PRINCE2 requires that the business case be formally verified by the project board at stage boundaries. This verification is essential to ensure that the project remains viable and desirable from a business perspective.
The project board's review of the business case at these critical points helps in deciding whether the project should continue into its next stage.References: PRINCE2 Foundation Manual

The purpose of the Risk Register in PRINCE2 is to act as a central record of information relating to the identified risks for the project. It includes details about each risk such as their description, probability, impact, and countermeasures. The register helps in monitoring and controlling risks efficiently throughout the project lifecycle.References: PRINCE2 7 Foundation handbook and documents.

A leadership activity within a PRINCE2 project involves several responsibilities, one of which is agreeing on work packages with team managers. This agreement includes specifying the tools and techniques to be used for product delivery. This activity is crucial for ensuring clarity and mutual understanding of how work is to be conducted, facilitating effective delegation and accountability within the project team.References:
(PRINCE2 7 Foundation Study Guide, AXELOS)

The purpose of the Communication Management Approach is to define how the project team will engage with and manage communications with stakeholders throughout the project. This includes planning the means and frequency of communication to ensure all stakeholders are appropriately informed and can provide necessary feedback on project progress and issues. References: PRINCE2 Foundation Guide,
"Communication Management Approach" section

The purpose of a Project Brief in PRINCE2 is to provide a solid basis for the initiation of the project. The Project Brief includes the outline of the project's objectives, scope, constraints, and assumptions. It is developed during the 'Starting up a Project' process and is used to ensure that a clear, concise, and feasible plan is proposed before the project is formally initiated.References: PRINCE2 7 Foundation resources and materials.