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# Positive scalar curvature with point singularities

Simone Cecchini  Department of Mathematics, Texas A&M University, College
Station, TX 77843, USA<EMAIL_ADDRESS>simonececchini.org ,  Georg Frenck
Institut für Mathematik, Universität Augsburg, Universitätsstraße 14, 86159
Augsburg, Bavaria, Germany<EMAIL_ADDRESS>frenck.net and  Rudolf Zeidler
University of Münster, Mathematisches Institut, Einsteinstr. 62, 48149
Münster, Germany<EMAIL_ADDRESS>www.rzeidler.eu

###### Abstract.

We show that in every dimension $n\geq 8$, there exists a smooth closed
manifold $M^{n}$ which does not admit a smooth positive scalar curvature
(“psc”) metric, but $M$ admits an $\mathrm{L}^{\infty}$-metric which is smooth
and has psc outside a singular set of codimension $\geq 8$. This provides
counterexamples to a conjecture of Schoen. In fact, there are such examples of
arbitrarily high dimension with only single point singularities. In addition,
we provide examples of $\mathrm{L}^{\infty}$-metrics on $\mathbb{R}^{n}$ for
certain $n\geq 8$ which are smooth and have psc outside the origin, but cannot
be smoothly approximated away from the origin by everywhere smooth Riemannian
metrics of non-negative scalar curvature. This stands in precise contrast to
established smoothing results via Ricci–DeTurck flow for singular metrics with
stronger regularity assumptions. Finally, as a positive result, we describe a
$\mathrm{KO}$-theoretic condition which obstructs the existence of
$\mathrm{L}^{\infty}$-metrics that are smooth and of psc outside a finite
subset. This shows that closed enlargeable spin manifolds do not carry such
metrics.

###### 2020 Mathematics Subject Classification:

53C21 (Primary) 57R65; 58J22 (Secondary)

Co-funded by the Deutsche Forschungs Gemeinschaft (SPP2026 “Geometry at
Infinity”)

Co-funded by the European Union (ERC Starting Grant 101116001 – COMSCAL).
Views and opinions expressed are however those of the author(s) only and do
not necessarily reflect those of the European Union or the European Research
Council. Neither the European Union nor the granting authority can be held
responsible for them.

Co-funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project numbers 523079177; 427320536; 390685587.

##  1\. Introduction

Studying weak lower curvature bounds in various singular settings has been a
core aspect of Riemannian geometry since a long time. While for spaces with
_scalar curvature_ bounded below such a theory is not yet fully developed,
various approaches with results in different directions have been proposed
recently (for instance [7, 25, 18, 4, 2, 41, 28, 15, 26, 39, 32]). In this
paper, we investigate scalar curvature lower bounds for Riemannian
$\mathrm{L}^{\infty}$-metrics in the sense of the following definition.

######  Definition 1.1 (cf. [28, Definition 1.3]).

Let $M$ be a smooth manifold. A _uniformly Euclidean_ metric (or for short “
_$\mathrm{L}^{\infty}$ -metric_”) is a measurable symmetric $(0,2)$-tensor $g$
on $M$ such that there exists a smooth Riemannian metric $g_{0}$ on $M$ and
$\Lambda>0$ with

$\Lambda^{-1}g_{0}(\xi,\xi)\leq g(\xi,\xi)\leq\Lambda g_{0}(\xi,\xi)$

for almost all $x\in M$ and all tangent vectors $\xi\in\mathrm{T}_{x}M$.

For uniformly Euclidean metrics which are smooth outside a singular subset of
sufficiently high codimension, it makes sense to study the behavior of lower
scalar curvature bounds on the regular subset of the metric. The first general
results in this direction have been obtained by [28], where the following
conjecture of Schoen was popularized.

######  Conjecture 1.2 (Schoen, cf. [28, Conjecture 1.5]).

Let $M$ be a closed manifold that does not support any smooth Riemannian
metric of positive scalar curvature (i.e. $M$ is Yamabe non-positive). Let
$S\subset M$ be a closed submanifold of codimension $\geq 3$ and let $g$ be an
$\mathrm{L}^{\infty}$-metric on $M$ which is smooth on $M\setminus S$ and
satisfies $\operatorname{scal}_{g}\geq 0$ on $M\setminus S$. Then $g$ is
smooth on all of $M$ and $\operatorname{Ric}_{g}=0$.

In other words, this conjecture predicts that subsets of codimension at least
three are invisible for the purposes of scalar curvature in a quite strong
sense. This differs from the setting of codimension one and two, where it is
well known that such statements do not hold without additional regularity
assumptions or boundary conditions, compare [28, §8.1–8.2]. Likewise, the
condition of being uniformly Euclidean cannot be dropped from Conjecture 1.2,
otherwise suitable Schwarzschild metrics would yield counterexamples with
point singularities, compare [28, §8.3].

