We prove that the Atiyah–Singer Dirac operator in L2 depends Riesz continuously on L∞ perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calderón’s first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.
On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that 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 away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.
On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that 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 away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.
Bioenergetic approaches are increasingly used to understand how marine mammal populations could be affected by a changing and disturbed aquatic environment. There remain considerable gaps in our knowledge of marine mammal bioenergetics, which hinder the application of bioenergetic studies to inform policy decisions. We conducted a priority-setting exercise to identify high-priority unanswered questions in marine mammal bioenergetics, with an emphasis on questions relevant to conservation and management. Electronic communication and a virtual workshop were used to solicit and collate potential research questions from the marine mammal bioenergetic community. From a final list of 39 questions, 11 were identified as 'key'questions because they received votes from at least 50% of survey participants. Key questions included those related to energy intake (prey landscapes, exposure to human activities) and expenditure (field metabolic rate, exposure to human activities, lactation, time-activity budgets), energy allocation priorities, metrics of body condition and relationships with survival and reproductive success and extrapolation of data from one species to another. Existing tools to address key questions include labelled water, animal-borne sensors, mark-resight data from long-term research programs, environmental DNA and unmanned vehicles. Further validation of existing approaches and development of new methodologies are needed to comprehensively address some key questions, particularly for cetaceans. The identification of these key questions can provide a guiding framework to set research priorities, which ultimately may yield more accurate information to inform policies and better conserve marine mammal populations.