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arxiv: 2605.18705 · v1 · pith:WIB5ZYGVnew · submitted 2026-05-18 · 🧮 math.AP · math.DG· math.SP

Nested nodal loops for sums of Laplace eigenfunctions

Robert Koirala This is my paper

Pith reviewed 2026-05-20 08:21 UTC · model grok-4.3

classification 🧮 math.AP math.DGmath.SP
keywords nodal setsLaplace eigenfunctionsnested loopsreal-analytic surfacesbiharmonic functionszero setsBoggio-Hadamard conjecturedouble nests
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The pith

Real-analytic surfaces admit a uniform bound on rooted double nests in nodal sets of sums of Laplace eigenfunctions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that on real-analytic closed surfaces, sums of Laplace eigenfunctions have nodal sets containing only a bounded number of rooted double nests, with the bound depending solely on the surface, a chosen root point, and the spectral cutoff. This settles a question of Logunov in the affirmative under the analyticity assumption. The authors demonstrate that analyticity is necessary by constructing a linear combination of eigenfunctions with eigenvalues zero and two on the smooth sphere that produces infinitely many rooted double nests. They further construct a planar biharmonic function whose nodal set contains a double nest and establish quantitative bounds on the nodal sets of entire biharmonic functions of polynomial growth, connecting the construction to the failure of the Boggio-Hadamard conjecture.

Core claim

In the real-analytic category there is a uniform bound for the number of rooted double nests in terms of the surface, the root, and the spectral cutoff. This bound is sharp, since a linear combination of eigenfunctions with eigenvalues 0 and 2 on the smooth sphere can have infinitely many rooted double nests, and the same phenomenon appears in a planar biharmonic function whose nodal set contains a double nest.

What carries the argument

Rooted double nests are configurations of nested nodal loops all sharing a common root point in the zero set; analyticity of the surface and functions allows control over their number via local power-series expansions and unique continuation.

If this is right

  • The bound holds uniformly for every choice of coefficients in the sum as long as the spectral cutoff is fixed.
  • Smoothness alone is insufficient to guarantee any finite bound, as shown by the sphere example with eigenvalues 0 and 2.
  • The same nodal-set phenomenon of double nests occurs for biharmonic functions in the plane.
  • Entire biharmonic functions of polynomial growth satisfy quantitative upper bounds on the number of double nests in their zero sets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result indicates that analytic regularity forces nodal sets of solutions to elliptic equations to have limited topological complexity near any point.
  • Analogous bounds may exist for sums of eigenfunctions of other elliptic operators on analytic manifolds.
  • The biharmonic construction supplies a concrete counterexample that can be used to test numerical methods for detecting nested nodal structures.
  • The connection to the Boggio-Hadamard conjecture suggests that nodal-set questions can serve as geometric probes for older potential-theoretic conjectures.

Load-bearing premise

The surface and the eigenfunctions must be real-analytic, without which the uniform bound on the number of rooted double nests can fail.

What would settle it

An explicit real-analytic sum of Laplace eigenfunctions on a real-analytic surface, with fixed spectral cutoff, whose nodal set contains more rooted double nests than any finite number depending only on the surface, root, and cutoff would disprove the claimed bound.

Figures

Figures reproduced from arXiv: 2605.18705 by Robert Koirala.

Figure 1
Figure 1. Figure 1: Nodal set of the biharmonic polynomial u(x, y) = Q(x, y)P(x, y), where P and Q are defined in (1.4). The nodal set contains an outer ellipse and an inner closed loop, giving a double nest. The model problem is already delicate. If a biharmonic function u vanishes on a circle ∂BR, then Almansi’s decomposition u(x, y) = h0(x, y) + (x 2 +y 2 )h1(x, y) and the maximum principle for harmonic functions imply u =… view at source ↗
Figure 2
Figure 2. Figure 2: A schematic of the disks U0, U1, U2 defined in (4.5) on S 2 . and Φ = Φ e ◦ h. By naturality of the Laplacian, −∆geΦ = 2 e Φ. The constant function 1 is a e 0-eigenfunction of (S 2 , ge). Therefore u := Φe − 1 is a linear combination of eigenfunctions with eigenvalues 2 and 0. Define Uej = h −1 (Uj ), γej = h −1 (4.8) (γj ). Then each γej is a connected component of u −1 (0), and ∂Uej = γej . Moreover, p ∈… view at source ↗
read the original abstract

