* [96] Anuja Nagpal.  L1 and L2 regularization methods, 2017.
  * [97] Andrew Y Ng.  Feature selection, L1 vs. L2 regularization, and rotational invariance.  In International Conference on Machine Learning, 2004.
  * [98] Zhaohui Zheng, Xiaoyun Wu, and Rohini Srihari.  Feature selection for text categorization on imbalanced data.  ACM SIGKDD Explorations Newsletter, 6(1):80–89, 2004.
  * [99] Vladimir Vapnik.  The nature of statistical learning theory.  Springer, New York, NY, USA, 1995.
  * [100] Jeremy Howard and Sylvain Gugger.  Fastai: A layered API for deep learning.  2020\.
  * [101] Tianqi Chen and Carlos Guestrin.  XGBoost: A scalable tree boosting system.  In International Conference on Knowledge Discovery, 2016.
  * [102] James Bergstra and Yoshua Bengio.  Random search for hyper-parameter optimization.  Journal of Machine Learning Research, 13(10):281–305, 2012.
  * [103] Galit Shmueli, Soumya Ray, Juan Manuel Velasquez Estrada, and Suneel Babu Chatla.  The elephant in the room: Predictive performance of PLS models.  Journal of Business Research, 69(10):4552–4564, 2016.
  * [104] John Venable, Jan Pries-Heje, and Richard Baskerville.  FEDS: A framework for evaluation in design science research.  European Journal of Information Systems, 25:77–89, 2016.
  * [105] Allen S. Lee and Geoffrey S. Hubona.  A scientific basis for rigor in information systems research.  MIS Quarterly: Management Information Systems, 2009.
  * [106] Fred D. Davis, Richard P. Bagozzi, and Paul R. Warshaw.  User acceptance of computer technology: A comparison of two theoretical models.  Management Science, 35(8):982–1003, 1989.
  * [107] David Gefen, Elena Karahanna, and Detmar W. Straub.  Trust and tam in online shopping: An integrated model.  MIS Quarterly: Management Information Systems, 27(1):51–90, 2003\.
  * [108] Viswanath Venkatesh, James Y.L. Thong, and Xin Xu.  Consumer acceptance and use of information technology: Extending the unified theory of acceptance and use of technology.  MIS Quarterly: Management Information Systems, 36(1):157–178, 2012\.
  * [109] IPropertyManagement.  Airbnb statistics, 2020.
  * [110] Macrotrends LLC.  Airbnb Revenue 2018-2022.  https://www.macrotrends.net/stocks/charts/ABNB/airbnb/revenue, 2022.
  * [111] Airbnb Community.  How to start an Airbnb, 2016.
  * [112] Airbnb Inc.  Set a price strategy, 2020.
  * [113] Guesty Inc.  2 Steps To Skyrocket Your Airbnb Pricing Strategy, 2018.
  * [114] Renting Your Place.  Airbnb Pricing Strategy, 2020.
  * [115] Omid Poursaeed, Tomáš Matera, and Serge Belongie.  Vision-based real estate price estimation.  Machine Vision and Applications, 29:667–676, 2018.
  * [116] Vlad Hosu, Hanhe Lin, Tamas Sziranyi, and Dietmar Saupe.  KonIQ-10k: An ecologically valid database for deep learning of blind image quality assessment.  Transactions on Image Processing, 9:4041–4056, 2020.
  * [117] Barney G. Glaser and Anselm L. Strauss.  The discovery of grounded theory: Strategies for qualitative research.  Aldine Publishing, Chicago, USA, 1967.

## Appendix A: Feature Set

Feature Name | Feature Description
---|---
MOS |  | The mean opinion score is a measure of perceived image quality.
---
We applied it to homes’ preview images. MOS values are between
1 (low quality) and 5 (high quality).
bathrooms | Number of bathrooms.
bedrooms | Number of bedrooms.
beds | Number of beds.
person_capacity | Maximum number of guests.
picture_count | Number of uploaded pictures of a home.
min_nights | Minimum number of overnight stays required for booking.
max_nights | Maximum number of overnight stays.
can_instant_book | Boolean value that signalizes whether a home can be booked instantly.
verified_card | Boolean value that signalizes whether the host has a verified profile.
en | Boolean value that signalizes whether the host speaks English.
preview_wifi | Boolean value that signalizes whether WiFi is a preview amenity.
preview_kitchen | Boolean value that signalizes whether a kitchen is a preview amenity.
preview_heating | Boolean value that signalizes whether heating is a preview amenity.
preview_air_conditioning | Boolean value that signalizes whether air conditioning is a preview amenity.
preview_free_parking | Boolean value that signalizes whether free parking is a preview amenity.
preview_washer | Boolean value that signalizes whether a washer is a preview amenity.
preview_pool | Boolean value that signalizes whether a pool is a preview amenity.
preview_hot_tub | Boolean value that signalizes whether a hot tub is a preview amenity.
CANCEL_FLEXIBLE | Boolean value that signalizes whether canceling is “flexible”.
CANCEL_MODERATE | Boolean value that signalizes whether canceling is “moderate”.
CANCEL_STRICT_14 | Boolean value that signalizes whether canceling is allowed within 14 days.
CANCEL_SUPER_STRICT_30 | Boolean value that signalizes whether canceling is allowed within 30 days.
CANCEL_SUPER_STRICT_60 | Boolean value that signalizes whether canceling is allowed within 60 days.
picture_bathroom | Boolean value that signalizes whether the preview picture shows a bathroom.
picture_bedroom | Boolean value that signalizes whether the preview picture shows a bedroom.
picture_dining_room | Boolean value that signalizes whether the preview picture shows a dining room.
picture_exterior | Boolean value that signalizes whether the preview picture shows an exterior.
picture_interior | Boolean value that signalizes whether the preview picture shows an interior.
picture_kitchen | Boolean value that signalizes whether the preview picture shows a kitchen.
picture_living_room | Boolean value that signalizes whether the preview picture shows a living room.
family_amenities |  | Boolean value that signalizes whether a home is equipped with amenities for families
---
(e.g., baby bath, babysitter recommendations, children’s dinnerware).
This feature results from hierarchical clustering of amenities.
basic_amenities |  | Boolean value that signalizes whether a home only comprises essential amenities
---
(e.g., WiFi, heating, smoke alarm). This feature results from hierarchical
clustering of amenities.
normal_amenities |  | Boolean value that signalizes whether a home comprises more than only
---
essential amenities. This feature results from hierarchical clustering of
amenities.
accessible_amenities |  | Boolean value that signalizes whether a home is equipped with amenities that indicate
---
accessibility (e.g., no stairs and steps, wide entrance, extra space around
bed).
This feature results from hierarchical clustering of amenities.
entire_home | Boolean value that indicates whether a listing is an entire home.
private_room | Boolean value that indicates whether a listing is a private room.
hotel_room | Boolean value that indicates whether a listing is a hotel room.
shared_room | Boolean value that indicates whether a listing is a shared room.
other_home |  | Boolean value that indicates whether a listing is an alternative home
---
(e.g., houseboat, tree house, tiny house).
sum_foreign_languages | Number of foreign languages supported by the host.
distance_to_center | Distance from home to the city center, measured in kilometers.
city_population | Population of the city that corresponds to a home.
population_density | Population density per square kilometer of the city that corresponds to a home.
beds_per_bedroom | Ratio of beds per bedroom.
persons_per_bedroom | Ratio of persons per bedroom.
persons_per_bathroom | Ratio of persons per bathroom.

## Appendix B: Interaction Effects

Here, we show a selection of conspicuous interaction effects from the XGBoost
model using dependence plots. These are scatter plots that show values of a
feature on the x-axis and the corresponding SHAP values on the y-axis.
Moreover, the plotted dots are colored according to the values of an
interacting feature.

Figure B1 depicts two plots, each showing the SHAP values of corresponding
numbers of persons per bathroom. While the left plot colors dots according to
the values of the binary feature “entire_home”, the right colors these
according to the values of “private_room”. The figure unveils that negative
SHAP values of “persons_per_bathroom” highly interact with private rooms,
whereas entire homes heavily interact with SHAP values that are either
positive or close to zero. This indicates that entire homes exert a positive
impact on prices per night, while private rooms tend to be less expensive.
Although this effect might seem trivial, it is highly suited to demonstrate
the appearance of feature interaction in dependence plots.

Figure B1: Interaction effects between persons per bathroom and home types

Figure B2 depicts interaction effects between “population_density” and (i)
whether a home’s listing uses the amenity “heating” as a preview as well as
(ii) whether the listing uses the amenity “free parking” as a preview. The
figure’s left plot unveils homes in cities with a high population density to
use heating more often as a preview compared to less densely populated cities.
However, this does not mean that homes with heating as a preview amenity are
more expensive because of heating—it is also conceivable that the increased
SHAP values are due to rental prices that tend to be higher in bigger cities.
It is also unclear if the value of “preview_heating” is actually a result of
higher population density or vice versa, as it might likewise be the case that
other characteristics of the corresponding cities cause the values of both
features. Moreover, the right plot of Figure B2 shows that free parking is not
used as a preview amenity in cities with a particularly high population
density. This effect is rather unsurprising, as we expect parking spots in
highly populated cities to be rarer. Yet, having only the information from the
figure’s right plot, we cannot confirm free parking to increase the SHAP
values of low “population_density” values considerably.

Figure B2: Interaction effects between population density and preview
amenities

Figure B3 depicts interaction effects of the binary variables
“picture_exterior” (i.e., whether the preview picture shows an exterior) and
“picture_bedroom” (i.e., whether the preview picture shows a bedroom). Here,
the left plot unveils that for homes with an exterior picture, higher
distances to city centers tend to contribute to an increase in a home’s SHAP
values and vice versa. This might be due to the fact that hosts use attractive
surroundings of secluded homes as preview pictures to compensate for their
high distances to city centers—or, these homes are deliberately far from
cities, because they aim to address guests who seek to stay in a place less
urban. The plot in the middle of the figure further shows that, for these
homes with an exterior picture, high population tends to increase prices. This
leads to the following assumption: Homes with an attractive surrounding at the
edge of a particularly large city seem to be more expensive. Finally, the
right plot shows that, for homes with a bedroom as preview picture, higher
population values interact with lower SHAP values and vice versa.

Figure B3: Interaction effects between preview pictures and location-based
features

Figure B4 depicts interactions of canceling policies. Interestingly, for homes
with a flexible canceling strategy, more bedrooms tend to decrease SHAP
values. Moreover, while the use of a pool as a preview amenity seems to
increase SHAP values of homes with a moderate canceling strategy, we observe
the opposite effect for homes with a strict canceling policy of 14 days.

Figure B4: Interaction effects of canceling policies


$\left|{\mathbb{E}}\left[\Delta\tilde{W}^{\delta}\,|\,\mathcal{F}_{s_{0}}\right]\right|\leq
C\beta^{3},\quad\left|{\mathbb{E}}\left[(\Delta\tilde{W}^{\delta})^{2}-q(\tilde{W}^{\delta}_{s_{0}})^{2}\Delta
s\,|\,\mathcal{F}_{s_{0}}\right]\right|\leq C\beta^{3},$ (6.5)

where $\Delta s=s_{1}-s_{0}$ and
$\Delta\tilde{W}^{\delta}=\tilde{W}^{\delta}_{s_{1}}-\tilde{W}^{\delta}_{s_{0}}$.
Roughly speaking, (6.5) corresponds to the discrete version of (6.3).

