Defining the spectral position of a Neumann domain
DDEFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN
RAM BAND , GRAHAM COX , SEBASTIAN K. EGGER Abstract.
A Laplacian eigenfunction on a two-dimensional Riemannian manifold pro-vides a natural partition into Neumann domains (a.k.a. Morse–Smale complexes). Thispartition is generated by gradient flow lines of the eigenfunction — these bound the so-called Neumann domains. We prove that the Neumann Laplacian ∆ defined on a singleNeumann domain is self-adjoint and possesses a purely discrete spectrum. In addition,we prove that the restriction of the eigenfunction to any one of its Neumann domains isan eigenfunction of ∆. As a comparison, similar statements for a nodal domain of aneigenfunction (with the Dirichlet Laplacian) are basic and well-known. The difficulty hereis that the boundary of a Neumann domain may have cusps and cracks, and hence is notnecessarily continuous, so standard results about Sobolev spaces are not available.Another very useful common fact is that the restricted eigenfunction on a nodal domainis the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumanndomain. Our results enable the investigation of the resulting spectral position problemfor Neumann domains, which is much more involved than its nodal analogue. Introduction and statement of results
Let M be a closed, connected, orientable Riemannian surface with a smooth Riemannianmetric g . It is well-known that the Laplace-Beltrami operator ∆ g is self-adjoint and has apurely discrete spectrum. We arrange the eigenvalues in increasing order0 = λ < λ ≤ λ ≤ · · · , (1.1)and let { f n } ∞ n =0 denote a corresponding complete system of orthonormal eigenfunctions, sothat ∆ g f n = λ n f n . (1.2)We assume throughout the paper that the eigenfunctions of ∆ g are Morse functions, as isgenerically the case [Uhl76]. In fact, most of our results (with the notable exception ofTheorem 1.3) are valid for arbitrary Morse functions, not just eigenfunctions.The main objects of study in this paper are the Neumann domains of a Morse function,to be defined next. Given a smooth function f on M , we let ϕ : R × M → M denote theflow along the gradient vector field, grad f , i.e. ϕ is the solution to ∂ t ϕ ( t, x ) = − grad f (cid:12)(cid:12) ϕ ( t, x ) , ϕ (0 , x ) = x . (1.3) Mathematics Subject Classification.
Key words and phrases.
Neumann domains, Neumann lines, nodal domains, Laplacian eigenfunctions,Morse–Smale complexes. a r X i v : . [ m a t h . SP ] S e p RAM BAND , GRAHAM COX , SEBASTIAN K. EGGER For a critical point x of f , we define its stable and unstable manifolds W s ( x ) := (cid:110) y ∈ M : lim t →∞ ϕ ( t, y ) = x (cid:111) and W u ( x ) := (cid:110) y ∈ M : lim t →−∞ ϕ ( t, y ) = x (cid:111) , (1.4)which are smoothly embedded submanifolds of M . If x is a non-degenerate critical pointof f then dim W u ( x ) = codim W s ( x ) equals the Morse index, i.e. the number of negativeeigenvalues of the Hessian of f at x . We denote the sets of minima, maxima and saddlesof f by Min( f ), Max( f ) and S ( f ), respectively. Definition 1.1 ([BF16]) . Let f be a Morse function on M .(1) Let p ∈ Min( f ) and q ∈ Max( 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 N ( f ) := (cid:91) r ∈ S ( f ) W s ( r ) ∪ W u ( r ) . (1.5)The above defines a partition of the manifold M , which we call the Neumann partition.Indeed, it is not hard to show that M equals the disjoint union of all Neumann domainsand the Neumann line set (under the assumption that N ( f ) (cid:54) = ∅ , see [BF16, Proposition1.3]). Figure 1.1 depicts the Neumann partition of a particular eigenfunction on the flattorus.It follows that the grad f is parallel to the boundary of any Neumann domain Ω (as theboundary is made up of gradient flow lines), so we conclude that the normal derivativevanishes, ∂ ν f = 0, assuming ∂ Ω is sufficiently smooth. This formal observation motivatesour study of the Neumann Laplacian on Ω, which we precisely define in Definition 4.1.While the Dirichlet Laplacian on any bounded open set has a purely discrete spectrum,the same is not necessarily true of the Neumann Laplacian. Indeed, the essential spectrummay be nonempty, and in fact can be an arbitrary closed subset of [0 , ∞ ), see [HSS91].Nevertheless, the Neumann Laplacian of a Neumann domain is well behaved. Theorem 1.2.
Let Ω be a Neumann domain of a Morse function f on M . Then theNeumann Laplacian ∆ on Ω is a non-negative, self-adjoint operator with purely discretespectrum, i.e. consisting only of isolated eigenvalues of finite multiplicity. The main difficulty in proving this theorem is due to possible cusps of the Neumanndomain’s boundary, ∂ Ω (see Proposition 2.5 and the preceding discussion). Such cuspsprevent the application of standard results on density and compact embeddings of Sobolevspaces. We overcome this difficulty in the proof of Theorem 1.2 by using some geometricproperties that the Neumann domain boundary possesses.It is well known that the restriction of f to any of its nodal domains is an eigenfunctionof the Dirichlet Laplacian. Similarly, we have Theorem 1.3.
Let Ω be a Neumann domain of a Morse eigenfunction f on M . Denotingby ∆ the Neumann Laplacian on Ω , we have f (cid:12)(cid:12) Ω ∈ D (∆) , and hence λ ∈ σ (∆) . EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 3
Figure 1.1.
Left: An eigenfunction corresponding to eigenvalue λ = 17 ofthe flat torus whose fundamental domain is [0 , π ] × [0 , π ]. Circles mark sad-dle points and triangles mark extremal points (maxima by triangles pointingupwards and vice versa for minima). The nodal set is drawn as dashed linesand the Neumann line set is marked by solid lines. The Neumann domainsare the domains bounded by the Neumann line set. Right: A magnificationof the marked square from the left figure, showing Neumann domains withand without cusps. (This figure was produced using [Tay18].)The theorem above is not trivial, since the Neumann Laplacian is abstractly defined interms of a quadratic form (see Section 4.1) and so its domain D (∆) is not immediatelyevident and one needs to verify that indeed f | Ω ∈ D (∆).The importance of Theorem 1.2 is that it allows Definition 1.4.
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 N Ω ( λ ) := |{ λ n ∈ σ (Ω) : λ n < λ }| , (1.6)where σ (Ω) := { λ n } ∞ n =0 is the Neumann spectrum of Ω (which is discrete by Theorem 1.2),containing multiple appearances of degenerate eigenvalues and including λ = 0.From Theorem 1.3 we in fact have λ = λ n for some n , and so we can equivalently write N Ω ( λ ) = min { n : λ n = λ } . In particular, if λ ∈ σ (∆) is simple, then λ = λ n for a unique n , and hence N Ω ( λ ) = n .This equality explains the terminology “spectral position” for N Ω ( λ ). RAM BAND , GRAHAM COX , SEBASTIAN K. EGGER The spectral position is a key notion for Neumann domains. Finding its value is a greatchallenge and is of major importance in studying Neumann domains and their properties[BF16, BET20, ABBE20]. As a comparison, the similar notion for a nodal domain is trivial:if Ξ is a nodal domain of f , then f | Ξ is the first eigenfunction of the Dirichlet Laplacianon Ξ. This is a basic observation which serves as an essential ingredient in many nodaldomain proofs. Structure of the paper.