Conjecture 1.2 has attracted considerable interest and it has been confirmed
in many specific cases. For instance, if $\dim(M)=3$, it holds in general by
[28, Corollary 1.6]. In the $4$-dimensional case, far-reaching results have
been established by [23]. In a closely related direction, a version of
Llarull’s theorem for metrics with $\mathrm{L}^{\infty}$ point singularities
was recently proved by [12]. Fill-in problems with $\mathrm{L}^{\infty}$ point
singularities in dimension three were studied by [31, §4]. We also note that
related results for conical singularities and the positive mass theorem have
been proved by [13, 14]. In sharp contrast to all of this, our main result
shows that $\mathrm{L}^{\infty}$-regularity is in general not enough to retain
positive scalar curvature (“psc”) obstructions on closed manifolds, even
assuming smoothness outside a single point.

######  Theorem A.

For every $n\geq 8$, there exists a closed smooth manifold $M$ of dimension
$n$ which does not admit a smooth psc metric, but which admits an
$\mathrm{L}^{\infty}$-metric which is smooth and of uniformly positive scalar
curvature on the complement of a connected submanifold $S\subseteq M$of
codimension $\geq 8$. Moreover, the metric we obtain is of regularity
$\mathrm{W}^{2,p/2}$, for every $p<\operatorname{codim}(S\subset M)$. If
$8\leq n\equiv 0,1,2,$ or $4\mod 8$, then there are examples where
$S=\\{\ast\\}$ is a single point.

In particular, Conjecture 1.2 is false in every dimension $n\geq 8$ and there
are counterexamples where the singular set has arbitrarily high codimension.

######  Remark 1.3.

Due to the Sobolev $\mathrm{W}^{2,p/2}$-regularity, in our examples the scalar
curvature even exists in the weak sense on all of $M$ as a uniformly positive
$\mathrm{L}^{p/2}$-function for all $p<\operatorname{codim}(S\subset M)$. In
particular, our metrics have positive distributional scalar curvature in the
sense of [25, Definition 2.1].

The counterexamples in Theorem A are obtained via classification results based
on surgery theory which allow to construct psc metrics with conical
singularities on all high-dimensional simply connected manifolds (see Section
2 for details). To contrast this with the existing positive results concerning
Conjecture 1.2, note that the proofs usually proceed by first blowing up the
singularity to a complete psc metric on the complement of the singular set.
Next, appropriate topological obstructions are used to exclude the existence
of such complete psc metrics, leading to a contradiction. The last step fails
precisely because this is unobstructed on punctured simply connected manifolds
in high dimensions.

A more analytic approach to study Conjecture 1.2, and lower (scalar) curvature
bounds on metrics with low regularity in general, is to regularize them using
geometric flows, for a selection of works in this direction see [2, 40, 4, 5,
24, 27, 9, 11]. However, all existing results following this line need
additional assumptions on the metric beyond being uniformly Euclidean—for
instance, additional Sobolev or $\operatorname{C}^{0}$ regularity near the
singular set or being sufficiently $\mathrm{L}^{\infty}$-close to a smooth
background metric. It stands to reason that Theorem A must impose definite
restrictions for such a procedure to work for the scalar curvature of
$\mathrm{L}^{\infty}$-metrics.

######  Remark 1.4.

Indeed, [11, Theorem 6.2] showed that uniformly Euclidean metrics of
$\mathrm{W}^{1,n}$-regularity can be smoothed via Ricci–DeTurck flow (see also
[24] who treated the $\mathrm{W}^{2,2}$ case in dimension four). Since our
counterexamples from Theorem A in the point singularity case are in
$\mathrm{W}^{2,p/2}\subset\mathrm{W}^{1,p}$ for all $p<n$, this means that
$\mathrm{W}^{1,n}$ is the sharp threshold on the Sobolev scale for such a
smoothing result in general.

In another direction, [5] recently showed that one may smooth away certain
$\mathrm{L}^{\infty}$ singularities on $\mathbb{R}^{n}$ using Ricci–DeTurck
flow while preserving non-negativity of the scalar curvature, provided that
the singular metric is sufficiently $\mathrm{L}^{\infty}$-close to the
Euclidean metric. Since the counterexamples in Theorem A are based on high-
dimensional simply connected manifolds which do not admit psc, at first glance
this may look like a subtle global phenomenon of the underlying closed
manifold. However, as we note in the following theorem, they actually lead to
examples of $\mathrm{L}^{\infty}$-metrics on $\mathbb{R}^{n}$ with a single
point singularity which are non-smoothable. In particular, this answers [5,
Question 3].

