Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture

Abstract

Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $$H^{n-1}(\{u=0 \} \cap B) \geq c.$$ We prove Nadirashvili’s conjecture as well as its counterpart on $C^\infty$-smooth Riemannian manifolds. The latter yields the lower bound in Yau’s conjecture. Namely, we show that for any compact $C^\infty$-smooth Riemannian manifold $M$ (without boundary) of dimension $n$, there exists $c>0$ such that for any Laplace eigenfunction $\varphi_\lambda$ on $M$, which corresponds to the eigenvalue $\lambda$, the following inequality holds: $c \sqrt \lambda \leq H^{n-1}(\{\varphi_\lambda =0\})$.

  • [Y] S. T. Yau, "Problem section," in Seminar on Differential Geometry, Princeton Univ. Press, Princeton, N.J., 1982, vol. 102, pp. 669-706.
    @INCOLLECTION{Y,
      author = {Yau, Shing Tung},
      title = {Problem section},
      booktitle = {Seminar on {D}ifferential {G}eometry},
      series = {Ann. of Math. Stud.},
      volume = {102},
      pages = {669--706},
      publisher = {Princeton Univ. Press, Princeton, N.J.},
      year = {1982},
      mrclass = {53Cxx (58-02)},
      mrnumber = {0645762},
      mrreviewer = {Yu. Burago},
      zblnumber = {},
      }
  • [DF] Go to document H. Donnelly and C. Fefferman, "Nodal sets of eigenfunctions on Riemannian manifolds," Invent. Math., vol. 93, iss. 1, pp. 161-183, 1988.
    @ARTICLE{DF,
      author = {Donnelly, Harold and Fefferman, Charles},
      title = {Nodal sets of eigenfunctions on {R}iemannian manifolds},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {93},
      year = {1988},
      number = {1},
      pages = {161--183},
      issn = {0020-9910},
      mrclass = {58G25 (35B60 35P05)},
      mrnumber = {0943927},
      mrreviewer = {P. Günther},
      doi = {10.1007/BF01393691},
      url = {http://dx.doi.org/10.1007/BF01393691},
      zblnumber = {0659.58047},
      }
  • [DF1] Go to document H. Donnelly and C. Fefferman, "Nodal sets for eigenfunctions of the Laplacian on surfaces," J. Amer. Math. Soc., vol. 3, iss. 2, pp. 333-353, 1990.
    @ARTICLE{DF1,
      author = {Donnelly, Harold and Fefferman, Charles},
      title = {Nodal sets for eigenfunctions of the {L}aplacian on surfaces},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {3},
      year = {1990},
      number = {2},
      pages = {333--353},
      issn = {0894-0347},
      mrclass = {58G25 (35P05)},
      mrnumber = {1035413},
      mrreviewer = {H.-B. Rademacher},
      doi = {10.2307/1990956},
      url = {http://dx.doi.org/10.2307/1990956},
      zblnumber = {0702.58077},
      }
  • [GL] Go to document N. Garofalo and F. Lin, "Monotonicity properties of variational integrals, $A_p$ weights and unique continuation," Indiana Univ. Math. J., vol. 35, iss. 2, pp. 245-268, 1986.
    @ARTICLE{GL,
      author = {Garofalo, Nicola and Lin, Fang-Hua},
      title = {Monotonicity properties of variational integrals, {$A_p$} weights and unique continuation},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana University Mathematics Journal},
      volume = {35},
      year = {1986},
      number = {2},
      pages = {245--268},
      issn = {0022-2518},
      mrclass = {35J20 (35J10 42B25)},
      mrnumber = {0833393},
      mrreviewer = {Stavros A. Belbas},
      doi = {10.1512/iumj.1986.35.35015},
      zblnumber = {0678.35015},
      }
  • [HS] Go to document R. Hardt and L. Simon, "Nodal sets for solutions of elliptic equations," J. Differential Geom., vol. 30, iss. 2, pp. 505-522, 1989.
    @ARTICLE{HS,
      author = {Hardt, Robert and Simon, Leon},
      title = {Nodal sets for solutions of elliptic equations},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {30},
      year = {1989},
      number = {2},
      pages = {505--522},
      issn = {0022-040X},
      mrclass = {58E05 (35J99)},
      mrnumber = {1010169},
      mrreviewer = {Fang Hua Lin},
      doi = {10.4310/jdg/1214443599},
      zblnumber = {0692.35005},
      }
  • [HL] Q. Han and F. -H. Lin, Nodal Sets of Solutions of Elliptic Differential Equations.
    @MISC{HL,
      author = {Han, Q. and Lin, F.-H.},
      title = {Nodal Sets of Solutions of Elliptic Differential Equations},
      note = {book in preparation},
      zblnumber = {},
      }
  • [LM1] A. Logunov and . E. Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three.
    @misc{LM1,
      author = {Logunov, A. and Malinnikova, {\relax Eu}.},
      title = {Nodal sets of {L}aplace eigenfunctions: estimates of the {H}ausdorff measure in dimension two and three},
      note = {preprint},
      zblnumber = {},
      }
  • [LM2] Go to document A. Logunov, "Nodal sets of Laplace eigenfunctions: polynomial upper bounds for the Hausdorff measure," Ann. of Math., vol. 187, iss. 1, pp. 221-239, 2018.
    @article{LM2,