In Section 2 we describe some geometric properties of Neumanndomains, emphasizing the potentially singular nature of their boundary. In Section 3 we usethis geometric structure to establish fundamental properties of Sobolev spaces on Neumanndomains, including non-standard density and compactness results. Finally, in Section 4 weuse these properties to study the Neumann Laplacian, proving Theorems 1.2 and 1.3.2.
Geometric properties of Neumann Domains
As above, we take M to be a closed, connected, orientable Riemannian surface with asmooth Riemannian metric g , and consider M to be the domain of all functions discussedbelow. Note that all of the statements in this section hold for arbitrary Morse functions,and not only for eigenfunctions. For convenience we recall the following definitions. Definition 2.1 ([BH04]) . Let f : M → R be a smooth function.(1) f is said to be a Morse function if the Hessian matrix, Hess f ( p ), is non-degenerateat every critical point p .(2) A Morse function f is said to be Morse–Smale if for all critical points p and q , thestable and unstable manifolds W s ( p ) and W u ( q ) intersect transversely.We now review some basic topological properties of Neumann domains. Theorem 2.2. [BF16, Theorem 1.4]
Let f be a Morse function with a non-empty set ofsaddle points, S ( f ) (cid:54) = ∅ . Let p ∈ Min( f ) , q ∈ Max( f ) with W s ( p ) ∩ W u ( q ) (cid:54) = ∅ , andlet Ω be a connected component of W s ( p ) ∩ W u ( q ) , i.e., Ω is a Neumann domain. Thefollowing 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. (3) The extremal points which belong to Ω are exactly p and 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. (5) If c ∈ R is such that f ( p ) < c < f ( q ) , then Ω ∩ f − ( c ) is a smooth, non-selfintersecting one-dimensional curve in Ω , with its two boundary points lying on ∂ Ω . Parts (2) and (4) of the theorem above motivate us to examine individual Neumann linesand their connectivity to the critical points of f . Definition 2.3.
EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 5 (1) A
Neumann line is the closure of a connected component of W s ( r ) \ { r } or W u ( r ) \{ r } for some r ∈ S ( f ).(2) For a critical point x of f , we define its degree, deg( x ), to be the number ofNeumann lines connected to x .Each Neumann line is thus the closure of a gradient flow line, connecting a saddle pointto another critical point. In fact, the connectivity of Neumann lines is directly related tothe Morse–Smale property of f . Lemma 2.4 ([ABBE20]) . On a two-dimensional manifold, a Morse function is Morse–Smale if and only if there is no Neumann line connecting two saddle points.
The following properties of Neumann lines will be used throughout the rest of the paper.
Proposition 2.5.
Let f be a Morse function and Ω one of its Neumann domains. (1) If c is a saddle point of f , then deg( c ) = 4 and the angle between each two adjacentNeumann lines which meet at c is π . (2) If c is an extremal point of f whose Hessian is not proportional to g , then any twoNeumann lines meet at c with angle , π , or π . (3) Let c be an intersection point of a nodal line and a Neumann line of f . If c is asaddle point, then the angle between those lines is π . Otherwise, the angle is π .Remark . More generally, if c is a saddle point and there exist coordinates ( x, y ) near c in which f is given by the homogeneous harmonic polynomial Re( x + iy ) k , then deg( c ) = 2 k .For a non-degenerate saddle the existence of such coordinates (with k = 2) is an immediateconsequence of the Morse lemma, so we obtain Proposition 2.5(1) as a special case of thisremark. Sufficient conditions for f to be written in this form are given in [Che76].The first and third parts of Proposition 2.5 were proven in [MF14], [BH04, Theorem 4.2]and [ABBE20, Proposition 4.1]. The second part of the proposition is proven below (seeRemark 2.9 after the proof), using the following version of Hartman’s theorem, which willalso be used in the proofs of Lemma 3.1 and Proposition 3.2. Proposition 2.7. [Har60]
Let E be an open neighbourhood of p ∈ R . Let F ∈ C ( E, R ) ,and let ϕ be the flow of the nonlinear system ∂ t ϕ ( t, x ) = F ( ϕ ( t, x )) . Assume that F ( p ) = and the Jacobian DF ( p ) is diagonalizable and its eigenvalues have non-zero real part.Then, there exists a C -diffeomorphism Φ : U → V of an open neighbourhood U of p ontoan open neighbourhood V of the origin, such that D Φ( p ) = I and for each x ∈ U the flowline through p is mapped by Φ to Φ( ϕ ( t, x )) = e DF ( p ) t Φ( x ) (2.1) for small enough t values.Remark . The textbook version of Hartman’s theorem in n dimensions (see, for in-stance, [Per01, p. 120]) only guarantees the existence of a homeomorphism Φ. For n = 2,the proposition above guarantees that Φ is a C -diffeomorphism, but for n > RAM BAND , GRAHAM COX , SEBASTIAN K. EGGER it suffices to assume that all of the eigenvalues of DF ( p ) are in the same (left or right) halfplane; see [Per01, p. 127]. That version of the theorem would be sufficient for our purposes,since we only apply Proposition 2.7 at non-degenerate extrema, where all eigenvalues havethe same sign. However, it is interesting to note that Proposition 2.7 also applies at saddlepoints in two dimensions. Proof of Proposition 2.5 (2) . Let c be an extremal point of f whose Hessian is not pro-portional to g . Since Hess f ( c ) is non-degenerate, both eigenvalues of Hess f ( c ) are eitherstrictly positive (if c is a minimum) or strictly negative (if c is a maximum). We choose nor-mal coordinates in an open neighbourhood (cid:101) E of c , with respect to which (cid:101) E is representedby an open subset E ⊂ R , c corresponds to the origin ∈ R , and g ij ( ) = δ ij .We now apply Proposition 2.7 to F = − grad f . Since DF ( ) = − Hess f ( ) is diago-nalizable and has nonzero eigenvalues, there exist U ⊂ E and V ⊂ R , both containingthe origin, and a C -diffeomorphism Φ : U → V , such that the gradient flow lines aremapped by Φ to the flow lines e − t Hess f ( ) Φ( x ) of the linearized system. In [MF14, Theorem3.1], [ABBE20, Proposition 4.1] it was shown that the angle between such flow lines atan extremal point is either 0, π , or π , under the assumption that Hess f ( ) is not a scalarmatrix. This assumption holds, as the Hessian is not proportional to the metric and wehave chosen coordinates with respect to which g ( ) is the identity.It is left to relate the meeting angle between the gradient flow lines in M and thecorresponding flow lines e − t Hess f ( ) Φ( x ) in V . Since the tangent map D Φ( ) : T U → T V is the identity, and g ij ( ) = δ ij , the meeting angle of any two curves at is preserved byΦ, hence this angle is either 0, π , or π . This completes the proof. (cid:3) Remark . The argument for Proposition 2.5(2) given in [ABBE20, Proposition 4.1] isincomplete and hence we have supplied a complete proof here. In particular, the Taylorexpansion argument used in the proofs of [MF14, Theorem 3.1] and [ABBE20, Proposition4.1] does not suffice here. The Taylor expansion of f in (1.3) gives (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) , (2.2)but this does not allow us to conclude that the flow may be approximated by (cid:18) x ( t ) y ( t ) (cid:19) ≈ e − t Hess f ( c ) · (cid:18) x (0) y (0) (cid:19) , due to the possible coupling of higher order terms in (2.2).From Proposition 2.5(2) we see that the boundary of a Neumann domain may possess acusp (when the meeting angle is 0) and so it can fail to be Lipschitz continuous. Further-more, it may even fail to be of class C . Recall that a boundary of a domain is said to be ofclass C if it can be locally represented as the graph of a continuous function; alternatively,if the domain has the segment property (see [EE87, Thm V.4.4] or [MP01] for details). To EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 7 demonstrate that this is a subtle property, we bring as an example the domainsΩ = (cid:8) ( x, y ) ∈ R : x < y < x (cid:9) , Ω = (cid:8) ( x, y ) ∈ R : − x < y < x , < x < (cid:9) , (2.3)which are shown in Figure 2.1. The domain Ω does not satisfy the segment propertyat the origin, and hence is not of class C , even though its boundary is the union of twosmooth curves. On the other hand, Ω (which contains Ω ) is of class C . This examplewill be important later, in the proof of Proposition 3.2.0 0 . . . . . . . .
