The Kato Square Root Problem on Vector Bundles with Generalised Bounded Geometry

Abstract

Abstract. We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call generalised bounded geometry. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions.

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric

“…This is a direct consequence of the fact that we are able to find a C-close smooth metric h , which is complete since h is complete, and for this metric h , there exists a sequence of smooth functions ρ k : N → [0, 1] with spt ρ k compact, with ρ k → 1 pointwise, and |dρ k | h ≤ C −1 1/k for almost-every x ∈ N (and hence |dρ k | h ≤ 1/k for almost-every x ∈ N ). See Proposition 2.3 in [6] or Proposition 1.3.5 in [15] for the existence of such a sequence. The aforementioned density is then simply a consequence of noting the formula / D *…”

“…Proof. The proof proceeds similar to Proposition 8.4 in [6], by replacing their Q B t with our Q t .…”

“…we use Lemma 8.3 in [6], which states that whenever r > 0 and {B j = B(x j , r)} is a disjoint collection of balls, then for every η ≥ 1,…”

Self-adjointness of the Gaffney Laplacian on Vector Bundles

“…This is a consequence of the well known fact that operators ∇ c and ∇ 2 are always densely-defined and closable (c.f. Proposition 2.2 in [4]). In particular, this means that Proposition 2.2 is valid even in this more general context.…”

Functional calculus and harmonic analysis in geometry

“…Theorem 3.3 (Theorem 1.1 in [13]). Let (M, g) be a smooth, complete Riemannian manifold and suppose there exists C < ∞ and κ > 0 with |Ric g | ≤ C and inj(M, g) ≥ κ > 0.…”