For $t>0$, let $F^{\delta}_{t}$ be the function defined on $V_{\delta}$ that
is $-\lambda_{0}$ on vertices to the right side of $\eta_{\delta}[0,t]$,
$+\lambda_{0}$ on the vertices to the left side of $\eta_{\delta}[0,t]$, equal
to the boundary data on $\partial V_{\delta}$, and is harmonic at all other
vertices in $V_{\delta}$. Suppose $v\in V_{\delta}$ such that the distance
between $v$ and $\partial R_{L}$ is at least $1/4$. For $k=0,1$, define
$X_{k}={\mathbb{E}}[\phi_{\delta}(v)\,|\,\mathcal{F}_{s_{k}}]$ and let
$\mathcal{A}_{k}$ be the event

$|X_{k}-F_{s_{k}}(v)|\geq\beta^{5}.$

By [SS09, Proposition 3.27], we have
${\mathbb{P}}(\mathcal{A}_{k})\leq\frac{1}{4}\alpha\beta^{5}$ if $\delta$ is
small enough. Consequently, we have

$\displaystyle|X_{0}-F_{s_{0}}(v)|$ $\displaystyle\leq\beta^{5},\quad\text{on
}\mathcal{A}_{0}^{c};$
$\displaystyle|{\mathbb{E}}[X_{1}-F_{s_{1}}(v)\,|\,\mathcal{F}_{s_{0}}]|$
$\displaystyle\leq\beta^{5}+O(1){\mathbb{P}}(\mathcal{A}_{1}\,|\,\mathcal{F}_{s_{0}}).$

Let $\mathcal{A}$ be the event
${\mathbb{P}}(\mathcal{A}_{1}\,|\,\mathcal{F}_{s_{0}})\geq\beta^{5}$, then
${\mathbb{P}}(\mathcal{A})\leq\frac{1}{4}\alpha$. Combing the above two
estimates and the fact that
${\mathbb{E}}[X_{1}\,|\,\mathcal{F}_{s_{0}}]=X_{0}$, we have

${\mathbb{E}}[F_{s_{1}}(v)\,|\,\mathcal{F}_{s_{0}}]-F_{s_{0}}(v)=O(\beta^{5}),\quad\text{on
}\mathcal{A}_{0}^{c}\cap\mathcal{A}^{c}.$

For $k=0,1$, let $H_{k}$ be the (continuous) function that is $-\lambda_{0}$
to the right side of $\eta_{\delta}[0,s_{k}]$, $+\lambda_{0}$ to the left side
of $\eta_{\delta}[0,s_{k}]$, equal to the boundary data on $\partial R_{L}$,
and is harmonic in $R_{L}\setminus\eta_{\delta}[0,s_{k}]$. Then
$H_{k}(v)-F_{s_{k}}(v)$ is small if $\delta$ is small enough and the distance
between $v$ and $\eta_{\delta}[0,s_{k}]$ is bounded from below. Precisely, let
$\mathcal{B}_{k}$ be the event $|H_{k}(v)-F_{s_{k}}(v)|\geq\beta^{5}$, we have
${\mathbb{P}}(\mathcal{B}_{k})\leq\frac{1}{4}\alpha\beta^{5}$ if $\delta$ is
small enough. Let $\mathcal{B}$ be the event
${\mathbb{P}}(\mathcal{B}_{1}\,|\,\mathcal{F}_{s_{0}})\geq\beta^{5}$, then
${\mathbb{P}}(\mathcal{B})\leq\frac{1}{4}\alpha$. From the above estimate, we
have

${\mathbb{E}}[H_{1}(v)\,|\,\mathcal{F}_{s_{0}}]-H_{0}(v)=O(\beta^{5}),\quad\text{on
}\mathcal{A}_{0}^{c}\cap\mathcal{A}^{c}\cap\mathcal{B}_{0}^{c}\cap\mathcal{B}^{c}.$
(6.6)

For $k=0,1$, denote by

$Z_{k}=\frac{2g^{\delta}_{s_{k}}(\varphi(v))-g^{\delta}_{s_{k}}(y^{L})-g^{\delta}_{s_{k}}(y^{R})}{g^{\delta}_{s_{k}}(y^{R})-g^{\delta}_{s_{k}}(y^{L})}.$

Then we have

$H_{k}(v)=\lambda_{0}-\frac{2\lambda_{0}}{\pi}\arg(Z_{k}-1)+\frac{2\lambda_{0}}{\pi}\arg(Z_{k}-\tilde{W}^{\delta}_{s_{k}})-\frac{2\lambda_{0}}{\pi}\arg(Z_{k}+1).$

By the calculation in [SS09, Proof of Proposition 4.2], we have

$\frac{\pi}{2\lambda_{0}}(H_{1}(v)-H_{0}(v))=\frac{y}{x^{2}+y^{2}}\left(\frac{x}{x^{2}+y^{2}}\left(q(\tilde{W}^{\delta}_{s_{0}})^{2}\Delta
s-(\Delta\tilde{W}^{\delta})^{2}\right)-\Delta\tilde{W}^{\delta}\right)+O(\beta^{3}),$

where $x=\Re(Z_{0}-\tilde{W}^{\delta}_{s_{0}})$ and $y=\Im(Z_{0})$. Plugging
into (6.6), with probability at least $1-\alpha$, we have

${\mathbb{E}}\left[\frac{x}{x^{2}+y^{2}}\left(q(\tilde{W}^{\delta}_{s_{0}})^{2}\Delta
s-(\Delta\tilde{W}^{\delta})^{2}\right)-\Delta\tilde{W}^{\delta}\,|\,\mathcal{F}_{s_{0}}\right]=O(\beta^{3}).$

By different choice of $v$, we obtain (6.5).

Recall that $W$ is driving function of $\text{SLE}_{4}(-2;-2)$. Denote by
$\tilde{W}$ the coordinate change of $W$ as in Section 6.1. With (6.5) at
hand, by arguments in [SS09, Section 4.4], we have the following conclusion.
For any fixed $\epsilon,\alpha>0$ small, there is $\delta_{0}>0$ such that, if
$\delta\leq\delta_{0}$, there is coupling between $\tilde{W}^{\delta}$ and
$\tilde{W}$ so that

${\mathbb{P}}\left(\sup_{s\in[0,T_{\epsilon}^{\delta}\wedge
T_{\epsilon}]}|\tilde{W}^{\delta}_{s}-\tilde{W}_{s}|\geq\alpha\right)\leq\alpha.$

Finally, by Lemma 6.2 and arguments in [SS09, Section 4.6], we obtain the
conclusion. ∎

###### Proof of Theorem 6.1.

Recall that $\eta_{\delta}$ is the level line of $\phi_{\delta}$, we
parameterize $\varphi(\eta_{\delta})$ by the half plane capacity, and we
denote by $W^{\delta}_{t}$ its driving function. Recall that
$\eta\sim\text{SLE}_{4}(-2;-2)$ in $\mathbb{H}$ with force points
$y^{L}=\varphi(a)<0$ and $y^{R}=\varphi(c)>0$, we denote by $W_{t}$ its
driving function. From Lemma 6.3, $W^{\delta}$ is close to $W$ in local
uniform topology. Combing with [SS09, Section 4.7 and Lemma 4.16],
$\varphi(\eta^{\delta})$ and $\eta$ are close in Hausdorff metric. Precisely,
for any fixed $\epsilon,\alpha>0$ small, there is $\delta_{0}>0$ such that, if
$\delta\leq\delta_{0}$, the interface $\eta_{\delta}$ can be coupled with
$\eta$ so that

${\mathbb{P}}\left(\sup_{t\in[0,T_{\epsilon}^{\delta}\wedge
T_{\epsilon}]}d_{H}(\varphi(\eta_{\delta}[0,t],\eta[0,t])\geq\alpha\right)\leq\alpha,$

where $d_{H}$ denotes the Hausdoff metric. For such argument to work, it is
important that $(\eta(t),0\leq t\leq T)$ hits $\mathbb{R}$ only at the two end
points $\eta(0)=0$ and $\eta(T)\in\\{y^{L},y^{R}\\}$. Then, using arguments in
[SS09, Section 4.8], we arrive at the following conclusion. For any fixed
$\epsilon,\alpha>0$ small, there is $\delta_{0}>0$ such that, if
$\delta\leq\delta_{0}$, the interface $\eta_{\delta}$ can be coupled with
$\eta$ so that

${\mathbb{P}}\left(\sup_{t\in[0,T_{\epsilon}^{\delta}\wedge
T_{\epsilon}]}|\varphi(\eta_{\delta}(t)-\eta(t)|\geq\alpha\right)\leq\alpha.$

It remains to get the convergence of the whole process. Suppose $\eta_{*}$ is
any subsequential limit of $\eta_{\delta}$ in Hausdorff metric. From the above
argument, we see that $\varphi(\eta_{*})$ has the same law as $\eta$ up to
$T_{\epsilon}$ for any $\epsilon>0$. Note that $\eta$ is continuous up to and
including $T$. We may conclude that $\varphi(\eta_{*})$ has the same law as
$\eta$ up to $T$. This gives the convergence of the whole process.
Consequently, we obtain the convergence of the crossing probability (combining
with (6.2))

$\displaystyle\lim_{\delta\to
0}\mathbb{P}\left([a_{\delta},b_{\delta})\stackrel{{\scriptstyle\phi_{\delta}\geq
0}}{{\longleftrightarrow}}[c_{\delta},d_{\delta})\right)={\mathbb{P}}(\eta(T)=y^{R})=\frac{-y^{L}}{y^{R}-y^{L}}.$

∎

We end this section by a discussion on the convergence of discrete GFF level
lines in Theorem 6.1 when $\lambda\neq\lambda_{0}$. By analyzing level lines
of continuous GFF as in [WW17], we believe that the level line $\eta_{\delta}$
of $\phi_{\delta}$ converges weakly to
$\eta\sim\text{SLE}_{4}(-2\rho,\rho-1;\rho-1,-2\rho)$ in $R_{L}$ from $b$ to
$d$ with force points $(a,b^{-};b^{+},c)$ where $\rho=\lambda/\lambda_{0}$.
However, the techniques in [SS09] do not apply to this general setting
directly to our knowledge. In particular, the authors in [SS09] derived the
convergence of driving function when $\lambda=0$: they proved that the driving
function of $\eta_{\delta}$ weakly converges to the driving function of
$\text{SLE}_{4}(-1;-1)$ in the local uniform topology; however, the
convergence in stronger topology is still missing. Assuming the convergence of
$\eta_{\delta}$ to $\eta\sim\text{SLE}_{4}(-1;-1)$ in Haudorff metric, we may
conclude that the crossing probability in (1.5) is convergent:

$\lim_{\delta\to
0}\mathbb{P}\left([a_{\delta},b_{\delta})\stackrel{{\scriptstyle\phi_{\delta}\geq
0}}{{\longleftrightarrow}}[c_{\delta},d_{\delta})\right)={\mathbb{P}}(\eta\text{
hits }[c,d)\text{ before }[d,a)).$

Whereas, to get an explicit formula as in (1.9) for the right-hand side is
another open question.

###  6.3 Scaling limits of level lines in metric graph GFF

Recall that $\tilde{\phi}_{\delta}$ is metric graph GFF on
$\tilde{V}_{\delta}$. Suppose the boundary condition is the following: (note
that this is different from the one in (1.3))

$\tilde{\phi}_{\delta}(v)=\begin{cases}\lambda,&v\in[a_{\delta},b_{\delta})\cup[c_{\delta},d_{\delta}),\\\
0,&v\in[b_{\delta},c_{\delta})\cup[d_{\delta},a_{\delta}).\end{cases}$ (6.7)

Recall that

$\tilde{\mathbb{A}}_{\delta,0}^{l}=\left\\{v\in\tilde{V}_{\delta}:\exists\text{
continuous path }\gamma\text{ connecting $v$ to $[a_{\delta},b_{\delta})$ such
that }\tilde{\phi}_{\delta}\geq 0\text{ on }\gamma\right\\}.$

Define the frontier of $\tilde{\mathbb{A}}_{\delta,0}^{l}$ as follows:
Consider the set of all the points in
$\partial\tilde{\mathbb{A}}_{\delta,0}^{l}$ that are connected in
$\tilde{\mathbb{A}}_{\delta,0}^{l}$ to $[a_{\delta},b_{\delta})$ and in
$\tilde{V}_{\delta}\setminus\tilde{\mathbb{A}}_{\delta,0}^{l}$ to
$[b_{\delta},c_{\delta})$. This set contains no vertices and the edges it
intersects in the dual graph give a path from the point
$b_{\delta}^{\diamond}$ to the point $(3\delta/2,1)$ (near $a$) or the point
$(L,3\delta/2)$ (near $c$). We call such path the frontier of
$\tilde{\mathbb{A}}_{\delta,0}^{l}$. Using conclusions in [ALS18, Section
5.2], we may conclude that, when $\lambda=2\lambda_{0}$, the frontier of
$\tilde{\mathbb{A}}_{\delta,0}^{l}$ weakly converges to
$\text{SLE}_{4}(-2;-2)$ in $R_{L}$ from $b$ to $d$ with force points $(a;c)$
in Hausdorff metric as $\delta=2^{-n}\to 0$. Consequently, we have the
convergence of the crossing probabilities: (see [ALS18, Remark 5.11])

$\lim_{\delta=2^{-n}\to
0}\mathbb{P}\left([a_{\delta},b_{\delta})\stackrel{{\scriptstyle\tilde{\phi}_{\delta}\geq
0}}{{\longleftrightarrow}}[c_{\delta},d_{\delta})\right)=\frac{(\varphi(b)-\varphi(a))(\varphi(d)-\varphi(c))}{(\varphi(c)-\varphi(a))(\varphi(d)-\varphi(b))},$
(6.8)

where $\varphi$ is any conformal map from $R_{L}$ onto the upper-half plane
$\mathbb{H}$ with $\varphi(a)<\varphi(b)<\varphi(c)<\varphi(d)$. We emphasize
that the metric graph GFF (6.8) has the boundary condition (6.7) with
$\lambda=2\lambda_{0}$ which is different from the one in (1.3).

