aa r X i v : . [ m a t h . A P ] N ov A remark on recent lower bounds for nodal sets
Dan Mangoubi
Abstract
Recently, two papers ([SZ, CM]) appeared which give lower bounds on the size of the nodal sets ofeigenfunctions. The purpose of this short note is to point out a third method to obtain a power lawlower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growthbound for eigenfunctions and a growth vs. volume relation we proved in [M].
Consider a C ∞ Riemannian manifold (
M, g ). Let ∆ be the Laplace-Beltrami operator on M . The eigen-functions are solutions of ∆ ϕ λ + λϕ λ = 0. We are interested in finding lower bounds on the size of the nodalset in the case where ( M, g ) is C ∞ but not real analytic.Yau’s Conjecture ([Y]) asserts that the size of the nodal set is comparable to λ / . Donnelly and Feffermanproved ([DF]) Yau’s conjecture in case ( M, g ) is real analytic. The real analyticity assumption is used in acrucial way: The eigenfunctions are analytically continued to holomorphic functions with bounded growth,and then the problem is reduced to a problem about polynomials.The history of the lower bounds in the C ∞ but non real-analytic case can be summarized as follows: Indimension two the lower bound in Yau’s conjecture was proved by Br¨uning in [B], and by Yau, independently.In dimension n = 3 it is known that the size of the nodal set is bounded away from 0 by a constantindependent of λ , due to the recent work of Colding and Minicozzi ([CM]). In dimensions n ≥ decreasing to λ . In fact, from [DF] and [HL] they were known to beexponentially decreasing. The recent developments by Sogge-Zelditch ([SZ]) and Colding-Minicozzi ([CM])give polynomially decreasing bounds. In this note (Theorem 2.3) we extract polynomially decreasing boundsin a few lines from our previous work in [M]. We recall three of the many innovative ideas proved in [DF], which frequently appear in the next sections.I. Let B ⊂ M be a metric ball. B is a concentric ball half the radius of B . Define the growth of ϕ λ in B by β ( ϕ λ ; B ) = log max B | ϕ λ | max B | ϕ λ | . Then, for every ball
B β ( ϕ λ ; B ) ≤ C ( M,g ) λ / . (1.1)This is true for any C ∞ -manifold.II. In the real analytic case, for each eigenvalue λ one can find disjoint balls of radius cλ − / , the totalvolume of which is at least C Vol( M ), and such that the growth of the eigenfunction in each of theseballs is at most β , where β is a constant independent of λ , and in addition the eigenfunction vanishesat the center of each such ball.III. There exists a relation between growth estimates and volume estimates: In each ball in which thegrowth of the eigenfunction is at most β , and in which the eigenfunction vanishes at a point of itsmiddle half the volume of the positive set, the volume of the negative set, and the volume of the ballare all comparable to each other. This relation is true in the general C ∞ -case.1rom II and III one obtains lower bounds on the size of the set { ϕ λ = 0 } ∩ B by the relative isoperimetricinequality ([F]): Let A , A ⊂ B be open subsets. ThenVol n − ( ∂A ∩ ∂A ) ≥ C min { (Vol n A ) n − n , (Vol n A ) n − n } , (1.2)where Vol n − is the Hausdorff measure. In our situation A = { ϕ λ > } ∩ B , A = { ϕ λ < } ∩ B . We getthat the ( n − { ϕ λ = 0 } in each ball of the collection in II is comparable to the ball’sboundary area. Finally, multiplying this estimate by the number of balls in the collection ( cλ n/ ) gives alower bound of cλ / . Our approach to lower bounds in the C ∞ case is to give an estimate on the positivity volume in every ballfor which