Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces
We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary.
1. Introduction and main results
1.1. Steklov problem
Let be a compact -dimensional manifold with boundary and unit exterior normal We consider the Steklov eigenvalue problem
The solutions are called Steklov eigenfunctions. Let denote the boundary restriction map. The boundary restrictions of the Steklov eigenfunctions, , are the eigenfunctions of the Dirichlet-to-Neumann (sometimes also referred to as the Poincaré–Steklov) operator . It is well-known that is a first-order homogeneous, self-adjoint, elliptic pseudodifferential operator where (see [Ta1]). Consequently, the Steklov spectrum, which coincides with the spectrum of , consists of an infinite sequence of eigenvalues with as
Recently, there has been a significant interest in the study of the nodal geometry for the Steklov problem, see [BL, Ze, WZ]. However, these results were concerned with the nodal sets of the Dirichlet-to-Neumann eigenfunctions , i.e. the boundary nodal sets. The present paper focusses on the interior nodal sets, i.e. the nodal sets of the Steklov eigenfunctions . With the exception of the recent preprint [SWZ] (see Remark 1.2.3), very little has been previously known on this subject. We also note that Courant type bounds were earlier obtained on the number of the interior (in any dimension) and boundary (in dimension two) nodal domains of Steklov eigenfunctions ([KS, AM, KKP], see also [GP, section 6.1]).
1.2. Main result
Motivated by a celebrated conjecture of Yau [Y1, Y2], it has been recently suggested in [GP, Open problem 11 (i)] that the ratio between the –dimensional Hausdorff measure of the nodal set of a Steklov eigenfunction and the corresponding Steklov eigenvalue is bounded above and below by some positive constants depending only on the geometry of the manifold. Our main result below gives a positive answer to this conjecture for real-analytic Riemannian surfaces. Note that, in this case, the -dimensional Hausdorff measure of a nodal set is simply equal to its total length, which we will denote by .
Let be a real-analytic compact Riemannian surface with boundary. Then there exist constants such that, for any Steklov eigenfunction with an eigenvalue , the total length of its nodal set satisfies:
This result may be viewed as an analogue of the Donnelly-Fefferman bound for the size of the nodal set of Laplace eigenfunctions on real-analytic manifolds [DF].
To illustrate the nodal bounds in Theorem 1.2.1, consider the Steklov eigenfunctions on a unit disk corresponding to the double eigenvalue , . They can be represented in polar coordinates by for some . The nodal set of each such eigenfunction is a union of diameters, of total length .
Our methods here are specific to the case of real-analytic surfaces. In higher dimensions, Steklov eigenfunctions on are much more complicated and their nodal structures are not well-understood. In the case of a general smooth -dimensional manifold with smooth boundary, a non-sharp lower bound
for the volume of the interior nodal sets has recently been proven in [SWZ]. It can be viewed as the Steklov analogue of the lower bounds on the size of the nodal sets of Laplace eigenfunctions obtained in [CM, SZ, HS, M, HW].
After the first version of the present paper was posted on the archive, an upper bound of order on the size of the nodal set of Steklov eigenfunctions on surfaces was obtained in [Zh2] using a quite different approach. While this bound is not sharp, it is valid for arbitrary compact smooth surfaces with boundary. More recently, while the present paper was under review, the sharp upper bound in Theorem 1.2.1 has been extended in [Zh3] to real-analytic Riemannian manifolds of arbitrary dimension.
1.3. Sketch of the proof
For any closed manifold with boundary , Steklov eigenfunctions decay rapidly into the interior of [HL, GPPS]. In order to analyze their nodal lengths, we must therefore consider a neighborhood of the boundary and its complement separately. The idea is to use quasimodes near the boundary and frequency function techniques in the interior.
We begin, in section 2, by constructing quasimodes for the Dirichlet-to-Neumann operator . In Proposition 2.3.4, we show that trigonometric functions are approximate eigenfunctions for , up to an exponentially decaying error term. The proof uses the surface assumption in a very strong way: from [GPPS], we conclude that the error term decays faster than any polynomial in , and the assumption of analyticity allows us to improve the decay of the error term to exponential. From Proposition 2.3.4 and some fairly standard linear algebra techniques, we show in Lemma 3.1.1 that the boundary Steklov eigenfunctions can be approximated by trigonometric functions plus an error which decays exponentially in in any norm.