Let us go back to the boundary condition (1.3) and to (1.6) in Theorem 1.2.
Suppose we have metric graph GFF with boundary condition (1.3). Using
conclusions in [ALS18], we may conclude that the crossing probabilities in
(1.6) are convergent. By Theorem 1.2, we know that the limit should be
different from the one in the case of discrete GFF. To derive the explicit
formula for this limit is an interesting question. We will give this explicit
formula when $\lambda=2\lambda_{0}$ in a forthcoming paper [LW20].

## References

  * [ALS18] J. Aru, T. Lupu, A. Sepúlveda.  First passage sets of the 2D continuum Gaussian free field: convergence and isomorphisms.  Preprint, arXiv:1805.09204.
  * [BS02] A. N. Borodin, P. Salminen.  Handbook of Brownian Motion - Facts and Formulae.  Birkhäuser-Verlag, 2002.
  * [BL76] H.J. Brascamp, E. H. Lieb.  On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation.  J. Functional Analysis. 22(4), 366-389, 1976.
  * [DL18] J. Ding, L. Li.  Chemical distances for percolation of planar gaussian free fields and critical random walk loop soups.  Comm. Math. Phys. 360(2), 523-553, 2018.
  * [DW19] J. Ding, M. Wirth.  Percolation for level-sets of Gaussian free fields on metric graphs.  Ann. Probab., to appear.
  * [Ke97] H. Kesten.  Hitting probabilities of random walks on $\mathbb{Z}^{d}$.  Stochastic Process. Appl. 25:165-184, 1987.
  * [LL10] G. F. Lawler, V. Limic.  Random Walk: A Modern Introduction.  Cambridge University Press, 2010.
  * [Lu16] T. Lupu.  From loop clusters and random interlacements to the free field.  Ann. Probab., 44(3):2117–2146, 2016.
  * [LW18] T. Lupu, W. Werner.  The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph.  Probab. Theory Relat. Fields 171(3-4): 775-818, 2018.
  * [LW20] M. Liu, H. Wu.  Scaling limits of crossing probabilities in metric graph GFF.  In preparation.
  * [Pr74] C.J. Preston.  A generalization of the FKG inequalities.  Comm. Math. Phys. 36, 233–241, 1974.
  * [Pi82] L. D. Pitt.  Positively correlated normal variables are associated.  Ann. Probab. 10, 496–499, 1982.
  * [RS05] S. Rohde, O. Schramm.  Basic properties of SLE.  Ann. of Math. (2), 161(2):883–924, 2005.
  * [RY99] D. Revuz, M. Yor.  Continuous Martingales and Brownian Motion.  Springer-Verlag Berlin Heidelberg, 1999.
  * [SS09] O. Schramm, S. Sheffield.  Contour lines of the two-dimensional discrete Gaussian free field.  Acta Math, 202(1):21-137, 2009.
  * [WW17] M. Wang, H. Wu.  Level lines of Gaussian Free Field I: Zero-boundary GFF.  Stochastic Process. Appl., 127(4):1045–1124, 2017.


# The Relative Heat Content for Submanifolds in Sub-Riemannian Geometry

Tommaso Rossi  Institut für Angewandte Mathematik, Universität Bonn, Bonn,
Germany<EMAIL_ADDRESS>
###### Abstract.

We study the small-time asymptotics of the relative heat content for
submanifolds in sub-Riemannian geometry. First, we prove the existence of a
smooth tubular neighborhood for submanifolds of any codimension, assuming they
do not have characteristic points. Next, we propose a definition of relative
heat content for submanifolds of codimension $k\geq 1$ and we build an
approximation of this quantity, via smooth tubular neighborhoods. Finally, we
show that this approximation fails to recover the asymptotic expansion of the
relative heat content of the submanifold, by studying an explicit example.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Tubular neighborhood for submanifolds
  4. 4 Induced measure on $S$
  5. 5 Heat content for submanifolds
  6. 6 An example: closed simple curve in $\mathbb{R}^{3}$

##  1\. Introduction

In this paper, we study the small-time behavior of the relative heat content
for submanifolds in sub-Riemannian geometry. Loosely speaking, the relative
heat content for a submanifold $S\subset M$ can be regarded as the total
amount of heat contained in $S$ at time $t$, corresponding to a uniform
initial temperature distribution concentrated on $S$. When $S=\Omega$ is an
open and bounded domain, the problem of finding an asymptotic expansion for
the heat content has been extensively studied, see for example [vdBLG94,
vdBG94, Sav98, vdBDG93, DG94, vdBG15] for the Euclidean and Riemannian case,
and [TW18, RR21, ARR21] for the sub-Riemannian case. In particular, in this
situation, in both the Riemannian and sub-Riemannian setting, the small-time
behavior of the heat content associated with $\Omega$ encodes geometrical
information of $\partial\Omega$, such as its perimeter or its mean curvature
(c.f. Theorem 2.3). Thus, we may expect that, in an analogous way, the
asymptotics of the relative heat content for a submanifold of higher
codimension detect its geometrical invariants.

Up to our knowledge, the relative heat content for submanifolds has never been
systematically studied, not even in Riemannian geometry, so we propose the
following definition. Let $M$ be a sub-Riemannian manifold, equipped with a
smooth measure $\omega$, and let $S\subset M$ be a smooth, compact submanifold
of codimension $k\geq 0$. Let $\omega$ be a smooth measure on $M$, and let
$\mu$ be a probability measure on $S$. Then, we consider $\mu$ as the initial
datum for the heat equation in the sense of distributions, and study the
associated Cauchy problem:

(1) $\displaystyle(\partial_{t}-\Delta)u(t,x)=0,$
$\displaystyle\forall(t,x)\in(0,\infty)\times M,$ $\displaystyle
u(t,\cdot)\xrightarrow{t\to 0}\mu,$ $\displaystyle\text{in
}\mathcal{D}^{\prime}(M),$

where $\Delta=\mathrm{div}_{\omega}\circ\nabla$ is the usual sub-Laplacian
associated with $\omega$. A solution to this problem, in the sense of
distribution, is given by

(2) $u(t,x)=\int_{S}p_{t}(x,y)d\mu(x),\qquad\forall\,(t,x)\in(0,\infty)\times
M,$

where $p_{t}(x,y)$ is the usual heat kernel associated with $\Delta$ and
$\omega$. We define the _relative heat content for a submanifold_ $S$ as:

(3) $H_{S}(t)=\int_{S}\int_{S}p_{t}(x,y)d\mu(x)d\mu(y),\qquad\forall\,t>0.$

Notice that, on the one hand if $S=\\{x_{0}\\}$ and $\mu=\delta_{x_{0}}$, we
obtain the trace heat kernel $p_{t}(x_{0},x_{0})$. On the other hand, if $S$
is a $0$-codimensional submanifold, i.e. $S$ is an open, relatively compact
and smooth set, and we choose $\mu=\mathds{1}_{S}\omega$ , then (3) coincides
with the usual relative heat content as defined in [ARR21]. Therefore, a
small-time asymptotic expansion of (3) would include many cases of interest,
ranging from the trace heat kernel asymptotics, see for example [Bar13,
CdVHT20], to the results of contained in [ARR21].

Our attempt to compute the asymptotics of (3) consists in building a suitable
approximation of it, using smooth tubular neighborhoods of $S$. When $S$ is a
smooth hypersurface, the existence of a tubular neighborhood is guaranteed
assuming that there are no characteristic points, cf. [ARR21, Def. 2.4].
Therefore, as a first step, we introduce a definition of a non-characteristic
submanifold, cf. Definition 3.1, generalizing the classical one of non-
characteristic hypersurface. Then, denoting by $\delta_{S}\colon
M\rightarrow[0,+\infty)$ the distance function from $S$, see (16) for the
precise definition, we prove the following.

######  Theorem 1.1.

Let $M$ be a sub-Riemannian manifold and $S\subset M$ be a compact smooth non-
characteristic submanifold of codimension $k\geq 1$. Then, there exists
$r_{0}>0$ such that, denoting by $S_{r_{0}}=\\{p\in M\mid
0<\delta_{\mathrm{S}}<r_{0}\\}$, the following conditions hold:

  * i)

$\delta_{\mathrm{S}}\colon S_{r_{0}}\rightarrow[0,\infty)$ is smooth and such
that $\|\nabla\delta_{\mathrm{S}}\|_{g}=1$;

  * ii)

there exists a diffeomorphism
$G\colon(0,r_{0})\times\\{\delta_{\mathrm{S}}=r_{0}\\}\rightarrow S_{r_{0}}$
such that:

(4) $\delta_{\mathrm{S}}(G(r,p))=r\qquad\text{and}\qquad
G_{*}\partial_{r}=\nabla\delta_{\mathrm{S}}.$

Theorem 1.1 generalizes the analogous result in [FPR20, Prop. 3.1], and
[PRS18, Lem. 23], to submanifolds of any codimension and is a key tool to
build both a canonical probability measure on $S$, cf. Lemma 4.1, and the
approximation of the relative heat content. Indeed, for any $\varepsilon\leq
r_{0}$, denoting by $S_{\varepsilon}$ the tubular neighborhood of $S$, of
radius $\varepsilon$, we consider the (rescaled) relative heat content
associated with $S_{\varepsilon}$, namely

(5)
$H^{\varepsilon}_{S}(t)=\frac{1}{\omega(S_{\varepsilon})^{2}}\int_{S_{\varepsilon}}\int_{S_{\varepsilon}}p_{t}(x,y)d\omega(x)d\omega(y),\qquad\forall\,t>0.$

We show that $H^{\varepsilon}_{S}$ converges point-wise to $H_{S}$, as
$\varepsilon\to 0$, cf. Proposition 5.1. Since now $H^{\varepsilon}_{S}$ is
the (rescaled) relative heat content associated with a non-characteristic open
and bounded set, for any $\varepsilon>0$, we can apply [ARR21, Thm. 1.1], see
also Theorem 2.3 for the statement, and try to deduce the asymptotics for the
limit as $\varepsilon\to 0$.

Unfortunately, the point-wise convergence of $H^{\varepsilon}_{S}$ is too weak
to recover any information regarding the small-time asymptotic expansion of
the limit. Indeed, by studying an explicit example, we show that the
approximation procedure using (5) fails to recover the asymptotic expansion of
$H_{S}$. More precisely, we consider a closed simple curve in
$\mathbb{R}^{3}$, equipped with the Euclidean metric and the Lebesgue measure.
In this case, on the one hand, by a standard application of the Laplace
method, it is possible to compute the asymptotic expansion of $H_{S}(t)$ as
$t\to 0$ at any order: the coefficients appearing in the expansion depend on
the curvature of the curve and its derivatives of _any order_. On the other
hand, following the approximation strategy described above, we obtain an
expansion in the limit which can’t possibly agree with the correct one since

  * •

the only geometrical invariant appearing in the coefficients is the curvature
of the curve _without_ its derivatives;

  * •

the orders of the expansion don’t agree.