the eigenfunction vanishes at its middle half. In this way we circumvent the need to estimate thenumber of balls in which the eigenfunction has bounded growth (cf. idea II. in Section 1.1).In [M] we have shown that in every ball B for which ϕ λ vanishes at B one hasVol( { ϕ λ > } ∩ B )Vol B ≥ Cβ ( ϕ λ ; B ) − ( n − . (2.1)Symmetrically, the same estimate is true also for the negativity set. The proof of (2.1) is based on an iterationprocedure which starts with an exponentially small lower bound. To explain the basic idea, we let u be a har-monic function in the unit ball. Suppose for simplicity u (0) >
0. We normalize u so that u (0) = 1. Suppose u < M = e β in B . Then the mean value property immediately gives that Vol( { u > } ) > C M − . Now weimprove this primary estimate by iteration: Consider the ball B / . If u ≤ M / on B / , then the sameargument as above gives Vol( { u > } ) > C M − / . Otherwise, there exists a point x such that | x | = 1 / u ( x ) > M / . Consider the ball B = B ( x, / B u ) /u ( x ) < M / , applying the above argumentto the ball B ( x, /
2) gives again Vol( { u > } ) < C M − / . Thus, in any case Vol( { u > } ) < C M − / .We can continue this sequence of improvements to obtain Vol( { u > } ) < C ε M − ε for all ε >
0. Optimizing,one gets in this way the bound C (log M ) − n = Cβ − n . A slight modification of this argument (see [M]) gives Cβ − ( n − . The case where u (0) = 0 is a little more involved, since we have to take into consideration thedifferent signs of u . We overcome this difficulty by applying the Harnack inequality. Finally, it turns outthat the proof for harmonic functions can be adapted to solutions of second order C ∞ elliptic equations.Plugging the estimate (2.1) for the positivity set, the same estimate for the negativity set and (1.1)in (1.2) we obtain Vol n − ( { ϕ λ = 0 } ∩ B )Vol n − ( ∂B ) ≥ Cλ − ( n − n (2.2)Finally, it is well known (and an easy fact) that for each λ one can find a set of disjoint balls of radius cλ − / such that the eigenfunction vanishes at the middle half of each such ball, and the total volume ofwhich is at least C Vol( M ). Hence, one multiplies the estimate (2.2) by the number of such balls ( Cλ n/ )and obtains Theorem 2.3.
Vol n − ( ϕ λ = 0) ≥ Cλ − ( n − n λ − n λ n/ = Cλ − n − n . Colding and Minicozzi give in [CM] a new argument that shows that on any C ∞ -Riemannian manifold onecan find a constant β and for each eigenvalue λ a disjoint set of balls of radius cλ − / such that the growthof ϕ λ in each such ball is bounded by β and such that the total L -norm of ϕ λ on the union of these balls G is at least k ϕ λ k L ( M ) , and in addition the eigenfunction vanishes at the center of each such ball. Thisshould be compared with idea II in Section 1.1. 2ow, one would like to estimate the number of balls in G . Since the L -norm of ϕ λ on G is big, we canapply H¨older’s inequality and upper L p -bounds for p >
2, in order to obtain a lower bound on the volume of G . The easiest such bounds are the Sobolev bounds: k ϕ λ k L p ≤ λ n ( − p ) k ϕ λ k L . The sharp L p bounds are Sogge estimates ([S, Ch. 5]) (which in the p = ∞ case reduce to the bound comingfrom local Weyl law): k ϕ λ k L p ( M ) ≤ λ δ ( p ) k ϕ λ k L ( M ) , where δ ( p ) = ( n − ( − p ) , ≤ p ≤ n +1) n − n ( − p ) − , n +1) n − ≤ p ≤ ∞ If we take p = 2( n + 1) / ( n − G :Vol( G ) > Cλ − ( n − / . Hence, the number of balls in G is at least λ ( n +1) / . Then we proceed as before to getVol n − ( { ϕ λ = 0 } ) ≥ Cλ ( n +1) / − n ) / = λ (3 − n ) / . Sogge and Zelditch were inspired in [SZ] by Dong’s formula ([D]). In particular, they prove: λ Z M | ϕ λ | d Vol = 2 Z { ϕ λ =0 } |∇ ϕ λ | d Area . (4.1)To the preceding formula one can join upper pointwise bounds on ∇ ϕ λ coming from the local Weylformula: |∇ ϕ λ | ≤ λ ( n +1) / . (4.2)Sogge’s L p -upper bounds on ϕ λ also give lower L -bounds. Indeed, by H¨older’s inequality:1 = k ϕ λ k L ≤ k ϕ λ k p − p − L k ϕ λ k pp − L p . (4.3)Thus, k ϕ λ k L ≥ k ϕ λ k − pp − L p ≥ λ − pδ ( p ) p − . If we choose p = 2( n + 1) / ( n −
