Yau’s nodal bounds: surfaces and higher dimensions

1 · The essentials

A vibrating shape can have places that stay at zero while the rest moves. These zero locations form the sets studied in this family of manuscripts. For a fixed smooth, closed, connected surface, the main manuscript shows that their total length is at most a constant times the square root of the vibration’s eigenvalue, a number associated with that vibration. Combined with an already known lower bound, this settles the stated version of Yau’s conjecture for surfaces in the manuscript. The companion manuscripts describe a different outcome in three and four dimensions: on certain fixed smooth shapes, no constant makes the corresponding upper bound hold for every vibration in the constructed sequence. In five dimensions, a further construction exceeds even a specified larger power of the eigenvalue. The higher-dimensional counterexample manuscripts have Lean formalizations; the supplied source does not identify the surface result as formalized.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine tracing every line of stillness in a vibrating sheet and measuring the total length of your tracing. The surface result limits that length in terms of the vibration’s eigenvalue, but this image does not capture the higher-dimensional zero sets or the precise geometric assumptions.

Why it matters

These results distinguish the surface case from the higher-dimensional smooth setting of Yau’s nodal upper-bound question. The counterexamples use one fixed metric for each sequence, so the failure does not depend on changing the geometry from one eigenfunction to the next. The five-dimensional construction also rules out the proposed remedy of allowing an arbitrarily small positive power loss.

2 · Going further

The precise surface problem concerns a nonzero real Laplace eigenfunction with eigenvalue $λ > 0$ on a fixed smooth closed connected Riemannian surface. Its nodal set is its zero set, and the main manuscript shows that its nodal length satisfies $length {u = 0} \leq C λ$ . Here the constant is for the fixed surface, rather than a single constant asserted for all surfaces. Together with the known lower bound cited in the abstract, this gives Yau’s conjecture in that setting, as shown in the manuscript.

The companion manuscript on dimensions three and four constructs a smooth metric on $S^{3}$ , arbitrarily close to the round metric in the smooth topology, and a smooth metric on $S^{2} \times T^{2}$ . For each of these fixed metrics, it constructs a sequence of exact real Laplace eigenfunctions for which the nodal measure divided by $λ$ is unbounded. Such a sequence rules out any constant $C$ giving the proposed upper bound for that metric. This manuscript is identified as Lean-formalized.

The five-dimensional companion constructs a smooth metric on $S^{4} \times S^{1}$ and real Laplace eigenfunctions whose nodal four-volume grows faster than $λ^{1/2 + ϵ_{0}}$ for one fixed $ϵ_{0} > 0$ . It is also identified as Lean-formalized. The abstracts describe the counterexample strategy through these fixed-metric constructions, but provide no further mechanism of construction and no argument outline for the surface estimate; a more detailed account of the methods would therefore go beyond the source.

Useful notions

Smooth surfaces and higher-dimensional spaces
Riemannian metrics
Laplace eigenfunctions and eigenvalues
Zero sets and their length or volume
Growth bounds along sequences

What was known before / what is new

Before

The main abstract cites a known lower bound for nodal length on the surfaces under consideration. It frames the matching upper bound as the remaining ingredient for Yau’s conjecture in this setting. The five-dimensional abstract also refers to a proposed upper bound allowing an arbitrarily small positive power loss.

New

The main manuscript shows the sharp $C λ$ upper bound on every fixed smooth closed connected surface. The Lean-formalized companion on dimensions three and four supplies fixed smooth metrics violating the corresponding upper bound, including metrics on $S^{3}$ arbitrarily close to round. The Lean-formalized five-dimensional companion supplies a stronger violation that exceeds $λ^{1/2 + ϵ_{0}}$ for some fixed positive $ϵ_{0}$ .

Limits and assumptions

The surface statement assumes a fixed smooth closed connected Riemannian surface and a nonzero real Laplace eigenfunction with positive eigenvalue; it does not assert a constant uniform over all metrics. The higher-dimensional statements are counterexamples on the specified spaces, not assertions that every smooth metric violates the bound. The five-dimensional exponent involves one fixed positive $ϵ_{0}$ , whose value is not supplied. Lean formalizations are identified for Manuscripts 2 and 3, not for the surface manuscript; the supplied abstracts do not specify their internal formalization scope. The source does not state what further cases remain open.

Mini glossary

Riemannian metric: A Riemannian metric specifies the geometry used to measure lengths and volumes on a space.
Laplace eigenfunction: A Laplace eigenfunction is a function satisfying the Laplace eigenvalue equation for a particular eigenvalue.
Nodal set: The nodal set of an eigenfunction is the set of points where it equals zero.
Nodal measure: Nodal measure is the size of a nodal set in the appropriate dimension, such as length on a surface or four-volume in the five-dimensional example.
Closed surface: A closed surface is a compact surface without boundary.
Positive power loss: A positive power loss replaces the proposed growth factor $λ^{1/2}$ with the larger power $λ^{1/2 + ϵ}$ for $ϵ > 0$ .

Keywords : Yau’s conjecture · Nodal sets · Laplace eigenfunctions

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves the sharp $C λ$ upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S³ arbitrarily close to round. In dimension five, nodal measure can grow faster than $λ^{1/2 + ε_{0}}$ for some fixed $ε_{0} > 0$ , ruling out even arbitrarily small power losses.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/350.md

Manuscripts in this family

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Sharp nodal length on smooth surfaces

23/09/2026

To be verified

We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most $C λ$ . Together with the known lower bound, this proves Yau's conjecture in this setting.

PDF GitHub folder

Smooth counterexamples to Yau's nodal upper bound in dimensions three and four

23/09/2026

Verified by Lean

We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on $S^{2} \times T^{2}$ 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.

PDF GitHub folder Lean formalization

Power-law violations of Yau's nodal upper bound

23/09/2026

Verified by Lean

We construct a smooth Riemannian metric on $S^{4} \times S^{1}$ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than $λ^{1/2 + ϵ_{0}}$ for one fixed $ϵ_{0} > 0$ . This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.

PDF GitHub folder Lean formalization