### Structure of the paper

In Section 2, we recall the basic definitions of sub-Riemannian geometry. In
Section 3, we give the definition of a non-characteristic submanifold and we
prove Theorem 1.1. In Section 4, we build the canonical probability measure on
$S$, which is induced by the outer measure $\omega$. In Section 5, we
introduce the definition of relative heat content for a submanifold $S$, we
build its approximation and we prove its point-wise convergence. Finally, in
Section 6, we show how the approximation fails to recover the asymptotic
expansion of the relative heat content of $S$.

### Acknowledgments.

This work was supported by the Grant ANR-18-CE40-0012 of the ANR, by the
Project VINCI 2019 ref. c2-1212 and by ERC Starting Grant 2020, project GEOSUB
(grant agreement No. 945655).

##  2\. Preliminaries

We recall some essential facts in sub-Riemannian geometry, following [ABB20].

### Sub-Riemannian geometry

Let $M$ be a smooth, connected finite-dimensional manifold. A sub-Riemannian
structure on $M$ is defined by a set of $N$ global smooth vector fields
$X_{1},\ldots,X_{N}$, called a _generating frame_. The generating frame
defines a _distribution_ of subspaces of the tangent spaces at each point
$x\in M$, given by

(6) $\mathcal{D}_{x}=\mathrm{span}\\{X_{1}(x),\ldots,X_{N}(x)\\}\subseteq
T_{x}M.$

We assume that the distribution is _bracket-generating_ , i.e. the Lie algebra
of smooth vector fields generated by $X_{1},\dots,X_{N}$, evaluated at the
point $x$, coincides with $T_{x}M$, for all $x\in M$. The generating frame
induces a norm on the distribution at $x$, namely

(7)
$g_{x}(v,v)=\inf\left\\{\sum_{i=1}^{N}u_{i}^{2}\mid\sum_{i=1}^{N}u_{i}X_{i}(x)=v\right\\},\qquad\forall\,v\in\mathcal{D}_{x},$

which, in turn, defines an inner product on $\mathcal{D}_{x}$ by polarization.
We use the shorthand $\|\cdot\|_{g}$ for the corresponding norm. We say that
$\gamma:[0,T]\to M$ is a _horizontal curve_ , if it is absolutely continuous
and

(8) $\dot{\gamma}(t)\in\mathcal{D}_{\gamma(t)},\qquad\text{for
a.e.}\,t\in[0,T].$

This implies that there exists $u:[0,T]\to\mathbb{R}^{N}$, such that

(9) $\dot{\gamma}(t)=\sum_{i=1}^{N}u_{i}(t)X_{i}(\gamma(t)),\qquad\text{for
a.e.}\,t\in[0,T].$

Moreover, we require that $u\in L^{2}([0,T],\mathbb{R}^{N})$. If $\gamma$ is a
horizontal curve, then the map $t\mapsto\|\dot{\gamma}(t)\|_{g}$ is integrable
on $[0,T]$, see [ABB20, Lemma 3.12]. We define the _length_ of a horizontal
curve as follows:

(10) $\ell(\gamma)=\int_{0}^{T}\|\dot{\gamma}(t)\|_{g}dt.$

The _sub-Riemannian distance_ is defined, for any $x,y\in M$, by

(11) $d_{\mathrm{SR}}(x,y)=\inf\\{\ell(\gamma)\mid\gamma\text{ horizontal
curve between $x$ and $y$}\\}.$

By Chow-Rashevskii Theorem, the bracket-generating assumption ensures that the
distance $d_{\mathrm{SR}}\colon M\times M\to\mathbb{R}$ is finite and
continuous. Furthermore it induces the same topology as the manifold one.

######  Remark 2.1.

The above definition includes all classical constant-rank sub-Riemannian
structures as in [Mon02, Rif14] (where $\mathcal{D}$ is a vector distribution
and $g$ a symmetric and positive tensor on $\mathcal{D}$), but also general
rank-varying sub-Riemannian structures. The same sub-Riemannian structure can
arise from different generating families.

### Geodesics and Hamiltonian flow

A _geodesic_ is a horizontal curve $\gamma:[0,T]\to M$, parametrized with
constant speed, and such that any sufficiently short segment is length-
minimizing. The _sub-Riemannian Hamiltonian_ is the smooth function
$H:T^{*}M\to\mathbb{R}$,

(12)
$H(\lambda)=\frac{1}{2}\sum_{i=1}^{N}\langle\lambda,X_{i}\rangle^{2},\qquad\lambda\in
T^{*}M,$

where $X_{1},\ldots,X_{N}$ is a generating frame for the sub-Riemannian
structure, and $\langle\lambda,\cdot\rangle$ denotes the action of covectors
on vectors. The _Hamiltonian vector field_ $\vec{H}$ on $T^{*}M$ is then
defined by $\varsigma(\cdot,\vec{H})=dH$, where
$\varsigma\in\Lambda^{2}(T^{*}M)$ is the canonical symplectic form.

Solutions $\lambda:[0,T]\to T^{*}M$ of the _Hamilton equations_

(13) $\dot{\lambda}(t)=\vec{H}(\lambda(t)),$

are called _normal extremals_. Their projections $\gamma(t)=\pi(\lambda(t))$
on $M$, where $\pi:T^{*}M\to M$ is the bundle projection, are locally length-
minimizing horizontal curves parametrized with constant speed, and are called
_normal geodesics_. If $\gamma$ is a normal geodesic with normal extremal
$\lambda$, then its speed is given by
$\|\dot{\gamma}\|_{g}=\sqrt{2H(\lambda)}$. In particular

(14) $\ell(\gamma|_{[0,t]})=t\sqrt{2H(\lambda(0))},\qquad\forall\,t\in[0,T].$

There is another class of length-minimizing curves in sub-Riemannian geometry,
called _abnormal_ or _singular_. As for the normal case, to these curves it
corresponds an extremal lift $\lambda(t)$ on $T^{*}M$, which however may not
follow the Hamiltonian dynamics (13). Here we only observe that an abnormal
extremal lift $\lambda(t)\in T^{*}M$ satisfies

(15)
$\langle\lambda(t),\mathcal{D}_{\pi(\lambda(t))}\rangle=0\quad\text{and}\quad\lambda(t)\neq
0,\qquad\forall\,t\in[0,T],$

that is $H(\lambda(t))\equiv 0$. A geodesic may be abnormal and normal at the
same time.

### Length-minimizers to a submanifold.

Let $S\subset M$ be a closed embedded submanifold of codimension $k\geq 0$ and
define the sub-Riemannian distance from $S$:

(16) $\delta_{\mathrm{S}}(p)=\inf\\{d_{\mathrm{SR}}(q,p),q\in
S\\},\qquad\forall p\in M.$

Let $\gamma:[0,T]\to M$ be a horizontal curve, parametrized with constant
speed, such that $\gamma(0)\in S$, $\gamma(T)=p\in M\setminus S$ and assume
$\gamma$ is a minimizer for $\delta_{\mathrm{S}}$, that is
$\ell(\gamma)=\delta_{\mathrm{S}}(p)$. In particular, $\gamma$ is a geodesic.
Any corresponding normal or abnormal lift, say $\lambda:[0,T]\to T^{*}M$, must
satisfy the transversality conditions, cf. [AS04, Thm 12.13],

(17) $\langle\lambda(0),v\rangle=0,\qquad\forall\,v\in T_{\gamma(0)}S.$

Equivalently, the initial covector $\lambda(0)$ must belong to the annihilator
bundle $\mathcal{A}(S)=\\{\lambda\in
T^{*}M\mid\langle\lambda,T_{\pi(\lambda)}S\rangle=0\\}$ of $S$.

### The heat equation and the relative heat content for a domain

Let $M$ be a sub-Riemannian manifold and let $\omega$ be a smooth measure on
$M$, i.e. defined by a positive tensor density. The _divergence_ of a smooth
vector field is defined by

(18)
$\mathrm{div}_{\omega}(X)\omega=\mathcal{L}_{X}\omega,\qquad\forall\,X\in\Gamma(TM),$

where $\mathcal{L}_{X}$ denotes the Lie derivative in the direction of $X$.
The _horizontal gradient_ of a function $f\in C^{\infty}(M)$, denoted by
$\nabla f$, is defined as the horizontal vector field (i.e. tangent to the
distribution at each point), such that

(19) $g_{x}(\nabla f(x),v)=v(f)(x),\qquad\forall\,v\in\mathcal{D}_{x},$

where $v$ acts as a derivation on $f$. In terms of a generating frame as in
(6), one has

(20) $\nabla f=\sum_{i=1}^{N}X_{i}(f)X_{i},\qquad\forall\,f\in C^{\infty}(M).$

The _sub-Laplacian_ is the operator $\Delta=\mathrm{div}_{\omega}\circ\nabla$,
acting on $C^{\infty}(M)$. Again, we may write its expression with respect to
a generating frame (6), obtaining

(21) $\Delta
f=\sum_{i=1}^{N}\left\\{X^{2}_{i}(f)+X_{i}(f)\mathrm{div}_{\omega}(X_{i})\right\\},\qquad\forall\,f\in
C^{\infty}(M).$

We denote by $L^{2}(M,\omega)$, or simply by $L^{2}$, the space of real
functions on $M$ which are square-integrable with respect to the measure
$\omega$. Let $\Omega\subset M$ be an open relatively compact set with smooth
boundary. This means that the closure $\bar{\Omega}$ is a compact manifold
with smooth boundary. We consider the _Cauchy problem for the heat equation_
on $\Omega$, that is we look for functions $u$ such that

(22) $\displaystyle\left(\partial_{t}-\Delta\right)u(t,x)$ $\displaystyle=0,$
$\displaystyle\forall(t,x)\in(0,\infty)\times M,$ $\displaystyle u(0,\cdot)$
$\displaystyle=\mathds{1}_{\Omega},$ $\displaystyle\text{in }L^{2}(M,\omega),$

where $u(0,\cdot)$ is a shorthand notation for the $L^{2}$-limit of $u(t,x)$
as $t\to 0$. Notice that $\Delta$ is symmetric with respect to the
$L^{2}$-scalar product and negative, moreover, if $(M,d_{\mathrm{SR}})$ is
complete as a metric space, it is essentially self-adjoint, see [Str86]. Thus,
there exists a unique solution to (22), and it can be represented as

(23) $u(t,x)=e^{t\Delta}\mathds{1}_{\Omega}(x),\qquad\forall\,x\in M,\ t>0,$

where $e^{t\Delta}\colon L^{2}\rightarrow L^{2}$ denotes the heat semi-group,
associated with $\Delta$. We remark that for all $\varphi\in L^{2}$, the
function $e^{t\Delta}\varphi$ is smooth for all $(t,x)\in(0,\infty)\times M$,
by hypoellipticity of the heat operator, see [Hör67]. Furthermore, there
exists a heat kernel associated with (22), i.e. a positive function
$p_{t}(x,y)\in C^{\infty}((0,+\infty)\times M\times M)$ such that:

(24)
$u(t,x)=\int_{M}p_{t}(x,y)\mathds{1}_{\Omega}(y)d\omega(y)=\int_{\Omega}p_{t}(x,y)d\omega(y).$

######  Definition 2.2 (Relative heat content).

Let $u(t,x)$ be the solution to (22). We define the _relative heat content_ ,
associated with $\Omega$, as

(25) $H_{\Omega}(t)=\int_{\Omega}u(t,x)d\omega(x),\qquad\forall\,t>0.$

In [ARR21], the authors proved the existence of a small-time asymptotic
expansion for $H_{\Omega}(t)$, provided that $\Omega$ is a non-characteristic
domain. Precisely, denoting by

(26)
$\delta_{\partial\Omega}(p)=\inf\\{d_{\mathrm{SR}}(q,p),q\in\partial\Omega\\},\qquad\forall
p\in M,$

the distance from the boundary of $\Omega$ and by $\sigma$ the induced sub-
Riemannian measure on $\partial\Omega$ (i.e. the one whose density is
$\sigma=|i_{\nu}\omega|_{\partial\Omega}$, where $\nu$ is the outward-pointing
normal vector field to $\Omega$), we have the following result. See Definition
3.1 for the notion of a characteristic point.