######  Theorem B.

For $n\geq 8$, $n\equiv 0,1,2,$ or $4\mod 8$, there exists a metric $g$ on
$\mathbb{R}^{n}$ which has regularity $\mathrm{W}^{2,{p/2}}_{\mathrm{loc}}$
for each $p<n$ such that

  * •

$\Lambda^{-1}g_{0}\leq g\leq\Lambda g_{0}$ a.e. for some fixed $\Lambda>0$ and
$g_{0}$ the Euclidean metric,

  * •

$g$ is smooth on $\mathbb{R}^{n}\setminus\\{0\\}$,

  * •

$\operatorname{scal}_{g}\geq\lambda|x|^{-2}>0$ on
$\mathbb{R}^{n}\setminus\\{0\\}$ for some $\lambda>0$,

but there exists no family $(g_{t})_{t\in(0,\varepsilon)}$ of smooth metrics
on $\mathbb{R}^{n}$ satisfying both

  1. (i)

$\operatorname{scal}_{g_{t}}\geq 0$ for all $t\in(0,\varepsilon)$,

  2. (ii)

$g_{t}\xrightarrow{\operatorname{C}^{\infty}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus\\{0\\})}g$
for $t\searrow 0$.

Theorems A and B suggest to us that the setting of
$\mathrm{L}^{\infty}$-metrics with small singular sets is in general
insufficient for studying lower scalar curvature bounds in a local sense.
Instead, the fact that Conjecture 1.2 does hold true in certain cases appears
to be more of a global topological phenomenon. Thus, as a companion to our
counterexamples which are based on simply connected spin obstructions, we note
a stronger global spin obstruction to psc which does enable a positive result
in the direction of Conjecture 1.2 for point singularities. To formulate it,
let $M$ be a closed $n$-dimensional spin manifold with fundamental group
$\Gamma$. Denote by $\alpha(M;1)\in\operatorname{KO}_{n}(\mathbb{R})$ the
Hitchin genus [21] of $M$, that is, the KO-theoretic index of the
$\mathrm{Cl}_{n}$-linear Dirac operator on $M$. Let $\mathrm{C}^{\ast}\Gamma$
be the maximal real group $\mathrm{C}^{\ast}$-algebra of $\Gamma$, and let
$\alpha(M;\Gamma)\in\operatorname{KO}_{n}(\mathrm{C}^{\ast}\Gamma)$ be the
Rosenberg index of $M$ [35, 34, 36], see also [43, §2] for a brief survey. The
inclusion $\mathbb{R}\hookrightarrow\mathrm{C}^{\ast}\Gamma$, sending
$\lambda\in\mathbb{R}$ into $\lambda\cdot 1_{\mathrm{C}^{\ast}\Gamma}$,
induces a map
$j\colon\operatorname{KO}_{n}(\mathbb{R})\to\operatorname{KO}_{n}(\mathrm{C}^{\ast}\Gamma)$.
We will use the condition that $j(\alpha(M;1))\neq\alpha(M;\Gamma)$. In other
words, this means that the Rosenberg index of $M$ does not come from the
trivial group.

######  Theorem C.

Let $M$ be a closed spin $n$-dimensional manifold with fundamental group
$\Gamma$. Suppose that $j(\alpha(M;1))\neq\alpha(M;\Gamma)$. Let $S\subset M$
be a finite set, and let $g$ be an $\mathrm{L}^{\infty}$-metric on $M$ which
is smooth outside of $S$. If $\operatorname{scal}_{g}\geq 0$ on $M\setminus
S$, then $\operatorname{Ric}_{g}\equiv 0$ on $M\setminus S$.

######  Remark 1.5.

The condition $j(\alpha(M;1))\neq\alpha(M;\Gamma)$ is satisfied for
enlargeable spin manifolds [19, 20]. This includes spin manifolds that admit a
non-zero degree map to the torus, in particular manifolds of the form
$M=\mathrm{T}^{n}\\#N$, where $N$ is an arbitrary spin manifold. See [6,
Remarks 1.10–1.12] for further discussions of examples. Moreover, all
$3$-manifolds not carrying psc metrics are enlargeable and thus satisfy this
condition, which is the case treated by [28, Theorem 1.6].

Conversely, by design this is never satisfied for simply connected manifolds
and so avoids the counterexamples of Theorem A. It is essentially the most
general such condition currently conceivable in the spin setting (compare [37,
Conjecture 1.5]).

We note that the proof of Theorem C follows the same lines as in the works of
[23, 28]. Our contribution is identifying the correct KO-theoretic obstruction
to complete psc metrics which already appeared in work of [6], see also [8,
§3.2]. Moreover, as in the classical setting, the positive mass theorem with
$\mathrm{L}^{\infty}$ point singularities can be reduced to a corresponding
psc obstruction statement for closed manifolds of the form
$\mathrm{T}^{n}\\#N$. Indeed, in [23, Proof of Corollary B], Kazaras extended
Lohkamp’s argument [30, Theorem 6.1] to uniformly Euclidean metrics.
Therefore, Theorem C directly implies the following.