81 0 0 . . . . − − . . Figure 2.1.
The region Ω (left) defined in (2.3) is not of class C , whereasΩ (right) is of class C .We add that there is very little known in general regarding the asymptotic behavior ofNeumann lines near cusps. In particular, methods to treat cusps in a spectral theoreticcontext as in, e.g., [JMS92, FP20, BET20] have to be generalized for our purpose.We end this section by examining Theorem 2.2 and its implications for the structureof Neumann domains. By the statement of the theorem, the boundary of a Neumanndomain always contains a minimum and a maximum (and no other extrema). It followsthat each Neumann domain of a Morse function must belong to one of following two types(illustrated in Figure 2.2): • a type-(i) Neumann domain has on its boundary a maximum and a minimum, eachof degree at least two (see Definition 2.3); • a type-(ii) Neumann domain has on its boundary an extremal point which is ofdegree one.Moreover, since the boundary is made up of Neumann lines, it must contain at leastone saddle point. If f is Morse–Smale the boundary contains at most two saddle points,by Theorem 2.2(4), but for a general Morse function it is possible to have more. Thepossible existence of additional saddle points is irrelevant for our analysis towards the RAM BAND , GRAHAM COX , SEBASTIAN K. EGGER proofs of Theorems 1.2 and 1.3, however, since the boundary is Lipschitz near these pointsby Proposition 2.5(1).Numerical observations suggest that generically Neumann domains are of type (i). How-ever, it is not hard to construct Morse functions having Neumann domains of type (ii); seeAppendix A. Theorems 1.2 and 1.3 apply to both types of domains, but in the proofs weneed to pay careful attention to domains of type (ii). In particular, any Neumann domainof type (ii) has at least one internal “crack” given a Neumann line, and hence is not ofclass C , as the domain lies on both sides of its boundary. Figure 2.2.
Possible types of Neumann domains for a Morse function. Sad-dle points are represented by balls, maxima by triangles pointing upwardsand vice versa for minima. If f is Morse–Smale, its Neumann domains mustlook like one of the first two examples, with either one or two saddle pointson the boundary. For the type-(ii) domain shown in the center, η is the onlyNeumann line connected to q , hence deg( q ) = 1. If f is not Morse–Smale, itsNeumann domains can have additional saddle points on the boundary, andcan have both extremal points of degree one, as shown on the right. (Notethat this last example has a Neumann line connecting two saddle points, andhence is not possible if f is Morse–Smale, by Lemma 2.4.) Remark . In summary, a Neumann domain may fail to be of class C for two reasons:1) a cusp on the boundary; or 2) a crack in the domain, i.e. a Neumann line contained inthe interior of Ω, as happens for type-(ii) domains. These are the main technical obstaclesto be overcome in proving Theorems 1.2 and 1.3.3. Sobolev spaces on Neumann domains
In this section we discuss properties of Sobolev spaces on Neumann domains. The mainresult is Proposition 3.2, which establishes density and embedding properties for the space W , (Ω). As described in the introduction, and indicated in Proposition 2.5(2) (see alsoRemark 2.10), the difficulty is that the boundary of a Neumann domain need not be ofclass C . EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 9
Definitions.