######  Theorem 2.3.

Let $M$ be a compact sub-Riemannian manifold, equipped with a smooth measure
$\omega$, and let $\Omega\subset M$ be an open subset whose boundary is smooth
and has no characteristic points. Then, as $t\to 0$,

(27)
$H_{\Omega}(t)=\omega(\Omega)-\frac{1}{\sqrt{\pi}}\sigma(\partial\Omega)t^{1/2}\\\
-\frac{1}{12\sqrt{\pi}}\int_{\partial\Omega}\left(2g(\nabla\delta_{\partial\Omega},\nabla(\Delta\delta_{\partial\Omega}))-(\Delta\delta_{\partial\Omega})^{2}\right)d\sigma
t^{3/2}+o(t^{2}).$

######  Remark 2.4.

The compactness assumption in Theorem 2.3 is technical and can be relaxed by
requiring, instead, global doubling of the measure and a global Poincaré
inequality. We refer to [ARR21] for more details.

##  3\. Tubular neighborhood for submanifolds

######  Definition 3.1 (Non-characteristic submanifold).

Let $M$ be a sub-Riemannian manifold and let $S\subset M$ be a smooth
submanifold of codimension $k\geq 0$. We say that a point $q\in S$ is _non-
characteristic_ if

(28) $\mathcal{D}_{q}+T_{q}S=T_{q}M.$

We say that $S$ is a _non-characteristic submanifold_ if (28) holds for any
point $q\in S$.

######  Remark 3.2.

Notice that Definition 3.1 includes the usual one for hypersurfaces, indeed if
$S\subset M$ is a submanifold of codimension $1$, it is easy to check that

(29) $\mathcal{D}_{q}\subset
T_{q}S\qquad\Leftrightarrow\qquad\mathcal{D}_{q}+T_{q}S\subsetneq T_{q}M.$

Under the assumption of non-characteristic submanifold, the distance from $S$,
$\delta_{\mathrm{S}}$ defined in (16), is smooth and it allows to build smooth
tubular neighborhoods of $S$.

######  Theorem 3.3.

Let $M$ be a sub-Riemannian manifold and $S\subset M$ be a compact smooth non-
characteristic submanifold of codimension $k\geq 1$. Then, there exists
$r_{0}>0$ such that, denoting by $S_{r_{0}}=\\{p\in M\mid
0<\delta_{\mathrm{S}}<r_{0}\\}$, the following conditions hold:

  * i)

$\delta_{\mathrm{S}}\colon S_{r_{0}}\rightarrow[0,\infty)$ is smooth and such
that $\|\nabla\delta_{\mathrm{S}}\|_{g}=1$;

  * ii)

there exists a diffeomorphism
$G\colon(0,r_{0})\times\\{\delta_{\mathrm{S}}=r_{0}\\}\rightarrow S_{r_{0}}$
such that:

(30) $\delta_{\mathrm{S}}(G(r,p))=r\qquad\text{and}\qquad
G_{*}\partial_{r}=\nabla\delta_{\mathrm{S}}.$

Before giving the proof of the theorem, we need a preliminary lemma, which can
be regarded as a partial generalization of [FPR20, Prop. 2.7].

######  Lemma 3.4.

Let $M$ be a sub-Riemannian manifold and $S\subset M$ be a smooth submanifold
of codimension $k\geq 1$. Let $\gamma\colon[0,1]\rightarrow M$ be a minimizing
geodesic such that

(31) $\gamma(0)\in S,\qquad\gamma(1)=p\in M\setminus
S,\qquad\delta_{\mathrm{S}}(p)=\ell(\gamma).$

If $\gamma$ is an abnormal geodesic, then $\gamma(0)$ is a characteristic
point of $S$.

###### Proof.

Let $\lambda\colon[0,1]\rightarrow T^{*}M$ be an abnormal lift of $\gamma$:
this means in particular that $\pi(\lambda(t))=\gamma(t)$ and

(32) $\langle\lambda(0),\mathcal{D}_{\gamma(0)}\rangle=0,\qquad\text{with
}\lambda(0)\neq 0,$

where $\langle\cdot,\cdot\rangle$ denotes the dual coupling. Moreover, since
$\gamma$ is a minimizing geodesic, any lift must necessarily satisfy the
transversality condition (17). Thus, since $\lambda(0)\neq 0$, conditions
(32), (17) imply that (28) fails at $q=\gamma(0)$. ∎

###### Proof of Theorem 3.3.

Let us consider the annihilator bundle of $S$, $\mathcal{A}S$, namely the
vector bundle of rank $k$, whose fibers are given by

(33) $\mathcal{A}_{q}S=\\{\lambda\in
T^{*}_{q}M\mid\langle\lambda,T_{q}S\rangle=0\\},\qquad\forall q\in S.$

At a point $q\in S$, let us fix a basis of the fiber $\mathcal{A}_{q}S$, say
$\\{\lambda_{1},\ldots,\lambda_{k}\\}$ and define, for any $j=1,\ldots,k$, the
element $v_{j}\in\mathcal{D}_{q}$ dual to $\lambda_{i}$ via the Hamiltonian
$H$, i.e.

(34)
$v_{j}=\pi_{*}\vec{H}(\lambda_{j})=\sum_{i=1}^{N}\langle\lambda_{j},X_{i}(q)\rangle
X_{i}(q)\qquad j=1,\ldots,k.$

_(Step 1)_ If $q$ is non-characteristic, then the set
$\\{v_{1},\ldots,v_{k}\\}$ is linearly independent.

Indeed assume there exists constants $\alpha_{i}$ for $i=1,\ldots,k$, such
that $\sum_{i=1}^{k}\alpha_{i}v_{i}=0$. Then,

(35)
$0=\sum_{j=1}^{k}\alpha_{j}v_{j}=\sum_{j=1}^{k}\alpha_{j}\sum_{i=1}^{N}\langle\lambda_{j},X_{i}(q)\rangle
X_{i}(q)=\sum_{i=1}^{N}\langle\sum_{j=1}^{k}\alpha_{j}\lambda_{j},X_{i}(q)\rangle
X_{i}(q)=\pi_{*}\vec{H}(\lambda),$

having set $\lambda=\sum_{j=1}^{k}\alpha_{j}\lambda_{j}\in\mathcal{A}_{q}S$.
Notice that, by the Lagrange multiplier rule, denoting by
$v_{\lambda}=\pi_{*}\vec{H}(\lambda)$, for any $\lambda\in T_{q}^{*}M$, we
have

(36) $\|v_{\lambda}\|^{2}_{g}=\inf\left\\{\sum_{i=1}^{N}u_{i}^{2}\mid
v_{\lambda}=\sum_{i=1}^{N}u_{i}X_{i}(q)\right\\}=\sum_{i=1}^{N}\langle\lambda,X_{i}(q)\rangle^{2}=2H(\lambda).$

Therefore, (35) implies that
$\|\pi_{*}\vec{H}(\lambda)\|^{2}_{g}=2H(\lambda)=0$, or equivalently:

(37) $\langle\lambda,\mathcal{D}_{q}\rangle=0.$

Since $\lambda\in\mathcal{A}_{q}S$ and $q$ is non-characteristic, by (37), we
deduce that $\lambda=0$. Thus:

(38)
$0=\lambda=\sum_{j=1}^{k}\alpha_{j}\lambda_{j}\qquad\Rightarrow\qquad\alpha_{j}=0,\quad\text{for
any }j=1,\ldots,k,$

since $\\{\lambda_{1},\ldots,\lambda_{k}\\}$ was a basis of the fiber of
$\mathcal{A}S$. This concludes the proof of the first step. Define now the
sub-Riemannian exponential map from $S$, i.e. the map

(39) $E\colon D\cap\mathcal{A}S\rightarrow M;\qquad E(\lambda)=\pi\circ
e^{\vec{H}}(\lambda),$

where $D\subset T^{*}M$ is the open set where the flow of $\vec{H}$ is defined
up to time $1$. Consider also the zero section of the annihilator bundle,
namely

(40) $i\colon S\rightarrow\mathcal{A}S;\qquad i(q)=(q,0)\in\mathcal{A}_{q}S.$

_(Step 2)_ $E$ is a local diffeomorphism at points of $i(S)$.

To prove the claim, we consider a point $(q,0)\in i(S)$ and verify that
$d_{(q,0)}E$ is invertible. Identifying $T_{(q,0)}(D\cap\mathcal{A}S)\cong
T_{q}S\oplus\mathcal{A}_{q}S$, we have, on the one hand $E\circ i=Id_{S}$,
therefore for a vector $v=(v,0)\in T_{q}S\oplus\mathcal{A}_{q}S$,

(41)
$d_{(q,0)}E(v)=\left.\frac{d}{dt}\right\rvert_{t=0}E(\lambda(t))=\left.\frac{d}{dt}\right\rvert_{t=0}E\circ
i(\gamma(t))=\left.\frac{d}{dt}\right\rvert_{t=0}\gamma(t)=v,$

since $\lambda(t)=(\gamma(t),0)$, with
$\gamma\colon(-\varepsilon,\varepsilon)\rightarrow S$, such that $\gamma(0)=q$
and $\dot{\gamma}(0)=v$. On the other hand, take an element
$\lambda=(0,\lambda)\in T_{q}S\oplus\mathcal{A}_{q}S$, then by definition, we
obtain

(42)
$d_{(q,0)}E(\lambda)=\left.\frac{d}{dt}\right\rvert_{t=0}E(q,t\lambda)=\left.\frac{d}{dt}\right\rvert_{t=0}E(q,t\lambda)=\left.\frac{d}{dt}\right\rvert_{t=0}\pi\circ
e^{t\vec{H}}(\lambda)=\pi_{*}\vec{H}(\lambda)=v_{\lambda}.$

Thus, choosing any basis for $T_{q}S$ and the basis
$\\{\lambda_{1},\ldots,\lambda_{k}\\}$ for $\mathcal{A}_{q}S$, as before, we
may write the $n\times n$ matrix representing the differential of $E$ as

(43) $d_{(q,0)}E=\pNiceArray{cc|c}[margin]\Block{2-2}{Id_{n-k}}&\\\
v_{1},\ldots,v_{k}\\\ \qquad\bm{0},$

where the vectors $v_{j}$ are defined in (34). Since, by the previous step,
the set $\\{v_{1},\ldots,v_{k}\\}$ is linearly independent in
$\mathcal{D}_{q}$, we conclude that $dE$ is invertible at $i(S)$.

_(Step 3)_ There exists $U\subset D\cap\mathcal{A}S$, such that $E\rvert_{U}$
is a diffeomorphism on its image. Moreover, $U$ can be chosen of the form:

(44)
$U=\\{\lambda\in\mathcal{A}S\mid\sqrt{2H(\lambda)}<r_{0}\\},\qquad\text{for
some }r_{0}>0.$

The proof of this step follows verbatim what has been done in [FPR20, Prop.
3.1], cf. also [PRS18, Lem. 23], once we have verified that $\sqrt{2H(\cdot)}$
is a fiber-wise norm on the annihilator bundle. Since $H$ is quadratic on
fibers, it immediately follows that $\sqrt{2H(\cdot)}$ is positive,
1-homogeneous and sub-additive. We are left to prove that, for
$\lambda\in\mathcal{A}_{q}S$,

(45) $\sqrt{2H(\lambda)}=0\qquad\Leftrightarrow\qquad\lambda=0.$

As already remarked in (37), an element $\lambda\in\mathcal{A}_{q}S$, such
that $\sqrt{2H((q,\lambda))}=0$, annihilates both the distribution and
$T_{q}S$, thus, being $q$ non-characteristic, $\lambda=0$.

_(Step 4)_ $E(U)=\\{p\in M\mid\delta_{\mathrm{S}}(p)<r_{0}\\}=S_{r_{0}}\cup S$
and, for elements $(q,\lambda)\in U$ we have
$\delta_{\mathrm{S}}(E(q,\lambda))=\sqrt{2H(\lambda)}$. In particular,
$\delta_{\mathrm{S}}\in C^{\infty}(S_{r_{0}})$.