######  Corollary D.

Let $(M,g)$ be a complete asymptotically flat $n$-dimensional spin Riemannian
manifold, where $g$ is an $\mathrm{L}^{\infty}$-metric which is smooth outside
a finite set $S\subset M$. If $\operatorname{scal}_{g}\geq 0$ on $M\setminus
S$, then the ADM mass of each asymptotically flat end is non-negative.

The paper is organized as follows. In Section 2 we present the counterexamples
to Schoen’s conjecture and prove Theorem A. Section 3 is devoted to the proof
of Theorem B. Finally, in Section 4 we prove Theorem C.

### Acknowledgements

We thank Otis Chodosh, Johannes Ebert and Sven Hirsch for insightful
conversations. SC gratefully acknowledges the hospitality of Mathematics
Münster during his visit as a Young Research Fellow, which significantly
contributed to the development of this paper.

##  2\. Counterexamples to Schoen’s conjecture

In this section, we prove Theorem A. The proof is based on the following
classification result which stands in contrast to [42, Theorem A] and
complements [17, Corollary C].

######  Proposition 2.1.

Every closed simply connected manifold $M$ of dimension $n\geq 5$ admits an
$\mathrm{L}^{\infty}$-metric $g$ which lies in $\mathrm{W}^{2,p/2}$ for every
$p<n$, is smooth away from a point $p\in M$, and whose scalar curvature is
uniformly positive on $M\setminus\\{p\\}$.

The idea of the proof is the following: First, we remove an open disk. On this
new manifold with boundary, there is a _collared_ psc metric $g$, that is,
$g=h+\mathrm{d}{t}^{2}$ near the boundary. Then we interpolate this to a
conical metric $(t/\lambda)^{2}h+\mathrm{d}{t}^{2}$ which has an
$\mathrm{L}^{\infty}$-singularity at $t=0$. We start with the following lemma.

######  Lemma 2.2.

Let $M$ be a closed simply connected spin manifold of dimension $n\geq 5$. Let
$\mathrm{D}^{n}\subset M$ be an embedded disk and
$M^{\prime}=M\setminus\mathrm{D}^{n}$. Then there exists a collared psc metric
$g$ on $M^{\prime}$.

###### Proof.

First, we perform $2$-surgeries on $M$ away from $\mathrm{D}^{n}$ to obtain
$M_{2}$ which is $2$-connected. This is possible as every embedded $2$-sphere
has trivial normal bundle because of the spin condition.

We write $M_{2}^{\prime}\coloneqq M_{2}\setminus\mathrm{D}^{n}$ and we observe
that the inclusion $\partial M_{2}^{\prime}=\mathrm{S}^{n-1}\hookrightarrow
M_{2}^{\prime}$ is $2$-connected by construction. If $n\geq 6$, the surgery
theorem of [17, 38] in its cobordism formulation [16, Theorem 1.5] implies
that there exists a psc metric $g_{2}$ on $M_{2}^{\prime}$ which is collared,
that is, $g_{2}=h+\mathrm{d}{t}^{2}$ on a collar $\partial
M_{2}^{\prime}\times[0,\varepsilon)\subset M_{2}^{\prime}$. Since the reverse
surgery of a $2$-surgery has codimension $3$, we can reverse the surgeries
from the beginning to obtain a collared psc metric $g$ on
$M^{\prime}=M\setminus\mathrm{D}^{n}$ which agrees with $g_{2}$ near $\partial
M^{\prime}=\partial M_{2}^{\prime}$.

If on the other hand $n=5$, $M$ itself already admits a psc metric by [42,
Theorem A]111This also follows directly from [17, Theorem B] and the fact that
$\Omega_{5}^{\mathrm{spin}}=0$. which can be deformed to equal a torpedo-
metric on an embedded $\mathrm{D}^{5}\subset M$ ([10, Theorem 1.1], see also
[16, Theorem 1.2]). Removing $\mathrm{D}^{5}$ will yield a psc metric on
$M^{\prime}$ which is collared. ∎

######  Remark 2.3.

The statement of Lemma 2.2 is also true if $M$ is simply connected and non-
spin: In this case, [17, Corollary C] yields a psc metric on $M$ which—as in
the proof above (using [Theorem 1.1]chernysh_on-the-homotopy-type-of[Theorem
1.2]ebertfrenck)—can be deformed to be a torpedo-metric on an embedded disk
$\mathrm{D}^{n}\subset M$. Deleting this disk with the torpedo-metric yields
the desired collared metric on the complement.

###### Proof of Proposition 2.1.

If $M$ does not admit a spin structure then by [17, Corollary C] there even
exists a smooth positive scalar curvature metric on $M$. Therefore, we may
assume that $M$ is spin.

