A finiteness theorem for the space of \(L^{p}\) harmonic sections
The authors prove some finite dimensionality results for subspaces of \(L^p\) harmonic sections of Riemannian manifold or Hermitian vector bundles over complete manifolds. Let \((M^n, g)\) be a connected complete Riemannian \(n\)-manifold and \(E\) a Riemannian (Hermitian) vector bundle of rank \(l\) over \(M\). Assume \(a(x) \in C^0(M), A \in {\mathbb R}, H \geq p >0\) with \(p \geq A+1\) and let \(V = V(a, A, p, H)\) be a vector subspace of the space of smooth sections on \(M\) with the following property: Every \(\xi\in V\) has the unique continuation property and the locally-Lipschitz function \(u = | \xi| \) satisfies \[ u(\Delta u + a(x) u) +A | \nabla u| ^2 \geq 0 \quad \text{weakly on}\quad M \] and \[ \int_{B_r} u^{2p} = o(r^2) \quad\text{as}\quad r \to \infty. \] With these assumptions, the authors prove that if there exists a solution \(0 < \varphi \in Lip_{loc}\) of the differential inequality \[ \Delta \varphi + H a(x) \varphi \leq 0\tag{*} \] weakly outside a compact set \(K \subset M\), then the dimension of \(V\) must be finite. The equation \((*)\) is intimately related with the spectral properties of the Schrödinger operator \(\mathcal L_H = -\Delta - H a(x)\). Thus, from the Bochner-Weitzenböck formula, one can deduce a result on finiteness of \(L^p\) harmonic sections. In other words, let \(\Delta_B = - trace(D^2)\) be the rough Laplacian and let \(\Delta_E = \Delta_B + \mathcal R\) for a smooth symmetric endomorphism of \(E\). Denote by \(L^{2p}\mathcal H(E)\) the space of \(L^{2p}\) \(\Delta_E\)-harmonic sections. In particular, let \(\mathcal L_H = - \Delta_E - H \mathcal R_-(x)\), where \(\mathcal R_-(x) = \sup_{| \xi| =1} \langle -\mathcal R \xi, \xi\rangle\). With these notations and assumptions, the authors deduce that if \(\text{Ind}(\mathcal L_h) < \infty\) for some \(H \geq 1\), then \(\dim(L^{2p}\mathcal H (E)) < \infty\) for every \(1 \leq p \leq H\). Applying the same arguments to a holomorphic line bundle over a complete Kähler manifold, the authors obtain that if the index of corresponding appropriate Schrödinger operators is finite, then the dimensions of the space of \(L^{4p}\) holomorphic \((k,0)\)-forms with values in \(E\), \(L^{4p}\Lambda^{(k,0)}(E)\) and the space of \(L^{2p}\) holomorphic sections of tensor powers of \(E\), \(L^{2p}H(\otimes^k E)\) for every \(0 < p \leq H\) are finite.
scientific article; zbMATH DE number 5307129