Neumann Domains on Graphs and Manifolds
NNEUMANN DOMAINS ON GRAPHS AND MANIFOLDS
LIOR ALON, RAM BAND, MICHAEL BERSUDSKY, SEBASTIAN EGGER
Abstract.
The nodal set of a Laplacian eigenfunction forms a partition of the under-lying manifold or graph. Another natural partition is based on the gradient vector fieldof the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on agraph). The submanifolds (or subgraphs) of this partition are called Neumann domains.This paper reviews the subject, as appears in [2, 6, 7, 46, 62] and points out some openquestions and conjectures. The paper concerns both manifolds and metric graphs andthe exposition allows for a comparison between the results obtained for each of them. Introduction
Given a Laplacian eigenfunction on a manifold or a metric graph, there is a naturalpartition of the manifold or the graph. The partition is dictated by the gradient vectorfield of the eigenfunction (on a manifold) or by the extremal points of the eigenfunction(on a graph). The submanifolds (or subgraphs) of such a partition are called Neumanndomains and the separating lines (or points in the case of a graph) are called Neumannlines (or points). The counterpart of this partition is the nodal partition (with the sameterminology of nodal domains, nodal lines and nodal points). This latter partition isextensively studied in the last two decades or so (though interesting results on nodaldomains appeared throughout all of the 20-th century and even earlier). When restrictingan eigenfunction to a single nodal domain one gets an eigenfunction of that domain withDirichlet boundary conditions. Similarly, when restricting an eigenfunction to a Neumanndomain, one gets a Neumann eigenfunction of that domain (Lemmata 3.1,8.1), whichexplains the name
Neumann domain and shows the most basic linkage between nodaldomains and Neumann domains.Neumann domains form a very new topic of study in spectral geometry. They werefirst mentioned in a paragraph of a manuscript by Zelditch [62]. Shortly afterwards (andindependently) a paper by McDonald and Fulling was dedicated to Neumann domains[46]. Since then two additional papers contributed to this topic; one of the authors withFajman [7] and two of the authors with Taylor [6]. The first part of the current manuscriptserves as an exposition of the known results for Neumann domains on two-dimensionalmanifolds, adding a few supplementary new results and proofs. The second part focuseson Neumann domains on metric graphs and reviews the results which appear in [2] . Weaim to point out similarities and differences between Neumann domains on manifolds andthose on graphs. For this purpose, each of the two parts of the papers is divided to exactlythe same subtopics: definitions, topology, geometry, spectral position and count. We also Mathematics Subject Classification.
Key words and phrases.
Neumann domains, Neumann lines, nodal domains, Laplacian eigenfunctions,Quantum graph, Morse-Smale complexes. While writing this manuscript, we became aware that there is an ongoing research on the related topic ofNeumann partitions on graphs. These works in progress are done by Gregory Berkolaiko, James Kennedy,Pavel Kurasov, Corentin L´ena and Delio Mugnolo. a r X i v : . [ m a t h . SP ] M a y LIOR ALON, RAM BAND, MICHAEL BERSUDSKY, SEBASTIAN EGGER include an appendix which contains a short review of relevant results in basic Morse theory,useful for the manifold part of the paper. The summary of the paper provides guidelinesfor comparison between the manifold results and the graph results. Such a comparison hadtaught us a great deal in what concerns to the field of nodal domains and yielded a wealthof new results both on manifolds and graphs. As an example we only mention the topic ofnodal partitions and refer the interested reader to [5, 13, 15, 16, 17, 20, 24, 36, 38] in orderto learn on the evolution of this research direction. In addition to that, we believe thatit is beneficial to compare problems between the fields of nodal domains and Neumanndomains. We point out such similarities and differences throughout the paper.Although new in spectral theory, Neumann domains were used in computational ge-ometry, where they are known as Morse-Smale complexes (see the book [64] or [18] foran extensive review). They are used as a tool to analyze sets of measurements on certainspaces and for getting a good qualitative and quantitative acquaintance with the measuredfunctions [23, 27, 28]. Another field of relevance is computer graphics, where Morse-Smalecomplexes of Laplacian eigenfunctions are applied for surface segmentation [26, 35, 51].
Part Neumann domains on two-dimensional manifolds Definitions
Let (
M, g ) be a two-dimensional, connected, orientable and closed Riemannian manifold.We denote by − ∆ the (negative) self-adjoint Laplace-Beltrami operator. Its spectrumis purely discrete since M is compact. We order the eigenvalues { λ n } ∞ n =0 increasingly,0 = λ < λ ≤ λ ≤ . . . , and denote a corresponding complete system of orthonormaleigenfunctions by { f n } ∞ n =0 , so that we have(2.1) − ∆ f n = λ n f n . We assume in the following that the eigenfunctions f are Morse functions, i.e. have nodegenerate critical points . We call such an f a Morse-eigenfunction . Eigenfunctions aregenerically Morse, as shown in [1, 58]. At this point, we refer the interested reader to theappendix, where some basic Morse theory which is relevant to the paper is presented.In order to define Neumann domains and Neumann lines we introduce the following con-struction based on the gradient vector field, ∇ f . This vector field defines the followingflow:(2.2) ϕ : R × M → M,∂ t ϕ ( t, x ) = −∇ f (cid:12)(cid:12) ϕ ( t, x ) ,ϕ (0 , x ) = x . The following notations are used throughout the paper. The set of critical points of f is denoted by C ( f ); the sets of saddle points and extrema of f are denoted by S ( f )and X ( f ); the sets of minima and maxima of f are denoted by M − ( f ) and M + ( f ),respectively.For a critical point x ∈ C ( f ), we define its stable and unstable manifolds by(2.3) W s ( x ) = { y ∈ M (cid:12)(cid:12) lim t →∞ ϕ ( t, y ) = x } and W u ( x ) = { y ∈ M (cid:12)(cid:12) lim t →−∞ ϕ ( t, y ) = x } , These are critical points where the determinant of the Hessian vanishes.
EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 3 respectively. Intuitively, these notions may be visualized in terms of surface topography;the stable manifold, W s ( x ), may be thought of as a dale (where falling rain droplets wouldflow and reach x ) and the unstable manifold, W u ( x ), as a hill (with opposite meaningin terms of water flow). An interesting scientific account on those appeared by Maxwellalready in 1870 [45]. Definition 2.1. [7] Let f be a Morse function.(1) Let p ∈ M − ( f ) , q ∈ M + ( f ), such that W s ( p ) ∩ W u ( q ) (cid:54) = ∅ . Each of theconnected components of W s ( p ) ∩ W u ( q ) is called a Neumann domain of f .(2) The Neumann line set of f is(2.4) N ( f ) := (cid:91) r ∈ S ( f ) W s ( r ) ∪ W u ( r ) . Note that the definition above may be applied to any Morse function and not necessarilyto eigenfunctions. Indeed, some of the results to follow do not depend on f being aneigenfunction. Yet, the spectral theoretic point of view is the one which motivates us toconsider the particular case of Laplacian eigenfunctions.It is not hard to see from basic Morse theory that Neumann domains are two-dimensionalsubsets of M , whereas the Neumann line set is a union of one dimensional curves on M (see appendix). Further properties of Neumann domains and Neumann lines are describedin the next section.Figure 2.1 shows an eigenfunction of the flat torus with its partition to Neumann do-mains.In the above and throughout the paper, we treat only manifolds without boundary,in order to avoid technicalities and ease the reading. It is possible to define Neumanndomains for manifolds with boundary and to prove analogous results for those. Theinterested reader is referred to [7] for such a treatment.3. Topology of Ω and topography of f | Ω Let f be an eigenfunction corresponding to an eigenvalue λ and let Ω be a Neumanndomain. The boundary, ∂ Ω, consists of Neumann lines, which are particular gradient flowlines (see appendix). As the gradient ∇ f is tangential to the Neumann lines we get thatˆ n · ∇ f | ∂ Ω = 0, where ˆ n is normal to ∂ Ω. As a consequence we have
Lemma 3.1. f | Ω is a Neumann eigenfunction of Ω and corresponds to the eigenvalue λ . This lemma is the reason for the name
Neumann domains.Next, we describe the topological properties of a Neumann domain Ω, as well as thetopography of f | Ω . By topography of a function, we mean the information on its levelsets and critical points. Theorem 3.2. [7, Theorem 1.4]
Let f be a Morse function with a non-empty set of saddle points, S ( f ) (cid:54) = ∅ .Let p ∈ M − ( f ) , q ∈ M + ( f ) with W s ( p ) ∩ W u ( q ) (cid:54) = ∅ .Let Ω be a connected component of W s ( p ) ∩ W u ( q ) , i.e., Ω is a Neumann domain.The following properties hold.(1) The Neumann domain Ω is a simply connected open set.(2) All critical points of f belong to the Neumann line set, i.e., C ( f ) ⊂ N ( f ) . LIOR ALON, RAM BAND, MICHAEL BERSUDSKY, SEBASTIAN EGGER
Figure 2.1.
Left: An eigenfunction corresponding to eigenvalue λ = 25of the flat torus whose fundamental domain is [0 , π ] × [0 , π ]. Red (blue)colors indicate positive (negative) values of the eigenfunction. Red (blue)points mark maximum (minimum) points and yellow points mark saddlepoints. The nodal set is drawn in grey and the Neumann line set in purple.The Neumann domains are the domains bounded by the Neumann line set.Right: A magnification of the marked square from the left figure. ThreeNeumann domains are marked by (s), (l) and (w) according to the threedistinguished Neumann domain types described in Section 4.1. (3) The extremal points which belong to Ω are exactly p , q , i.e., X ( f ) ∩ ∂ Ω = { p , q } .(4) If f is a Morse-Smale function then ∂ Ω consists of Neumann lines connectingsaddle points with p or q . In particular, ∂ Ω contains either one or two saddlepoints (see also Proposition A.7).(5) Let c ∈ R . such that f ( p ) < c < f ( q ) . Ω ∩ f − ( c ) is a smooth, non-self intersectingone-dimensional curve in Ω , with boundary points lying on ∂ Ω . This last theorem contains different properties of Neumann domains: claim (1) concernsthe topology, claims (2),(3),(4) the critical points, and claim (5) the level sets. A specialemphasize should be made for the case when f is a Morse function which is also aneigenfunction. For Laplacian eigenfunctions we have that maxima are positive and minimaare negative, i.e., f ( p ) < , f ( q ) >
0, in the notation of the theorem. Hence we maychoose c = 0 in claim (5) above and obtain a characterization of the nodal set which iscontained within a Neumann domain.Figure 3.1 shows the two possible schematic shapes of Neumann domains of a Morse-Smale eigenfunction, as implied from the properties above. We complement the figure bynoting that there exist Morse functions with Neumann domains of type (ii) but numericalexplorations have not revealed any eigenfunction with a Neumann domain of this type.Let us compare the results above with similar properties of nodal domains. Nodaldomains are not necessarily simply connected. On the contrary, it was recently found that See appendix for the definition of a Morse-Smale function.
EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 5
Figure 3.1.
Two possible types of Neumann domains for a Morse-Smaleeigenfunction. Red (blue) discs mark maximum (minimum) points and yel-low discs mark saddle points. The nodal set is drawn in grey.random eigenfunctions may have nodal domains of arbitrarily high genus [52]. Also, therein no upper bound on the number of critical points in a nodal domain. A particular nodaldomain may have either minima or maxima (but not both) in its interior and saddle pointsboth in its interior or at its boundary.4.
Geometry of
Ω4.1.
Angles.
The angles between Neumann lines meeting at critical points are discussedin [46]. The first two parts of the next proposition summarize the content of theorems 3.1and 3.2 in [46] and further generalize their result from the Euclidean case to an arbitrarysmooth metric. The third part of the proposition is new and concern the angles betweenNeumann lines and nodal lines. The proof of the first two parts is almost the same as theone in [46] and we bring it here for completeness.
Proposition 4.1.
Let f be a Morse function on a two dimensional manifold with a smoothRiemannian metric g .(1) Let c be a saddle point of f . Then there are exactly four Neumann lines meetingat c with angles π / .(2) Let c be an extremal point of f whose Hessian is not proportional to g . Then anytwo Neumann lines meet at c with either angle , π , or π / .(3) Further assume that f is a Morse eigenfunction.Let c be an intersection point of a nodal line and a Neumann line of f .If c is a saddle point then the angle between those lines is π / .Otherwise, this angle is π / .Proof. We start by some preliminaries that are relevant to proving all parts of the propo-sition. Let c be an arbitrary critical point of f . We may find a local coordinate system( x, y ) around c , such that c = (0 ,
0) and ∂ x , ∂ y is an orthonormal basis for the tangentspace T c M with respect to the metric g at c . This means, in particular, that in thosecoordinates, g at c is the identity. Thus, we get that the angle between any two vectors, u, v ∈ T c M is given by the usual Euclidean inner product, (cid:104) u, v (cid:105) R .Next, we analyze the Neumann lines which start or end at c . To do that, we keep inmind that Neumann lines are gradient flow lines which start or end at a saddle point (seeappendix), so we first seek for gradient flow lines. Using [10, Lemma 4.4] we deduce thatthe first (matrix-valued) coefficient in the Taylor expansion of ∇ f is Hess f | c . Hence, thegradient flow equations, (2.2), written in this local coordinate system, satisfy(4.1) (cid:18) x (cid:48) ( t ) y (cid:48) ( t ) (cid:19) = − Hess f | c · (cid:18) x ( t ) y ( t ) (cid:19) + O (cid:0) (cid:107) ( x ( t ) , y ( t )) (cid:107) R (cid:1) . LIOR ALON, RAM BAND, MICHAEL BERSUDSKY, SEBASTIAN EGGER
As the Hessian is symmetric, we may diagonalize it by an orthonormal change of thecoordinates and get Hess f | c = (cid:18) α x α y (cid:19) , where α x , α y are both non-zero since f is a Morse function. In those new coordinates, g at c is still the identity. Hence, the assumption in the second part of the proposition, thatthe Hessian is not proportional to g , is equivalent to α x (cid:54) = α y . In the vicinity of c thegradient flow equations, (4.1), may now be approximated by (cid:18) x (cid:48) ( t ) y (cid:48) ( t ) (cid:19) = (cid:18) − α x x ( t ) − α y y ( t ) (cid:19) , where we abuse notation by using ( x, y ) again to denote the new coordinates which diag-onalize the Hessian. The solutions of the above are(4.2) (cid:18) x ( t ) y ( t ) (cid:19) = (cid:18) a x e − α x t a y e − α y t (cid:19) , with a x , a y , t ∈ R . Consider first the case of α x (cid:54) = α y both positive, i.e., c is a minimum point. In this case,all the flow lines (4.2) asymptotically converge to c as t → ∞ . Recall that α x (cid:54) = α y byassumption. This allows to assume without loss of generality that α y > α x >
0. If a x (cid:54) = 0,we get that asymptotically as t → ∞ (cid:18) x ( t ) y ( t ) (cid:19) = e − α x t (cid:18) a x a y e − ( α y − α x ) t (cid:19) ∼ e − α x t (cid:18) a x (cid:19) . Any such flow line is tangential to the ± ˆ x direction at c . This gives a continuous familyof gradient flow lines, some of which are actually also Neumann lines (this depends onwhether or not there is a saddle point at their other end, t → −∞ ). Hence, the possibleangles between any of those Neumann lines at c are either 0 or π . In addition, if a x = 0,we get a gradient flow line which is tangential to the ± ˆ y direction at c . This gradient flowline (which is not necessarily a Neumann line) makes an angle of π / with all others. Thisproves the second part of the proposition if c is a minimum point. The case of a maximumis proven in exactly the same manner.Next we prove the first part of the proposition. If c is a saddle point, then α x , α y areof different signs. The only gradient flow lines, (4.2), which start or end at c are thosefor which either a x = 0 or a y = 0. At c , these lines are either tangential to ˆ x (if a y = 0)or tangential to ˆ y (if a x = 0). These are indeed Neumann lines, as they are connected toa saddle point ( c ). There are four such Neumann lines, corresponding to all possible signchoices ( a x = 0 and a y is positive \ negative or a y = 0 and a x is positive \ negative). Theangles between any neighbouring two lines out of the four is therefore π / .Finally, we prove the third part of the proposition. If c is a critical point, with ∇ f | c = 0,and f ( c ) = 0 then it must be a saddle point, since maxima of a Laplacian eigenfunctionare positive and minima are negative. As f is a Laplace-Beltrami eigenfunction, we get(4.3) 0 = − λf ( c ) = ∆ f ( c ) = traceHess f | c . The sum of Hessian eigenvalues is therefore zero and we may denote those by ± α . Choosinga coordinate system which diagonalizes the Hessian at c = (0 , f ( x, y ) = 12 (cid:0) αx − αy (cid:1) + O (cid:0) (cid:107) ( x ( t ) , y ( t )) (cid:107) R (cid:1) . This shows that the nodal lines of f at c may be approximated by y = ± x . We havealready seen in the previous part of the proof that the Neumann lines which are connected EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 7 to a saddle point, c , are tangential to either the ˆ x or the ˆ y axis and this gives an angle of π / between neighbouring Neumann and nodal lines.If c is not a critical point then ∇ f | c (cid:54) = 0 and we may write d f ( v ) = (cid:104)∇ f | c , v (cid:105) R forevery v ∈ T c M . By taking v in the direction of the nodal line, we get that the anglebetween the Neumann line and the nodal line at c is (cid:104)∇ f | c , v (cid:105) R , as g is the identity at c .Now, since f is constant along the nodal line we have d f ( v ) = 0, and get that the anglebetween the nodal line and the Neumann line is π / . (cid:3) Remark.
It is also stated in [46, theorem 3.1] that an angle of π / between Neumann linesat an extremal point is non-generic (or “unstable special case”, citing [46]). The proof ofthe first part of the proposition clarifies why it is so.The angles between Neumann lines may be observed in Figures 2.1 and 3.1. The exactangles in Figure 2.1 are better seen when zooming in (see right part of the figure).Proposition 4.1 allows to classify Neumann domains to three distinguished types, aswas suggested in [6]. Each Neumann domain has one maxima and one minima on itsboundary. Assume that the Neumann domain is of type ( i ) as depicted in Figure 3.1, i.e.,it does not have an extremal point which is connected only to a single Neumann line. Wecall a Neumann domain • star-like if both angles at its extremal points are 0, • lens-like if both angles at its extremal points are π , • wedge-like if one of those angles is 0 and the other is π .Those three types of domains are indicated in Figure 2.1(Right) by (s), (l), (w), corre-spondingly.Note that this classification requires a couple of genericity assumptions: that the Hessianat the extremal points is not proportional to the metric and that Neumann lines do notmeet perpendicularly at an extremal point (see remark after Proposition 4.1). Indeed, ournumeric explorations reveal that Neumann domains are categorized into those three types[6].4.2. Area to perimeter ratio.Definition 4.2. [29] Let f be a Morse eigenfunction corresponding to the eigenvalue λ and let Ω be a Neumann domain of f . We define the normalized area to perimeter ratioof Ω by ρ (Ω) := | Ω || ∂ Ω | √ λ, with | Ω | being the area of Ω and | ∂ Ω | the total length of its perimeter.This parameter was introduced in [29] in order to quantify the geometry of nodal do-mains. A related quantity, √ | Ω || ∂ Ω | , is a classical one, and it is known to be bounded fromabove by √ π (isoperimetric inequality [30]). The value | Ω || ∂ Ω | has also an interesting geo-metric meaning - it is the mean chord length of the two-dimensional shape Ω. The meanchord length is defined as follows: consider all the parallel chords in a chosen directionand take their average length. The mean chord length is then the uniform average overall directions of that average length . We thank John Hannay for pointing out this interesting geometrical meaning to us.