We would like to use the quasimode approximation to estimate nodal length of a Steklov eigenfunction near the boundary, but from the example of the annulus (see Example 1.4), we know that it is possible for to itself be exponentially small near one or more boundary components. Indeed, this seems to be the generic situation. Near these “residual” boundary components, the error in the quasimode approximation can be larger than the quasimode itself and hence the approximation is not particularly useful. Our proof avoids this difficulty via harmonic extension of the Steklov eigenfunctions across residual boundary components. By controlling the -norm of the extended eigenfunctions (see Lemma 4.3.1), one can effectively treat a neighborhood of the “residual” boundary components in the same way as the interior of
To illustrate the proof, consider a Steklov eigenfunction on the annular domain in Figure 1. Suppose that is a non-residual, i.e. “dominant”, boundary component for and that is a residual boundary component. In section 3.2, we extend our boundary quasimode approximation given by Lemma 3.1.1 into the interior. The exponential decay of the error means that the approximation is effective in a -independent neighborhood of . A direct comparison of nodal sets (section 4.2) then gives us upper and lower bounds on the nodal length of in a small neighborhood of , denoted by the dotted line in Figure 1.
To treat the interior and a neighborhood of the residual boundary components, we extend (section 2.4) to a small -independent neighborhood of , with boundary denoted by the dotted circle. In this neighborhood, is still exponentially small in (section 4.3). Now let be two sets as in Figure 1. In section 4.4, we use standard frequency function techniques to bound the nodal length of in from above by the “renormalized Almgren frequency function” of the larger domain . Then, using the quasimode approximation near the boundary and the fact that a portion of coincides with , we bound this frequency function by a multiple of . A covering argument extends these upper bounds to all of outside a neighborhood of , including a neighborhood of . Combining these upper bounds with the two-sided bounds near completes the proof.
A particularly novel aspect of the problem is the exponential decay into the interior of the eigenfunctions, namely that for some positive constants and depending only on the geometry of ,
This is an immediate consequence of Lemma 3.2.9 and the exponential decay of the interior quasimode approximations. A similar estimate holds for the derivatives of order of , provided we multiply the right-hand side by . Our problem therefore resembles the question of estimating the size of nodal sets in forbidden regions for eigenfunctions of Schrödinger operators (see [HZZ, CT]). A different method of proving (1.3.2) has been communicated to us by M. Taylor [Ta2].
It also follows from our results (see Proposition 3.1.3) that for real-analytic surfaces, the error term in the eigenvalue approximation of [GPPS, Theorem 1.4] in fact decays exponentially as the index of the eigenvalue increases. In particular, for a simply connected real-analytic surface , there exists a constant depending on the geometry of such that
1.4. Example: Steklov problem on an annulus
First we compare its Steklov spectrum to the spectrum of a union of two disks, of radii and , which consists of the double eigenvalue zero, as well as the double eigenvalues and for each . Let us show that the difference between these eigenvalues and the corresponding eigenvalues of is in fact exponentially vanishing in . Indeed, as computed in [D] (see also [GP, Section 4.2]), the nonzero Steklov eigenvalues of the annulus are given by the roots of the quadratic polynomial
for each (each root corresponds to a double eigenvalue). We could compute the eigenvalues directly, but it is easier to compare to the polynomial
which has roots and . In fact , where
Since , a straightforward calculation shows that the roots of differ from and by as well. The error is exponentially decreasing in , which agrees with the eigenvalue approximation result in Proposition 3.1.3.
Steklov eigenfunctions on surfaces also exhibit exponential decay at certain boundary components, which motivates our definition of “dominant” and “residual” boundary components (see Definition 4.1.1). To illustrate this, consider an eigenfunction of the annulus corresponding to an eigenvalue . As computed in [GP],
where is a unit norm linear combination of and and is an appropriate normalizing constant. Since , we see that , that the -norm of on is roughly , and that the -norm on is - which is exponentially decaying in . So, in this case, the circle of radius is a dominant boundary component, while the circle of radius is a residual boundary component. A similar analysis holds for the eigenfunctions corresponding to eigenvalues of the form , with the roles of the boundary components reversed.
The authors are grateful to M. Sodin and M. Taylor for useful discussions, and to the anonymous referee for many helpful suggestions on the presentation of the paper. The research of I.P. was partially supported by NSERC, FRQNT and Canada Research Chairs Program. The research of D.S was partially supported by NSF EMSW21-RTG 1045119. The research of J.T. was partially supported by NSERC and FRQNT. I.P. and J.T. were also supported by the French National Research Agency project Gerasic-ANR-13-BS01-0007-0.