As above, we assume that (
M, g ) is a smooth, closed, connected, ori-ented Riemannian manifold. For an open submanifold N ⊂ M , the Sobolev space W j, ( N )is defined to be the completion of C ∞ ( N ) with respect to the norm (cid:107) f (cid:107) W j, ( N ) := j (cid:88) i =0 (cid:90) N |∇ i f | g d g x , (3.1)where ∇ denotes the covariant derivative and d g x is the Riemannian volume form. Thenorm depends on the choice of Riemannian metric g , but since M is compact, it is wellknown that different metrics will produce equivalent norms. We will sometimes take ad-vantage of this fact and compute the Sobolev norm using a metric ˜ g defined in a localcoordinate chart to have components ˜ g ij = δ ij (so that covariant derivatives become par-tial derivatives, the Riemannian volume form reduces to the Euclidean one, etc.). Thisallows us to apply standard methods in Sobolev space theory to Lipschitz domains in M ,[McL00].Assume that N ⊂ M is an open submanifold of M , such that the boundary ∂N isLipschitz. We will later choose N to be a Neumann domain Ω, or a proper subset thereof(see Section 3.3) if ∂ Ω has a cusp. We then define the boundary Sobolev spaces H s ( ∂N )for | s | ≤
1, following [McL00], so that the dual space is given by H s ( ∂N ) ∗ = H − s ( ∂N ).Moreover, for any open subset Γ ⊂ ∂N we let, see [McL00, pp. 66, 77 and 99], H s (Γ) := (cid:8) f (cid:12)(cid:12) Γ : f ∈ H s ( ∂N ) (cid:9)(cid:101) H s (Γ) := (cid:8) f ∈ H s ( ∂N ) : supp f ⊂ Γ (cid:9) (3.2)and observe that H s (Γ) ∗ = (cid:101) H − s (Γ), by [McL00, Theorems 3.29, 3.30, and p. 99]. Inparticular, we have φ = 0 in (cid:101) H − s (Γ) ⇐⇒ φ ( f ) = 0 ∀ f ∈ H s (Γ) . (3.3)We define for 0 ≤ s ≤ · dual : L (Γ) → H s (Γ) ∗ ,w dual ( v ) := (cid:104) v, w (cid:105) L (Γ) , v ∈ H s (Γ) , (3.4)observing that the L inner product is well defined because H s (Γ) ⊂ L (Γ) for 0 ≤ s ≤ ϕ ∈ H s (Γ) ∗ on v ∈ H s (Γ), i.e. we will write ϕ ( v ) = (cid:90) Γ ϕv even when ϕ is not in the range of the map · dual ; see in particular Green’s formula (3.5)below.For a Lipschitz domain N the trace map ·| ∂N : W , ( N ) → H ( ∂N ) is continuous andthe (outward) normal derivative ∂ ν : W , ( N ) → H ( ∂N ) is continuous as well. Green’s , GRAHAM COX , SEBASTIAN K. EGGER formula (cid:90) N (cid:104) grad ψ, grad h (cid:105) = (cid:90) N (∆ g ψ ) h + (cid:90) ∂N h ( ∂ ν ψ ) (3.5)then holds for all ψ ∈ W , ( N ) and h ∈ W , ( N ). By (3.4), the boundary integralis equivalent to the pairing ( ∂ ν ψ ) dual ( h | ∂N ). Green’s formula then admits the follow-ing important generalization: For any ψ ∈ W , ( N ) with ∆ g ψ ∈ L ( N ), there exists aunique ∂ ν ψ ∈ H − ( ∂N ) = H ( ∂N ) ∗ such that (3.5) holds for all h ∈ W , ( N ), [McL00,Lemma 4.3, Theorem 4.4]. The boundary term now has to be understood as the action of ∂ ν ψ ∈ H − ( ∂N ) on h | ∂N ∈ H ( ∂N ), i.e., ( ∂ ν ψ )( h | ∂N ), but to simply the presentation westill use the integral notation of (3.5). Finally, we remark that ∆ g ψ ∈ L ( N ) means thatthere exists f ∈ L ( N ) such that (cid:90) N (cid:104) grad ψ, grad h (cid:105) = (cid:90) N f h (3.6)for all h ∈ C ∞ ( N ), or equivalently for all h ∈ W , ( N ), in which case we set ∆ g ψ = f .3.2. Dissections of Neumann domains.
The boundary of a type-(ii) Neumann domaincannot be of class C , whether or not there is a cusp on the boundary, due to the Neumannline η contained in the interior of Ω, see Figure 2.2. We deal with this by dissecting sucha Neumann domain into two pieces, as shown in Figure 3.1, where one piece has Lipschitzboundary, and the other has boundary that is Lipschitz except possibly at a cusp point,i.e. it has the same regularity as a type-(i) Neumann domain. For type-(ii (cid:48) ) domains as inFigure 2.2 an analogous statement holds as the proof for that case is essentially the same.The dissection thus reduces many of the proofs for type-(ii) domains to the correspondingtype-(i) results.This dissection is made possible by the following lemma. Lemma 3.1.
Assume f is a Morse function, and let γ be a Neumann line. Then γ has afinite length l γ < ∞ and γ allows an arc-length parametrization with γ ∈ C ([0 , l γ ]) , i.e.,boundary points are included.Proof. We decompose γ = γ ∪ γ ∪ γ , where γ is defined in a small neighborhood of theinitial endpoint of γ and γ is defined in a small neighborhood of the terminal endpoint.Then it is enough to prove the corresponding statement for γ , γ and γ . The result for γ follows by standard results for flows of smooth vector fields. If the initial endpoint (whichwe label c ) is a saddle, then the result for γ follows, e.g., by [BH04, Theorem 4.2, p. 94].On the other hand, if c is an extremum we use the map Φ from Proposition 2.7. Then Φ ◦ γ is a flow line generated by e − Hess f ( c ) t , and hence satisfies the properties of the claim, i.e. itis C up to the endpoint and has finite length. As Φ − is a C map and γ = Φ − ◦ (Φ ◦ γ )is a composition of C functions, the claim for γ follows. The proof for γ is identical. (cid:3) Now suppose that Ω is a type-(ii) Neumann domain. The type-(ii (cid:48) ) case in Figure 2.2can be treated analogously. Denote by q the extrema in the interior of Ω, and let η be the EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 11
Figure 3.1.
The dissection of a type-(ii) Neumann domain, as given in(3.7). The Neumann line η is extended to a Lipschitz curve η ∪ ˜ η , so that Ω l is a Lipschitz domain and Ω r possesses a cusp at p .Neumann line attached to q ; see Figure 3.1. Using Lemma 3.1 we may dissect Ω into twoparts (Ω l and Ω r in Figure 3.1) so thatΩ \ ˜ η = Ω l ·∪ Ω r , (3.7)where ˜ η is a Lipschitz curve in Ω joining q with a non-cusp point of ∂ Ω, hence η ∪ ˜ η is aLipschitz curve by Lemma 3.1. This dissection induces an isometric embedding W , (Ω) → W , (Ω l ) ⊕ W , (Ω r ) ,ϕ (cid:55)→ (cid:0) ϕ | Ω l , ϕ | Ω r (cid:1) . (3.8)3.3. Truncated Neumann domains.
To deal with potential cusps at the maximum andminimum of f , we introduce truncated versions of Ω and suitable localizing functions.Denoting by p ∈ Min( f ) and q ∈ Max( f ) the minimum and maximum of f in Ω, we defineΩ t := { x ∈ Ω : f ( x ) < tf ( q ) } , q is a cusp , p is not a cusp , { x ∈ Ω : tf ( p ) < f ( x ) } , p is a cusp , q is not a cusp , { x ∈ Ω : tf ( p ) < f ( x ) < tf ( q ) } , q is a cusp , p is a cusp , Ω , otherwise , (3.9)for any 0 < t <
1. For a truncated Neumann domain Ω t we denote its complement in Ωby Ω ct := Ω \ Ω t .To (3.9) we associate smooth cut-off functions κ t,(cid:15) ∈ C ∞ (Ω) using the generating eigen-function f for the Neumann domain Ω. For suitable (cid:15) >
0, we define κ t,(cid:15) ( x ) := ( α t,(cid:15) ◦ f )( x ) , (3.10)where α t,(cid:15) ∈ C ∞ ( R ) is chosen such that 0 ≤ α t,(cid:15) ≤ κ t,(cid:15) ( x ) = ( α t,(cid:15) ◦ f )( x ) = (cid:40) , x ∈ Ω t − (cid:15) , , x ∈ Ω ct . (3.11) , GRAHAM COX , SEBASTIAN K. EGGER It follows that grad κ t,(cid:15) ( x ) = 0 for all x ∈ Ω t − (cid:15) ∪ Ω ct . More generally, we have | grad κ t,(cid:15) ( x ) | ≤ K (3.12)for all x ∈ Ω, where K = K ( t, (cid:15) ) is a constant depending on t and (cid:15) .By (3.10) it follows that f and κ t,(cid:15) f possess the same non-zero level-lines but usuallywith different level values if α t,(cid:15) is locally non-constant at the level value of f .The boundary of Ω t can be decomposed as ∂ Ω t = γ ± ,t ∪ γ ,t , where γ ± ,t are one or twolevel lines defined by { f ( x ) = tf ( q ) } and { f ( x ) = tf ( p ) } , respectively, and γ ,t ⊂ ∂ Ω.Note that γ ,t (cid:54) = ∅ , and Proposition 2.5(3) implies that γ ± ,t meets ∂ Ω perpendicularly,except for a finite number of exceptional times where γ ± ,t meets ∂ Ω at a saddle point, inwhich case the angle is π , see Figure 3.2. Figure 3.2.