LIOR ALON, RAM BAND, MICHAEL BERSUDSKY, SEBASTIAN EGGER
There are some numerical explorations, which were performed to study the values of ρ for Neumann domains. In [6] the numerics was done for random eigenfunctions on theflat torus, where the eigenvalues are highly degenerate. More specifically, for a particulareigenvalue, many random eigenfunctions were chosen out of the corresponding eigenspaceand the ρ value was numerically computed for all their Neumann domains. The obtainedprobability distribution of ρ for three different eigenvalues is shown in Figure 4.1,(i).A few interesting observations can be made from those plots. First, it seems that theprobability distribution does not depend on the eigenvalue. Furthermore, in Figure 4.1,(ii)the distribution was drawn separately for each of the three types of Neumann domainsmentioned in the previous subsection (star, lens and wedge). The lens-like domains tend toget higher ρ values, star-like domains get lower values and the wedge-like are intermediate.Another conclusion which may be drawn from these plots is related to the spectral positionof the Neumann domains, which is described in detail in the next section. .
25 0 .
50 0 .
75 1 .
00 1 . ρ . . . . ρλ = 65 λ = 325 λ = 925 .
25 0 .
50 0 .
75 1 .
00 1 . P D F P D F star-likelens-likewedge-like Figure 4.1. (i): A probability distribution function of ρ -values of Neu-mann domains for three different eigenvalues, (ii): A probability distribu-tion function of ρ -values of Neumann domains for λ = 925 for lens-like,wedge-like and star-like domains. The vertical black line marks the value ρ ≈ . . · Neumann domains.We may compare those results with the ones obtained for the distribution of ρ fornodal domains [29]. It is shown in [29] that for nodal domains of separable eigenfunctions π < ρ < π . Furthermore, it is numerically observed there that these bounds are satisfiedwith probability 1 for random eigenfunctions. Also, the calculated probability distributionof ρ for nodal domains looks qualitatively different when comparing to Figure 4.1 (see forexample figures 1,2,6 in [29]).5. Spectral position of
ΩConsider a nodal domain Ξ of some eigenfunction f corresponding to an eigenvalue λ .It is known that f | Ξ is the first eigenfunction (ground-state) of Ξ with Dirichlet boundary EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 9 conditions [25]. Equivalently, λ is the lowest eigenvalue in the Dirichlet spectrum ofΞ. This observation is fundamental in many results concerning nodal domains and theircounting. In this section we consider the analogous statement for Neumann domains. Ourstarting point is Lemma 3.1, according to which an eigenvalue λ appears in the Neumannspectrum of each of its Neumann domains. This allows the following definition. Definition 5.1.
Let f be a Morse eigenfunction of an eigenvalue λ and let Ω be a Neumanndomain of f . We define the spectral position of Ω as the position of λ in the Neumannspectrum of Ω. It is explicitly given by(5.1) N Ω ( λ ) := |{ λ n ∈ Spec(Ω) : λ n < λ }| , where Spec(Ω) := { λ n } ∞ n =0 is the Neumann spectrum of Ω, containing multiple appear-ances of degenerate eigenvalues and including λ = 0. Remark. (1) It can be shown (see [6]) that if Ω is a Neumann domain, then its Neumannspectrum is purely discrete. This makes the above well-defined.(2) If λ is a degenerate eigenvalue of Ω, then by this definition the spectral position isthe lowest position of λ in the spectrum.(3) For any Neumann domain, N Ω ( λ ) >
0. Indeed, N Ω ( λ ) = 0 is possible only for λ = 0, but the zero eigenvalue corresponds to the constant eigenfunction and thisdoes not have Neumann domains at all.A qualitative feeling on the value of N Ω ( λ ) might be given by Theorem 3.2. Thistheorem implies that the topography of f | Ω cannot be too complex; its domain, Ω, issimply connected domain; f | Ω has no critical points in the interior of Ω; and its zero set ismerely a single simple non-intersecting curve. These observations suggest that f | Ω mightnot lie too high in the spectrum of Ω. Such a belief is also apparent in [62], where it iswritten that possibly, the spectral position of Neumann domains ’often’ equals one, justas in the case of nodal domains. Our task is to study the possible values of N Ω ( λ ) forvarious eigenfunctions and their Neumann domains and to investigate to what extent λ is indeed the first non trivial eigenvalue of Ω ( N Ω ( λ ) = 1). We proceed by relating thespectral position and the area to perimeter ratio (Definition 4.2).5.1. Connecting spectral position and area to perimeter ratio.
The spectral po-sition may be used to bound from above the area to perimeter ratio. This holds as thearea to perimeter ratio may be written as ρ (Ω) = (cid:112) | Ω || ∂ Ω | (cid:112) | Ω | λ, where the first factor is bounded from above by the classical geometric isoperimetricinequality √ | Ω || ∂ Ω | ≤ √ π [30], and the second factor is bounded from above by the spectralisoperimetric inequality, once the spectral position is known. We state below the exactresult, whose proof is given in [6]. Proposition 5.2. [6]
Let f be a Morse eigenfunction corresponding to eigenvalue λ . Let Ω be a Neumann domain of f . We have(1) ρ (Ω) ≤ √ N Ω ( λ ) .(2) if N Ω ( λ ) = 1 then ρ (Ω) ≤ j ≈ . (3) if N Ω ( λ ) = 2 then ρ (Ω) ≤ j √ ≈ . ,where j denotes the first zero of the derivative of the J Bessel function.
The bounds above may be used to gather information on the spectral position. Thecalculation of ρ (Ω) is easier (either numerically or sometimes even analytically) than thisof N Ω ( λ ). As an example, we bring the probability distribution of ρ given in Figure 4.1,(i).The distribution was calculated numerically for random eigenfunctions on the torus. Itis easy to observe that a substantial proportion of the Neumann domains have a ρ valuewhich is larger than 0 . ii ). Hence, all thoseNeumann domains have spectral position which is larger than one, N Ω ( λ ) >
1. We notethat those results seem to be independent of the particular eigenvalue, as the ρ distributionitself seem not to depend on the eigenvalue. Those results are somewhat counter-intuitive,due to what is written above (see discussion after Definition 5.1). Furthermore, whencalculating the ρ distribution separately for each of the three different types of Neumanndomains (Figure 4.1,(ii)), the higher ρ values of lens-like domains suggest that the spectralposition of those domains is higher. These results call for some further investigation ofthe spectral position dependence on the shape of the Neumann domains.5.2. Separable eigenfunctions on the torus.
The general problem of analytically de-termining the spectral position is quite involved. Yet, there are some interesting resultsobtained for separable eigenfunctions on the torus, which we review next. We considerthe flat torus with fundamental domain R / Z equipped with the Laplace operator. Theeigenvalues are λ a,b : = π (cid:18) a + 1 b (cid:19) , (5.2)where(5.3) a := 14 m x , b := 14 m y , for m x , m y ∈ N . We consider in the following only the separable eigenfunctions, which may be writtenas(5.4) f a,b ( x, y ) = sin (cid:16) π a x (cid:17) cos (cid:16) π b y (cid:17) . Half of the Neumann domains of this eigenfunction are star-like and congruent to eachother and the other half are lens-like and also congruent (Figure 5.1). We denote those do-mains by Ω star a,b (Figure 5.1(ii)) and Ω lens a,b (Figure 5.1(iii)), respectively, and in the followingwe investigate their spectral position.First, we may consider only the case b ≤ a thanks to the symmetry of the problem.Second, the spectral position of either Ω star a,b or Ω lens a,b depends only on the ratio ba , asrescaling both a and b by the same factor amounts to an appropriate rescaling of theNeumann domain together with the restriction of the eigenfunction to it. The next theoremsummarizes results on the spectral positions of Ω star a,b and Ω lens a,b from [6] and [7]. Theorem 5.3. [6, 7](1) The set of spectral positions of the lens -like domains (cid:110) N Ω lens a,b ( λ a,b ) (cid:111) a,b is un-bounded. In particular, N Ω lens a,b ( λ a,b ) → ∞ for ab → ∞ EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 11
Figure 5.1. (i): Grey lines indicate the nodal set and purple lines indicatethe Neumann set of a torus eigenfunction f ( x, y ) = sin(2 πx ) cos(4 πy ). (ii)and (iii): the star-like and lens-like Neumann domains of a separable eigen-function (5.4), with the typical lengths a, b marked as dashed lines. Saddlepoints are marked by yellow points and extrema by blue and red points.(2) There exists c > ab > c then the spectral position of the star -likedomains is one, i.e., N Ω star a,b ( λ a,b ) = 1. In addition, λ a,b is a simple eigenvalue ofΩ star a,b . Remark.
