Lower bounds on the Hausdorff measure of nodal sets II

TL;DR

This paper presents a concise argument that leverages a key identity to establish the strongest known lower bounds on the Hausdorff measure of nodal sets in higher dimensions, building on previous foundational work.

Contribution

It provides a simplified proof connecting earlier identities to the best existing lower bounds for nodal set measures in dimensions three and above.

Findings

Established the best known lower bounds for nodal set measures in dimensions n≥3.

Simplified the proof technique using a main identity from previous work.

Connected earlier theoretical results to current optimal bounds.

Abstract

We give a very short argument showing how the main identity from our earlier paper (Sogge and Zelditch, 2011) immediately leads to the best lower bound currently known (Colding and Minicozzi II, 2011) for the Hausdorff measure of nodal sets in dimensions $n\ge 3$.