Firstly, we recall that, for an element $\lambda\in U$, the length of the
curve

(46) $[0,1]\ni t\mapsto\pi\circ e^{t\vec{H}}(\lambda)\in M$

is equal to $\sqrt{2H(\lambda)}<r_{0}$, as one can check using (36). Thus,
$E(U)\subset S_{r_{0}}\cup S$. Secondly, we prove the opposite inclusion: up
to restricting $r_{0}$, we may assume that $S_{r_{0}}\subset K$, for a compact
set $K\subset M$. Therefore, for an element $p\in S_{r_{0}}$, there exists a
minimizing geodesic $\gamma\colon[0,1]\rightarrow M$ such that

(47) $\gamma(0)=q\in
S,\qquad\gamma(1)=p\qquad\text{and}\quad\ell(\gamma)=\delta_{\mathrm{S}}(p).$

Applying Lemma 3.4, we deduce that $\gamma$ is not an abnormal geodesic,
meaning that there exists a unique normal lift for $\gamma$, with initial
covector given by $\lambda\in T^{*}_{q}M$, which implies

(48) $\gamma(t)=\pi\circ e^{t\vec{H}}(\lambda),$

and in particular, $E(q,\lambda)=p$. Moreover, $\lambda\in U$ as, by
optimality, it satisfies the transversality condition (17), and also

(49) $\ell(\gamma)=\sqrt{2H(\lambda)}<r_{0},$

being $p\in S_{r_{0}}$. Finally, we conclude that $p\in E(U)$ and
$\delta_{\mathrm{S}}(E(q,\lambda))=\sqrt{2H(\lambda)}$, by (49). Since
$\sqrt{2H(\cdot)}$ is smooth, as long as $H(\lambda)\neq 0$, we also have that
$\delta_{\mathrm{S}}$ is smooth on the set $E(U\setminus i(S))=S_{r_{0}}$.

_(Step 5)_ There exists a diffeomorphism
$G\colon(0,r_{0})\times\\{\delta_{\mathrm{S}}=r_{0}\\}\rightarrow S_{r_{0}}$
satisfying item $(ii)$ of the statement. Moreover,
$\|\nabla\delta_{\mathrm{S}}\|_{g}=1$ in $S_{r_{0}}$.

Once again, this part of the proof follows verbatim [FPR20, Prop. 3.1]. ∎

######  Remark 3.5.

Consider the set $U=\mathcal{A}S\cap\\{\sqrt{2H(\cdot)}<r_{0}\\}$ defined in
(44). What we proved in the previous Theorem is that $E$ defines a
diffeomorphism between $U$ and $S_{r_{0}}\cup S$. In particular, choosing a
local trivialization of the annihilator bundle, this means that

(50) $S_{r_{0}}\cup
S\cong\mathcal{A}S\cap\left\\{\sqrt{2H(\cdot)}<r_{0}\right\\}\underset{locally}{\cong}S\times
B^{H}_{r_{0}}(0),$

where $B^{H}_{r_{0}}(0)$ denotes the ball of radius $r_{0}$, centered at the
origin of the Euclidean space $\left(\mathbb{R}^{k},\sqrt{2H(\cdot)}\right)$.
Of course, in general, the annihilator bundle will not be globally
trivializable, however, this is the case when $S$ is the boundary of an open
set and we are able to extend (50) to the whole submanifold.

Whenever $S$ is a boundary of an open set, we can refine Theorem 3.3 building
a double-sided tubular neighborhood of $S$, in which we are able to
distinguish the inside and the outside of the open set. This is done using the
signed distance function. We recall here its definition.

######  Definition 3.6 (Signed distance).

Let $M$ be a sub-Riemannian manifold and $\Omega\subset M$ be an open subset.
Define $\delta\colon M\rightarrow\mathbb{R}$ to be the _signed distance
function_ from $\partial\Omega$, i.e.

(51) $\delta(p)=\begin{cases}\delta_{\partial\Omega}(p)&p\in\Omega,\\\
-\delta_{\partial\Omega}(p)&p\in M\setminus\Omega,\end{cases}$

where $\delta_{\partial\Omega}\colon M\rightarrow[0,+\infty)$ denotes the
usual distance function from the boundary of $\Omega$.

######  Theorem 3.7 (Double-sided tubular neighborhood).

Let $M$ be a sub-Riemannian manifold and $\Omega\subset M$ be an open,
relatively compact subset, whose boundary is smooth and has no characteristic
points. Denote by $\Omega_{-r_{0}}^{r_{0}}=\\{p\in M\mid-
r_{0}<\delta<r_{0}\\}$. Then, there exists $r_{0}>0$ such that, the following
conditions hold:

  * i)

$\delta\colon\Omega_{-r_{0}}^{r_{0}}\rightarrow\mathbb{R}$ is smooth and such
that $\|\nabla\delta\|_{g}=1$;

  * ii)

there exists a diffeomorphism
$G\colon(-r_{0},r_{0})\times\partial\Omega\rightarrow\Omega_{-r_{0}}^{r_{0}}$
such that:

(52) $\delta(G(t,p))=t\qquad\text{and}\qquad G_{*}\partial_{t}=\nabla\delta.$

######  Remark 3.8.

The main differences with respect to Theorem 3.3 are that $\delta$ is smooth
up to the boundary of $\Omega$, and the diffeomorphism is built starting from
$\partial\Omega$.

###### Proof of Theorem 3.7.

By Theorem 3.3, applied with $S=\partial\Omega$, the sub-Riemannian
exponential map from $\partial\Omega$ is a diffeomorphism for small covectors,
namely there exists $r_{0}>0$, such that:

(53)
$E\colon\mathcal{A}(\partial\Omega)\cap\left\\{\sqrt{2H(\lambda)}<r_{0}\right\\}\overset{\cong}{\longrightarrow}\Omega_{-r_{0}}^{r_{0}}$

and $|\delta(E(q,\lambda))|=\sqrt{2H(\lambda)}$. Now, since $\Omega$ is an
open set with smooth boundary, $\mathcal{A}(\partial\Omega)$ is trivializable,
i.e. there exists a never-vanishing and inward-pointing smooth section

(54)
$\lambda^{+}\colon\partial\Omega\rightarrow\mathcal{A}(\partial\Omega);\qquad
q\mapsto\lambda^{+}_{q}.$

Furthermore, by non-characteristic assumption, $\sqrt{2H(\cdot)}$ is a fiber-
wise norm on the annihilator bundle, hence we may assume without loss of
generality that

(55) $\sqrt{2H(\lambda^{+}_{q})}=1,\qquad\forall\,q\in S.$

Thus, we find a unique smooth function $\xi(\lambda)\in
C^{\infty}(\mathcal{A}(\partial\Omega))$ such that

(56)
$\lambda=\xi(\lambda)\lambda^{+}_{q},\qquad\lambda\in\mathcal{A}_{q}(\partial\Omega).$

Hence, the annihilator bundle is trivializable via the map $\xi$, i.e.

(57)
$F\colon\mathcal{A}(\partial\Omega)\overset{\cong}{\longrightarrow}\partial\Omega\times\mathbb{R};\qquad
F(\lambda)=\left(\pi(\lambda),\xi(\lambda)\right).$

Notice that, by definition, $|\xi(\lambda)|=\sqrt{2H(\lambda)}$. Moreover,
$\xi(\lambda)>0$, whenever $E(q,\lambda)\in\Omega$, by definition of
$\lambda^{+}$, $\xi(0)=0$ and negative otherwise. Therefore, having defined
the signed distance such that it is positive inside of $\Omega$, we obtain
that

(58)
$\delta(E(q,\lambda))=\xi(\lambda),\qquad\forall\,\lambda\in\left\\{\sqrt{2H(\lambda)}<r_{0}\right\\},$

proving the smoothness of $\delta$ on the set $\Omega_{-r_{0}}^{r_{0}}$.
Finally, define $G$ as the composition of $E\circ F^{-1}$ restricted to the
set $(-r_{0},r_{0})\times\partial\Omega$. Since $E$ and $F$ are
diffeomorphisms, also $G$ is and moreover,

(59)
$G(t,q)=E(q,t\lambda^{+}_{q})\qquad\forall\,(t,q)\in(-r_{0},r_{0})\times\partial\Omega,$

therefore
$\delta(G(t,q))=\delta(E(q,t\lambda^{+}_{q}))=\xi(t\lambda^{+}_{q})=t$. This
concludes the proof. ∎

##  4\. Induced measure on $S$

Let $\omega$ be a smooth measure on $M$. We define a measure on $S$ induced by
$\omega$, assigning a tensor density. This construction specializes to the
sub-Riemannian perimeter measure, when $S$ is the boundary of an open set.
Recall that, by Theorem 3.3, there exists $r_{0}>0$ such that (50) holds
locally and define $\mathrm{vol}_{H}$ as the Riemannian measure associated
with $\big{(}\mathbb{R}^{k},\|\cdot\|_{\perp}\big{)}$, where
$\|\cdot\|_{\perp}$ is a shorthand notation for
$\sqrt{2H(\cdot)}\rvert_{\mathcal{A}_{q}S}$. In particular, $\mathrm{vol}_{H}$
is well-defined since $\|\cdot\|_{\perp}$ is induced by the fiber-wise
bilinear form

(60)
$(\lambda_{1},\lambda_{2})_{\perp}=\sum_{i=1}^{N}\langle\lambda_{1},X_{i}\rangle\langle\lambda_{2},X_{i}\rangle,\qquad\forall\lambda_{1},\lambda_{2}\in\mathcal{A}_{q}S,\
q\in S,$

where $\langle\cdot,\cdot\rangle$ denotes the dual coupling.

######  Lemma 4.1.

Let $M$ be a sub-Riemannian manifold and $S\subset M$ be a compact smooth non-
characteristic submanifold of codimension $k\geq 1$. Then, there exists a
unique smooth probability measure $\mu_{\mathrm{S}}$ on $S$, such that,

(61) $\int_{M}h(p)\omega^{\varepsilon}(p)\xrightarrow{\varepsilon\to
0}\int_{S}h(q)\mu_{\mathrm{S}}(q),$

for any $h\in C_{c}(M)$, where,

(62)
$\omega^{\varepsilon}=\frac{1}{\omega(S_{\varepsilon})}\mathds{1}_{S_{\varepsilon}},\qquad\forall\,\varepsilon>0,$

###### Proof.

Proceeding with hindsight, we are going to define explicitly the measure
$\mu_{\mathrm{S}}$ and then prove the convergence. We may define
$\mu_{\mathrm{S}}$ locally, hence, fix an open coordinate chart $V\subset S$
for $S$ and a local trivialization of $\mathcal{A}S$ over $V$, so that

(63) $\mathcal{A}S\rvert_{V}\cong V\times\mathbb{R}^{k}.$

By Theorem 3.3, we have that, denoting by
$V_{r_{0}}=E(\mathcal{A}S\rvert_{V}\cap\\{\sqrt{2H(\cdot)}<r_{0}\\})$,

(64)
$V_{r_{0}}\cong\mathcal{A}S\rvert_{V}\cap\left\\{\sqrt{2H(\cdot)}<r_{0}\right\\}\cong
V\times B^{H}_{r_{0}}(0).$

Consider on $V_{r_{0}}$, coordinates $(x,z)$ where $(x_{1},\ldots,x_{n-k})$
are coordinates on $V$ and $S\cap V_{r_{0}}=\\{(x,z)\mid z=0\\}$. Thus, since
$\omega$ is smooth, we have

(65) $d\omega(x,z)=\omega(x,z)dxdz,\qquad\text{with }\omega(\cdot)\in
C^{\infty}(V_{r_{0}}),$

where $dx$ and $dz$ are the Lebesgue measures in coordinates. Moreover, since
(60) is a metric along the fibers, we can define canonically a volume
associated with $H$, which in coordinates is given by

(66) $d\mathrm{vol}_{H}(z)=\sqrt{\det H_{q}(z)}dz,\qquad\forall\,q\in S,$

with never-vanishing density. Therefore, we may rewrite $\omega$ in terms of
$\mathrm{vol}_{H}$, obtaining

(67) $d\omega(x,z)=\omega(x,z)dxdz=\frac{\omega(x,z)}{\sqrt{\det
H_{q}(z)}}dxd\mathrm{vol}_{H}(z)$