The condition ab > c in the second part of the theorem is equivalent to thecondition m y m x > c (see (5.3)). As m x , m y ∈ N , this means that the claim in the secondpart of the theorem is valid for a particular proportion of the separable eigenfunctions onthe torus. In particular, combining both parts of the theorem, there is a range of valuesfor a, b for which N Ω star a,b ( λ a,b ) = 1, but N Ω lens a,b ( λ a,b ) is as large as we wish.The proofs of the two parts of this theorem are of different nature. The proof of (1)appears in [7]. It shows by means of contradiction that fixing the value of a and letting b tend to zero the spectral positions { N Ω lens a,b ( λ a,b ) } a,b cannot be bounded. This is done byproving that bounded spectral positions would imply a too rapid growth of the numberof Neumann domains. This contradicts the actual growth of the number of Neumanndomains, which is explicitly known for those eigenfunctions.The proof of (2) appears in [6]. It is based on three main ingredients. The first is thesymmetry of the domain Ω star a,b along a horizontal axis and a vertical axis. The second isa non-standard rearrangement technique using a sector as an intermediate domain [43,44] and the third is the solution of a suitable geometric isoperimetric problem with aconstraint.The motivation which stands behind Theorem 5.3 is the following. As already mentionedabove, it was very natural to believe that generically the spectral position equals one,just as in the case of nodal domains. The first part of the theorem shows that thisbelief is extremely violated in a particular case. The second part somewhat revives thisbelief, by showing that this violation which occurs for half of the Neumann domains issomewhat compensated by the other half. We wonder whether this compensation holdsfor all manifolds. For example, can it be that for any manifold, there exists a constant0 < p ≤
1, such that each eigenfunction would have at least a p proportion of its Neumanndomains with spectral position equals to one? (see Lemma 6.3, where a similar assumptionis employed). Neumann domain count
A wealth of results exists on the number of nodal domains. We start this section bybounding the number of Neumann domains from below in terms of the number of nodaldomains. Denote the number of Neumann domains of some eigenfunction f by µ ( f )and the number of its nodal domains by ν ( f ). Observe that Theorem 3.2,(5) implies thateach Neumann domain intersects with exactly two nodal domains (see discussion followingTheorem 3.2). This allows to conclude. Corollary 6.1. [7](6.1) µ ( f ) ≥ ν ( f ) . Next, we equip the Neumann lines with a graph structure which we call the Neumann setgraph. This allows to provide further estimates on the number of the Neumann domains.Let f be a Morse function on a closed two-dimensional manifold and consider its Neumannset graph obtained by taking the vertices ( V ) to be all critical points, the edges ( E ) arethe Neumann lines connecting critical points and the faces ( F ) are the Neumann domains.Define the valency of a critical point , val ( x ), as the number of Neumann lines which areconnected to x . Proposition 6.2. [7]
We have (6.2) | E | ≤ | S ( f ) | , (6.3) µ ( f ) ≤ | S ( f ) | , where S ( f ) is the set of saddle points of f . If we further assume a Morse-Smale functionwe get equalities in both (6.2) and (6.3). In addition we have µ ( f ) = 12 (cid:88) x ∈ X ( f ) val ( x ) ≥ | X ( f ) | = 12 ( χ ( M ) + | S ( f ) | ) , (6.4) where χ ( M ) is the Euler characteristic of the manifold. The proof of this proposition is done by combining Euler’s formula and Morse inequalitiesfor the Neumann set graph.
The ratio µ n n . The most fundamental result for the nodal domain count is Courant’sbound ν n n ≤
1, where ν n is the nodal count of the n th eigenfunction [25]. Following this,Pleijel had shown that lim sup n →∞ ν n n ≤ (cid:16) j , (cid:17) , where j , is the first zero of the J Bessel function, [48]. Many modern works concern the generalizations or improvements ofPleijel’s result, as well as the distribution of the ratio ν n n [11, 19, 21, 22, 33, 37, 42, 50, 54].The study of the distribution of ν n n was initiated in [19]. This distribution was presentedthere for separable eigenfunctions on the rectangle and the disc. Later, in [33], a moregeneral calculation of the distribution of ν n n was performed. It was done there for theSchr¨odinger operator on separable systems of any dimension.In the following, we consider the analogous quantity, µ n n , the number of Neumann do-mains of the n th eigenfunction divided by n . We start by pointing out the connectionbetween µ n n , and the spectral position. EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 13
Lemma 6.3.
Let ( M, g ) be a two-dimensional, connected, orientable and closed Riemann-ian manifold. Assume that there exists < C ≤ such that (6.5) (cid:88) Ω s.t. N Ω ( λ n )=1 | Ω | > C | M | . for all λ n in the spectrum of M , where the sum above is over all Neumann domains (ofan eigenfunction) of λ n whose spectral position equals one. Then (6.6) lim inf n →∞ µ n n ≥ C (cid:18) j (cid:19) . Proof.
The Szeg¨o-Weinberger inequality [55, 60] is λ (Ω) | Ω | ≤ πj , where j is the firstzero of the derivative of the J Bessel function. Consider an eigenfunction f n of M corre-sponding to an eigenvalue λ n . For each Neumann domain Ω of f n , for which N Ω ( λ n ) = 1,we have λ n = λ (Ω). Combining the Szeg¨o-Weinberger inequality with the assumption inthe lemma gives µ n πj ≥ (cid:88) Ω s.t. N Ω ( λ n )=1 πj ≥ (cid:88) Ω s.t. N Ω ( λ n )=1 λ n | Ω | > Cλ n | M | . Applying Weyl asymptotics [61] we get (6.6). (cid:3)
Such a result is interesting since it shows that the Neumann count tends to infinity.Similar problems are investigated for the nodal count. It was asked a few years ago byHoffmann-Ostenhof whether lim sup n →∞ ν n = ∞ holds for any manifold [59]. Followingthis, Ghosh, Reznikov and Sarnak proved that the number of nodal domains of Maassforms tends to infinity with the eigenvalue [32]. Shortly afterwards, Jung and Zelditchhave shown that for negatively curved compact surfaces with some orientation-reversingisometric involution, the number of nodal domains tends to infinity for a density onesub-sequence of the eigenfunctions [40]. Sequentially, they improved upon this result byshowing the same asymptotics for non-positively surfaces without the need of an involution[41]. The most recent result is by Zelditch who provided a logarithmic lower bound forthe nodal count of eigenfunctions on the first class of manifolds mentioned above [63].The validity of the inequality (6.6) (and hence the validity of the assumption (6.5)) maybe checked by investigating the distribution of µ n n , which is our next task.We consider the separable eigenfunctions of the flat torus T with fundamental domain R / Z . For those eigenfunctions we calculate the limiting probability distribution of µ n n .Given a couple of natural numbers m x , m y ∈ N , we have that(6.7) f m x ,m y ( x, y ) = sin (2 πm x x ) cos (2 πm y y ) , is a separable eigenfunction of the following eigenvalue(6.8) λ m x ,m y := 4 π (cid:0) m x + m y (cid:1) , (as in (5.2),(5.4)). Note that the functions cos (2 πm x x ) cos (2 πm y y ), cos (2 πm x x ) sin (2 πm y y ),sin (2 πm x x ) sin (2 πm y y ) together with (6.7) are linearly independent eigenfunctions whichbelong to the eigenvalue (6.8). The set of all those separable eigenfunctions for all possiblevalues of m x , m y ∈ N form an orthogonal complete set of eigenfunctions on T .We further note that the four eigenfunctions above which correspond to a particulareigenvalue λ m x ,m y are equal on the torus up to a translation. Hence, all four have the same number of Neumann domains as f m x ,m y and we denote this number by µ m x ,m y . Withthis we may define the following cumulative distribution function(6.9) F λ ( c ) := 4 N T ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( m x , m y ) ∈ N : λ m x ,m y < λ , µ m x ,m y N T ( λ m x ,m y ) < c (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) , where N T ( λ ) is the spectral position of λ in the torus T , as in (5.1), and the factor 4 standsfor the four eigenfunctions which correspond to λ m x ,m y . In words, F λ ( c ) is the proportionof the separable eigenfunctions with eigenvalue less than λ , whose normalized Neumanncount is smaller than c . Its limiting distribution is given by the following. Proposition 6.4.
For c < π (6.10) lim λ →∞ F λ ( c ) = 12 (cid:90) c (cid:112) − ( π x ) dx and for c ≥ π lim λ →∞ F λ ( c ) = 1 . Proof.
The proof consists of a reduction to a lattice counting problem, which allows toderive the limiting distribution. First, observe that the number of Neumann domains of f m x ,m y is µ m x ,m y = 8 m x m y . This holds since f m x ,m y is Morse-Smale, so that there is anequality in (6.3), and the number of saddle points of f m x ,m y is the number nodal crossingswhich is easily shown to be 4 m x m y . The symmetry between m x and m y in the expressionfor µ m x ,m y motivate us to define the set W := (cid:8) ( m x , m y ) ∈ N : m x < m y (cid:9) , and observe ∀ λ (cid:12)(cid:12)(cid:8) ( m x , m y ) ∈ N : λ m x ,m y < λ (cid:9)(cid:12)(cid:12) =2 (cid:12)(cid:12)(cid:8) ( m x , m y ) ∈ W : λ m x ,m y < λ (cid:9)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:8) ( m x , m y ) ∈ N : m x = m y and λ m x ,m y < λ (cid:9)(cid:12)(cid:12) (6.11)Plugging (6.11) in (6.9) and taking the limit λ → ∞ gives(6.12) lim λ →∞ F λ ( c ) = lim λ →∞ N T ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( m x , m y ) ∈ W : λ m x ,m y < λ , µ m x ,m y N T ( λ m x ,m y ) < c (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) , where we use the Weyl asymptotics, lim λ →∞ N T ( λ ) = λ π [61] and that the second term inthe right hand side of (6.11) grows like √ λ and hence drops when taking the limit.We analyze (6.12) geometrically. First, N T ( λ ) counts the number of Z points withnon-zero coordinates, that lie inside a disc of radius √ λ around the origin. Hence, it maybe written as(6.13) N T ( λ m x ,m y ) = π ( m x + m y ) + Err ( m x + m y ) , where Err ( m x + m y ) = o ( m x + m y ) [39]. In addition, the point ( m x , m y ) ∈ W may becharacterized by the angle it makes with the x -axis, i.e., m y m x = tan θ m x, ,m y , so that(6.14) 2 m x m y m x + m y = 2 cos θ m x, ,m y · sin θ m x, ,m y = sin 2 θ m x, ,m y . EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 15
With (6.13) and (6.14) we may write µ m x ,m y N T ( λ m x ,m y ) = 8 m x m y π ( m x + m y ) (cid:0) Err ( m x + m y ) /π ( m x + m y ) (cid:1) = 1 (cid:0) Err ( m x + m y ) /π ( m x + m y ) (cid:1) π · sin 2 θ m x, ,m y . Let ε >
0. Since
Err ( m x + m y ) = o ( m x + m y ), there exists Λ > m x , m y ) ∈ W satisfying 4 π (cid:0) m x + m y (cid:1) > Λ, the following holds(6.15) 11 + ε π sin 2 θ m x, ,m y < µ m x ,m y N T ( λ m x ,m y ) < − ε π sin 2 θ m x, ,m y . The limiting cumulative distribution (6.12) may be slightly rewritten aslim λ →∞ F λ ( c ) = lim λ →∞ N T ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( m x , m y ) ∈ W : Λ < λ m x ,m y < λ , µ m x ,m y N T ( λ m x ,m y ) < c (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) , where the additional condition Λ < λ m x ,m y removes only a finite number of points from theset and does not affect the limit. We may now use (6.15) to get the following inequalitiesby set inclusionlim λ →∞ F λ ( c ) ≤ (6.16)lim λ →∞ N T ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( m x , m y ) ∈ W : Λ < λ m x ,m y < λ and θ m x, ,m y <
12 arcsin (cid:18) πc (1 + ε )4 (cid:19)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) , andlim λ →∞ F λ ( c ) ≥ (6.17)lim λ →∞ N T ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( m x , m y ) ∈ W : Λ < λ m x ,m y < λ and θ m x, ,m y <
12 arcsin (cid:18) πc (1 − ε )4 (cid:19)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) , where in the above we assume that 0 ≤ c < π and ε is small enough so that πc (1+ ε ) / ≤ πc (1 + ε ) /
4) is well defined.We notice that the right hand sides of (6.16) and (6.17) correspond to counting integerlattice points which are contained within a certain sector. This number of points growslike the area of the corresponding sector [39], i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( m x , m y ) ∈ W : Λ < λ m x ,m y < λ , θ m x ,n y <
12 arcsin (cid:18) πc (1 ± ε )4 (cid:19)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) == 14 arcsin (cid:18) πc (1 ± ε )4 (cid:19) λ − Λ4 π (cid:124) (cid:123)(cid:122) (cid:125) area of a sector + o ( λ ) . (6.18)Plugging (6.18) in the bounds (6.16),(6.17) and using (6.13) gives2 π arcsin (cid:18) πc (1 − ε )4 (cid:19) ≤ lim λ →∞ F λ ( c ) ≤ π arcsin (cid:18) πc (1 + ε )4 (cid:19) . As ε > ∀ c < π lim λ →∞ F λ ( c ) = 2 π arcsin (cid:16) πc (cid:17) , = 12 (cid:90) c (cid:112) − ( π x ) dx, which proves (6.10). Finally note that we have lim c → π lim λ →∞ F λ ( c ) = 1, and as F λ ( c ) isa cumulative distribution function we get lim λ →∞ F λ ( c ) = 1 for c ≥ π . (cid:3) Remark.
