Low Energy Limit for the Resolvent of Some Fibered Boundary Operators

Abstract

For certain Dirac operators $\eth _{\phi }$ associated to a fibered boundary metric $g_{\phi }$, we provide a pseudodifferential characterization of the limiting behavior of $(\eth _{\phi }+k\gamma )^{-1}$ as $k\searrow 0$, where $\gamma$ is a self-adjoint operator anti-commuting with $\eth _{\phi }$ and whose square is the identity. This yields in particular a pseudodifferential characterization of the low energy limit of the resolvent of $\eth _{\phi }^2$, generalizing a result of Guillarmou and Sher about the low energy limit of the resolvent of the Hodge Laplacian of an asymptotically conical metric. As an application, we use our result to give a pseudodifferential characterization of the inverse of some suspended version of the operator $\eth _{\phi }$. One important ingredient in the proof of our main theorem is that the Dirac operator $\eth _{\phi }$ is Fredholm when acting on suitable weighted Sobolev spaces. This result has been known to experts for some time and we take this as an occasion to provide a complete explicit proof.

Appendix A. Blow-ups in Manifolds with Corners

In this appendix, we will establish the commutativity of blow-ups of two p-submanifolds used in Lemma 4.4 to show that our two ways of constructing the $k,\phi$-double space are equivalent. Indeed, this result definitely requires a proof, especially since it does not seem to follow from standard results like the commutativity of nested blow-ups or the commutativity of blow-ups of transversal p-submanifolds.

Lemma A.1

Let W be a manifold with corners. Suppose that X and Y are two p-submanifolds such that their intersection $Z= X\cap Y$ is also a p-submanifold with the property that for every $w\in Z$, there is a coordinate chart

$$\begin{aligned} \varphi : {\mathcal {U}}\rightarrow {\mathbb {R}}^{n_1}_{k_1}\times {\mathbb {R}}^{n_2}_{k_2}\times {\mathbb {R}}^{n_3}_{k_3}\times {\mathbb {R}}^{n_4}_{k_4} \end{aligned}$$

(A.1)

sending w to the origin such that

$$\begin{aligned} \begin{aligned} \varphi ({\mathcal {U}}\cap X)&= \{0\}\times \{0\}\times {\mathbb {R}}^{n_3}_{k_3}\times {\mathbb {R}}^{n_4}_{k_4}, \\ \varphi ({\mathcal {U}}\cap Y)&= \{0\}\times {\mathbb {R}}^{n_2}_{k_2}\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4}, \\ \varphi ({\mathcal {U}}\cap Z)&= \{0\}\times \{0\}\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4}. \end{aligned} \end{aligned}$$

(A.2)

Then the identity map in the interior extends to a diffeomorphism

$$\begin{aligned} {[}W;X,Y,Z]\rightarrow [W;Y,X,Z]. \end{aligned}$$

(A.3)

Proof

Since we blow up Z last, notice first that this result does not quite follows from the commutativity of nested blow-ups. Now, clearly, away from Z, the blow-ups of X and Y commute since they do not intersect. Thus, to establish (A.3), it suffices to establish it near Z. Let $w\in Z$ be given and consider a coordinate chart $({\mathcal {U}},\varphi )$ as in (A.2). Let $x=(x_1,\ldots , x_{n_1})$, $y=(y_1,\ldots ,y_{n_2})$, $z=(z_1,\ldots ,z_{n_3})$ and $w=(w_1,\ldots ,w_{n_4})$ be the canonical coordinates for the factors ${\mathbb {R}}^{n_1}_{k_1}$, ${\mathbb {R}}^{n_2}_{k_2}$, ${\mathbb {R}}^{n_3}_{k_3}$ and ${\mathbb {R}}^{n_4}_{k_4}$ respectively.

