The $\overline{\partial }$-Neumann Problem

Abstract

Here we study a boundary problem arising in the theory of functions of several complex variables. A function u on an open domain $\Omega \subset {\mathbb{C}}^{n}$ is holomorphic if $\bar{\partial }u = 0$, where

$$\bar{\partial }u ={ \sum \limits_{j}} \frac{\partial u} {\partial \bar{{z}}_{j}}\ d\bar{{z}}_{j},$$

(0.1)

with $d\bar{{z}}_{j} = d{x}_{j} - id{y}_{j}$ and

$$\frac{\partial u} {\partial \bar{{z}}_{j}} = \frac{1} {2}\left (\frac{\partial u} {\partial {x}_{j}} + i \frac{\partial u} {\partial {y}_{j}}\right).$$

(0.2)

In the study of complex function theory on Ω, one is led to consider the equation

$$\bar{\partial }u = f,$$

(0.3)

with $f = \sum \nolimits {f}_{j}\ d\bar{{z}}_{j}$. More generally, one studies (0.3) as an equation for a (0, q)-form u, given a (0, q + 1)-form f; definitions of these terms are given in §1.

1 B Complements on the Levi form

In this appendix we will give further formulas and other results for the Levi form on a hypersurface in $\mathbb{C}^{n}$. As a preliminary, we reexamine the formulas (2.7) and (2.8) in terms of the complex vector fields

$$Z = \sum \nolimits {u}_{k} \frac{\partial } {\partial {z}_{k}},\quad \overline{Z} = \sum \nolimits {\overline{u}}_{k} \frac{\partial } {\partial \bar{{z}}_{k}}. $$

(B.1)

We assume that at each $p \in \partial \Omega,\ u(p) = ({u}_{1},\ldots,{u}_{n})$ belongs to ${\mathfrak{H}}_{p}(\partial \Omega)$, defined by (2.13). As noted in §2, this hypothesis is equivalent both to Zρ = 0 and to $\overline{Z}\rho = 0$ on ∂Ω. Then (2.7) simply says $\overline{Z}Z\rho = 0$ on ∂Ω. Also, of course, $Z\overline{Z}\rho = 0$ on ∂Ω, and hence

$$[Z,\overline{Z}]\rho = 0\quad \text{ on }\ \partial \Omega,$$

but this is not the content of (2.8). To restate (2.8), note that

$$[Z,\overline{Z}] ={ \sum \limits_{j,k}}\left ({u}_{k}\frac{\partial {\overline{u}}_{j}} {\partial {z}_{k}} \frac{\partial } {\partial \bar{{z}}_{j}} -{\overline{u}}_{j}\frac{\partial {u}_{k}} {\partial \bar{{z}}_{j}} \frac{\partial } {\partial {z}_{k}}\right). $$

(B.2)

Now let us apply the operator J that gives the complex structure of $\mathbb{C}^{n}$, so

$$J \frac{\partial } {\partial {x}_{j}} = \frac{\partial } {\partial {y}_{j}},\quad J \frac{\partial } {\partial {y}_{j}} = - \frac{\partial } {\partial {x}_{j}}, $$

(B.3)

and hence

$$J \frac{\partial } {\partial {z}_{j}} = i \frac{\partial } {\partial {z}_{j}},\quad J \frac{\partial } {\partial \bar{{z}}_{j}} = -i \frac{\partial } {\partial \bar{{z}}_{j}}. $$

(B.4)

We have

$$W = J[Z,\overline{Z}] = -i{\sum \limits_{j,k}}\left ({u}_{k}\frac{\partial {\overline{u}}_{j}} {\partial {z}_{k}} \frac{\partial } {\partial \bar{{z}}_{j}} +{ \overline{u}}_{k}\frac{\partial {u}_{j}} {\partial \bar{{z}}_{k}} \frac{\partial } {\partial {z}_{j}}\right), $$

(B.5)

where to get the last term in parentheses from J applied to (B.2), we have interchanged the roles of j and k. Hence

$$W\rho = -2i\text{ Re }\left ({\sum \limits_{j,k}}{\overline{u}}_{k}\frac{\partial {u}_{j}} {\partial \bar{{z}}_{k}} \frac{\partial \rho } {\partial {z}_{j}}\right). $$

(B.6)