The calculation in the proof above may be considered as a particular case ofthose done in [33]. The proof here is explicitly tailored for the purpose of the currentpaper.The next figure shows the probability distribution given in (6.10) and compares it to anumerical examination of the probability distribution of µ n n for the separable eigenfunctionson the torus. Figure 6.1.
Orange curve : the probability distribution of µ n n as given in(6.10). Blue curve : a numerical calculation of this distribution as calculatedfor the first 2 · torus eigenfunctions.Examining the µ n n distribution leads to the following. First, we note that µ n n may getarbitrarily low values for a positive proportion of the eigenfunctions. This is in contra-diction with (6.6) and therefore we conclude that the separable eigenfunctions on the flattorus do not satisfy assumption (6.5) in Lemma 6.3. Indeed, it can be checked directly inthis case that the total area of all star domains goes to zero as the eigenvalue λ a,b tendsto infinity. Therefore their total area does not satisfy (6.5).Turning our attention to the higher values of µ n n we notice that µ n n > µ n ≤ h ( n ) exists, with h being possibly a linear function. If so, a possible upper boundmight be suggested by the separable eigenfunctions, for which h ( n ) = π n , serves as a EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 17 bound. Furthermore, the Courant-like bound on the Neumann count for metric graphs(11.2), to be discussed in the next part, is a suggestive evidence for the existence of aCourant-like bound in the case of manifolds . Part Neumann domains on metric graphs Definitions
Discrete graphs and graph topologies.
We denote by Γ = ( V , E ) a connectedgraph with finite sets of vertices V and edges E . We allow the graph edges to connecteither two distinct vertices or a vertex to itself. In the latter case, such an edge is calleda loop.For a vertex v ∈ V , its degree, d v , equals the number of edges connected to it. The setof graph vertices of degree one turns out to be useful and we denote it by ∂ Γ := { v ∈ V : d v = 1 } . We call the vertices in ∂ Γ, boundary vertices and the rest of the vertices, V\ ∂ Γ, are called interior vertices .An important topological quantity of graphs is the first Betti number (dimension of thefirst homology group) given, for a connected graph, by(7.1) β := |E | − |V| + 1 . The value of β is the number cycles needed to span the space of cycles on the graph. Bydefinition a graph is simply connected when β = 0, and such a graph is called a tree graph.Two particular examples of trees are star graphs and path graphs. A star graph is a graphwith one interior vertex which is connected by edges to the other |V|− |V| − Spectral theory of metric graphs. A metric graph is a discrete graph for whicheach edge, e ∈ E , is identified with a one-dimensional interval [0 , L e ] of a positive finitelength L e . We assign to each edge e ∈ E a coordinate, x e , which measures the distancealong the edge from one of the two boundary vertices of e .A function on the graph is described by its restrictions to the edges, { f | e } e ∈E , where f | e : [0 , L e ] → C . We equip the metric graphs with a self-adjoint differential operator,(7.2) − ∆ : f | e ( x e ) (cid:55)→ − d d x e f | e ( x e ) , While writing this manuscript we became aware of a work in progress by Buhovski, Logunov, Nazarovand M. Sodin, which might disprove the existence of such a bound. which is the Laplacian . It is most common to call this setting of a metric graph and anoperator by the name quantum graph.To complete the definition of the operator we need to specify its domain. We considerfunctions which belong to the following direct sum of Sobolev spaces(7.3) H (Γ) := (cid:77) e ∈E H ([0 , L e ]) . In addition we require some matching conditions on the graph vertices. A function f ∈ H (Γ) is said to satisfy the Neumann vertex conditions at a vertex v if(1) f is continuous at v ∈ V , i.e.,(7.4) ∀ e , e ∈ E v f | e (0) = f | e (0) , where E v is the set of edges connected to v , and for all e ∈ E v , x e = 0 at v .(2) The outgoing derivatives of f at v satisfy(7.5) (cid:88) e ∈E v d f d x e (cid:12)(cid:12)(cid:12)(cid:12) e (0) = 0 . Requiring these conditions at each vertex leads to the operator (7.2) being self-adjoint andits spectrum being real and bounded from below [14]. In addition, since we only considercompact graphs, the spectrum is discrete. We number the eigenvalues in the ascendingorder and denote them by { λ n } ∞ n =0 and their corresponding eigenfunctions by { f n } ∞ n =0 . Asthe operator is both real and self-adjoint, we may choose the eigenfunctions to be real,which we will always do.In this paper, we only consider graphs whose vertex conditions are Neumann at allvertices, and call those standard graphs. A special attention should be given to vertices ofdegree two. Introducing such a vertex at the interior of an existing edge (thus splittingthis edge into two) and requiring Neumann conditions at this vertex does not change theeigenvalues and eigenfunctions of the graph. The same holds when removing a degreetwo vertex and uniting two existing edges into one. This spectral invariance allows usto assume in the following that standard graphs do not have any vertices of degree two.Furthermore, the only graph, all of whose vertices are of degree two (or equivalently hasno vertices at all) is the single loop graph. We assume throughout the paper that ourgraphs are different than the single loop graph and call those nontrivial graphs .The spectrum of a standard graph is non-negative, which means that we may representthe spectrum by the non-negative square roots of the eigenvalues, k n = √ λ n . For conve-nience we abuse terminology and call also { k n } ∞ n =0 the eigenvalues of the graph. Most ofthe results and proofs in this part are expressed in terms of those eigenvalues. A Neumanngraph has k = 0 with multiplicity which equals the number of graph components. Thecommon convention is that if an eigenvalue is degenerate (i.e. non simple) it appears morethan once in the sequence { k n } ∞ n =0 . For any such degenerate eigenvalue, we pick a basis forits eigenspace and all members of this basis appear in the sequence { f n } ∞ n =0 . Obviously,this makes the choice of the sequence { f n } ∞ n =0 non unique. It is important to note that allthe statements to follow hold for any choice of { f n } ∞ n =0 . More general operators appear in the literature. See for example [14, 34].
EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 19
Neumann points and Neumann domains.
For metric graphs, the nodal pointset of a function is the set of points at which the function vanishes. Removing the nodalpoint set from the graph, splits it into connected components and those are called nodaldomains. The Neumann set and Neumann domains are similarly defined, but before doingso we need to restrict to particular classes of functions.
Definition 7.1.
Let Γ be a nontrivial standard graph and f be an eigenfunction of Γ.(1) We call f a Morse eigenfunction if for each edge e , f | e is a Morse function. Namely,at no point in the interior of e both the first and the second derivatives of f vanish.(2) We call an eigenfunction f generic if it is a Morse eigenfunction and in additionsatisfies all of the following:(a) f corresponds to a simple eigenvalue.(b) f does not vanish at any vertex.(c) f has no extremal points at interior vertices.An equivalent characterization of a Morse eigenfunction is Lemma 7.2.
Let f be a non-constant eigenfunction. f is Morse if and only if there existsno edge e such that f | e ≡ .Proof. First, observe that a non-constant eigenfunction of the Laplacian vanishes at aninterior point of an edge if and only if the second derivative vanishes at that point. There-fore, if f is a Morse eigenfunction then there is no interior point at which both the functionand its derivative vanish. This means that a Morse eigenfunction cannot vanish entirelyat a graph edge. As for the converse, if f is a non-Morse eigenfunction then there exists x , an interior point of an edge e , such that f | (cid:48) e ( x ) = f | (cid:48)(cid:48) e ( x ) = 0. By the same argumentas above, this means that either f | e ( x ) = 0 or f | e is the constant eigenfunction. The van-ishing of f | e and its first derivative at the same point, together with f | e being a solutionof an ordinary differential equation of second order implies f | e ≡ (cid:3) We complement this lemma and note that the constant eigenfunction, correspondingto k = 0 is not a Morse function. This, together with the lemma, implies that a Morseeigenfunction may vanish only at isolated points of the graph; the same holds for itsderivative. This quality allows the following. Definition 7.3.