2. Steklov problem for real-analytic planar domains
2.1. Koebe uniformization
We first prove Theorem 1.2.1 in the case where is a real-analytic planar domain and explain the fairly minor modifications needed to treat the general case of compact Riemann surfaces with boundary in section 4.5. The two-dimensional case is quite special. Indeed, unlike the higher-dimensional case, for surfaces the Steklov operator agrees with the square root of the Laplacian on the various boundary components modulo a smoothing operator. Upon reparametrization by arclength, the latter operator is consequently (modulo smoothing) just the Fourier multiplier acting on the component boundary circles. Although not necessary, one can see the former directly by applying conformal mapping. Indeed, by the Koebe uniformization theorem [HS], one can conformally map to a planar domain , where is the disk of radius 1 with a finite number of interior disks removed. In the cases where is simply-connected, this reduces to the Riemann mapping theorem. When is real-analytic, it follows by regularity up to the boundary in the associated Dirichlet problem for the Green’s function [MN] that the conformal map extends to a real-analytic map of to Moreover, maps boundary to boundary in a univalent fashion. In particular, the induced boundary restriction is a -diffeomorphism.
The boundary consists of a union of circles, which we denote by , , with radii and centers respectively. The corresponding boundary components of are denoted by We let be the usual angle coordinate on for each , and let the arc length coordinate be . Let be an arc length coordinate on which coincides with when we restrict attention to . Finally, let and be the measures on and respectively induced by the Euclidean measure on . As in [Ed], the Steklov problem in (1.1) is conformally mapped to the problem
with analytic for and . By conformal mapping, - so without loss of generality, it suffices to work with the conformal model (2.1) and we will do so here. We will abuse notation somewhat and denote the corresponding Steklov operator in the conformal model (2.1) by as well, and its eigenfunctions by . We henceforth make the normalization that
Since we consistently work with the model case in (2.1), this should not lead to confusion.
Given we complexify to a Grauert tube (ie. an annulus)
We choose here to be the analytic modulus of ; that is, the maximal tube radius for which has a holomorphic extension to .
We also note that the length of is , and define a new coordinate on by . Let be the corresponding coordinate on all of . Note that since is analytic and strictly positive, this is an analytic reparametrization with analytic inverse.
2.2. Potential layer formulas and the Steklov operator
We briefly review the characterization of the Steklov operator in terms of potential layer operators. This material here is well-known and further details can be found in [Ta1, Sec. 7.1]. Here, we assume that is a bounded domain with -boundary. Let be the ambient free Green’s function for in . Consider the single and double layer operators and given by
Corresponding to and are the boundary layer operators and given by
These operators are classical pseudodifferential with , and is elliptic. Given a function and let and denote the limits of as from and respectively. The layer potential operators in (2.2) and the induced boundary operators in (2.2) are linked via the boundary jumps equations
Now let be the Dirichlet-to-Neumann operator for . Consider the Dirichlet problem
A straightforward application of Green’s formula gives
Since and is elliptic, it follows from (2.2.5) by a parametrix construction in the standard pseudodifferential calculus that and is also elliptic.
In the case where is real-analytic, is also analytic pseudodifferential in the sense of [BK] (see also [Tr, Ch. 5]), and we write To see this, we first note integration against in is a pseudodifferential operator with full symbol in the usual coordinates. As the subprincipal symbols are all zero, the symbol satisfies Cauchy estimates and so, . Since is real-analytic, the Fermi coordinates near are also real-analytic. In terms of these local coordinates, is thus an analytic symbol; hence so is . It is easy to check that the symbols of and are given, respectively, by and respectively, and hence are also analytic. So , By constructing a parametrix for in the analytic pseudodifferential calculus (see [Tr, Ch. 5] for details), then multiplying (2.2.5) by this parametrix on the left and rearranging, it follows that
In the following, we say that an operator is analytic smoothing if its Schwartz kernel , and we write
In what follows, we use the following notation: given a set and two non-negative functions and , throughout the paper we write if there exists a constant such that for all In addition, will mean that both and are satisfied.
2.3. Quasimodes for the Steklov operator
where , with the Fourier multiplier defined by
Note that . As in [GPPS], let be the sequence and be the union of these sequences, with multiplicity, arranged in the appropriate order. We write
It follows that as We abuse notation somewhat and denote the corresponding orthonormal basis of eigenfunctions of on by , so that each is the restriction of to . Although the operator acting on is non-local, it clearly maps each of the boundary components to itself; that is, for any with supp
Denote the -normalized eigenfunctions of on the boundary component by where
We also let be the function which is on and 0 on the other boundary components, with an analogous definition for .