Now the quantity in parentheses here is precisely the left side of (2.8). Since the right side of (2.8) is clearly real-valued, we have

$$\frac{1} {2i}{\Bigl\langle J[Z,\overline{Z}],d\rho \Bigr \rangle} ={ \sum \limits_{j,k}}{\mathcal{L}}_{jk}{u}_{j}{\overline{u}}_{k}\quad \text{ on }\ \partial \Omega. $$

(B.7)

Let α = J^t dρ, so the left side of (B.7) is 1 ∕ 2i times $\alpha ([Z,\overline{Z}])$. Since

$$(d\alpha)(Z,\overline{Z}) = Z \cdot \alpha (\overline{Z}) -\overline{Z} \cdot \alpha (z) - \alpha ([Z,\overline{Z}])$$

and $\alpha (\overline{Z}) = d\rho (J\overline{Z}) = -i\overline{Z}\rho = 0$, we have $(d\alpha)(Z,\overline{Z}) = -\alpha ([Z,\overline{Z}])$, so (B.7) implies

$${\sum \limits_{j,k}}{\mathcal{L}}_{jk}{u}_{j}{\overline{u}}_{k} = -\frac{1} {2i}(d\alpha)(Z,\overline{Z}), $$

(B.8)

another useful formula for the Levi form.

It is also useful to write these formulas in terms of

$$X ={ \sum \limits_{k}}\left ({f}_{k} \frac{\partial } {\partial {x}_{k}} + {g}_{k} \frac{\partial } {\partial {y}_{k}}\right),\quad {u}_{k} = {f}_{k} + i{g}_{k}, $$

(B.9)

where ${f}_{k} = \text{ Re }{u}_{k},\ {g}_{k} = \text{ Im }{u}_{k}$. The hypothesis Zρ = 0 is still in effect, so, for $p \in \partial \Omega,\ X(p) \in {\mathfrak{H}}_{p}(\partial \Omega) \subset {T}_{p}\partial \Omega \subset {\mathbb{R}}^{2n}$. Note that

$$2Z = X - iJX,\quad 2\overline{Z} = X + iJX. $$

(B.10)

Thus (B.8) implies

$$4{\sum \limits_{j,k}}{\mathcal{L}}_{jk}{u}_{j}{\overline{u}}_{k} = -(d\alpha)(X,JX). $$

(B.11)

Following the trail (B.7) ⇒ (B.8) backward, we note that $(d\alpha)(X,JX) = X \cdot \alpha (JX) - (JX) \cdot \alpha (X) - \alpha ([X,JX])$ and $\alpha (X) = 0 = \alpha (JX)$, so $(d\alpha)(X,JX) = -\alpha ([X,JX])$, and hence

$$4{\sum \limits_{j,k}}{\mathcal{L}}_{jk}{u}_{j}{\overline{u}}_{k} = \alpha ([X,JX]) ={\Bigl \langle J[X,JX],d\rho \Bigr \rangle}. $$

(B.12)

Note also that, by direct calculation,

$$4{\sum \limits_{j,k}}{\mathcal{L}}_{jk}{u}_{j}{\overline{u}}_{k} = H(X,X) + H(JX,JX), $$

(B.13)

where H is the (2n) × (2n) real Hessian matrix of second-order partial derivatives of ρ with respect to $({x}_{1},\ldots,{x}_{n},{y}_{1},\ldots,{y}_{n})$.

We can recast the Levi form in the following more invariant way, as done in [HN]. For a local section X of $\mathfrak{H}(\partial \Omega)$, we will define

$${\mathcal{L}}_{p}(X,X) \in {\mathfrak{H}}_{p}^{0}(\partial \Omega), $$

(B.14)

where ${\mathfrak{H}}_{p}^{0}(\partial \Omega) \subset {T}_{p}^{\ast}(\partial \Omega)$ is the annihilator of ${\mathfrak{H}}_{p}(\partial \Omega) \subset {T}_{p}(\partial \Omega)$. To do this, we set

$$\begin{array}{rlrlrl}4\mathcal{L}(X,X)(\alpha)& =\langle [X,JX],\alpha \rangle & & \cr & = -(d\alpha)(X,JX).& \cr \end{array} $$

(B.15)

When α = J^t dρ, this coincides with (B.11)–(B.12). This object is clearly invariant under conjugation by biholomorphic maps (i.e., under biholomorphic changes of coordinates). The property of positivity of ℒ is invariantly defined, since the real line bundle ${\mathfrak{H}}_{p}^{0}(\partial \Omega)$ has a natural orientation, defined by declaring that J^t dρ > 0. Thus we have the following:

Proposition B.1.