Let f be a Morse eigenfunction.(1) A Neumann point of f is an extremal point (maximum or minimum) not locatedat a boundary vertex. Namely, the set of Neumann points is(7.6) N ( f ) := { x ∈ Γ \ ∂ Γ : x is an extremal point of f } . Note that we reuse here the notation for the Neumann lines in the manifold case,(2.4).(2) A Neumann domain of f is a closure of a connected component of Γ \N ( f ). Theclosure is done by adding vertices of degree one at the open endpoints of theconnected component.Figure 7.1 shows the Neumann point and Neumann domains of a particular eigenfunction. Remark. (1) The definition implies that a Neumann point is either a point x ∈ Γ \V at someinterior of an edge such that f (cid:48) ( x ) = 0, or it is a vertex v ∈ V such that all outgoing (i) (ii) (iii) Figure 7.1. (i) A graph Γ (ii) An eigenfunction f of Γ, with its single Neu-mann point marked (iii) A decomposition of Γ into the Neumann domainsof f .derivatives of f at that vertex vanish. The latter possibility does not occur if f isgeneric.(2) From the proof of Lemma 7.2 we learn that no point can be both a nodal pointand a Neumann point.All the results to follow concerning Neumann points and Neumann domains are statedfor either Morse or generic eigenfunctions. We start by stating what proportion of theeigenfunctions are Morse and which proportion of the Morse ones are generic. In orderto do so, we need to assume that the set of edge lengths is linearly independent over thefield Q . We call such lengths rationally independent and we will assume this for the graphedge lengths in some of the propositions to follow. Proposition 7.4. [2, 3]
Let Γ be a nontrivial standard graph, with rationally independentedge lengths { L e } e ∈E . Let { f n } ∞ n =0 be a complete set of eigenfunctions of Γ .(1) The proportion of Morse eigenfunctions is given by (7.7) d (cid:126)L := lim N →∞ |{ n ≤ N : f n is Morse }| N = 1 − (cid:80) e ∈E L L e (cid:80) e ∈E L e , where E is the set of graph edges and E L is a subset of E consisting of all edgeswhich form loops (edges which connect a vertex to itself ).(2) The proportion of generic eigenfunctions out of the Morse ones is (7.8) lim N →∞ |{ n ≤ N : f n is generic }||{ n ≤ N : f n is Morse }| = 1 d (cid:126)L lim N →∞ |{ n ≤ N : f n is generic }| N = 1 . Namely, almost all Morse eigenfunctions are generic.Remark. (1) The limits in (7.7) and (7.8) exist even without assuming that the edge lengths arerationally independent. This assumption is needed to obtain the exact values ofthose limits.(2) From the proposition we get that at least half of the eigenfunctions are Morse, andif a graph has no loops, almost all eigenfunctions are Morse and generic.(3) The proof of (7.7) is similar to the proof of proposition A.1 in [3]. The proof of(7.8) appears in [2].8.
Topology of Ω and topography of f | Ω Let Γ be a nontrivial standard graph and f an eigenfunction of Γ corresponding to theeigenvalue k . Formally, every Neumann domain Ω of f may be considered as a subgraphof Γ, if we add degree two vertices to Γ at all the Neumann points of f (see discussion EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 21 on those vertices in Section 7.2). In particular, a Neumann domain is a closed set (byDefinition 7.3). This difference from the manifold case (where Neumann domains are opensets) is technical and serves our need to consider Ω as a metric graph on its own. Being ametric graph, we take the usual Laplacian on Ω and impose Neumann vertex conditionsat all of its vertices, so that Ω is considered as a standard graph. Note that the restrictionof f | Ω to the edges of Ω trivially satisfies f (cid:48)(cid:48) = − k f . It also obeys Neumann vertexconditions at all vertices of Ω, as each vertex is either a vertex of Γ or a point x ∈ Γ inan interior of an edge for which f (cid:48) ( x ) = 0. This gives the following, which is analogous toLemma 3.1. Lemma 8.1. f | Ω is an eigenfunction of the standard graph Ω and corresponds to theeigenvalue k .Remark. Furthermore, it can be proved that if f is a generic eigenfunction and Ω is a treegraph then f | Ω is also generic [2].8.1. Possible topologies for Neumann domains.
In this subsection we discuss whichgraphs may be obtained as a Neumann domain. The next lemma shows that if we consideran eigenfunction, f , whose eigenvalue is high enough, each of its Neumann domains iseither a path graph or a star graph. A star Neumann domain contains an interior vertexof the graph, and a path Neumann domain is contained in a single edge of the graph (seeFigure 8.1).(i) (ii) Figure 8.1. (i) A graph Γ with Neumann points (in purple) of a giveneigenfunction (ii) The decomposition of the graph to the corresponding Neu-mann domains.
Lemma 8.2.
Let Γ be a nontrivial standard graph. Let f be an eigenfunction correspond-ing to an eigenvalue k > πL min , where L min is the minimal edge length of Γ . Let Ω be aNeumann domain of f . (1) If Ω contains a vertex v ∈ V of degree d v > star graph with deg ( v )edges.(2) If Ω does not contain a vertex v ∈ V of degree d v > path graph, oflength πk .Proof. For any edge e ∈ E we have that f | e ( x ) = B e cos ( kx + ϕ e ), where B e , ϕ e are someedge dependent real parameters. This together with k > πL min implies that the derivativeof f vanishes at least once at the interior of each edge. Hence, the set of Neumann points, N ( f ) contains at least one point on each edge. It follows that each Neumann domaincontains at most one vertex of Γ. Thus, there are two types of Neumann domains: if aNeumann domain, Ω, contains a vertex with deg( v ) > star graph , whosenumber of edges is d v ; otherwise Ω is a path graph . A Neumann domain which is a path graph can be parameterized as Ω = [0 , l ]. Since f (cid:48) (0) = 0 we get that f | Ω ( x ) = cos ( kx )up to a multiplicative constant . Using f (cid:48) ( l ) = 0 and that f (cid:48) does not vanish in the interiorof Ω we conclude l = πk . (cid:3) Remark.
Only finitely many eigenvalues do not satisfy the condition k > πL min in thelemma. The number of those eigenvalues is bounded by (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) n ∈ N : 0 ≤ k n ≤ πL min (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | Γ | L min , where | Γ | = (cid:80) e ∈E L e is the total sum of all edge lengths of Γ. This can be shown using(8.1) ∀ n ∈ N , k n ≥ π | Γ | ( n + 1) , which is the statement of Theorem 1 in [31].To complement the lemma above, we note that there are also Neumann domains which arenot simply connected. Indeed, consider the graph Γ depicted in Figure 8.2(i). It has aneigenfunction with no Neumann points, so that the eigenfunction has a single Neumanndomain which is the whole of Γ and in particular, it is not simply connected (Figure8.2(ii)).(i) l l l l (ii) l l l l Figure 8.2. (i) A graph Γ with (ii) An eigenfunction whose single Neu-mann domain is not simply connected.8.2.
Critical points and nodal points - number and position.
In the following weconsider the critical points and nodal points of f | Ω . Note that, by definition, a Morsefunction on a one dimensional interval cannot have a saddle point. Hence, all criticalpoints of a Morse eigenfunction of a graph are extremal points. We reuse the notationsfrom the manifold part: X ( f ) for extremal points of f and M + ( f ) ( M − ( f )) for maxima(minima). Denote by φ ( f | Ω ) the number of nodal points of f | Ω , by E Ω the number ofedges of Ω, by V Ω the number of its vertices, and by ∂ Ω the vertices of Ω which are ofdegree one.
Proposition 8.3. [2]
Let f be a generic eigenfunction and Ω a Neumann domain of f .Then (1) The extremal points of f | Ω , which are located on Ω are exactly the boundary ofΩ, i.e., X ( f ) ∩ Ω = ∂ Ω(2) 1 ≤ | M + ( f ) ∩ ∂ Ω | ≤ | ∂ Ω | − | M − ( f ) ∩ ∂ Ω | ).(3) 1 ≤ φ ( f | Ω ) ≤ E Ω − V Ω + | ∂ Ω | . EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 23
Remark.
Note that when Ω is a path graph the proposition implies that it has exactly onemaximum, one minimum and one nodal point. Also, when Ω is a tree graph, the last partof the proposition gives 1 ≤ φ ( f | Ω ) ≤ | ∂ Ω | − Geometry of
ΩSimilarly to the manifold case we use the normalized area to perimeter ratio to quantifythe geometry of a Neumann domain. The following is to be compared with Definition 4.2.
Definition 9.1.
