On the Kato inequality in Riemannian geometry (Q2765844)

Let \(E\to M\) be a Riemannian vector bundle, \(\xi\) a smooth section. Then, for points outside the zero set of \(\xi\), NEWLINE\[NEWLINE|d|\xi\|\leq |\nabla\xi|\tag{1}NEWLINE\]NEWLINE which is the classical Kato inequality. Equality is achieved at a point if and only if NEWLINE\[NEWLINE\nabla\xi= \alpha\otimes \xi.NEWLINE\]NEWLINE Following J.-P. Bourguignon, consider a section \(\xi\) lying in the kernel of a natural first-order operator \(P\) on \(E\). Any such operator is the composition of the covariant derivative followed by projection \(\Pi\) on one (or more) irreducible components of the bundle \(T^* M\otimes E\), and its symbol reads: \(\sigma(P)= \sigma(\Pi\circ\nabla)= \Pi\). Assume (1) is optimal at some point. The discussion above shows that \(\nabla\xi= \alpha\in\xi\) at that point. But NEWLINE\[NEWLINE0\in P\xi= \Pi\circ \nabla\xi= \Pi(\alpha\otimes \xi).NEWLINE\]NEWLINE Thus, optimality is possible if and only if \(P\) is not an elliptic operator. Conversely, one might guess that any elliptic operator \(P\) gives rise, for any section \(\xi\) in its kernel, to a refined Kato inequality NEWLINE\[NEWLINE|d\xi|\leq k_P|\nabla\xi|NEWLINE\]NEWLINE with a constant \(k_P\) depending only on the operator \(P\) involved.NEWLINENEWLINENEWLINEConsider an irreducible natural vector bundle \(E\) over a Riemannian manifold \((M,g)\) of dimension \(n\), with scalar product \(\langle\cdot,\cdot\rangle\) and a metric (not necessarily Levi-Civita) connection \(\nabla\). By assumption, \(E\) is associated to an irreducible representation \(\lambda\) of the group \(\text{SO}(n)\) (resp. \(\text{Spin}(n)\) if necessary). The tensor product of \(\lambda\) with the standard representation \(\tau\) splits into irreducible components as \(\tau\otimes\lambda= \oplus^N_{j=1} \mu_j\). Write NEWLINE\[NEWLINET^* M\otimes E= \bigoplus^N_{j=1} F_j.NEWLINE\]NEWLINE Projection on the \(j\) th summand will be denoted by \(\Pi_j\). Apart from the exceptional case where \(T^* M\otimes E\) contains two irreducible components for \(\text{SO}(n)\) whose sum is an irreducible representation for \(\text{O}(n)\), each \(F_j\) is an eigenspace for the endomorphism \(B\) of \(T^* M\otimes E\) defined as NEWLINE\[NEWLINEB(\alpha\otimes v)= \sum^n_{i=1} e_i\otimes(e_i\wedge \alpha)\cdot v,NEWLINE\]NEWLINE where the dot means the action of \({\mathfrak s}{\mathfrak o}(n)\) on the representation space \(E\). The endomorphism \(B\) plays an important role in conformal geometry. Its eigenvalues are called the conformal weights, denoted \(w_j\): NEWLINE\[NEWLINEw_j= \textstyle{{1\over 2}} (C({\mathfrak s}{\mathfrak o}(n),\mu_j)- C({\mathfrak s}{\mathfrak o}(n), \lambda)- C({\mathfrak s}{\mathfrak o}(n),\tau)).NEWLINE\]NEWLINE denote by \(\widetilde w_j= w_j+ (n-1)/2\) the modified conformal weights. Natural first order differentials are indexed by subsets \(I\) of \(\{1,\dots, N\}\). They all are of the following form NEWLINE\[NEWLINEP_I= \sum_{i\in I} a_i\Pi_i\circ\nabla;NEWLINE\]NEWLINE any such operator is said to be (injectively, or overdetermined) elliptic if its symbol \(\Pi= \sum_{i\in I} a_i\Pi_i\) does not vanish on any decomposable element \(\alpha\otimes v\) of \(T^* M\otimes E\).NEWLINENEWLINENEWLINEThen the authors prove the following theorem.NEWLINENEWLINENEWLINETheorem 1. Let \(I\) a subset of \(\{1,\dots, N\}\) corresponding to an elliptic operator \(P_I\) acting on \(E\). Then a refined Kato inequality \(|d|\xi\|\leq k_I|\nabla\xi|\) holds for any section \(\xi\) in the kernel of \(P_I\), outside the zero set of \(\xi\). If \(N\) is odd, then NEWLINE\[NEWLINEk^2_I= 1- \inf_{J\in{\mathcal N}{\mathcal E}} \Biggl(\sum_{i\in I} {\prod_{j\in J\setminus\{i\}}(\widetilde w_i+ \widetilde w_j)\over \prod_{j\in\widehat J\setminus\{i\}}(\widetilde w_i- \widetilde w_j)} \varepsilon_i(J)\Biggr).NEWLINE\]NEWLINE These results are sharp except if \(n\) is odd where some ``bad cases'' may appear. If \(N\) is even, then NEWLINE\[NEWLINEk^2_I= 1-\inf_{J\in{\mathcal N}{\mathcal E}}\Biggl(\sum_{i\in I} \Biggl(\widetilde w_i-{1\over 2}\Biggr) {\prod_{j\in J\setminus\{i\}}(\widetilde w_i+ \widetilde w_j)\over \prod_{j\in\widehat J\setminus\{i\}} (\widetilde w_i- \widetilde w_j)} \varepsilon_i(J)\Biggr).NEWLINE\]NEWLINE This result is always sharp.NEWLINENEWLINENEWLINEHere \({\mathcal N}{\mathcal E}\) and \(\varepsilon_i(J)\) come from \textit{Th. Branson's} work [Math. Res. Lett. 7, No. 2-3, 245-261 (2000; Zbl 1039.53033)].NEWLINENEWLINEFor the entire collection see [Zbl 0973.00041].