When we blow up X, this coordinate chart is replaced by the one induced by the coordinates

$$\begin{aligned} (\omega _{x,y}=(\frac{x}{r},\frac{y}{r}), r=\sqrt{|x|^2+|y^2|},z,w)\in {\mathbb {S}}^{n_1+n_2-1}_{k_1+k_2}\times [0,\infty )_r\times {\mathbb {R}}^{n_3}_{k_3}\times {\mathbb {R}}^{n_4}_{k_4}. \end{aligned}$$

In this coordinate chart, the lifts of Y and Z corresponds to

$$\begin{aligned} (\{0\}\times {\mathbb {S}}^{n_2-1}_{k_2})\times [0,\infty )_r\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4} \end{aligned}$$

and

$$\begin{aligned} {\mathbb {S}}^{n_1+n_2-1}_{k_1+k_2}\times \{0\}\times \{0\} \times {\mathbb {R}}^{n_4}_{k_4}, \end{aligned}$$

where $\{0\}\times {\mathbb {S}}^{n_2-1}_{k_2}\subset {\mathbb {R}}^{n_1}_{k_1}\times {\mathbb {R}}^{n_2}_{k_2}$ is seen as a p-submanifold of ${\mathbb {S}}^{n_1+n_2-1}_{k_1+k_2}$ seen as the unit sphere in ${\mathbb {R}}^{n_1}_{k_1}\times {\mathbb {R}}^{n_2}_{k_2}$. To blow up Y, this suggests to consider the smaller coordinate chart induced by the coordinates

$$\begin{aligned} (x,\omega _y=\frac{y}{|y|},r,z,w)\in {\mathbb {R}}^{n_1}_{k_1}\times {\mathbb {S}}^{n_2-1}_{k_2}\times [0,\infty )_{r}\times {\mathbb {R}}^{n_3}_{k_3}\times {\mathbb {R}}^{n_4}_{k_4} \end{aligned}$$

in which the lift of Y corresponds to

$$\begin{aligned} \{0\}\times {\mathbb {S}}^{n_2-1}_{k_2}\times [0,\infty )_r\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4} \end{aligned}$$

and the lift of Z to

$$\begin{aligned} {\mathbb {R}}^{n_1}_{k_1}\times {\mathbb {S}}^{n_2-1}_{k_2}\times \{0\}\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4}. \end{aligned}$$

Hence, blowing up Y, we obtain a coordinate chart on [W; X, Y] by considering the one induced by the coordinates

$$\begin{aligned}&(\omega _{x,z}=(\frac{x}{\rho },\frac{z}{\rho }), \rho =\sqrt{|x|^2+|z|^2}, \omega _y=\frac{y}{|y|},r=\sqrt{|x|^2+|y|^2},w)\in {\mathbb {S}}^{n_1+n_3-1}_{k_1+k_3}\\&\quad \times [0,\infty )_{\rho } \times {\mathbb {S}}^{n_2-1}_{k_2}\times [0,\infty )_r\times {\mathbb {R}}^{n_4}_{k_4} \end{aligned}$$

in which the lift of Z corresponds to

$$\begin{aligned} ({\mathbb {S}}^{n_1-1}_{k_1}\times \{0\})\times [0,\infty )_{\rho }\times {\mathbb {S}}^{n_2-1}_{k_2}\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4} \end{aligned}$$

with ${\mathbb {S}}^{n_1-1}_{k_1}\times \{0\}\subset {\mathbb {R}}^{n_1}_{k_1}\times {\mathbb {R}}^{n_3}_{k_3}$ seen as a p-submanifold of ${\mathbb {S}}^{n_1+n_3-1}_{k_1+k_3}$. To blow up Z, this suggests to consider the coordinate charts induced by the coordinates

$$\begin{aligned}&(z,\omega _x=\frac{x}{|x|}, \rho =\sqrt{|x|^2+|z|^2}, \omega _y=\frac{y}{|y|}, r=\sqrt{|x|^2+|y|^2},w)\in {\mathbb {R}}^{n_3}_{k_{3}}\times {\mathbb {S}}^{n_1-1}_{k_1}\\&\quad \times [0,\infty )_{\rho }\times {\mathbb {S}}^{n_2-1}_{k_2}\times [0,\infty )_{r}\times {\mathbb {R}}^{n_4}_{k_4} \end{aligned}$$

in which the lift of Z corresponds to

$$\begin{aligned} \{0\}\times {\mathbb {S}}^{n_1-1}_{k_1}\times [0,\infty )_{\rho }\times {\mathbb {S}}^{n_2-1}_{k_2}\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4}. \end{aligned}$$