Let f be a Morse eigenfunction corresponding to the eigenvalue k . LetΩ be a Neumann domain of f , whose edge lengths are { l j } E Ω j =1 . We define the normalizedarea to perimeter ratio of Ω to be ρ (Ω) := | Ω || ∂ Ω | k, where | Ω | = (cid:80) E Ω j =1 l j and | ∂ Ω | is the number of boundary vertices of Ω.For graphs we are able to obtain global bounds on ρ (Ω). Proposition 9.2. [2]
Let Ω be a Neumann domain. We have (9.1) 1 | ∂ Ω | ≤ ρ (Ω) π ≤ E Ω | ∂ Ω | . If Ω is a star graph then we have a better upper bound ρ (Ω) π ≤ − | ∂ Ω | .If Ω is a path graph then ρ (Ω) = π . Next, we study the probability distribution of ρ . We find that for this purpose, it isuseful to consider separately only the Neumann domains containing a particular vertex.Let Γ be a nontrivial standard graph and let f n be its n th eigenfunction. Assume that f n is generic. Then, for any vertex v ∈ V there is a unique Neumann domain of f n whichcontains v and we denote it by Ω ( v ) n . Proposition 9.3. [2]
Let v ∈ V of degree d v > . The value of π ρ on { Ω ( v ) n } ∞ n =1 isdistributed according to (9.2) lim N →∞ (cid:12)(cid:12)(cid:12)(cid:110) n ≤ N : f n is generic and π ρ (cid:16) Ω ( v ) n (cid:17) ∈ ( a, b ) (cid:111)(cid:12)(cid:12)(cid:12) |{ n ≤ N : f n is generic }| = (cid:90) ba ζ ( v ) ( x ) d x, where ζ ( v ) is a probability distribution supported on [ d v , − d v ] .Furthermore, it is symmetric around , i.e. ζ ( v ) ( x ) = ζ ( v ) (1 − x ) .Remark. If d v = 1 then Ω ( v ) n is a path graph for all n , so that by Proposition 9.2 we getthat ζ ( v ) is a Dirac measure ζ ( v ) ( x ) = δ (cid:0) x − (cid:1) .As is implied by choice of notation, the distribution ζ ( v ) indeed depends on the particularvertex v ∈ V . We demonstrate this in Figure 9.1,(iii) where we compare between the prob-ability distributions of two vertices of different degrees from the same graph. In addition,Figure 9.1,(vi) shows a comparison between the probability distributions of two verticesof the same degree from different graphs. The numerics suggest that the distributions aredifferent, which implies that ζ ( v ) may depend on the graph connectivity and not only onthe degree of the vertex. It is of interest to further investigate this distribution, ζ ( v ) , andin particular its dependence on the graph’s properties. (i) Γ v u (ii) Γ w (iii)(iv) Figure 9.1. (i) Γ , with vertices v, u of degrees 5 ,
3, correspondingly. (ii)Γ , with vertex w of degree 5. (iii) A probability distribution function of ρπ -values for the Γ Neumann domains which contain v (i.e., ζ ( v ) in (9.2))compared with ζ ( u ) . (iv) Similarly, ζ ( v ) compared with ζ ( w ) .All the numerical data was calculated for the first 10 eigenfunctions andfor a choice of rationally independent lengths.10. Spectral position of
ΩBy Lemma 8.1, a graph eigenvalue k appears in the spectrum of each of its Neumanndomains. Exactly as in Definition 5.1 for manifolds, we define the spectral position of aNeumann domain Ω, as the position of k in the spectrum of Ω and denote it by N Ω ( k ).Also, as in the manifold case, we have that N Ω ( k ) ≥ f | Ω . EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 25
Lemma 10.1. [2]
Let Γ be a nontrivial standard graph, f be a generic eigenfunction of Γ corresponding to an eigenvalue k and let Ω be a Neumann domain of f , which is a treegraph. Then(1) N Ω ( k ) = φ ( f | Ω ) .(2) N Ω ( k ) ≤ | ∂ Ω | − .In particular if Ω is a path graph then N Ω ( k ) = 1 . The statement in (1) was proven in [12, 49, 53] under the assumption that f | Ω is generic.This is indeed the case since f itself is generic and Ω is a tree graph (see remark afterLemma 8.1). The statement in (2) follows as a combination of (1) with Proposition 8.3,(3).We further remark on the applicability of the lemma above; it applies for almost allNeumann domains. Indeed, for any given graph, all Neumann domains except finitelymany are star graphs or path graphs (by Lemma 8.2), and those are particular cases oftree graphs.Next, we show that the value of the spectral position implies bounds on the value of ρ ,just as we had for manifolds (Proposition 5.2). For manifolds we got upper bounds on ρ ,whereas for graphs we get bounds from both sides. Proposition 10.2. [2]
Let Γ be a nontrivial standard graph, f be an eigenfunction of Γ corresponding to an eigenvalue k and let Ω be a Neumann domain of f . Then (10.1) ρ (Ω) π ≥ | ∂ Ω | (cid:18) N Ω ( k ) + 12 (cid:19) . If Ω is a star graph then we further have the upper bound (10.2) ρ (Ω) π ≤
12 + 1 | ∂ Ω | (cid:18) N Ω ( k ) − (cid:19) . Remark.
Note that if N Ω ( λ ) > N Ω ( λ ) < | ∂ Ω | −
1, then the bound(10.2) improves the upper bound given in Proposition 9.2 for star graphs.Next, we show that the spectral position has a well-defined probability distribution. Asin the previous section (Proposition 9.2), we find that this distribution is best describedwhen one focuses on Neumann domains containing a particular graph vertex.
Proposition 10.3. [2]
Let v ∈ V of degree d v . We have that the spectral position proba-bility, (10.3) P ( N Ω ( v ) = j ) := lim N →∞ (cid:12)(cid:12)(cid:12)(cid:110) n ≤ N : f n is generic and N Ω ( v ) n ( k n ) = j (cid:111)(cid:12)(cid:12)(cid:12) |{ n ≤ N : f n is generic }| is well defined. If we further assume that d v > then(1) P ( N Ω ( v ) = j ) is supported in the set j ∈ { , ..., d v − } .(2) P ( N Ω ( v ) = j ) is symmetric around d v , i.e., P ( N Ω ( v ) = j ) = P ( N Ω ( v ) = d v − j ) .If d v = 1 then P ( N Ω ( v ) = j ) = δ j, . By the proposition the support and the symmetry of the spectral position probabilitydepend on the degree of the vertex. Yet, vertices of the same degree, but from dif-ferent graphs may have different probability distributions as is demonstrated in Figure ρ (Ω) depends on the value of the spectral position N Ω (compare with thebounds (10.1),(10.2)).(i) Γ v u (ii) Γ w (iii)(iv) Figure 10.1. (i) Γ , with vertex v of degree 5. (ii) Γ , with vertex w ofdegree 5. (iii) The spectral position probability P ( N Ω ( v ) = j ) for v of Γ compared with P ( N Ω ( w ) = j ) for w of Γ . (iv) A probability distributionfunction of ρπ -values for the Γ Neumann domains which contain v , condi-tioned on the value of the spectral position N Ω ( v ) n .All the numerical data was calculated for the first 10 eigenfunctions for achoice of rationally independent lengths.11. Neumann count
In this section we present bounds on the number of Neumann points and provide someproperties of the probability distribution of this number.
EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 27
Definition 11.1.
Let Γ be a nontrivial standard graph and { f n } ∞ n =0 a complete set of itseigenfunctions. Denote by µ n := µ ( f n ) and φ n := φ ( f n ) the numbers of Neumann pointsand nodal points respectively. We call the sequences { µ n } , { φ n } the Neumann count andnodal count, and the normalized quantities ω n := µ n − n, σ n := φ n − n are called theNeumann surplus and nodal surplus. Proposition 11.2. [2]
Let Γ be a nontrivial standard graph. Let f n be the n th eigenfunctionof Γ and assume it is generic. We have the following bounds: (11.1) 1 − β ≤ σ n − ω n ≤ β − | ∂ Γ | , and (11.2) 1 − β − | ∂ Γ | ≤ ω n ≤ β − , where β = |E | − |V| + 1 is the first Betti number of Γ . Moreover, both quantities σ n − ω n and ω n have well defined probability distributions,as stated in what follows. Proposition 11.3. [2] (1) The difference between the Neumann and nodal surplus has a well defined probabilitydistribution given by (11.3) P ( σ − ω = j ) = lim N →∞ |{ n ≤ N : f n is generic and σ n − ω n = j }||{ n ≤ N : f n is generic }| . Furthermore, it is symmetric around | ∂ Γ | ,i.e., P ( σ − ω = j ) = P ( σ − ω = | ∂ Γ | − j ) . (2) The Neumann surplus has a well defined probability distribution which is symmetricaround ( β − | ∂ Γ | ) . This proposition is in the spirit of the recently obtained result for the distribution ofthe nodal surplus [3]. It was shown in [3] that the nodal surplus, σ , has a well definedprobability distribution which is symmetric around β . The proof of Proposition 11.3 usessimilar techniques to the proof of this latter result and appears in [2].The proposition above also has an interesting meaning in terms of inverse problems.It is common to ask what one can deduce on a graph out of its nodal count sequence, { φ n } [8, 9, 47]. It was found in [4] that the nodal count distinguishes tree graphs fromothers. The result already mentioned in [3] took a step further by showing that the nodalsurplus distribution reveals the graph’s first Betti number, as twice the expected value ofthe nodal surplus. However, it should be noted that all tree graphs have the same nodalcount, so that one cannot distinguish between different trees in terms of the nodal count.Proposition 11.3 shows that the Neumann count, { µ n } contains information on the size ofthe graph’s boundary, | ∂ Γ | . In particular, this enables the distinction between some treegraphs, which was not possible before.Different tree graphs with the same boundary size, | ∂ Γ | , have the same expected valuefor their Neumann count and are not distinguishable in this sense. Nevertheless, wemay wonder whether the boundary size of a tree graph fully determines the probabilitydistribution of its Neumann count. We do not have an answer to this question yet andcarry on this exploration. We end this section by noting that the bounds obtained in (11.2) on the Neumannsurplus ω n seem not to be strict, as we observed in many examples. Furthermore, weconjecture the following sharper bounds on ω n . Conjecture 11.4.