Finally, on the fiber, we can choose an orthonormal (w.r.t.
$\sqrt{2H(\cdot)}$) basis of smooth local sections
$\\{\lambda_{1},\ldots,\lambda_{k}\\}$, so that
$\mathrm{vol}_{H}(\lambda_{1},\ldots,\lambda_{k})=1$, and define
$\tilde{\mu}_{S}$ in coordinates $(x,z)$, to be the contraction of (67) along
these covectors, restricted to $S$, namely

(68) $\tilde{\mu}_{S}=\frac{\omega(x,0)}{\sqrt{\det
H_{q}(0)}}dxd\mathrm{vol}_{H}(\lambda_{1},\ldots,\lambda_{k})=\frac{\omega(x,0)}{\sqrt{\det
H_{q}(0)}}dx.$

One can check that this procedure defines a smooth measure on $S$,
independently on the choice of the coordinates. We can now verify the
convergence, using a partition of the unity argument. Fix a covering of $S$
with a finite number of open charts $\\{V_{i}\\}_{i=1}^{L}$ and consider the
associated covering $\\{V_{r_{0}}^{i}\\}$ of $S\cup S_{r_{0}}$, defined by

(69)
$V_{r_{0}}^{i}=E\left(\mathcal{A}S\rvert_{V_{i}}\cap\\{\sqrt{2H(\cdot)}<r_{0}\\}\right),\qquad\forall\,i=1,\ldots,L.$

Then, consider $\\{\rho_{i}\\}_{i=1}^{L}$ to be a partition of unity
subordinate to the covering $\\{V_{r_{0}}^{i}\\}$ of $S\cup S_{r_{0}}$.
Exploiting the coordinate expression of $\tilde{\mu}_{S}$, we have, for any
$\varepsilon\leq r_{0}$:

(70)
$\begin{split}\omega(S_{\varepsilon})&=\int_{S_{\varepsilon}}\sum_{i=1}^{L}\rho_{i}(q)d\omega(q)=\sum_{i=1}^{L}\int_{V_{i}}\int_{B_{\varepsilon}^{H}(x)}\rho_{i}(x,z)\frac{\omega(x,z)}{\sqrt{\det
H_{q}(z)}}d\mathrm{vol}_{H}(z)dx\\\
&=\varepsilon^{k}\sum_{i=1}^{L}\int_{V_{i}}\int_{0}^{1}\int_{\mathbb{S}^{k-1}}\rho_{i}(x,\varepsilon
r,\theta)\frac{\omega(x,\varepsilon r,\theta)}{\sqrt{\det H_{q}(\varepsilon
r,\theta)}}r^{k-1}d\theta drdx,\end{split}$

having expressed the volume $\mathrm{vol}_{H}$ in polar coordinates
$r^{k-1}d\theta dr$. Therefore, up to a factor $\varepsilon^{k}$, we see that

(71)
$\frac{\omega(S_{\varepsilon})}{\varepsilon^{k}}\xrightarrow{\varepsilon\to
0}\varpi_{k}\sum_{i=1}^{L}\int_{V_{i}}\rho_{i}(x,0)d\tilde{\mu}_{S}(x)=\varpi_{k}\int_{S}d\tilde{\mu}_{S}$

where $\varpi_{k}$ is the volume of the standard unit ball in
$\mathbb{R}^{k}$. Finally, reasoning as above, since for any $h\in C_{c}(M)$,
we are able to extract a factor $\varepsilon^{k}$ from the integral of $h$
over $S_{\varepsilon}$, we obtain the convergence in the weak-star topology
(61), having normalized $\tilde{\mu}_{S}$ to obtain a probability measure
$\mu_{\mathrm{S}}$. ∎

######  Remark 4.2.

Since $\omega^{\varepsilon}$, for any $\varepsilon\leq r_{0}$, has compact
support which is contained in $S_{r_{0}}$, we can extend the convergence (61)
to any continuous function on $M$.

##  5\. Heat content for submanifolds

Let $M$ be a sub-Riemannian manifold, equipped with a smooth measure $\omega$,
and let $S\subset M$ be a smooth, compact submanifold of codimension $k\geq
1$. We may consider $\mu$ a smooth probability measure on $S$ as initial datum
for the heat equation, in the sense of distributions, and study the associated
Cauchy problem:

(72) $\displaystyle(\partial_{t}-\Delta)u(t,x)=0,$
$\displaystyle\forall(t,x)\in(0,\infty)\times M,$ $\displaystyle
u(t,\cdot)\xrightarrow{t\to 0}\mu,$ $\displaystyle\text{in
}\mathcal{D}^{\prime}(M).$

A solution to this problem, in the sense of distribution, is given by

(73) $u(t,x)=\int_{S}p_{t}(x,y)d\mu(x),\qquad\forall\,(t,x)\in(0,\infty)\times
M,$

which, by hypoellipticity, is a smooth function for positive times. Recall
that, by definition of the relative heat content associated with an open set
$\Omega\subset M$, we have

(74)
$H_{\Omega}(t)=\int_{\Omega}\int_{\Omega}p_{t}(x,y)d\omega(x)d\omega(y),\qquad\forall\,t>0.$

Thus, a suitable generalization of (74) for a submanifold $S$ seems to be:

(75) $H_{S}(t)=\int_{S}\int_{S}p_{t}(x,y)d\mu(x)d\mu(y),\qquad\forall\,t>0.$

Moreover, when $S$ is non-characteristic, Lemma 4.1 provides with a canonical
probability measure on $S$, induced by $\omega$, i.e. $\mu_{\mathrm{S}}$.
Henceforth, we assume $S$ non-characteristic and fix $\mu=\mu_{\mathrm{S}}$.
In this setting, we can hope to obtain an asymptotic expansion of (75).

######  Proposition 5.1.

Let $M$ be a sub-Riemannian manifold, equipped with a smooth measure $\omega$,
let $S\subset M$ be a smooth, compact and non-characteristic submanifold of
codimension $k\geq 1$ and fix the probability measure $\mu_{\mathrm{S}}$ on
$S$. Define, for any $\varepsilon\leq r_{0}$,

(76)
$H^{\varepsilon}_{S}(t)=\int_{M}\int_{M}p_{t}(x,y)d\omega^{\varepsilon}(x)d\omega^{\varepsilon}(y),\qquad\forall\,t>0.$

Then, for any $t>0$,

(77) $H^{\varepsilon}_{S}(t)\xrightarrow{\varepsilon\to 0}H_{S}(t).$

###### Proof.

Firstly, notice that, applying Lemma 4.1, we have that

(78)
$u^{\varepsilon}(t,x)=\int_{M}p_{t}(x,y)d\omega^{\varepsilon}(y)=\langle\omega^{\varepsilon},p_{t}(x,\cdot)\rangle\xrightarrow{\varepsilon\to
0}\langle\mu_{\mathrm{S}},p_{t}(x,\cdot)\rangle,$

for any $t>0$ and $y\in M$. Secondly, since the heat kernel $p_{t}$ is smooth
on $M\times M$, there exists a constant $C(t)>0$ depending on $t$, such that:

(79) $\left\|p_{t}(\cdot,\cdot)\right\|_{L_{\mathrm{loc}}^{\infty}(M\times
M)}\leq C(t),$

and we remark that the constant $C(t)$ explodes as $t\to 0$. Therefore, the
convergence (78) is locally uniform with respect to $x\in M$. In conclusion,

(80)
$\begin{split}|H^{\varepsilon}_{S}(t)-H_{S}(t)|&=|\langle\omega^{\varepsilon},u^{\varepsilon}(t,\cdot)\rangle-\langle\mu_{\mathrm{S}},u(t,\cdot)\rangle|\\\
&\leq|\langle\omega^{\varepsilon},u^{\varepsilon}(t,\cdot)-u(t,\cdot)\rangle|+|\langle\omega^{\varepsilon},u(t,\cdot)\rangle-\langle\mu_{\mathrm{S}},u(t,\cdot)\rangle|\\\
&\leq\|u^{\varepsilon}(t,\cdot)-u(t,\cdot)\|_{L^{\infty}(S_{r_{0}})}|\langle\omega^{\varepsilon},1\rangle|+|\langle\omega^{\varepsilon}-\mu_{\mathrm{S}},u(t,\cdot)\rangle|,\end{split}$

and taking the limit as $\varepsilon\to 0$ in the last line proves the desired
result. ∎

######  Remark 5.2.

The convergence (77) is never uniform as $t\to 0$, being the constant $C(t)$
in (79) not bounded as $t\to 0$. This suggests that, while
$H_{S}^{\varepsilon}$ seems to be the best possible approximation of the heat
content associated with $S$, using such a strategy to deduce the asymptotics
of $H_{S}(t)$ is not correct. Indeed, we can show that the coefficients of the
expansion of $H_{S}^{\varepsilon}$ can not approximate those of $H_{S}(t)$, in
general.

##  6\. An example: closed simple curve in $\mathbb{R}^{3}$

In $\mathbb{R}^{3}$ equipped with the Euclidean scalar product and the
Lebesgue measure, let us consider a biregular closed simple curve,
parametrized by arc-length, $\gamma\colon[0,\ell]\rightarrow\mathbb{R}^{3}$,
where $\ell$ denotes the length of $\gamma$. Recall that a smooth curve
$\gamma\colon I\rightarrow\mathbb{R}^{3}$ is biregular if

(81) $\dot{\gamma}(s)\wedge\ddot{\gamma}(s)\neq 0,\qquad\forall\,s\in I,$

where $\wedge$ denotes the cross product in $\mathbb{R}^{3}$. In this setting,
define $S=\gamma([0,\ell])\subset\mathbb{R}^{3}$, submanifold of codimension
$2$. The tubular neighborhood of $S$ given by Theorem 3.3 coincides with the
usual Euclidean tubular neighborhood, which can be conveniently described by
the Frenet-Serret moving frame along $\gamma$, it being biregular. In
particular, denoting by $\\{T(s),N(s),B(s)\\}$ the Frenet-Serret frame for
$s\in[0,\ell]$, we have,

(82) $S_{\varepsilon}=\\{\gamma(s)+r(\cos\theta N(s)+\sin\theta B(s))\mid
s\in[0,\ell],\ \theta\in(0,2\pi],\
r\in(0,\varepsilon)\\},\qquad\forall\varepsilon\leq r_{0}.$

Thus, in coordinates $(s,r,\theta)$, the Lebesgue measure is

(83) $dxdydz=r(1-rk(s)\cos\theta)dsdrd\theta,$

where $k(s)=\|\ddot{\gamma}(s)\|$ is the curvature of $\gamma$, and the
procedure of Lemma 4.1 gives the probability measure
$\mu_{\mathrm{S}}=ds/\ell$. Following the discussion of Section 5, we define
the heat content associated with $S$ as

(84) $H_{S}(t)=\frac{1}{\ell^{2}}\frac{1}{(4\pi
t)^{3/2}}\int_{0}^{\ell}\int_{0}^{\ell}e^{-\frac{|\gamma(s)-\gamma(\tau)|^{2}}{4t}}dsd\tau,\qquad\forall
t>0.$

### Asymptotic expansion of $H_{S}(t)$

Denote by $\phi_{\tau}(s)=|\gamma(s)-\gamma(\tau)|^{2}$. We can explicitly
compute the asymptotic expansion of $H_{S}$ applying the Laplace method to the
integral

(85)
$I_{\tau}(\lambda)=\int_{0}^{\ell}e^{-\phi_{\tau}(s)\lambda}ds,\qquad\text{as
}\lambda\to+\infty.$

In particular, since $\gamma$ is a simple curve, the phase $\phi_{\tau}(s)$
has a strict minimum at $s=\tau$, thus there exists
$\varepsilon=\varepsilon(\tau)$ such that $\phi_{\tau}^{\prime}(s)>0$ for any
$s\in[\tau-\varepsilon,\tau+\varepsilon]\setminus\\{\tau\\}$. A direct
computation, building upon $\|\dot{\gamma}\|=1$, yields:

(86) $\displaystyle\phi_{\tau}^{\prime}(\tau)=0,$
$\displaystyle\phi_{\tau}^{\prime\prime}(\tau)=2,$
$\displaystyle\phi_{\tau}^{(5)}(\tau)=-5\partial_{\tau}\left(k(\tau)^{2}\right),$
$\displaystyle\phi_{\tau}^{\prime\prime\prime}(\tau)=0,$
$\displaystyle\phi_{\tau}^{(4)}(\tau)=-2k(\tau)^{2},$
$\displaystyle\phi_{\tau}^{(6)}(\tau)=-9\partial_{\tau}^{2}\left(k(\tau)^{2}\right)+2\|\dddot{\gamma}(\tau)\|^{2}.$