Neumann domains and their truncations, with the dotted lineindicating the curve γ ± ,t . The top two figures show domains of type (i) andtype (ii) for t close to 1. The bottom left figure shows an exceptional valueof t , where γ ± ,t meets ∂ Ω at angle π , and the bottom right figure shows asmaller value of t .3.4. Density and embedding results.
We can now state and prove the main result ofthis section.
Proposition 3.2.
Let ( M, g ) be a compact, orientable and connected Riemannian surface.If Ω ⊂ M is a Neumann domain of a Morse function f , the following hold: EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 13 (1) the embedding W , (Ω) → L (Ω) is compact; (2) if Ω is of type (i), then C (Ω) is dense in W , (Ω) ; (3) if Ω is of type (ii), then there exists t ∈ (0 , such that the set of functions (cid:8) ψ ∈ W , (Ω) : ψ | Ω ct ∈ C (Ω ct ) (cid:9) (3.13) is dense in W , (Ω) . The result is known if ∂ Ω is of class C (see [MP01]) but, as noted above, the boundaryof a Neumann domain doesn’t need to have this property. The key ingredient in the proofis the following lemma, which says that in a neighborhood of a cusp point, functions canbe extended to a larger domain which still has a cusp but is of class C ; c.f. the domainsΩ and Ω in Figure 2.1. Lemma 3.3. [MP01, Lemma 5.4.1/1, p. 285]
Consider the domain ˜Ω = (cid:8) ( x, y ) ∈ R : c ϕ ( x ) < y < c ϕ ( x ) , < x < (cid:9) for some c < c , where ϕ ∈ C , ([0 , is an increasing function with ϕ (0) = 0 and ϕ (cid:48) ( t ) → as t → , and define G = (cid:8) ( x, y ) ∈ R : | y | < M ϕ ( x ) , < x < (cid:9) (3.14) for M ≥ max {| c | , | c |} . Then there exists a continuous extension operator E : W , ( ˜Ω) → W , ( G ) . In the following proof we will apply this lemma with ϕ ( x ) = x α for some α > Proof of Proposition 3.2.
We first prove (1) and (2) for type-(i) Neumann domains. Onlythe behavior near the cusps has to be investigated, as they are the only possible non-Lipschitz points on ∂ Ω. A cusp is either a maxima or a minima by Proposition 2.5.Without loss of generality, let c ∈ Max( f ) be the only cusp on ∂ Ω.We first localize at c by the smooth function κ c := 1 − κ t,(cid:15) with suitable (cid:15) > < t <
1, so that κ c equals 1 near c but vanishes outside a small neighborhood Ω c := Ω ct − (cid:15) . Nowtake φ ∈ W , (Ω). We write φ = κ c φ +(1 − κ c ) φ and observe that κ c φ, (1 − κ c ) φ ∈ W , (Ω).Thus, it is sufficient to prove the statements for both functions separately. For the latterfunction the observation that the boundary of its support is Lipschitz implies both (1) and(2) in Proposition 3.2.For the former space we choose t − (cid:15) close to 1 and employ Proposition 2.7. Let Φbe the resulting C -diffeomorphism and define ˜Ω c = Φ(Ω c ). Owing to (2.1), the twoboundary curves meeting at c of ˜Ω c are flow lines obeying ∂ t γ = − Hess f ( c ) γ . These aregenerated by e − t Hess f ( c ) x where x is a suitable point on γ . We assume without loss ofgenerality that c = . An analogous calculation as in [MF14, Section 3] and [ABBE20,Proof of Prop. 4.1] shows that the flow lines near c may be parametrized in suitablecoordinates by γ ( x ) = ( x, cx α ), where α > f ( c )(in fact only on their ratio). This implies that near c = , the domain ˜Ω c is described by( x, y ) ∈ ˜Ω c ⇐⇒ c x α < y < c x α and x > . (3.15) , GRAHAM COX , SEBASTIAN K. EGGER We can assume that α >
1. (If α = 1, then ˜Ω c is in fact Lipschitz near c , so thereis nothing to prove; if α < x and y to obtain a similar description of theboundary with α replaced by 1 /α .) Now Lemma 3.3 says that there exists a continuousextension operator E : W , ( ˜Ω c ) → W , ( G ) where ˜Ω c ⊂ G and near the domain G ischaracterized by ( x, y ) ∈ G ⇐⇒ | y | < M x α and x > , (3.16)with M large enough. Since the boundary ∂G is of class C , we can now infer by [EE87,Theorem 4.17, p. 267] and [MP01, Theorem 1.4.2/1, p.28] that W , ( G ) satisfies statements(1) and (2) of the proposition. In particular, W , ( G ) → L ( G ) is compact and C ( G ) isdense in W , ( G ).Since Φ is a C -diffeomorphism, it is easily shown that the pull-back mapΦ ∗ : W , ( ˜Ω c ) → W , (Ω c ) , φ (cid:55)→ φ ◦ Φ , (3.17)is well defined and bijective, with1 C (cid:48) (cid:107) φ (cid:107) W , (˜Ω c ) < (cid:107) φ ◦ Φ (cid:107) W , (Ω c ) < C (cid:48) (cid:107) φ (cid:107) W , (˜Ω c ) for some C (cid:48) >
