Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 350-02. Smooth counterexamples to Yau's nodal upper bound in dimensions three and four
Manuscript map · lines 9145–9154
OpenAI@2026-10-07#rem:oai-p-350-02
Authors: OpenAI. Family 350.
Title: Smooth counterexamples to Yau's nodal upper bound in dimensions three and four.
Abstract: We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on $S{}2{}times{}mathbb T{}2$ for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue. Each sequence belongs to one fixed metric. Thus the upper-bound part of Yau's nodal conjecture fails for smooth metrics in dimensions three and four.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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