Therefore the phase has a Taylor expansion at its minimum and we can apply the
Laplace method, which gives a full asymptotic expansion, cf. [Olv97, Thm.
8.1],

(87) $I_{\tau}(\lambda)\sim
e^{-\lambda\phi_{\tau}(\tau)}\sum_{i=0}^{\infty}\Gamma\left(\frac{i+1}{2}\right)\frac{a_{i}(\tau)}{\lambda^{\frac{i+1}{2}}}\qquad\text{as
}\lambda\to\infty,$

where $\Gamma$ is the Euler Gamma function, and the $a_{i}(\tau)$ are given by
explicit formulas in terms of the derivatives of $\phi_{\tau}$ at its minimum.
Moreover, since, for any $\tau\in[0,\ell]$, the phase has an interior minimum,
the odd coefficients of the expansion vanish. For the even-order coefficients,
we have

(88) $a_{0}=1,\qquad a_{2}=\frac{1}{8}k(\tau)^{2},\qquad
a_{4}=\frac{1}{1152}\left(36\partial_{\tau}^{2}\left(k(\tau)^{2}\right)+35k(\tau)^{2}-8\|\dddot{\gamma}(\tau)\|^{2}\right).$

To conclude, we have to integrate with respect to $\tau$ the asymptotic
expansion of (85). In general, the expansion may not be uniform in
$\tau\in(0,\ell)$, however, since $\gamma$ is uniformly continuous on
$[0,\ell]$ and can be extended by periodicity on the whole real line, the
choice of $\varepsilon>0$ such that $\phi_{\tau}^{\prime}(s)>0$ for any
$s\in[\tau-\varepsilon,\tau+\varepsilon]\setminus\\{\tau\\}$ can be made
uniform, providing uniform estimates of the remainder. In particular, we have

(89) $H_{S}(t)\sim\frac{1}{\ell^{2}}\frac{1}{(4\pi
t)^{3/2}}\int_{0}^{\ell}\int_{\tau-\varepsilon}^{\tau+\varepsilon}e^{-\frac{\phi_{\tau}(s)}{4t}}dsd\tau,\qquad\text{as
}t\to 0,$

where $\varepsilon>0$ is chosen uniformly with respect to $\tau$. Hence, we
conclude that:

(90) $\begin{split}H_{S}(t)&\sim\frac{1}{\ell^{2}}\frac{1}{(4\pi
t)^{3/2}}\int_{0}^{\ell}\sum_{i=0}^{\infty}\Gamma\left(\frac{2i+1}{2}\right)a_{2i}(\tau)(4t)^{\frac{2i+1}{2}}d\tau\\\
&=\frac{1}{\ell^{2}}\frac{1}{4\pi
t}\sum_{i=0}^{\infty}\alpha_{i}t^{i},\end{split}$

as $t\to 0$, where the coefficients $\alpha_{i}$’s are defined by

(91)
$\alpha_{i}=2^{2i-1}(2i+1)\int_{0}^{\ell}a_{2i}(\tau)d\tau,\qquad\forall\,i\geq
0,$

and are given explicitly by (88), up to $i=2$.

### Approximation via tubular neighborhoods

We compute the asymptotic expansion of the approximation defined in
Proposition 5.1, when $S=\gamma([0,\ell])$, and we compare its coefficients
with those obtained in (90). Recall that, by (76), we set

(92)
$H^{\varepsilon}_{S}(t)=\frac{1}{|S_{\varepsilon}|^{2}}\int_{S_{\varepsilon}}\int_{S_{\varepsilon}}\frac{1}{(4\pi
t)^{3/2}}e^{-\frac{|x-y|^{2}}{4t}}dxdy,\qquad\forall\,t>0.$

By Theorem 2.3, there exists an asymptotic expansion up to order $4$ in
$\sqrt{t}$, of the form

(93)
$H^{\varepsilon}_{S}(t)=\frac{1}{|S_{\varepsilon}|^{2}}\left(\alpha_{0}^{\varepsilon}+\alpha_{1}^{\varepsilon}t^{1/2}+\alpha_{3}^{\varepsilon}t^{3/2}+o(t^{2})\right),$

where $\alpha_{0}^{\varepsilon}=|S_{\varepsilon}|$ and

(94)
$\alpha_{1}^{\varepsilon}=-\frac{1}{\sqrt{\pi}}\sigma^{\varepsilon}(\partial
S_{\varepsilon}),\qquad\alpha_{3}^{\varepsilon}=-\frac{1}{12\sqrt{\pi}}\int_{\partial
S_{\varepsilon}}\left(2\nabla\delta_{\mathrm{S}}\cdot\nabla(\Delta\delta_{\mathrm{S}})-(\Delta\delta_{\mathrm{S}})^{2}\right)d\sigma^{\varepsilon}.$

Notice that, in tubular coordinates $(s,r,\theta)$,
$\nabla\delta_{\mathrm{S}}=\partial_{r}$, and, since (83) holds,

(95)
$d\sigma^{r}=r(1-rk(s)\cos\theta)dsd\theta,\qquad\Delta\delta_{\mathrm{S}}=\frac{1}{r}+\partial_{r}\left(\log(1-rk(s)\cos\theta)\right).$

Thus, we can explicitly compute the coefficients (94):

(96)
$\begin{split}\alpha_{1}^{\varepsilon}&=-\frac{\varepsilon}{\sqrt{\pi}}\int_{0}^{\ell}\int_{0}^{2\pi}(1-\varepsilon
k(s)\cos\theta)dsd\theta=-2\varepsilon\ell\sqrt{\pi}\\\
\alpha_{3}^{\varepsilon}&=-\frac{\varepsilon}{12\sqrt{\pi}}\int_{0}^{\ell}\int_{0}^{2\pi}\left(-\frac{3}{\varepsilon^{2}}+\frac{A_{1}(s,\varepsilon,\theta)}{\varepsilon}+A_{0}(s,\varepsilon,\theta)\right)(1-\varepsilon
k(s)\cos\theta)dsd\theta,\end{split}$

where $A_{0},A_{1}$ are smooth functions defined by

(97)
$\begin{split}A_{0}(s,r,\theta)&=2\partial^{2}_{r}\left(\log(1-rk(s)\cos\theta)\right)-(\partial_{r}\left(\log(1-rk(s)\cos\theta)\right))^{2},\\\
A_{1}(s,r,\theta)&=-2\partial_{r}\left(\log(1-rk(s)\cos\theta)\right).\end{split}$

### Comparison between the two approaches

Let us compare the asymptotics of the two quantities in exam: for a fixed
$\varepsilon>0$ and as $t\to 0$, we have from (90) and (93)

(98)
$\begin{split}H^{\varepsilon}_{S}(t)&=\frac{1}{|S_{\varepsilon}|^{2}}\left(\alpha_{0}^{\varepsilon}+\alpha_{1}^{\varepsilon}t^{1/2}+\alpha_{3}^{\varepsilon}t^{3/2}+o(t^{2})\right),\\\
H_{S}(t)&=\frac{1}{\ell^{2}}\frac{1}{4\pi
t}\left(\alpha_{0}+\alpha_{1}t+\alpha_{3}t^{2}+o(t^{2})\right),\end{split}$

where the coefficients are given by (96) and (88), respectively. At this
stage, on the one hand, we notice that the order in $t$ of the expansions
doesn’t agree. On the other hand, the coefficients
$\alpha_{1}^{\varepsilon},\alpha_{3}^{\varepsilon}$ do not contain fine
geometrical information of $S$, indeed, the functions $A_{0}$, $A_{1}$ depend
only the curvature of $\gamma$, as opposed to (88), where derivatives of
$k(s)$ appear. Moreover, at a formal level, $\alpha_{1}^{\varepsilon}\to 0$ as
$\varepsilon\to 0$, whereas $\alpha_{3}^{\varepsilon}$ explodes: it is
possible to give meaning to these limits, taking into account the parabolic
scaling between the space and time variables of the heat equation and formally
replacing $t\mapsto\varepsilon^{2}t$, in (93), however, we do not recover any
geometrical meaning.

######  Remark 6.1.

The approximating relative heat content is too coarse a tool to detect the
geometry of a submanifold of high codimension and a different strategy is
needed. Inspired by the construction of the tubular neighborhood for $S$, we
may study the asymptotic expansion of $H_{S}(t)$ using a perturbative approach
similarly to what has been done in [Bar13], presenting the sub-Laplacian of
$S$ as a perturbation of a simpler operator. This will be object of future
research.

## References

  * [ABB20] A. Agrachev, D. Barilari, and U. Boscain.  A comprehensive introduction to sub-Riemannian geometry, volume 181 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2020.
  * [ARR21] A. Agrachev, L. Rizzi, and T. Rossi.  Relative heat content asymptotics for sub-riemannian manifolds.  arXiv preprint arXiv:2110.03926, 2021.
  * [AS04] A. Agrachev and Y. L. Sachkov.  Control theory from the geometric viewpoint, volume 87 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2004.  Control Theory and Optimization, II.
  * [Bar13] D. Barilari.  Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry.  J. Math. Sci. (N.Y.), 195(3):391–411, 2013.  Translation of Sovrem. Mat. Prilozh. No. 82 (2012).
  * [CdVHT20] Y. Colin de Verdière, L. Hillairet, and E. Trélat.  Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results.  arXiv preprint arXiv:2004.06461, 2020.
  * [DG94] S. Desjardins and P. Gilkey.  Heat content asymptotics for operators of Laplace type with Neumann boundary conditions.  Math. Z., 215(2):251–268, 1994.
  * [FPR20] V. Franceschi, D. Prandi, and L. Rizzi.  On the essential self-adjointness of singular sub-Laplacians.  Potential Anal., 53(1):89–112, 2020.
  * [Hör67] L. Hörmander.  Hypoelliptic second order differential equations.  Acta Math., 119:147–171, 1967.
  * [Mon02] R. Montgomery.  A tour of subriemannian geometries, their geodesics and applications, volume 91 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2002.
  * [Olv97] F. W. J. Olver.  Asymptotics and special functions.  AKP Classics. A K Peters, Ltd., Wellesley, MA, 1997.  Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)].
  * [PRS18] D. Prandi, L. Rizzi, and M. Seri.  Quantum confinement on non-complete Riemannian manifolds.  J. Spectr. Theory, 8(4):1221–1280, 2018.
  * [Rif14] L. Rifford.  Sub-Riemannian geometry and optimal transport.  SpringerBriefs in Mathematics. Springer, Cham, 2014.
  * [RR21] L. Rizzi and T. Rossi.  Heat content asymptotics for sub-Riemannian manifolds.  J. Math. Pures Appl. (9), 148:267–307, 2021.
  * [Sav98] A. Savo.  Uniform estimates and the whole asymptotic series of the heat content on manifolds.  Geom. Dedicata, 73(2):181–214, 1998.
  * [Str86] R. S. Strichartz.  Sub-Riemannian geometry.  J. Differential Geom., 24(2):221–263, 1986.
  * [TW18] J. Tyson and J. Wang.  Heat content and horizontal mean curvature on the Heisenberg group.  Comm. Partial Differential Equations, 43(3):467–505, 2018.
  * [vdBDG93] M. van den Berg, S. Desjardins, and P. Gilkey.  Functorality and heat content asymptotics for operators of Laplace type.  Topol. Methods Nonlinear Anal., 2(1):147–162, 1993.
  * [vdBG94] M. van den Berg and P. B. Gilkey.  Heat content asymptotics of a Riemannian manifold with boundary.  J. Funct. Anal., 120(1):48–71, 1994.
  * [vdBG15] M. van den Berg and P. Gilkey.  Heat flow out of a compact manifold.  J. Geom. Anal., 25(3):1576–1601, 2015.
  * [vdBLG94] M. van den Berg and J.-F. Le Gall.  Mean curvature and the heat equation.  Math. Z., 215(3):437–464, 1994.