Relate the sharp heat codimension coefficient to blow-down geometry

Characterize the relationship between the sharp large-time heat codimension coefficient of a metric measure Dirichlet space and the geometry of its blow-downs.

Background

The paper introduces a heat scalar–codimension condition whose macroscopic parameter q controls the large-time behavior of the propagated pointed Nash entropy. The authors note that the estimate q < 2 obtained in the positive-scalar-curvature setting is only a uniform lower bound and need not be sharp for an individual space.

For products such as K^k × R^{N−k}, the sharp large-time coefficient is explicitly tied to the Euclidean factor and hence to the geometry of the space. The unresolved problem is to understand whether, and under what conditions, the sharp coefficient can generally be recovered from geometric information about blow-down limits.