Hence, blowing up the lift of Z, we see that [W; X, Y, Z] admits a coordinate chart induced by the coordinates

$$\begin{aligned}&(\omega _{z,r}=(\frac{z}{s},\frac{r}{s}), s= \sqrt{|z|^2+r^2}, \omega _x= \frac{x}{|x|}, \omega _y=\frac{y}{|y|}, \rho =\sqrt{|x|^2+ |y|^2},w) \nonumber \\&\quad \in {\mathbb {S}}^{n_3}_{k_3+1}\times [0,\infty )_{s}\times {\mathbb {S}}^{n_1-1}_{k_1}\times {\mathbb {S}}^{n_2-1}_{k_2}\times [0,\infty )_{\rho }\times {\mathbb {R}}^{n_4}_{k_4}. \end{aligned}$$

(A.4)

In this chart, Z lifts to

$$\begin{aligned} {\mathbb {S}}^{n_3}_{k_3+1}\times \{0\}\times {\mathbb {S}}^{n_1-1}_{k_1}\times {\mathbb {S}}^{n_2-1}_{k_2}\times [0,\infty )_{\rho }\times {\mathbb {R}}^{n_4}_{k_4}, \end{aligned}$$

Y lifts to

$$\begin{aligned} {\mathbb {S}}^{n_3}_{k_3+1}\times [0,\infty )_s\times {\mathbb {S}}^{n_1-1}_{k_1}\times {\mathbb {S}}^{n_2-1}_{k_2}\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4} \end{aligned}$$

and X lifts to

$$\begin{aligned} ({\mathbb {S}}^{n_3-1}_{k_3}\times \{0\})\times [0,\infty )_s\times {\mathbb {S}}^{n_1-1}_{k_1}\times {\mathbb {S}}^{n_2-1}_{k_2}\times [0,\infty )_{\rho }\times {\mathbb {R}}^{n_4}_{k_4}, \end{aligned}$$

where ${\mathbb {S}}^{n_3-1}_{k_3}\times \{0\}\subset {\mathbb {R}}^{n_3}_{k_3}\times [0,\infty )_r$ is seen as a p-submanifold of the unit sphere ${\mathbb {S}}^{n_3}_{k_3+1}\subset {\mathbb {R}}^{n_3}_{k_3}\times [0,\infty )_{r}$.

Since we are interested in the commutativity of the blow-ups of X and Y when the blow-up of Z is subsequently performed, we can consider instead a smaller coordinate chart on [W; X, Y, Z] in a neighborhood of the lift of X which is induced by the coordinates

$$\begin{aligned}&(\omega _x=\frac{x}{|x|}, \omega _y=\frac{y}{|y|}, \omega _z=\frac{z}{|z|}, r=\sqrt{|x|^2+|y|^2},\nonumber \\&\rho =\sqrt{|x|^2+|z|^2},s=\sqrt{|x|^2+|y|^2+|z|^2},w) \nonumber \\&\quad \in {\mathbb {S}}^{n_1-1}_{k_1}\times {\mathbb {S}}^{n_2-1}_{k_2}\times {\mathbb {S}}^{n_3-1}_{k_3}\times [0,\infty )_{r}\times [0,\infty )_{\rho }\times [0,\infty )_{s}\times {\mathbb {R}}^{n_4}_{k_4}. \end{aligned}$$

(A.5)

This chart is defined near the intersection of the lifts of X, Y and Z, which corresponds to ${\mathbb {S}}^{n_1-1}_{k_1}\times {\mathbb {S}}^{n_2-1}_{k_2}\times {\mathbb {S}}^{n_3-1}_{k_3}\times \{0\}\times \{0\}\times \{0\}\times {\mathbb {R}}^{n_4}_{k_4}$.

The definition of this system of coordinates is symmetric with respect to X and Y, namely, considering instead [W; Y, X, Z], we would have obtain the same coordinate system valid near the intersection of the lifts of X, Y and Z. This indicates that in this region, the identity map in the interior naturally extends to a diffeomorphism. Since this clear elsewhere, the result follows. $\quad \square$