The Neumann surplus is bounded by − − | ∂ Γ | ≤ ω n ≤ β + 1 . Proving the bounds (11.2) on ω n is done by combining the bounds on σ n − ω n (11.1)with the bounds 0 ≤ σ n ≤ β [17]. The bounds on both σ n − ω n and σ n are known to bestrict. Hence, if indeed the bounds on ω n are not strict, it implies that the nodal surplus, σ n , and the Neumann surplus, ω n , are correlated when considered as random variables,which is an interesting result on its own. Part Summary
In this part we summarize the main results of this paper and focus on the comparisonbetween analogous statements on graphs and manifolds. This is emphasized by usingcommon terminology and notations for both graphs and manifolds.Let f be an eigenfunction corresponding to the eigenvalue λ and Ω be a Neumanndomain of f . On manifolds, we have that Ω and f | Ω are of a rather simple form; Ω issimply connected; f | Ω has only two nodal domains and its critical points are all locatedon ∂ Ω (Theorem 3.2). On graphs, the situation is similar, as almost all Neumann domainsare either star graphs or path graphs; it is possible to have other Neumann domains, andeven non simply connected ones, only if λ is small enough (Lemma 8.2). For graphs, f | Ω has two nodal domains if Ω is a path graph, but otherwise may have more, with a globalbound on this number (Proposition 8.3,(3)).The most basic property of Neumann domains is that f | Ω is a Neumann eigenfunctionof Ω (Manifolds - Lemma 3.1; Graphs - Lemma 8.1). The eigenvalue of f | Ω is also λ and the interesting question is to find out what is the position of λ in the spectrum ofΩ - a quantity which we denote by N Ω ( λ ) (Definition 5.1). The intuitive feeling at thebeginning of the Neumann domain study was that generically, N Ω ( λ ) = 1 or that at leastthe spectral position gets low values.The general problem of determining the spectral position is quite hard for manifolds.The most general result we are able to provide for manifolds (Proposition 5.2) is a lowerbound given in terms of the geometric quantity ρ , which is a normalized area to perimeterratio (Definition 4.2). Interestingly, this result allows to estimate the spectral positionnumerically; a numerical calculation of ρ is rather easy compared to the involved calcu-lation of the spectrum of an arbitrary domain, which is needed to determine a spectralposition. This numerical method allows to refute the belief that for manifolds, generically, N Ω ( λ ) = 1. For graphs, the quantity ρ (Definition 9.1)) allows to bound the spectralposition from both sides, for almost all Neumann domains (Proposition 10.2). Two addi-tional results we have for the spectral position on graphs (but not for manifolds) are asfollows. First, the spectral position of Ω is given explicitly by the nodal count of f | Ω , andthis yields an upper bound on the spectral position (Lemma 10.1). Second, the spectralposition has a limiting distribution which is symmetric (Proposition 10.3). Another pointof comparison is that an upper bound on the spectral position, which we have for graphs,does not exist for manifolds. We show by means of an example that the spectral position EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 29 is unbounded in the manifold case. This example is given in terms of separable eigenfunc-tions on the torus. For this example, we show that although the spectral position of half ofthe Neumann domains is unbounded, it equals one for the other half (Theorem 5.3). Thisfinding might imply that even though N Ω ( λ ) = 1 does not hold generically, there mightbe a substantial proportion of Neumann domains, for which it does hold (see e.g., (6.5)).This is indeed the case for graphs where the spectral position of each path graph Neumanndomain equals one, and all of those form a substantial proportion of all Neumann domains(their number as well as their total length increase with the eigenvalue).Finally, we discuss the Neumann domain count. On manifolds we count the number ofNeumann domains, while on graphs we count the number of Neumann points. There isalso a connection between the Neumann count and the nodal count. On manifolds, wehave that the difference between the Neumann count and half the nodal count is non-negative (Corollary 6.1). On graphs, the difference between the Neumann count and thenodal count is bounded from both sides (Proposition 11.2). As for the Neumann countitself, it makes sense to consider it with a normalization: µ ( f n ) n on manifolds and µ ( f n ) − n on graphs. For graphs we provide general bounds on ω n = µ ( f n ) − n (Proposition 11.2),but believe that those are not sharp and conjecture sharper bounds (Conjecture 11.4).The validity of the conjecture would also imply a correlation between the nodal and theNeumann counts. In addition, ω n possesses a limiting probability distribution which issymmetric (Proposition 11.3). The expected value of this distribution stores informationon the size of the graph’s boundary, | ∂ Γ | ; an information that is absent from the nodalcount. Which other graph properties may be revealed by this distribution is still to befound. Turning back to manifolds, we treat separable eigenfunctions on the torus and forthose derive the probability distribution of µ ( f n ) n (Proposition 6.4). This is to be viewed asthe beginning of the analysis of Neumann count on manifolds. Two approaches in whichsome progress can be made are the following. One is getting a Courant-like bound of theform µ ( f n ) ≤ h ( n ), with h being possibly a linear function. The second would be studyingthe asymptotic behaviour, and for example showing that lim sup n →∞ µ ( f n ) = ∞ . Bothapproaches are related to analogous results on nodal domains. The first is tied to theCourant bound for nodal domains (whose strict version does not hold for the Neumanncount). The second is based on a series of works on asymptotic growth of the nodal count[32, 40, 41, 63] (see full description in Section 6) together with the basic bound (6.1) whichrelates the nodal count and the Neumann count. Acknowledgments
We thank Alexander Taylor for providing the python code [56] we used to generate thefigures of Neumann domains on manifolds and calculate the distribution of ρ . The authorswere supported by ISF (Grant No. 494/14). Appendix A. Basic Morse Theory
This section brings some basic statements in Morse theory which are useful for under-standing the first part of the paper. For a more thorough exposition, we refer the reader to[10]. Throughout the appendix we take (
M, g ) be a compact smooth Riemannian manifoldof a finite dimension. At some points of the appendix we specialize for the two-dimensionalcase and mention explicitly when we do so.
Definition A.1.
Let f : M → R be a smooth function. (1) f is a Morse function if at every critical point, p ∈ C ( f ), the Hessian matrix,Hess f | p , is non-degenerate, i.e., it does not have any zero eigenvalues.(2) The Morse index λ p of a critical point p ∈ C ( f ) is the number of negative eigen-values of the Hessian matrix, Hess f | p .The following three propositions may be found in [10]. Proposition A.2. [10, Lemma 3.2 and Corollary 3.3] If f is a Morse function then the critical points of f are isolated and f has only finitelymany critical points. Next, we consider the gradient flow ϕ : R × M → M defined by (2.2). For a particular x ∈ M we call the image of ϕ : R × x → M , a gradient flow line. Note that a gradientflow line, { ϕ ( t ; x ) } ∞ t = −∞ has a natural direction dictated by the order of the t values. Proposition A.3. [10, Propositions 3.18, 3.19] (1) Any smooth real-valued function f decreases along its gradient flow lines. Thedecrease is strict at noncritical points.(2) Every gradient flow line of a Morse function f begins and ends at a critical point.Namely, for all x ∈ M both limits lim t →±∞ ϕ ( t, x ) exist and they are both criticalpoints of f . Proposition A.4 (Stable/Unstable Manifold Theorem for a Morse Function) . [10, The-orem 4.2] Let f be a Morse function and p ∈ C ( f ) . Then the tangent space at p splits as T p M = T s p M ⊕ T u p M, where the Hessian is positive definite on T s p M and negative definite on T u p M .Moreover, the stable and unstable manifolds, (2.3), are surjective images of smoothembeddings T s p M → W s ( p ) ⊆ MT u p M → W u ( p ) ⊆ M. Therefore, W u ( p ) is a smoothly embedded open disk of dimension λ p and W s ( p ) is asmoothly embedded open disk of dimension m − λ p , where m is the dimension of M . Let us examine the implications of the results above in the particular case of Morsefunctions on a two-dimensional manifold. • If q is a maximum then λ q = 2 and so W u ( q ) is a two-dimensional open and simplyconnected set and W s ( q ) = { q } . • If p is a minimum then λ p = 0 and so W s ( p ) is a two-dimensional open and simplyconnected set and W u ( p ) = { p } . • If r is a saddle point then λ r = 1 and so both W s ( r ) and W u ( r ) are one-dimensional curves. Note that W s ( r ) ∩ W u ( r ) = { r } and so we get that W s ( r ) isa union of two gradient flow lines (actually even Neumann lines) which end at r .Similarly, W u ( r ) is a union of two gradient flow lines (Neumann lines) which startat r .By Definition 2.1 we get that Neumann domains are open two-dimensional sets and thatthe Neumann line set is a union of one dimensional curves. Moreover, those sets are com-plementary. Namely, the union of all Neumann domains together with the Neumann line EUMANN DOMAINS ON GRAPHS AND MANIFOLDS 31 set gives the whole manifold [7, Proposition 1.3].Next, we focus on a subset of the Morse functions, known as Morse-Smale functions,described by the following two definitions.
Definition A.5.
We say that two sub-manifolds M , M ⊂ M intersect transversally andwrite M (cid:116) M if for every x ∈ M ∩ M the tangent space of M at x equals the sum oftangent spaces of M and M at x , i.e.(A.1) T x M = T x M + T x M . This is also called the transversality condition . Definition A.6.
A Morse function such that for all of its critical points p , q ∈ C ( f ) thestable and unstable sub-manifolds intersect transversely, i.e., W s ( q ) (cid:116) W u ( p ) is called a Morse-Smale function.Let us assume now that M is a two-dimensional manifold and provide a necessary andsufficient condition for a Morse function to be a Morse-Smale function. First, for twocritical points p , q ∈ C ( f ), the intersection W s ( p ) ∩ W u ( q ) may be non-empty only forthe following cases:(1) if p = q ,(2) if p is a minimum and q is a maximum,(3) if p is a minimum and q is a saddle point,(4) if p is a saddle point and q is a maximum, or(5) if both p and q are saddle points.In the first four cases, it is straightforward to check that the transversality condition issatisfied. In the last case we have that if W s ( p ) ∩ W u ( q ) (cid:54) = ∅ then W s ( p ) ∩ W u ( q ) equalsto the gradient flow line (also Neumann line in this case) which asymptotically starts at q and ends at p . In such a case we get that for all x ∈ W s ( p ) ∩ W u ( q ), the tangent spacesobey T x W s ( p ) = T x W u ( q ) and those are one-dimensional, so their sum cannot be equalto the two-dimensional T x M . Therefore, in this case the transversality condition, (A.1) isnot satisfied and as a conclusion we get Proposition A.7.
On a two-dimensional manifold, a Morse function is Morse-Smale ifand only if there is no Neumann line connecting two saddle points.
By the Kupka-Smale theorem (see [10]) Morse-Smale gradient vector fields are genericamong the set of all vector fields. Currently, there is no similar genericity result regardingeigenfunctions of elliptic operators which are Morse-Smale (in the spirit of [57, 58]). Ourpreliminary numerics suggest that Morse-Smale eigenfunctions are indeed generic.
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