0. Therefore, the inclusion W , (Ω c ) → L (Ω c ) can be written as thecomposition of a compact operator W , (Ω c ) (Φ − ) ∗ −−−−→ W , ( ˜Ω c ) E −−→ W , ( G ) −→ L ( G )and a bounded operator L ( G ) −→ L ( ˜Ω c ) Φ ∗ −−→ L (Ω c )(where the first map is restriction), and hence is compact. This completes the proof of (1)for type-(i) Neumann domains.To prove (2), let φ ∈ W , (Ω c ), so that φ ◦ Φ − ∈ W , ( ˜Ω c ) and E ( φ ◦ Φ − ) ∈ W , ( G ).For any δ >
0, there exists ψ ∈ C ( G ) with (cid:107) ψ − E ( φ ◦ Φ − ) (cid:107) W , ( G ) < δ , and hence (cid:13)(cid:13) ψ (cid:12)(cid:12) ˜Ω c ◦ Φ − φ (cid:13)(cid:13) W , (Ω c ) ≤ C (cid:48) (cid:13)(cid:13) ψ (cid:12)(cid:12) ˜Ω c − φ ◦ Φ − (cid:13)(cid:13) W , (˜Ω c ) ≤ C (cid:48) (cid:13)(cid:13) ψ − E ( φ ◦ Φ − ) (cid:13)(cid:13) W , ( G ) < C (cid:48) δ . Since ψ (cid:12)(cid:12) ˜Ω c ◦ Φ ∈ C (Ω c ), this completes the proof of (2).We next prove (1) for type-(ii) Neumann domains, using the decomposition (3.8). Moreprecisely, using Lemma 3.1 we may dissect Ω as in (3.7) and, without loss of generality,assume that the cusp is located on the boundary of Ω r , as in Figure 2.2. Note that W , (Ω) → W , (Ω l ) ⊕ W , (Ω r ) → L (Ω l ) ⊕ L (Ω r ) = L (Ω)and so it is enough to prove compactness of the embedding W , (Ω i ) → L (Ω i ) for i = l , r.For i = l this follows from the Lipschitz property of ∂ Ω l . For i = r we observe that ∂ Ω r isLipschitz except at the cusp, and so the proof given above for type-(i) Neumann domainsapplies. EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 15
Finally, we prove (3). For 0 < t (cid:48) < ct (cid:48) ⊂ Ω j for either j = l or r (the case j = r is shown in Figure 2.2(a)), so we choose t sufficiently close to1 and (cid:15) > c = Ω ct − (cid:15) ⊂ Ω j for some j . Now let φ ∈ W , (Ω). Given δ >
0, there exists by (2) a function φ δ ∈ C (Ω c ) such that (cid:107) φ − φ δ (cid:107) W , (Ω c ) < δ . Defining˜ φ δ = κ t,(cid:15) φ + (1 − κ t,(cid:15) ) φ δ , we see that κ t,(cid:15) φ ∈ W , (Ω) because κ t,(cid:15) is smooth on Ω. Moreover,since supp(1 − κ t,(cid:15) ) ⊂ Ω c , we also have (1 − κ t,(cid:15) ) φ δ ∈ W , (Ω) and hence ˜ φ δ ∈ W , (Ω).Writing φ = κ t,(cid:15) φ + (1 − κ t,(cid:15) ) φ , we compute (cid:107) φ − ˜ φ δ (cid:107) W , (Ω) = (cid:107) (1 − κ t,(cid:15) )( φ − φ δ ) (cid:107) W , (Ω c ) ≤ (1 + K ) (cid:107) φ − φ δ (cid:107) W , (Ω c ) < (1 + K ) δ , (3.18)where K is the constant from (3.12). Finally, since supp κ t,(cid:15) ⊂ Ω t , we have˜ φ δ (cid:12)(cid:12) Ω ct = (1 − κ t,(cid:15) ) φ δ (cid:12)(cid:12) Ω ct ∈ C (Ω ct ) . This completes the proof. (cid:3) The Neumann Laplacian on a Neumann domain
In this section we define the Neumann Laplacian on a Neumann domain Ω, and establishsome of its fundamental properties, in particular proving Theorems 1.2 and 1.3. This relieson the technical results of the previous section, in particular Proposition 3.2.4.1.
Definition and proof of Theorem 1.2.
To keep notation simple we slightly abusethe symbol of the pure Laplace–Beltrami operator and introduce
Definition 4.1.
The Neumann Laplacian on Ω, denoted ∆, is the unique self-adjointoperator corresponding to the bilinear form a ( ψ, φ ) := (cid:90) Ω (cid:104) grad ψ, grad φ (cid:105) g d g x , D ( a ) := W , (Ω) . (4.1)More precisely, ∆ is an unbounded operator on L (Ω) with domain D (∆) = (cid:8) ψ ∈ W , (Ω) : ∃ f ψ ∈ L (Ω) with a ( ψ, φ ) = (cid:104) f ψ , φ (cid:105) L (Ω) ∀ φ ∈ W , (Ω) (cid:9) (4.2)and for any ψ ∈ D (∆) we have ∆ ψ = f ψ . The existence and uniqueness of such anoperator follows immediately from the completeness of the form domain D ( a ) = W , (Ω)and standard theory of self-adjoint operators, [RS72, Theorem VIII.15]. If ψ ∈ D (∆), then(4.2) implies (cid:90) N (cid:104) grad ψ, grad φ (cid:105) = (cid:90) N (∆ ψ ) φ for all φ ∈ W , (Ω), and hence ∆ g ψ = ∆ ψ . That is, ∆ acts as the weak Laplace–Beltramioperator ∆ g defined in (3.6). , GRAHAM COX , SEBASTIAN K. EGGER The next result is nontrivial, and relies on the special geometric structure of Neumanndomains.
Proposition 4.2.
Let Ω be a Neumann domain for a Morse function, and define ∆ asabove. Then ∆ has compact resolvent, and hence has purely discrete spectrum satisfying σ (∆) ⊂ [0 , ∞ ) .Proof. Proposition 3.2(1) says that the form domain W , (Ω) is compactly embedded in L (Ω), so the result follows from [RS78, Theorem XIII.64]. (cid:3) Domain of the Neumann Laplacian: Proof of Theorem 1.3.
