25 research outputs found
Quadratic estimates for perturbed Dirac type operators on doubling measure metric spaces
We consider perturbations of Dirac type operators on complete, connected
metric spaces equipped with a doubling measure. Under a suitable set of
assumptions, we prove quadratic estimates for such operators and hence deduce
that these operators have a bounded functional calculus. In particular, we
deduce a Kato square root type estimate.Comment: To appear in Proceedings of the AMSI International Conference on
Harmonic Analysis and Applications, Proc. Centre Math. Appl. Austral. Nat.
Uni
Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions
On a smooth complete Riemannian spin manifold with smooth compact boundary,
we demonstrate that the Atiyah-Singer Dirac operator
in depends Riesz continuously on
perturbations of local boundary conditions . The Lipschitz bound
for the map depends on Lipschitz smoothness
and ellipticity of and bounds on Ricci curvature and its first
derivatives as well as a lower bound on injectivity radius. More generally, we
prove perturbation estimates for functional calculi of elliptic operators on
manifolds with local boundary conditions.Comment: Final versio
Continuity of solutions to space-varying pointwise linear elliptic equations
We consider pointwise linear elliptic equations of the form Lα uα = ŋα on a smooth compact manifold where the operators Lα are in divergence form with real, bounded, measurable coefficients that vary in the space variableα. We establish L2-continuity of the solutions at α whenever the coefficients of Lα are L∞ -continuous at α and the initial datum is L2 -continuous at α. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a time evolving family of metrics gt that is tangential to the Ricci flow almost-everywhere along geodesics when starting with a smooth initial metric. Under the assumption that our initial metric is a rough metric on ʍ with a C1 heat kernel on a "non-singular" nonempty open subset Ɲ, we show that α à gt (α) is continuous whenever α € Ɲ
Geometric singularities and a flow tangent to the Ricci flow
We consider a geometric flow introduced by Gigli and Mantegazza which, in the
case of smooth compact manifolds with smooth metrics, is tangen- tial to the
Ricci flow almost-everywhere along geodesics. To study spaces with geometric
singularities, we consider this flow in the context of smooth manifolds with
rough metrics with sufficiently regular heat kernels. On an appropriate non-
singular open region, we provide a family of metric tensors evolving in time
and provide a regularity theory for this flow in terms of the regularity of the
heat kernel.
When the rough metric induces a metric measure space satisfying a Riemannian
Curvature Dimension condition, we demonstrate that the distance induced by the
flow is identical to the evolving distance metric defined by Gigli and
Mantegazza on appropriate admissible points. Consequently, we demonstrate that
a smooth compact manifold with a finite number of geometric conical
singularities remains a smooth manifold with a smooth metric away from the cone
points for all future times. Moreover, we show that the distance induced by the
evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by
Gigli-Mantegazza.Comment: Fixed proof of Lemma 5.4, updated references to published work
Boundary value problems for general first-order elliptic differential operators
We study boundary value problems for first-order elliptic differential
operators on manifolds with compact boundary. The adapted boundary operator
need not be selfadjoint and the boundary condition need not be pseudo-local. We
show the equivalence of various characterisations of elliptic boundary
conditions and demonstrate how the boundary conditions traditionally considered
in the literature fit in our framework. The regularity of the solutions up to
the boundary is proven. We show that imposing elliptic boundary conditions
yields a Fredholm operator if the manifold is compact. We provide examples
which are conveniently treated by our methods.Comment: Link to video abstract and material on the Fredholm property adde