In this case let $g$ be a collared psc metric on
$M^{\prime}=M\setminus\mathrm{D}^{n}$ provided by Lemma 2.2, where
$\mathrm{D}^{n}$ is some embedded disk around the point $p\in M$. Denote the
restriction of $g$ to the boundary $\partial M^{\prime}$ by $h$. Then $h$ has
uniformly positive scalar curvature due to compactness. Let
$C\coloneqq\inf_{x\in\partial M^{\prime}}\operatorname{scal}_{h}(x)>0$. Now,
let $R>1$ be such that there exists a smooth, monotonically decreasing
function $\alpha\colon\mathbb{R}\to[0,1]$ (see Figure 1) such that
$\alpha|_{(-\infty,-R+\varepsilon]}=1$ and $\alpha|_{[-\varepsilon,\infty)}=0$
with

$-\frac{1}{2}\sqrt{\frac{C}{4(n-1)(n-2)}}\leq\alpha^{\prime}(t)\leq
0\quad\text{ and }\quad|\alpha^{\prime\prime}(t)|\leq\frac{C}{8(n-1)}.$

Furthermore, let $\lambda\in\mathbb{R}$ be such that

$\lambda\geq\max\left\\{R,2\sqrt{\frac{4(n-1)(n-2)}{C}}\right\\}>1.$

$-R$$-R+\varepsilon$$-\varepsilon$$1$$f$$\alpha$

Figure 1. The auxiliary function $\alpha\colon\mathbb{R}\to[0,1]$ and the
warping function $f\colon\mathbb{R}\to(-\infty,1]$.

We define $f(t)\coloneqq\alpha(t)\frac{t+R}{\lambda}+(1-\alpha(t))$ and
consider the conical metric

$g_{\mathrm{cone}}=f(t)^{2}h+\mathrm{d}{t}^{2}$

on $\mathrm{S}^{n-1}\times(-R,0]$. Its scalar curvature is given by

(2.1) $\displaystyle\operatorname{scal}_{g_{\mathrm{cone}}}$
$\displaystyle=\frac{1}{f(t)^{2}}\left(\operatorname{scal}_{h}-2(n-1)f(t)f^{\prime\prime}(t)-(n-1)(n-2)f^{\prime}(t)^{2}\right).$

We note that, by our choices, we have for $t>-R$

$\displaystyle f(t)$
$\displaystyle=-\underbrace{\alpha(t)(1-\frac{t+R}{\lambda})}_{\in[0,1)}+1\in(0,1]$
$\displaystyle|f^{\prime}(t)|$
$\displaystyle=|\alpha^{\prime}(t)(\frac{t+R}{\lambda}-1)+\alpha(t)\frac{1}{\lambda}|\leq|\alpha^{\prime}(t)|\cdot\underbrace{|\frac{t+R}{\lambda}-1|}_{\leq
1}+\underbrace{|\alpha(t)|}_{\leq 1}\cdot|\frac{1}{\lambda}|$
$\displaystyle\leq-\alpha^{\prime}(t)+\frac{1}{\lambda}\leq\sqrt{\frac{C}{4(n-1)(n-2)}}$
(2.2) $\displaystyle\Rightarrow(n-1)(n-2)f^{\prime}(t)^{2}\leq\frac{C}{4}$
$\displaystyle f^{\prime\prime}(t)$
$\displaystyle=\alpha^{\prime\prime}(t)\left(\frac{t+R}{\lambda}-1\right)+\underbrace{\frac{2\alpha^{\prime}(t)}{\lambda}}_{\leq
0}\leq|\alpha^{\prime\prime}(t)|\left(1-\frac{t+R}{\lambda}\right)$
$\displaystyle\leq|\alpha^{\prime\prime}(t)|\leq\frac{C}{8(n-1)}$ (2.3)
$\displaystyle\Rightarrow 2(n-1)f(t)f^{\prime\prime}(t)\leq
2(n-1)f^{\prime\prime}(t)\leq\frac{C}{4}.$

Inserting Equation 2.2 and Equation 2.3 into Equation 2.1, we obtain:

$\operatorname{scal}_{g_{\mathrm{cone}}}=\frac{1}{f(t)^{2}}\Bigl{(}\operatorname{scal}_{h}-2(n-1)f(t)f^{\prime\prime}(t)-(n-1)(n-2)f^{\prime}(t)^{2}\Bigr{)}\geq\frac{C}{2}>0.$

Note that $f(-R)=0$ and, since $g_{\mathrm{cone}}=h+\mathrm{d}{t}^{2}$ for
$t>-\varepsilon$, we can extend the metric $g$ on $M^{\prime}$ by
$g_{\mathrm{cone}}$ to a metric $\hat{g}$ on $M$ which in general has a true
singularity at the point $p\in M$ corresponding to $t=-R$. A visualization of
$\hat{g}$ can be found in Figure 2. Near the singularity we have
$\operatorname{scal}_{\hat{g}}>\frac{C}{2}$, so $\hat{g}$ has uniformly
positive scalar curvature on all of $M\setminus\\{p\\}$.