If $\bar{\Omega }$ is strongly pseudoconvex at p ∈ ∂Ω and if $F : \mathcal{O}\,\rightarrow \,U$ $\subset {\mathbb{C}}^{n}$ is a biholomorphic map defined on a neighborhood of p, then $F(\mathcal{O}\cap \bar{ \Omega }) =\widetilde{ \Omega }$ is strongly pseudoconvex at $\tilde{p} = F(p)$.

It follows readily from (B.13) that $\bar{\Omega }$ is strongly pseudoconvex at any p ∈ ∂Ω at which $\bar{\Omega }$ is strongly convex. By Proposition B.1 we see then that any (local) biholomorphic image of a strongly convex $\bar{\Omega } \subset {\mathbb{C}}^{n}$ is (locally) strongly pseudoconvex.

We can also relate the Levi form to the second fundamental form of ∂Ω as a hypersurface of ℝ²ⁿ, using the following:

Lemma B.2.

If II is the second fundamental form of ∂Ω ⊂ ℝ ²ⁿ , and if X is a section of $\mathfrak{H}(\partial \Omega)$ , then

$$II(X,X) = -{P}_{N}J\ {\nabla }_{X}(JX) = -J{P}_{JN}\ {\nabla }_{X}(JX). $$

(B.16)

Here, ∇ is the Levi–Civita connection on ∂Ω, P_N is the orthogonal projection of ℝ²ⁿ onto the span of $N = -\nabla \rho$ (the sign chosen so N points inward), and P_JN is the orthogonal projection of ℝ²ⁿ onto the span of JN. We denote the span of JN by ${\mathfrak{H}}^{\perp }(\partial \Omega)$, which is isomorphic to ${\mathfrak{H}}^{0}(\partial \Omega)$, via the Riemannian metric on ∂Ω.

To prove the lemma, recall from §4 of Appendix C (Connections and Curvature) that if X and Y are tangent to ∂Ω, then II(X, Y) = P_N D_X Y, where D_X denotes the standard flat connection on ℝ²ⁿ. Of course, also $II(X,Y) = {D}_{X}Y -{\nabla }_{X}Y$. Note that D_X(JY) = JD_X Y, so $II(JX,X) = II(X,JX) = {P}_{N}{D}_{X}(JX) = {P}_{N}J({D}_{X}X)$. Hence

$$II(JX,X) = {P}_{N}J\ II(X,X) + {P}_{N}J\ {\nabla }_{X}X. $$

(B.17)

Similarly, $II(JX,JX) = {P}_{N}J{D}_{JX}X = {P}_{N}J\ II(JX,X) + {P}_{N}J\ {\nabla }_{JX}X$, and substituting (B.17) yields

$$II(JX,JX) = {P}_{N}J{P}_{N}J\ II(X,X) + {P}_{N}J{P}_{N}J\ {\nabla }_{X}X + {P}_{N}J\ {\nabla }_{JX}X.$$

Now, $J{P}_{N}J = -{P}_{JN}$, which is orthogonal to P_N, so P_N JP_N J = 0, and we have

$$II(JX,JX) = {P}_{N}J\ {\nabla }_{JX}X = J{P}_{JN}\ {\nabla }_{JX}X. $$

(B.18)

Replacing X by JX hence yields (B.16) and proves the lemma.

We can add (B.16) and (B.18), obtaining

$$\begin{array}{lll} II& (X,X) + II(JX,JX) & \\ & = {P}_{N}J{\Bigl [{\nabla }_{JX}X -{\nabla }_{X}(JX)\Bigr ]} = {P}_{N}J\ [JX,X].\end{array}$$

(B.19)

Comparing this with (B.12) and using the notation $II(X,Y) =\widetilde{ II}(X,Y)N$, as in (4.15) of Appendix C, we see that

$$4{\sum \limits_{j,k}}{\mathcal{L}}_{jk}{u}_{j}{\overline{u}}_{k} =\widetilde{ II}(X,X) +\widetilde{ II}(JX,JX). $$

(B.20)

This can also be obtained from (B.13), plus formula (4.25) of Appendix C.

