The Witten Laplacian on negatively curved simply connected manifolds. (English) Zbl 0799.53046

Elworthy, K. D.; Rosenberg, Steven

Tokyo J. Math. 16, No. 2, 513-524 (1993).

The Hopf conjecture stated that \((-1)^ n\chi(M)>0\) for the Euler characteristic

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of a compact Riemannian \(2n\)-manifold \(M\) of negative sectional curvature. The Singer’s approach using the \(L^ 2\)-index theorem depends on the vanishing of \(L^ 2\)-harmonic forms. Using the Witten Laplacian \(\square_ \tau=d_ \tau d^*_ \tau+d^*_ \tau d_ \tau\) where \(d_ \tau=e^{\tau h}de^{-\tau h}\) is a modification of the exterior derivative

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\(d\), the author derives certain estimates of the infimum of the spectrum on \(L^ 2\Omega^ q\) (the space of \(q\)- forms) in terms of eigenvalues of the Hessian \(\nabla^ 2h\). If in particular \(h=h(x)\) is the distance function

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from a fixed point, if moreover \(\pi_ 1(M)=0\), \(\text{Sect}_ M\leq k^ 2<0\), and \(R^ q\) is bounded from below (where \(R^ q\) is the curvature term in the Weitzenbock formula \(\triangle=\nabla^*\nabla+R^ q\)), then there exists \(\tau\geq 0\) such that any harmonic form \(\varphi\) with \(e^{\tau h}\varphi\in L^ 2\) vanishes identically. The result indicates the difficulty of the Hopf conjecture since such forms \(\varphi\) cannot be induced from embeddings of hyperbolic \(n\)-spaces thought of as a unit ball into some \({\mathbb{R}}^ n\).

Reviewer: J.Chrastina (Brno)

MSC:

53C20	Global Riemannian geometry, including pinching
58J50	Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

Euler characteristic; negative sectional curvature; Witten Laplacian; harmonic form