[Paper Review] Positive scalar curvature on foliations

Weiping Zhang|arXiv (Cornell University)|Aug 19, 2015

Advanced Operator Algebra Research21 references3 citations

TL;DR

This paper generalizes Lichnerowicz and Hitchin's theorems on positive scalar curvature to foliated spin manifolds by introducing an adiabatic limit approach and sub-Dirac operators on almost isometric foliations. It proves that if a closed spin manifold admits a foliation with positive leafwise scalar curvature, then its $̂{\mathcal{A}}$-genus vanishes, and as a consequence, no torus admits such a foliation—extending the classical non-existence result for positive scalar curvature metrics.

ABSTRACT

We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the famous theorem of Schoen-Yau and Gromov-Lawson on the non-existence of metrics of positive scalar curvature on torus to the case of foliations. Moreover, our method, which is partly inspired by the analytic localization techniques of Bismut-Lebeau, also applies to give a new proof of the celebrated Connes vanishing theorem without using noncommutative geometry.

Motivation & Objective

• To extend classical results on positive scalar curvature metrics on spin manifolds to the setting of foliated manifolds.
• To address the open question of whether positive leafwise scalar curvature on a foliation implies positive scalar curvature on the total manifold.
• To provide a new analytic proof of Connes' vanishing theorem without using noncommutative geometry.
• To establish a vanishing result for the $̂{\mathcal{A}}$-genus under positive leafwise scalar curvature on integrable subbundles of spin manifolds.

Proposed method

• Utilizes the adiabatic limit technique to construct an almost isometric fibration over the foliated manifold.
• Introduces sub-Dirac operators associated with spin integrable subbundles and their twisted versions on the Connes fibration.
• Applies the Atiyah-Singer index theorem to compute the mod 2 index of a zeroth-order elliptic pseudodifferential operator.
• Employs analytic localization techniques inspired by Bismut-Lebeau to control the kernel of the sub-Dirac operator.
• Constructs a family of operators $\widehat{P}_{R,\beta,\varepsilon}$ whose kernel dimension mod 2 computes the $\alpha$-invariant.
• Uses the vanishing of the kernel dimension for large $R, \beta, \varepsilon$ to deduce the vanishing of the $\alpha$-invariant and $\u0302{\mathcal{A}}$-genus.

Experimental results

Research questions

• RQ1 Does the existence of a metric with positive leafwise scalar curvature on a foliation of a closed spin manifold imply that the $\u0302{\mathcal{A}}$-genus of the total manifold vanishes?
• RQ2 Can the classical non-existence result for positive scalar curvature on tori be extended to the setting of foliations?
• RQ3 Is it possible to prove Connes' vanishing theorem for foliations without relying on noncommutative geometry or cyclic cohomology?
• RQ4 What are the topological obstructions to the existence of positive leafwise scalar curvature on integrable subbundles of spin manifolds?
• RQ5 Can the index-theoretic methods used here be generalized to include higher-order Pontrjagin classes of the normal bundle?

Key findings

• If a closed spin manifold $M$ admits a foliation $F$ with positive leafwise scalar curvature, then $\widehat{\mathcal{A}}(M) = 0$, generalizing Lichnerowicz and Hitchin's theorems.
• There is no foliation of positive leafwise scalar curvature on any torus $T^n$, extending the Schoen-Yau and Gromov-Lawson non-existence result.
• A new analytic proof of Connes' vanishing theorem is established without using noncommutative geometry or cyclic cohomology.
• The mod 2 index of a certain zeroth-order elliptic pseudodifferential operator $\widehat{P}_{R,\beta,\varepsilon}$ vanishes for large parameters, implying $\alpha(M) = 0$.
• For any integer $k \geq 0$, the pairing $\left\langle \widehat{A}(F) p(TM/F) e(TM/F)^k, [M] \right\rangle = 0$ holds under the positive leafwise scalar curvature condition.
• In particular, $\left\langle \widehat{A}(F) e(TM/F), [M] \right\rangle = 0$, and for $\dim M = 6$, $\operatorname{rk}(F) = 4$, it follows that $\left\langle e(TM/F)^3, [M] \right\rangle = 0$, implying no positive leafwise scalar curvature codimension-two foliation on $\mathbb{C}P^3$.