We will consider one more formula for the Levi form, in terms of the geometry of $\mathfrak{H}(\partial \Omega)$ as a subbundle of the trivial bundle $\partial \Omega \times {\mathbb{R}}^{2n} \approx \partial \Omega \times {\mathbb{C}}^{n}$. Associated to this subbundle there is a second fundamental form $I{I}_{\mathfrak{H}}$, defined as in (4.40) of Appendix C. A formula for $I{I}_{\mathfrak{H}}$ can be given as follows. Let $\mathfrak{K}(\partial \Omega)$ denote the orthogonal complement of $\mathfrak{H}(\partial \Omega)$; this can be viewed as a real vector bundle of rank 2, generated by N and JN, or as a complex line bundle generated by N. If ${P}_{\mathfrak{K}}$ denotes the orthogonal projection of ℝ²ⁿ onto $\mathfrak{K}$, then we have

$$I{I}_{\mathfrak{H}}(X,Y) = {P}_{\mathfrak{K}}{D}_{X}Y, $$

(B.21)

when X and Y are sections of $\mathfrak{H}(\partial \Omega)$.

We want to relate $I{I}_{\mathfrak{H}}$ to the Levi form. It is convenient to use the previous analysis of II. Since ${P}_{\mathfrak{K}} = {P}_{N} + {P}_{JN}$, we have

$$I{I}_{\mathfrak{H}}(X,X) = II(X,X) + {P}_{JN}{D}_{X}X.$$

As noted in the proof of (B.16), II(JX, X) = P_N J D_X X, which is equal to JP_JN D_X X, so we have

$$I{I}_{\mathfrak{H}}(X,X) = II(X,X) - J\ II(JX,X). $$

(B.22)

Substituting JX for X, we have

$$I{I}_{\mathfrak{H}}(JX,JX) = II(JX,JX) + J\ II(JX,X), $$

(B.23)

and adding this to (B.22) and using (B.20), we obtain

$$I{I}_{\mathfrak{H}}(X,X) + I{I}_{\mathfrak{H}}(JX,JX) = 4{\Bigl ({\sum \limits_{j,k}}{\mathcal{L}}_{jk}{u}_{j}{\overline{u}}_{k}\Bigr)}N. $$

(B.24)

2 C The Neumann operator for the Dirichlet problem

Let $\overline{M}$ be a compact Riemannian manifold with boundary ∂M = X. Then X has an induced Riemannian metric, and $X\hookrightarrow \overline{M}$ has a second fundamental form, with associated Weingarten map

$${A}_{N} : {T}_{x}X\rightarrow {T}_{x}X, $$

(C.1)

defined as in §4 of Appendix C, Connections and Curvature. We take N to be the unit normal to X, pointing into M.

Both $\overline{M}$ and X have Laplace operators, which we denote Δ and Δ_X, respectively. The Neumann operator $\mathcal{N}$ is an operator on $\mathcal{D}'(X)$ defined as follows:

$$\mathcal{N}f = \frac{\partial u} {\partial N},\quad u = \text{ PI}\,f, $$

(C.2)

where to say u = PI f is to say

$$\Delta u = 0\ \text{ on }\ M,\quad {u\Bigr |}_{\partial M} = f. $$

(C.3)

As shown in §§11 and 12 of Chap. 7, $\mathcal{N}$ is a negative-semidefinite, self-adjoint operator, and also an elliptic operator in OPS¹(X). It is fairly easy to see that

$$\mathcal{N} = -\sqrt{-{\Delta }_{X}}\quad \text{ mod }OP{S}^{0}(X). $$

(C.4)

Our main purpose here is to capture the principal part of the difference.

Proposition C.1.

The Neumann operator $\mathcal{N}$ is given by

$$\mathcal{N} = -\sqrt{-{\Delta }_{X}} + B\quad { mod }OP{S}^{-1}(X), $$

(C.5)

where B ∈ OPS ⁰ (X) has principal symbol

$${\sigma }_{B}(x,\xi) = \frac{1} {2}\left (\text{Tr }{A}_{N} -\frac{\langle {A}_{N}^{\ast}\xi,\xi \rangle } {\langle \xi,\xi \rangle } \right). $$

(C.6)

Here, A_N^∗ : T_x^∗ X → T_x^∗ X is the adjoint of (C.1), and ⟨, ⟩ is the inner product on T_x^∗ X arising from the given Riemannian metric.

