Spherical cones: classification and a volume minimization principle

TL;DR

This paper establishes a variational principle linking volume minimization to conical Calabi-Yau structures on horospherical cones, enabling explicit computations and revealing many irregular examples from symmetric spaces.

Contribution

It introduces a variational approach to connect volume minimization with Calabi-Yau structures on horospherical cones, including explicit examples from symmetric spaces.

Findings

Existence of many irregular horospherical cones.

Explicit computations on rank-two symmetric space examples.

Equivalence between volume minimization and Calabi-Yau structures.

Abstract

Using a variational approach, we establish the equivalence between a weighted volume minimization principle and the existence of a conical Calabi-Yau structure on horospherical cones with mild singularities. This allows us to do explicit computations on the examples arising from rank-two symmetric spaces, showing the existence of many irregular horospherical cones.