If Ω is a type-(i)Neumann domain, then its truncation Ω t is Lipschitz for any t , so we can define the normalderivative and use Green’s formula (3.5), as in Section 3.For type-(ii) Neumann domains a modification of the normal derivative is needed, sinceΩ t is not Lipschitz because of the Neumann line η ; see Figure 2.2. Here we use the dissectionmethod in Subsection 3.2. Since η ∪ ˜ η has a Lipschitz neighborhood in both Ω l and Ω r , wehave for the trace maps ϕ Ω l | ˜ η = ϕ Ω r | ˜ η for ϕ ∈ W , (Ω) (4.3)and so the map W , (Ω) → H ( η o ) ⊕ H ( η o ) ⊕ H (˜ η ) ,ϕ (cid:55)→ (cid:0) ϕ Ω l | η o , ϕ Ω r | η o , ϕ | ˜ η (cid:1) , (4.4)is well defined, where ϕ | ˜ η denotes the common value in (4.3). We now claim that (cid:90) Ω (cid:104) grad f, grad φ (cid:105) g d g x = (cid:104) ∆ g f, φ (cid:105) L (Ω) ∀ φ ∈ C ∞ (Ω)= ⇒ ( ∂ ν ( f | Ω l )) (cid:12)(cid:12) ˜ η = − ( ∂ ν ( f | Ω r )) (cid:12)(cid:12) ˜ η , in ˜ H − (˜ η ) , (4.5)where we recall that we put Γ ≡ ˜ η and ∂N = Ω l ∩ Ω t , respectively, ∂N = Ω r ∩ Ω t for t close enough to 1 in (3.2). Indeed, (4.5) together with Green’s formula (3.5) implies (cid:90) ˜ η (cid:0) ∂ ν ( f | Ω l ) + ∂ ν ( f | Ω r ) (cid:1)(cid:12)(cid:12) ˜ η v = 0for all v ∈ C ∞ (˜ η ), and hence for all v ∈ H (˜ η ). Since H (˜ η ) = H (˜ η ) (see, for instance,[McL00, Theorem 3.40]), the claim follows from (3.3).Furthermore, when integrating functions that do not have compact support in Ω, wemust take into account the fact that Ω l and Ω r need not be Lipschitz; see Figure 3.1, whereΩ r has a cusp on its boundary. In this case we combine the dissection and truncation ofSections 3.2 and 3.3, respectively. The resulting domains are shown in Figure 4.1. Notethat the boundaries of Ω t ∩ Ω l and Ω t ∩ Ω r can be partitioned into three parts: γ ± ,t comingfrom the truncation; ˜ η coming from the dissection; and γ ,t , coming from the originaldomain Ω. · o refers to the interior in γ ,t ∪ ˜ η EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 17
We emphasize here that the dissection (3.7) is an auxiliary construction, and our analysisdoes not depend on the specific choice of ˜ η . Figure 4.1.
The dissected and truncated domains appearing in the proofof Proposition 4.3. Here γ − ,t is a result of the truncation, ˜ η is from thedissection, and γ ,t is the part of the original boundary, ∂ Ω, that remainsafter the truncation.
Proposition 4.3.
Let Ω be a Neumann domain of a Morse eigenfunction f . (1) (a) For type-(i) Neumann domains the operator domain D (∆) satisfies D (∆) ⊂ (cid:8) ψ ∈ W , (Ω) : ∆ g ψ ∈ L (Ω) and ∂ ν ( ψ | Ω t ) (cid:12)(cid:12) γ ,t o = 0 in ˜ H − ( γ ,t o ) for all < t < (cid:111) . (4.6)(b) For type-(ii) Neumann domains the operator domain D (∆) satisfies D (∆) ⊂ (cid:110) ψ ∈ W , (Ω) : ∆ g ψ ∈ L (Ω) , ∂ ν ( ψ | Ω t ∩ Ω l ) (cid:12)(cid:12) γ ,t o = 0 , and ∂ ν ( ψ | Ω t ∩ Ω r ) (cid:12)(cid:12) γ ,t o = 0 in ˜ H − ( γ ,t o ) for all < t < (cid:111) . (4.7)(2) f | Ω ∈ D (∆) , hence, f | Ω is an eigenfunction of ∆ with the same eigenvalue.Proof. We first prove (1)(b); the proof of (a) for type-(i) domains is similar but less involved,so we omit it. We use the dissection (3.7). Then, for any ψ ∈ D (∆) and any test function ϕ ∈ W , (Ω) with supp ϕ ∩ ˜ η = ∅ and that vanishes in Ω ct (and in particular vanishes along , GRAHAM COX , SEBASTIAN K. EGGER γ ± ,t and ˜ η ), we have (cid:104) ∆ g ψ, ϕ (cid:105) L (Ω) = (cid:90) Ω (cid:104) grad ψ, grad ϕ (cid:105) g d g x = (cid:90) Ω t ∩ Ω l (cid:104) grad ψ, grad ϕ (cid:105) g d g x + (cid:90) Ω t ∩ Ω r (cid:104) grad ψ, grad ϕ (cid:105) g d g x = (cid:104) ∆ g ψ, ϕ (cid:105) L (Ω) + (cid:90) ∂ (Ω t ∩ Ω l ) ( ∂ ν ( ψ | Ω t ∩ Ω l )) ϕ d g σ + (cid:90) ∂ (Ω t ∩ Ω r ) ( ∂ ν ( ψ | Ω t ∩ Ω r )) ϕ d g σ , (4.8)where in the first line we used the defining property of ∆ and in the last line we appliedGreen’s formula (3.5) to the domains Ω t ∩ Ω l and Ω t ∩ Ω r , which are both Lipschitz. Assupp ϕ | ∂ Ω t ⊂ γ ,t we are left with (cid:90) ( ∂ (Ω t ∩ Ω l ) ∩ γ ,t ) ( ∂ ν ( ψ | Ω t ∩ Ω l )) ϕ d g σ + (cid:90) ∂ (Ω t ∩ Ω r ) ∩ γ ,t ( ∂ ν ( ψ | Ω t ∩ Ω r )) ϕ d g σ = 0 . The image of the trace map from { ϕ ∈ W , (Ω) : supp ϕ ∩ ˜ η } is dense in H (cid:0) ( ∂ (Ω t ∩ Ω i ) ∩ γ ,t ) o (cid:1) = H (cid:0) ( ∂ (Ω t ∩ Ω i ) ∩ γ ,t ) o (cid:1) for i = l , r, [McL00, Theorem 3.40], so we use (3.3) toobtain ∂ ν ( ψ | Ω t ∩ Ω i ) = 0 ∈ ˜ H − (cid:0) ( ∂ (Ω t ∩ Ω i ) ∩ γ ,t ) o (cid:1) . This completes the proof of claim (1).To prove claim (2) we again only consider type-(ii) Neumann domains, as the type-(i) case is analogous but simpler. Let ∆ g f = λf on M and denote ˜ f = f | Ω , then byelliptic regularity we have f ∈ C ∞ ( M ) and hence ˜ f ∈ C ∞ (Ω). That in turn implies thatthe normal derivatives ∂ ν ( ˜ f | Ω l ) ∈ L ( ∂ Ω l ) and ∂ ν ( ˜ f | Ω r ) ∈ L ( ∂ Ω r ), and the distributionis given by (3.4), i.e., by corresponding proper integrals in (4.8). In particular, we cansplit the integrals’ domains into suitable sub-intervals consisting of ˜ η and γ ,t . For theintegrations on ˜ η we use again (4.5). We use Proposition 3.2(3) to approximate W , (Ω)functions by functions that are C near the cusp point. Thus, let ϕ ∈ W , (Ω) be an testfunction that is C in Ω ct for t sufficiently close to 1. Using (3.7) and computing as in (4.8),we have (cid:90) Ω t (cid:104)∇ ˜ f , ∇ ϕ (cid:105) g d g x = (cid:90) Ω t (∆ g ˜ f ) ϕ d g x + (cid:90) ∂ (Ω t ∩ Ω l ) ( ∂ ν ( ˜ f | Ω l )) ϕ d g σ + (cid:90) ∂ (Ω t ∩ Ω r ) ( ∂ ν ( ˜ f | Ω r )) ϕ d g σ . EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 19
Taking the limit t →
1, we find that a ( ˜ f , ψ ) = (cid:90) Ω (∆ g ˜ f ) ϕ d g x + lim t → (cid:90) ∂ Ω t \ ∂ Ω ( ∂ ν ( ˜ f | Ω t )) ϕ | Ω t d g σ = (cid:90) Ω (∆ g ˜ f ) ϕ d g x . (4.9)where in the first line we have used the fact that (cid:104)∇ ˜ f , ∇ ϕ (cid:105) g and (∆ g ˜ f ) ϕ are in L (Ω),so their integrals over Ω t converge to their integrals over Ω as t → f on η , and (4.3) for ϕ , so the corresponding boundary integrals on η ∪ ˜ η cancel out. The last line uses the factthat both ∂ ν ( ˜ f | Ω t ) and ϕ are bounded on ∂ Ω t \ ∂ Ω, so that the boundary integral vanishesin the limit. Proposition 3.2(3) then implies a ( ˜ f , ϕ ) = (cid:90) Ω (∆ g ˜ f ) ϕ d g x ∀ ϕ ∈ W , (Ω) , so it follows from the definition of ∆, in particular (4.2), that ˜ f ∈ D (∆), with ∆ ˜ f = ∆ g ˜ f = λ ˜ f . (cid:3) Acknowledgments.