It remains to verify that $\hat{g}$ is uniformly Euclidean and satisfies the
desired Sobolev regularity on all of $M$. Since the metric is smooth away from
the singularity, it suffices to show that the metric has these properties on
some neighborhood including the singularity, where the metric is precisely
conical. This follows from a direct local computation which we record in Lemma
2.4 below.

$\underbrace{\quad\qquad\qquad\qquad\qquad}_{M\setminus\mathrm{D}^{n}}$
$\underbrace{\quad\qquad\qquad\qquad\qquad}_{\mathrm{S}^{n-1}\times[-R+\varepsilon,0]}$
${\mathrm{S}^{n-1}\times(-R,-R+\varepsilon]}$ $g$$g_{\mathrm{cone}}$transition
metric$\left(\frac{t+R}{\lambda}\right)^{2}h+\mathrm{d}{t}^{2}$ Figure 2. The
metric $\hat{g}$ on $M$: On $M\setminus\mathrm{D}^{n}$ we have a psc metric
$g$ which is collared, then there is a transition region
$\mathrm{S}^{n-1}\times[-R+\varepsilon,0]$ and on
$\mathrm{S}^{n-1}\times(-R,R+\varepsilon]$ the metric $g_{\mathrm{cone}}$ is
the linear scaling of $h$.

∎

######  Lemma 2.4.

Let $h$ be a Riemannian metric on $\mathrm{S}^{n-1}$. Then the metric
$g=t^{2}h+\mathrm{d}{}t^{2}$ on
$\mathrm{D}^{n}\setminus\\{0\\}\cong\mathrm{S}^{n-1}\times(0,1]$, identified
via $x\mapsto(x/\lvert x\rvert,\lvert x\rvert)$, is uniformly Euclidean on the
entire disk $\mathrm{D}^{n}$ and its coefficients lie in
$\mathrm{W}^{2,p/2}(\mathrm{D}^{n})\cap\mathrm{L}^{\infty}(\mathrm{D}^{n})$
for every $p<n$.

###### Proof.

Since $\mathrm{S}^{n-1}$ is compact, the metric $h$ is “uniformly round”, that
is, there exists $\Lambda_{\circ}>1$ such that
$\Lambda_{\circ}^{-1}h_{\circ}\leq h\leq\Lambda_{\circ}h_{\circ}$ for
$h_{\circ}$ the round metric on $\mathrm{S}^{n-1}$. Then
$g_{\circ}=t^{2}h_{\circ}+\mathrm{d}{t}^{2}$ corresponds to the standard
Euclidean metric on $\mathrm{D}^{n}$ and we have

$\displaystyle\Lambda_{\circ}^{-1}g_{\circ}\leq\Lambda_{\circ}^{-1}t^{2}h_{\circ}+\mathrm{d}{t}^{2}\leq
t^{2}h+\mathrm{d}{t}^{2}=g\leq\Lambda_{\circ}t^{2}h_{\circ}+\mathrm{d}{t}^{2}\leq\Lambda_{\circ}g_{\circ},$

so $g$ is uniformly Euclidean on $\mathrm{D}^{n}$.

To verify the Sobolev regularity, let
$\Phi\colon\mathrm{D}^{n}\setminus\\{0\\}\to\mathrm{S}^{n-1}\times(0,1]$
denote the diffeomorphism sending $x=(x^{1},\dotsc,x^{n})$ to $(x/\lvert
x\rvert,\lvert x\rvert)$. Then

$\frac{\partial\Phi}{\partial x^{i}}=\frac{1}{\lvert
x\rvert}\left(\frac{\partial}{\partial x^{i}}-x^{i}\frac{\mathbf{x}}{\lvert
x\rvert^{2}},x^{i}\right),$

where $\mathbf{x}=\sum_{k=1}^{n}x^{k}\frac{\partial}{\partial{x^{k}}}$ is the
identity viewed as vector field. Thus