To prove this, we choose coordinates $x = ({x}_{1},\ldots,{x}_{m-1})$ on an open set in X (if dim M = m) and then coordinates (x, y) on a neighborhood in $\overline{M}$ such that y = 0 on X and |∇ y| = 1 near X while y > 0 on M and such that x is constant on each geodesic segment in $\overline{M}$ normal to X. Then the metric tensor on $\overline{M}$ has the form

$${\Bigl ({g}_{jk}(x,y)\Bigr)} = \left (\begin{array}{c@{\quad }c} {h}_{jk}(x,y)\quad &0\\ 0 \quad &1 \end{array} \right), $$

(C.7)

where, in the first matrix, 1 ≤ j, k ≤ m, and in the second, 1 ≤ j, k ≤ m − 1. Thus the Laplace operator Δ on $\overline{M}$ is given in local coordinates by

$$\begin{array}{lll} \Delta u& = {g}^{-1/2}{\partial }_{j}{\Bigl ({g}^{1/2}{g}^{jk}{\partial }_{k}u\Bigr)} & \\ & = {h}^{-1/2}{\partial }_{y}({h}^{1/2}{\partial }_{y}u) + {h}^{-1/2}{\partial }_{j}{\Bigl ({h}^{1/2}{h}^{jk}{\partial }_{k}u\Bigr)}& \\ & = {\partial }_{y}^{2}u + \frac{1} {2} \frac{{h}_{y}} {h} {\partial }_{y}u + L(y,x,{D}_{x})u,\end{array}$$

(C.8)

where, as usual,

$$g = \text{ det}({g}_{jk}),\quad h = \text{ det}({h}_{jk}); $$

(C.9)

we set ${h}_{y} = \partial h/\partial y$, and L(y) = L(y, x, D_x) is a family of Laplace operators on X, associated to the family of metrics (h_jk(y)) on X, so L(0) = Δ_X. In other words,

$$\Delta u = {\partial }_{y}^{2}u + a(y){\partial }_{ y}u + L(y)u,\quad a(y) = \frac{1} {2} \frac{{h}_{y}} {h}.$$

(C.10)

We will construct smooth families of operators A_j(y) ∈ OPS¹(X) such that

$${\partial }_{y}^{2} + a(y){\partial }_{ y} + L(y) ={\Bigl ({\partial }_{y} - {A}_{1}(y)\Bigr)}{\Bigl ({\partial }_{y} + {A}_{2}(y)\Bigr)}, $$

(C.11)

modulo a smoothing operator. The principal parts of A₁(y) and A₂(y) will be $\sqrt{ -L(y)}$. It will follow that

$$\mathcal{N} = -{A}_{2}(0)\ \text{ mod }\ OP{S}^{-\infty }(X), $$

(C.12)

and we can then read off (C.5)–(C.6).

To construct A_j(y), we compute that the right side of (C.11) is equal to

$${\partial }_{y}^{2} - {A}_{ 1}(y){\partial }_{y} + {A}_{2}(y){\partial }_{y} + {A'}_{2}(y) - {A}_{1}(y){A}_{2}(y), $$

(C.13)

so we need

$$\begin{array}{lll}{A}_{2}(y) - {A}_{1}(y) = a(y), \\- {A}_{1}(y){A}_{2}(y) + {A'}_{2}(y) = L(y).\end{array}$$

(C.14)

Substituting ${A}_{2}(y) = {A}_{1}(y) + a(y)$ into the second identity, we get an equation for A₁(y):

$${A}_{1}{(y)}^{2} + {A}_{ 1}(y)a(y) - {A'}_{1}(y) = -L(y) + a'(y). $$

(C.15)

Now set

$${A}_{1}(y) = \Lambda (y) + B(y),\quad \Lambda (y) = \sqrt{-L(y)}. $$

(C.16)

We get an equation for B(y):

$$\begin{array}{rcl} 2B(y)\Lambda (y) + [\Lambda (y),B(y)] + B{(y)}^{2} - B'(y) + B(y)a(y)& & \\ = \Lambda '(y) - \Lambda (y)a(y) + a'(y).\end{array}$$

(C.17)

Granted that B(y) is a smooth family in OPS⁰(X), the principal part B₀(y) must satisfy $2{B}_{0}(y)\Lambda (y) = \Lambda '(y) - a(y)\Lambda (y)$, or

$${B}_{0}(y) = \frac{1} {2}\Lambda '(y)\Lambda {(y)}^{-1} -\frac{1} {2}a(y)\quad \text{ mod }OP{S}^{-1}(X). $$

(C.18)

We can inductively obtain further terms B_j(y) ∈ OPS^{− j}(X) and establish that, with B(y) ∼ ∑_{j ≥ 0} B_j(y), the operators

$${A}_{1}(y) = \sqrt{-L(y)} + B(y),\quad {A}_{2}(y) = \sqrt{-L(y)} + B(y) + a(y)$$

do yield (C.11) modulo a smoothing operator. Details are similar to those arising in the decoupling procedure in §12 of Chap. 7.

