Relaxation of the product-structure and boundary-smoothness assumptions

Establish precise relaxations of the product-structure assumption near the boundary of the Riemannian manifold and determine the behavior of the renormalization analysis when the boundary has limited smoothness.

Background

The analysis of boundary singularities assumes that a neighborhood of the boundary has an exact product structure, with a metric independent of the normal coordinate. The author notes that this assumption substantially simplifies the calculations and that the appropriate weaker hypothesis has not been identified.

The conclusion also raises the unresolved issue of manifolds whose boundaries have limited smoothness. Determining suitable geometric assumptions in both directions would clarify the scope of the renormalization and gluing results beyond the particularly regular product setting used in the paper.

References

One of the interesting questions is the generalization of the properties of the manifold $\mathcal{M}$. In Section \ref{sec:g1}, an assumption was formulated about the product structure near the boundary. This fact has greatly simplified the calculation of boundary singularities. The author could not find a precise relaxation of this condition. The situation with the limited smoothness of the boundary is also not entirely clear.

— Renormalization, cutoff, and gluing for a quartic model with boundary  (2608.23069 - Ivanov, 24 Aug 2026) in Conclusion, Section 4