$g_{ij}(x)=g_{x}\left(\frac{\partial\Phi}{\partial
x^{i}},\frac{\partial\Phi}{\partial x^{j}}\right)=h_{x/\lvert
x\rvert}\left(\frac{\partial}{\partial x^{i}}-x^{i}\frac{\mathbf{x}}{\lvert
x\rvert^{2}},\frac{\partial}{\partial x^{j}}-x^{j}\frac{\mathbf{x}}{\lvert
x\rvert^{2}}\right)+\frac{x^{i}x^{j}}{\lvert x\rvert^{2}}$

where we used the identification $t=\lvert x\rvert$ under the diffeomorphism
$\Phi$. We deduce that we can write $g_{ij}(x)=f_{ij}(x/\lvert x\rvert)$ for
some smooth functions $f_{ij}\colon\mathbb{R}^{n}\to\mathbb{R}$. Hence,
$g_{ij}\in\operatorname{\mathcal{O}}(1)$,
$\partial_{k}g_{ij}\in\operatorname{\mathcal{O}}(t^{-1})$ and
$\partial_{l}\partial_{k}g_{ij}\in\operatorname{\mathcal{O}}(t^{-2})$ as $t\to
0$. This shows that each $g_{ij}$ is in
$\mathrm{W}^{2,p/2}(\mathrm{D}^{n})\cap\mathrm{L}^{\infty}(\mathrm{D}^{n})$,
for every $p<n$. ∎

To prove Theorem A, we need closed simply connected manifolds which do not
support positive scalar curvature. In dimensions $\geq 5$ such examples must
be spin and they exist whenever the receptacle for the Hitchin
$\alpha$-invariant [21] does not vanish. The latter is the group
$\operatorname{KO}_{n}$ which is $\mathbb{Z}$ if $n\equiv 0,4\mod 8$,
$\mathbb{Z}/2\mathbb{Z}$ if $n\equiv 1,2\mod 8$ and vanishes otherwise.

######  Remark 2.5.

If $4\leq n\equiv 0,1,2,$ or $4\mod 8$, then there exists a simply connected
closed spin $n$-manifold $B$ which does not admit a smooth positive scalar
curvature metric because $\alpha(B;1)\neq 0\in\operatorname{KO}_{n}$. Indeed,
the Atiyah–Bott–Shapiro shapiro orientation yields a surjective homomorphism
from the spin cobordism group
$\Omega^{\mathrm{spin}}_{n}\to\operatorname{KO}_{n}$ for $n\geq 0$ and for
$n\geq 4$ we can use surgery to find simply connected representatives. More
concretely, for $n\equiv 0$ or $4\mod 8$ we can just take $B$ to be a product
of $\mathrm{K}3$ surfaces to obtain a simply connected spin manifold with non-
vanishing $\widehat{\mathrm{A}}$-genus. In the cases $8\leq n\equiv 1,2\mod 8$
one can find exotic spheres with non-vanishing Hitchin genus in
$\mathbb{Z}/2\mathbb{Z}$. This follows from classical computations of [1, 33],
see also the discussion in [22, §4].

###### Proof of Theorem A.

In the cases $8\leq n\equiv 0,1,2,$ or $4\mod 8$ there exists a simply
connected spin $n$-manifold $B$ which does not admit a smooth positive scalar
curvature as explained in Remark 2.5 above. By Proposition 2.1 however, there
exists an $\mathrm{L}^{\infty}$-metric of regularity $\mathrm{W}^{2,p/2}$ on
$B$ which is smooth away from a point and has uniformly positive scalar
curvature. This metric is already the desired counterexample.

The cases of other dimension parities follow from the fact that the product of
a smooth metric on another manifold $N$ and an $\mathrm{L}^{\infty}$-metric of
regularity $\mathrm{W}^{k,p}$ on $B$ which is smooth on $B\setminus\\{\ast\\}$
yields an $\mathrm{L}^{\infty}$-metric on $B\times N$ which is of regularity
$\mathrm{W}^{k,p}$ and smooth on $B\times N\setminus(\\{\ast\\}\times N)$.
Therefore, on the one hand, $M\coloneqq B\times\mathrm{T}^{k}$ admits an
$\mathrm{L}^{\infty}$-metric which is smooth with uniformly positive scalar
curvature outside $S=\\{\ast\\}\times\mathrm{T}^{k}\subset M$. Moreover, such
metric has regularity $\mathrm{W}^{2,p/2}$ for every $p<n$. Note that the
codimension of $S$ in $M$ is equal to $\dim(B)=n\geq 8$. On the other hand,
$\alpha(B\times\mathrm{T}^{k};\mathbb{Z}^{k})\neq 0$ (see for instance [44,
Appendix B]) and so $M$ does not support a smooth metric of positive scalar
curvature. ∎

##  3\. Non-smoothability of $\mathrm{L}^{\infty}$-metrics on $\mathbb{R}^{n}$
with positive scalar curvature outside a point

Here we prove Theorem B by a similar construction as in the previous section.