1) we obtain k ϕ λ k L ≥ Cλ − ( n − / . (4.4)From (4.2), (4.4) and (4.1) one obtains λ ( n +1) / Vol n − ( { ϕ λ = 0 } ) ≥ Cλ · λ (1 − n ) / , and after rearranging Vol n − ( { ϕ λ = 0 } ) ≥ Cλ (7 − n ) / . Conclusion
We conclude by a short summary of the three methods discussed above:The idea in [CM] is closest in spirit to the work of [DF]: The number of disjoint balls of the wavelengthradius centered on the nodal set and in which the growth of the eigenfunction is bounded is estimated usingSogge’s estimates. Since the size of the nodal set in each such ball is comparable to the size of the boundaryof the ball, a lower bound on the size of the nodal set is obtained. This method gives the best known boundstoday.Our approach from [M] gives an estimate of the size of the nodal set in any ball in terms of the growth ofthe eigenfunction in the ball. It uses inequality (1.1) to bound the growth in the worst case. In particular, wecircumvent the estimate of the number of balls with bounded growth. Our estimates are not sharp. Hence,it seems that room for strengthening the result is still left.The method of [SZ] is based on expressing the L -norm of the eigenfunction as an integral of the gradientover the nodal set. This is close in spirit to [D]. Pointwise gradient estimates from the local Weyl law anda sharp L -lower bound are applied. Acknowledgements
I would like to thank Iosif Polterovich for drawing my attention to this problem and for useful discussions.I thank Joseph Bernstein for an illuminating discussion. I am grateful to Leonid Polterovich for valuableremarks on a preliminary version of this note. This work was partially supported by ISF grant no. 225/10.
References [B] J. Br¨uning, ¨Uber Knoten von Eigenfunktionen des Laplace-Beltrami-Operators , Math. Z. (1978), no. 1, 15–21.[CM] T. H. Colding and W. P. Minicozzi II,
Lower bounds for nodal sets of eigenfunctions , to appear in Comm. Math. Phys.,available at arXiv:math/1009.4156 .[D] R.-T. Dong,
Nodal sets of eigenfunctions on Riemann surfaces , J. Differential Geom. (1992), no. 2, 493–506.[DF] H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds , Invent. Math. (1988), no. 1,161–183.[F] H. Federer, Geometric measure theory , Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-VerlagNew York Inc., New York, 1969.[HL] Q. Han and F. Lin,
Nodal sets of solutions of elliptic differential equations , 2007, in preparation.[M] D. Mangoubi,
Local asymmetry and the inner radius of nodal domains , Comm. Partial Differential Equations (2008),no. 7-9, 1611–1621.[S] C. D. Sogge, Fourier integrals in classical analysis , Cambridge Tracts in Mathematics, vol. 105, Cambridge UniversityPress, Cambridge, 1993.[SZ] C. D. Sogge and S. Zelditch,
Lower bounds on the Hausdorff measure of nodal sets , to appear in Math. Res. Lett.,available at arXiv:math/1009.3573 .[Y] S.-T. Yau,
Problem section , Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press,Princeton, N.J., 1982, pp. 669–706.
Dan Mangoubi,The Hebrew University of Jerusalem, Givat-Ram,Jerusalem 91904,Israel [email protected]@math.huji.ac.il