We study nested loops in zero sets of sums of Laplace eigenfunctions on closed surfaces. In the real-analytic category, answering a question of Logunov, we prove a uniform bound for the number of rooted double nests in terms of the surface, the root, and the spectral cutoff. We show that this analyticity hypothesis is sharp: on a smooth sphere, a linear combination of eigenfunctions with eigenvalues \(0\) and \(2\) can have infinitely many rooted double nests. We also answer a question of Logunov and Nadirashvili by constructing a planar biharmonic function whose nodal set contains a double nest, and we prove a quantitative bound for entire biharmonic functions of polynomial growth. The biharmonic construction gives a nodal-set manifestation of the failure of the Boggio--Hadamard conjecture from the 1900s.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves a uniform bound, depending only on the surface, a chosen root, and the spectral cutoff, for the number of rooted double nests in the zero sets of sums of Laplace eigenfunctions on real-analytic closed surfaces. It shows this bound is sharp by exhibiting a linear combination of eigenfunctions for eigenvalues 0 and 2 on the smooth sphere that produces infinitely many such nests. The authors also construct a planar biharmonic function whose nodal set contains a double nest (answering a question of Logunov and Nadirashvili) and establish a quantitative bound on the number of double nests for entire biharmonic functions of polynomial growth, providing a nodal-set illustration of the failure of the Boggio-Hadamard conjecture.

Significance. If the results hold, the work supplies a positive answer to a question of Logunov on nodal nesting for eigenfunction sums in the analytic category, together with an explicit sharpness counterexample that isolates the role of analyticity. The biharmonic construction and growth bound give a concrete nodal manifestation of a classical conjecture from the 1900s. The explicit counterexample on the sphere and the quantitative estimates are particular strengths.

minor comments (3)
  1. Abstract: the term 'rooted double nests' appears without a brief definition or forward reference to its precise meaning in §2; adding one sentence would improve accessibility for readers outside the immediate subfield.
  2. §4 (biharmonic construction): the statement of the quantitative bound for entire biharmonic functions of polynomial growth should make the dependence on the growth degree explicit in the theorem (currently described only at high level).
  3. Introduction, paragraph following the statement of the main theorem: the dependence of the constant on the analyticity radius of the surface and the root point is asserted but not quantified; a short remark on how this radius enters the holomorphic-extension argument would clarify uniformity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and detailed summary of our manuscript, as well as for the favorable significance assessment and recommendation of minor revision. We are pleased that the work is viewed as providing a positive answer to Logunov's question on nodal nesting in the analytic category, along with the sharpness counterexample and biharmonic construction. No specific major comments or requested changes were enumerated in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript establishes a uniform bound on the number of rooted double nests for sums of Laplace eigenfunctions on real-analytic closed surfaces by invoking analyticity to obtain control through local holomorphic extensions and standard nodal-set properties of eigenfunctions. The bound is stated to depend only on the fixed surface, chosen root, and spectral cutoff. Sharpness is demonstrated via an explicit counterexample on the smooth sphere using a linear combination of eigenfunctions for eigenvalues 0 and 2, which produces infinitely many nests. The biharmonic constructions and quantitative bounds for entire functions of polynomial growth are likewise presented as direct constructions answering prior questions, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. All steps rest on external analytic techniques and are self-contained against independent benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all results are stated as theorems or constructions without visible ad-hoc choices.

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Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    B\'erard, Pierre and Helffer, Bernard , TITLE =. Doc. Math. , FJOURNAL =. 2018 , PAGES =

    work page 2018

  2. [2]

    B\'erard, Pierre and Helffer, Bernard , TITLE =. Mosc. Math. J. , FJOURNAL =. 2020 , NUMBER =. doi:10.17323/1609-4514-2020-20-1-1-25 , URL =

    work page doi:10.17323/1609-4514-2020-20-1-1-25 2020

  3. [3]

    B\'erard, Pierre and Helffer, Bernard , TITLE =. Expo. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1016/j.exmath.2018.10.002 , URL =

    work page doi:10.1016/j.exmath.2018.10.002 2020

  4. [4]

