Refined Kato Inequalities and Conformal Weights in Riemannian Geometry

Abstract

We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These constants are shown to be optimal and are computed explicitly in most practical cases. Academic Press

The Huber theorem for non-compact conformally flat manifolds

“…The next ingredient is the refined Kato-like inequality proven in [6]; for Ric 0 is in the kernel of the elliptic first order operator d D + δ D acting on trace-free symmetric tensors, we have…”

Refined Kato inequalities in riemannian geometry

“…More recently, S. Bando, A. Kasue and H. Nakajima [1] proved that the Weyl curvature of a Ricci flat manifold satisfies the inequality \d\W\\ < ^/j^[|Vn (6) To give a last example, one may also quote the work of J. Rade on Yang Mills fields on R 4 [II], where the following inequality for the curvature F of such a field appears:…”

“…For sake of simplicity, we will forget here the exceptional situation and argue only for the general case. The interested reader will find information on the exceptional case, its occurrences and its consequences in [6,7].…”

“…We shall try below to give a very rough idea of the strategy we used and the techniques employed. A more detailed version appears in [7] and full information is of course given in the original paper [6].…”

On a characteristic of the first eigenvalue of the Dirac operator on compact spin symmetric spaces with a Kähler or Quaternion-Kähler structure

“…Furthermore, we already note in lemma 3.4, that ρ * (C 2 ) acts on each irreducible component as a scalar multiple of identity with the same eigenvalue. Finally, to prove our holonomy criterion, we only have to examine the eigenvalues of ρ * (C 4 ) on the irreducible components of the spin representation of K. Those eigenvalues may be computed with the help of a formula given in [CGH00]. Denoting by (e 1 , e 2 , e 3 , e 4 ) the standard basis of R 4 , the root system of F 4 is the set of elements x = 4 i=1 x i e i with integer or half-integer coordinates in R 4 such that x 2 = 1 or 2, [Hum72], [BMP85].…”