The following lemma is central to the proof of Theorem 1.2.1 and shows that when is analytic, the functions are quasimodes for to exponential error in
Suppose that is analytic, with , the conformal map , and the Dirichlet-to-Neumann operator as in (2.1). Then:
The remainder in (2.3.1) has Schwartz kernel with analytic modulus at least in each variable separately. Here, depends on the analytic modulus of the conformal multiplier and the geometry of .
In the last line, denotes a non-negative term that is with constant depending on
Our proof uses the boundary integral equation for . For , , we continue to let be the free Green’s kernel for , which is . Using (2.2.5) (see also [Sh, Ro]), the functions satisfy the boundary jumps equation:
Change variables in the first integral to integrate in terms of :
We now claim:
where , and with given by
Recall that is the space of operators with Schwartz kernels. We also claim:
If we extend to act on by letting it be zero when acting on constants,
Assuming both claims, we now prove Proposition 2.3.4. The claims combine to show
Given , we can write with . By linearity, the boundedness of and , and the rapid decay of and of , it follows that
Since and standard theory of analytic pseudodifferential operators [Tr, Ch. 5] implies that , with
Now, we choose . For any , by contour deformation we have
where the choice of is determined by the sign of . The same is true for each , so for all ,
The same is therefore true for for any . Since is a (classical) pseudodifferential operator, it is bounded from for each . From this and Sobolev embedding we conclude that for any . Combining this with (2.3.10) completes the proof of the proposition. ∎
We now prove both claims.
Proof of Claim 2.3.7.
Assume that . The portions of the integrals in (2.3.6) over can be absorbed into and , since for and therefore the integrands are real-analytic functions of . Using the symbol to denote equivalence up to terms of the form with and , we have
For notational simplicity, we now suppress all subscripts. The last term in (2.3.12) can be absorbed, because the normal derivative is itself real-analytic in To see this, observe that analyticity away from the diagonal is clear. To establish the analyticity near we insert the Taylor expansion into the formula for in (2.3.5). Here denotes the curvature at the point . The result is that
and so, since is real-analytic with as .
Finally, all we need to do to complete the proof is to show that
is analytic. However, the reparametrization is analytic with analytic inverse, and Taylor expansion of the quotient on the right-hand side shows that it is analytic and nonzero at (cf. ([TZ, formula (8.8)])). Therefore, its logarithm is also analytic there. Analyticity away from is automatic. ∎
Proof of Claim 2.3.8.
By scaling, it suffices to prove the claim for and . We will show
The claim follows since is a Fourier basis. To prove (2.3.13), consider the harmonic function on the disk whose Neumann data is . On the one hand, this is . On the other hand, the Poisson kernel for the Neumann problem is , so
Restricting to shows (2.3.13).∎
2.4. Harmonic extension of interior eigenfunctions across the boundary
It is well known that a harmonic function on a domain can be extended harmonically across a real analytic boundary (see [LM]). The following lemma, which will be used in the proof of Theorem 1.2.1, provides an explicit estimate on the -norms of the harmonic extension of a Steklov eigenfunction across a boundary component in terms of the -norm of its trace on .
Let be a Steklov eigenfunction. Then, for every connected component of the boundary, and with defined in (3.2.1), there is a harmonic continuation of (also denoted ) across the boundary component to an annulus of width . Moreover, for one has the exterior estimate
From Proposition we know where . We use this characterization of to estimate Fourier coefficients of along the boundary circle To simplify the writing in the following we assume that and that the center of the disc is the origin. Also, we abuse notation somewhat and simply write for in (2.4)-(2.4.4) below.
From (2.4), we observe
It follows that holomorphically continues to the strip and so holomorphically continues to an annular neighbourhood of of any width In terms of the parametrizing coordinates one holomorphically continues to the annulus
Without loss of generality, we assume here that the set where (denoted by ) is the part of the annulus lying inside the domain and (denoted by ) is the part lying outside. Note that is independent of It also follows from (2.4.3) that the holomorphic continuation, of the boundary Steklov eigenfunction satisfies
with an appropriate constant
Now we need to construct the harmonic continuation of the interior Steklov eigenfunction By Green’s formula,
Since the harmonic continuation of both terms in (2.4.6) is carried out in the same way, we consider here the second term and then just indicate the minor changes necessary to deal with the first one. Since the multiplicative factor is irrelevant to the continuation, we just consider