorname{Sym}(T^{

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Conjecture
  • Theorem 1.8
  • Remark 1.9
  • ...and 38 more