Given this, we have (C.5) with

$$-B = {B}_{0}(0) + a(0) = \frac{1} {2}\Lambda '(0)\Lambda {(0)}^{-1} + \frac{1} {2}a(0)\quad \text{ mod }OP{S}^{-1}(X). $$

(C.19)

In turn, since $\Lambda (y) = \sqrt{-L(y)}$, we have

$$\Lambda '(0)\Lambda {(0)}^{-1} = \frac{1} {2}L'(0)L{(0)}^{-1}\quad \text{ mod }OP{S}^{-1}(X). $$

(C.20)

Hence

$$-B = \frac{1} {4}{\Bigl (L'(0)L{(0)}^{-1} + \frac{{h}_{y}} {h} (0,x)\Bigr)}. $$

(C.21)

To compute the symbol of B, note that

$${\sigma }_{L'(0)}(x,\xi) = -\sum \nolimits {\partial }_{y}{h}^{jk}(0,x){\xi }_{ j}{\xi }_{k}, $$

(C.22)

while, of course, ${\sigma }_{L(0)}(x,\xi) = -\sum \nolimits {h}^{jk}(0,x){\xi }_{j}{\xi }_{k} = -\langle \xi,\xi \rangle$. Now, (4.68)–(4.70) of Appendix C, Connections and Curvature, we have

$$\sum \nolimits {\partial }_{y}{h}_{jk}(0,x){U}_{j}{V }_{k} = -2\langle {A}_{N}U,V \rangle, $$

(C.23)

so

$$\sum \nolimits {\partial }_{y}{h}^{jk}(0,x){\xi }_{ j}{\xi }_{k} = 2\langle {A}_{N}^{\ast}\xi,\xi \rangle. $$

(C.24)

Thus,

$${\sigma }_{L'(0)L{(0)}^{-1}}(x,\xi) = 2\frac{\langle {A}_{N}^{\ast}\xi,\xi \rangle } {\langle \xi,\xi \rangle }. $$

(C.25)

Next, for $h = \text{ Det}({h}_{jk}) = \text{ Det }H$, we have ${h}_{y} = h\text{ Tr}({H}^{-1}{H}_{y})$. Looking in a normal coordinate system on X, centered at x₀, we have

$$\frac{{h}_{y}} {h} (0,{x}_{0}) ={ \sum \limits_{j}}{\partial }_{y}{h}_{jj}(0,{x}_{0}) = -2\text{ Tr }{A}_{N}, $$

(C.26)

the last identity by (C.23). Combining (C.25) and (C.26) yields the desired formula (C.6).

The following alternative way of writing (C.6) is useful. We have

$${\sigma }_{B}(x,\xi) = \frac{1} {2}\text{ Tr}\,({A}_{N}^{\ast}{P}_{ \xi }^{\perp }), $$

(C.27)

where, for nonzero ξ ∈ T_x^∗ X, P_ξ^⊥ is the orthogonal projection of T_x^∗ X onto the orthogonal complement of the linear span of ξ. Another equivalent formula is

$${\sigma }_{B}(x,\xi) = \frac{1} {2}\text{ Tr}\,({A}_{N}{P}_{\xi }^{0}), $$

(C.28)

where P_ξ⁰ is the orthogonal projection of T_x X onto the subspace annihilated by ξ.

To close, we mention the special case where $\overline{M}$ is the closed unit ball in ℝ^m, so $\partial M = {S}^{m-1}$. It follows from (4.5)–(4.6) of Chap. 8 that

$$\mathcal{N} = -\sqrt{-{\Delta }_{X } + {c}_{m }^{2}} + {c}_{m},\quad {c}_{m} = \frac{m - 2} {2}, $$

(C.29)

in this case. Note that, in this case, A_N = I, so this formula is consistent with (C.5)–(C.6).

We mention that calculations of the symbol of $\mathcal{N}$ in a similar spirit (but for a different purpose) were done in [LU]. Another approach was taken in [CNS].