G.C. acknowledges the support of NSERC grant RGPIN-2017-04259.R.B. and S.K.E. were supported by ISF (grant No. 844/19).
Appendix A. Morse–Smale functions with type-(ii) Neumann domains
In this appendix we construct Morse–Smale functions having type-(ii) Neumann do-mains. As in the rest of the paper, we assume M is a smooth, closed, connected orientablesurface. Theorem A.1.
Let f be a Morse–Smale function on M and Ω a Neumann domain of f .Then there exists a Morse–Smale function ˜ f that has a type-(ii) Neumann domain ˜Ω ⊂ Ω . We will see in the proof that ˜ f can be chosen to agree with f outside an arbitrary openset U ⊂ Ω. However, the difference ˜ f − f may be large inside U . The existence of ˜ f isgiven by the following general lemma. Lemma A.2.
Let U ⊂ M be an open subset, and f : U → R a smooth function having nocritical points. There exists a smooth function ˜ f : U → R , with supp( ˜ f − f ) ⊂ U , whoseonly critical points are a non-degenerate maximum and a non-degenerate saddle.Proof. Since f has no critical points in U , we can invoke the canonical form theorem forsmooth vector fields and find local coordinates ( x, y ) with respect to which f ( x, y ) = Ax + , GRAHAM COX , SEBASTIAN K. EGGER − Ax x Figure A.1.
The function α ( x ) used in the proof of Lemma A.2. B , for ( x, y ) ∈ ( − , × ( − , α ( x ) with supp α ⊂ ( − , (cid:90) − α ( x ) dx = 0 , (A.1)so that there exist points − < x < x < α ( x ) > − A, − < x < x , = − A, x = x ,< − A, x < x < x , = − A, x = x ,> − A, x < x < , (A.2)as shown in Figure A.1.We define ˜ f ( x, y ) = f ( x, y ) + β ( x ) γ ( y ) , (A.3)where β ( x ) = (cid:82) x − α ( t ) dt and γ ( y ) = exp {− / (1 − y ) } . Note that γ is a non-negativebump function supported in ( − ,
1) with γ (cid:48) (0) = 0 and γ (cid:48)(cid:48) (0) <
0. It follows that ∂ ˜ f∂x = A + α ( x ) γ ( y ) and ∂ ˜ f∂y = β ( x ) γ (cid:48) ( y ) , EFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN 21 and so the only critical points of ˜ f in U are ( x ,
0) and ( x , ∂ ˜ f∂x ( x ,
0) = α (cid:48) ( x ) γ (0) < ,∂ ˜ f∂x ( x ,
0) = α (cid:48) ( x ) γ (0) > ,∂ ˜ f∂y ( x i ,
0) = β ( x i ) γ (cid:48)(cid:48) (0) < , and conclude that ( x ,
0) and ( x ,
0) are a non-degenerate maximum and a non-degeneratesaddle, respectively. (cid:3)
Proof of Theorem A.1.
If Ω is of type (ii) we simply choose ˜ f = f and there is nothing toprove. Therefore we assume that Ω is of type (i), and hence its closure contains exactlyfour critical points, all of which are on the boundary: a maximum q , a minimum p , andsaddle points r and r ; see Figure 2.2.Now choose ˜ f according to Lemma A.2, for some open set U (cid:98) Ω. By construction, ˜ f has two critical points in Ω: a maximum q ∗ and a saddle point r ∗ . Since ˜ f is a Morsefunction, r ∗ has degree four, i.e. there are four Neumann lines connected to r ∗ . We obtainthe result by studying the endpoints of these lines depicted in Figure A.2. Since ˜ f agrees Figure A.2.
The dotted line represents the boundary of the support con-taining set U of ˜ f − f ; left: the generated type-(ii) Neumann domain, purplelines represent Neumann lines; right: The hypothetical red Neumann lineintersects with another Neumann line; note that the intersection of two suchNeumann lines is unavoidable as W s ( r ∗ ) and W u ( r ∗ ) are smooth submani-folds and two consecutive Neumann lines meeting at a non-degenerate saddlecan not meet the same extremal point .with f in a neighborhood of ∂ Ω, the invariant manifolds W s ( r i ) and W u ( r i ) are unchangedby the perturbation. As a result, it is not possible for any of the Neumann lines comingfrom r ∗ to end at r or r . Therefore, the four Neumann lines from r ∗ can only end at q , p or q ∗ , so it follows from Lemma 2.4 that ˜ f is Morse–Smale. The two lines along which ˜ f is decreasing must end at p , since it is the only minimum in Ω. This means the two lines , GRAHAM COX , SEBASTIAN K. EGGER along which f is increasing are connected to either q or q ∗ . We claim that there is oneNeumann line connected to each maximum.Suppose instead that both ended at q . Then the union of these Neumann lines formsa closed loop. Similarly, the union of the two lines ending at p is a closed loop. Bothloops intersect at r ∗ , where they are orthogonal by Proposition 2.5(1). Since Ω is simplyconnected, this can only happen if the loops also intersect at a point other than r ∗ , butthis is impossible since gradient flow lines cannot cross. The same argument shows thatthese lines cannot both be connected to q ∗ , hence one must end at each maximum.Since all of the Neumann lines in Ω have been accounted for, this means q ∗ has degreeone, hence the Neumann domain with q ∗ on its boundary is of type (ii). (cid:3) References [ABBE20] L. Alon, Ram Band, M. Bersudsky, and S. Egger,
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