    Schr\"odinger operators, spectral analysis and number theory , SERIES =

    B\'erard, Pierre and Helffer, Bernard , TITLE =. Schr\"odinger operators, spectral analysis and number theory , SERIES =. [2021] 2021 , ISBN =. doi:10.1007/978-3-030-68490-7\_4 , URL =

    work page doi:10.1007/978-3-030-68490-7 2021

  5. [5]

    B\'erard, Pierre and Charron, Philippe and Helffer, Bernard , TITLE =. J. Anal. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s11854-021-0189-9 , URL =

    work page doi:10.1007/s11854-021-0189-9 2022

  6. [6]

    , title =

    Boggio, T. , title =. Rend. Acc. Lincei , volume =. 1901 , pages =

    work page 1901

  7. [7]

    , title =

    Boggio, T. , title =. Rend. Circ. Mat. Palermo , volume =. 1905 , pages =

    work page 1905

  8. [8]

    Buhovsky, Lev and Logunov, Alexander and Sodin, Mikhail , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2020 , NUMBER =. doi:10.1093/imrn/rnz181 , URL =

    work page doi:10.1093/imrn/rnz181 2020

  9. [9]

    Chanillo, Sagun and Logunov, Alexander and Malinnikova, Eugenia and Mangoubi, Dan , TITLE =. J. Differential Geom. , FJOURNAL =. 2024 , NUMBER =. doi:10.4310/jdg/1707767334 , URL =

    work page doi:10.4310/jdg/1707767334 2024

  10. [10]

    Cheng, Shiu Yuen , TITLE =. Comment. Math. Helv. , FJOURNAL =. 1976 , NUMBER =. doi:10.1007/BF02568142 , URL =

    work page doi:10.1007/bf02568142 1976

  11. [11]

    and Hilbert, D

    Courant, R. and Hilbert, D. , TITLE =. 1953 , PAGES =

    work page 1953

  12. [12]

    Donnelly, Harold , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1994 , NUMBER =. doi:10.2307/2160300 , URL =

    work page doi:10.2307/2160300 1994

  13. [13]

    Duffin, R. J. , TITLE =. J. Math. Physics , FJOURNAL =. 1949 , PAGES =

    work page 1949

  14. [14]

    Garabedian, P. R. , TITLE =. Pacific J. Math. , FJOURNAL =. 1951 , PAGES =

    work page 1951

  15. [15]

    Javier Gómez-Serrano and Robert Koirala and Alexander Logunov , title =

    work page

  16. [16]

    Grunau, Hans-Christoph and Robert, Fr\'ed\'eric , TITLE =. Arch. Ration. Mech. Anal. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00205-009-0230-0 , URL =

    work page doi:10.1007/s00205-009-0230-0 2010

  17. [17]

    , title =

    Hadamard, J. , title =. 1968 , pages =

    work page 1968

  18. [18]

    A maximum principle \`a

    Hedenmalm, H. A maximum principle \`a. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 1999 , NUMBER =. doi:10.1016/S0764-4442(99)80308-5 , URL =

    work page doi:10.1016/s0764-4442(99)80308-5 1999

  19. [19]

    arXiv preprint arXiv:2511.11570 , year=

    Structure Theory of Parabolic Nodal and Singular Sets , author=. arXiv preprint arXiv:2511.11570 , year=

    work page arXiv

  20. [20]

    , TITLE =

    Hardt, Robert M. , TITLE =. Invent. Math. , FJOURNAL =. 1976/77 , NUMBER =. doi:10.1007/BF01403128 , URL =

    work page doi:10.1007/bf01403128 1976

  21. [21]

    , TITLE =

    Hardt, Robert M. , TITLE =. Amer. J. Math. , FJOURNAL =. 1980 , NUMBER =. doi:10.2307/2374240 , URL =

    work page doi:10.2307/2374240 1980

  22. [22]

    Discrete Contin

    Lin, Fanghua and Liu, Dan , TITLE =. Discrete Contin. Dyn. Syst. , FJOURNAL =. 2016 , NUMBER =. doi:10.3934/dcds.2016.36.4517 , URL =

    work page doi:10.3934/dcds.2016.36.4517 2016

  23. [23]

    and Tegmark, Max , TITLE =

    Shapiro, Harold S. and Tegmark, Max , TITLE =. SIAM Rev. , FJOURNAL =. 1994 , NUMBER =. doi:10.1137/1036005 , URL =

    work page doi:10.1137/1036005 1994