      author = {Logunov, A.},
      title = {Nodal sets of {L}aplace eigenfunctions: polynomial upper bounds for the {H}ausdorff measure},
      journal={Ann. of Math.},
      VOLUME={187},
      number={1},
      year={2018},
      pages={221--239},
      doi={10.4007/annals.2018.198.1.4},
      zblnumber = {},
      }
  • [CM] Go to document T. H. Colding and W. P. Minicozzi II, "Lower bounds for nodal sets of eigenfunctions," Comm. Math. Phys., vol. 306, iss. 3, pp. 777-784, 2011.
    @ARTICLE{CM,
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Lower bounds for nodal sets of eigenfunctions},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {306},
      year = {2011},
      number = {3},
      pages = {777--784},
      issn = {0010-3616},
      mrclass = {58J50 (28A78 35P15 35P20)},
      mrnumber = {2825508},
      mrreviewer = {Julie Rowlett},
      doi = {10.1007/s00220-011-1225-x},
      url = {http://dx.doi.org/10.1007/s00220-011-1225-x},
      zblnumber = {1238.58020},
      }
  • [SZ] Go to document C. D. Sogge and S. Zelditch, "Lower bounds on the Hausdorff measure of nodal sets II," Math. Res. Lett., vol. 19, iss. 6, pp. 1361-1364, 2012.
    @ARTICLE{SZ,
      author = {Sogge, Christopher D. and Zelditch, Steve},
      title = {Lower bounds on the {H}ausdorff measure of nodal sets {II}},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {19},
      year = {2012},
      number = {6},
      pages = {1361--1364},
      issn = {1073-2780},
      mrclass = {58C40 (28A78 35P15 35R01)},
      mrnumber = {3091613},
      mrreviewer = {Nelia Charalambous},
      doi = {10.4310/MRL.2012.v19.n6.a14},
      url = {http://dx.doi.org/10.4310/MRL.2012.v19.n6.a14},
      zblnumber = {1283.58020},
      }
  • [B] J. Brüning, "Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators," Math. Z., vol. 158, iss. 1, pp. 15-21, 1978.
    @ARTICLE{B,
      author = {Brüning, Jochen},
      title = {{Ü}ber {K}noten von {E}igenfunktionen des {L}aplace-{B}eltrami-{O}perators},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {158},
      year = {1978},
      number = {1},
      pages = {15--21},
      }
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977, vol. 224.
    @BOOK{GT,
      author = {Gilbarg, David and Trudinger, Neil S.},
      title = {Elliptic Partial Differential Equations of Second Order},
      series = {Grundlehren Math. Wiss.},
      volume = {224},
      publisher = {Springer-Verlag, Berlin-New York},
      year = {1977},
      pages = {x+401},
      isbn = {3-540-08007-4},
      mrclass = {35-02 (35J25 35J65)},
      mrnumber = {0473443},
      mrreviewer = {O. John},
      zblnumber = {0361.35003},
      }
  • [L] Go to document F. Lin, "Nodal sets of solutions of elliptic and parabolic equations," Comm. Pure Appl. Math., vol. 44, iss. 3, pp. 287-308, 1991.
    @ARTICLE{L,
      author = {Lin, Fang-Hua},
      title = {Nodal sets of solutions of elliptic and parabolic equations},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {44},
      year = {1991},
      number = {3},
      pages = {287--308},
      issn = {0010-3640},
      mrclass = {58G11 (35J05 35K05 58G03)},
      mrnumber = {1090434},
      mrreviewer = {Robert McOwen},
      doi = {10.1002/cpa.3160440303},
      url = {http://dx.doi.org/10.1002/cpa.3160440303},
      zblnumber = {0734.58045},
      }
  • [M] Go to document D. Mangoubi, "The effect of curvature on convexity properties of harmonic functions and eigenfunctions," J. Lond. Math. Soc. (2), vol. 87, iss. 3, pp. 645-662, 2013.
    @ARTICLE{M,
      author = {Mangoubi, Dan},
      title = {The effect of curvature on convexity properties of harmonic functions and eigenfunctions},
      journal = {J. Lond. Math. Soc. (2)},
      fjournal = {Journal of the London Mathematical Society. Second Series},
      volume = {87},
      year = {2013},
      number = {3},
      pages = {645--662},
      issn = {0024-6107},
      mrclass = {58J50 (35J15 35P20 35R01 53C21 58E20)},
      mrnumber = {3073669},
      mrreviewer = {Tanya J. Christiansen},
      doi = {10.1112/jlms/jds067},
      url = {http://dx.doi.org/10.1112/jlms/jds067},
      zblnumber = {1316.35220},
      }
  • [N] Go to document N. Nadirashvili, "Geometry of nodal sets and multiplicity of eigenvalues," Curr. Dev. Math., pp. 231-235, 1997.
    @ARTICLE{N,
      author = {Nadirashvili, N.},
      title = {Geometry of nodal sets and multiplicity of eigenvalues},
      journal = {Curr. Dev. Math.},
      year = {1997},
      pages = {231--235},
      zblnumber = {},
      doi = {10.4310/CDM.1997.v1997.n1.a16},
      }

Authors

Alexander Logunov

School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
Chebyshev Laboratory, St. Petersburg State University, Saint Petersburg, Russia
Institute for Advanced Study, Princeton, NJ, USA