###### Proof of Theorem B.

Let $n$ be as specified in the statement of the theorem. Then by Remarks 2.5
and 2.2 there exists a closed $n$-manifold $M$ which does not admit psc but
there exists a psc metric on $g_{M}$ on $M\setminus\mathrm{D}^{n}$ that is
collared near the boundary. In particular, we can arrange that
$g_{M}=h+\mathrm{d}t^{2}$ near
$\partial(M\setminus\mathrm{D}^{n})=\mathrm{S}^{n-1}$ for some metric $h$ on
$\mathrm{S}^{n-1}$ with $\operatorname{scal}_{h}\geq 1$. We define an
$\mathrm{L}^{\infty}$-metric on $\mathbb{R}^{n}$ by

$g\coloneqq\left(\frac{t}{\lambda}\right)^{2}h+\mathrm{d}t^{2}$

where $\lambda\coloneqq\sqrt{2(n-1)(n-2)}$. As in the proof of Lemma 2.4,
since any metric on $\mathrm{S}^{n-1}$ is “uniformly round” and the Euclidean
metric is given by $g_{\circ}=t^{2}h_{\circ}+\mathrm{d}{t}^{2}$, this means
that there exists a constant $\Lambda>1$ such that $\Lambda^{-1}g_{\circ}\leq
g\leq\Lambda g_{\circ}$. The metric $g$ is smooth on
$\mathbb{R}^{n}\setminus\\{0\\}$ and the desired
$\mathrm{W}^{2,p/2}_{\mathrm{loc}}$-regularity near $0$ also follows from
Lemma 2.4. Its scalar curvature on $\mathbb{R}^{n}\setminus\\{0\\}$ is given
by

$\operatorname{scal}_{g}=\left(\frac{\lambda}{t}\right)^{2}\left(\operatorname{scal}_{h}-\frac{(n-1)(n-2)}{\lambda^{2}}\right)\geq\frac{1}{2}\lambda^{2}t^{-2}>0.$

Furthermore, the induced metric on
$\mathrm{S}^{n-1}_{\lambda}=\\{t=\lambda\\}\subset\mathbb{R}^{n}$ is precisely
given by $h$. The mean curvature222Using the convention that the mean
curvature of the unit sphere with respect to the Euclidean metric is equal to
$n-1>0$. $\operatorname{H}_{\lambda}$ of $\mathrm{S}^{n-1}_{\lambda}$
satisfies

$\operatorname{H}_{\lambda}=(n-1)\frac{1}{\lambda}>0.$

Let us now proceed by contradiction and assume the existence of the family of
smooth metrics $g_{t}$ with non-negative scalar curvature and converging to
$g$ in $\operatorname{C}^{\infty}_{\mathrm{loc}}(M\setminus\\{0\\})$. Then
there exists an $\varepsilon>0$ such that
$\operatorname{scal}_{g_{\varepsilon}}(p)\geq\frac{1}{4}\lambda^{2}$ for all
$\lambda/2\leq|p|\leq\lambda$, the metric $h_{\varepsilon}$ induced by
$g_{\varepsilon}$ on $\mathrm{S}^{n}_{\lambda}$ satisfies
$\operatorname{scal}_{h_{\varepsilon}}\geq\frac{1}{2}>0$ and the mean
curvature of $\mathrm{S}^{n}_{\lambda}$ is
$\operatorname{H}_{\lambda}\geq(n-1)/(2\lambda)$. Consider now the restriction
of $g_{\varepsilon}$ to $\mathrm{D}^{n}_{\lambda}$. It has non-negative scalar
curvature overall, positive scalar curvature near the mean convex boundary
$\partial\mathrm{D}^{n}_{\lambda}$ and the induced metric on the boundary
equals $h_{\varepsilon}$ and has positive scalar curvature. Thus we can assume
that $g_{\varepsilon}$ is collared while retaining non-negative scalar
curvature and without changing the restriction to the boundary
$\mathrm{S}_{\lambda}^{n-1}$, again by [3, Corollary 4.9]333Even though it is
not stated in loc.cit. the deformation principle they prove leaves the metric
on the boundary fixed ([3, Theorem 3.7 (b)]). In fact the proof of [3, Theorem
4.8] also works for a fixed psc metric $h$ on the boundary and then [3,
Theorem 4.5] with $\mathscr{X}=\\{h\\}$..

Furthermore, if $\varepsilon$ is small enough, $h_{\varepsilon}$ and $h$ are
isotopic through psc metrics, since the space of psc metrics is an open
subspace of the space of all metrics in the (weak)
$\operatorname{C}^{\infty}$-topology. Hence, there is a concordance from $h$
to $h_{\varepsilon}$ by [17, Lemma 3] (see also [16, Lemma 2.5]), that is,
there is a psc metric $G$ on the cylinder $\mathrm{S}^{n-1}\times[0,1]$ which
is collared near both boundaries and restricting to
$h_{\varepsilon}+\mathrm{d}t^{2}$ and $h+\mathrm{d}t^{2}$ on the respective
boundary components.

Gluing $G$ onto $g_{\varepsilon}|_{\mathrm{D}^{n}_{\lambda}}$, we obtain a
metric on $\mathrm{D}^{n}$ of non-negative scalar curvature with collared
boundary that restricts to $h$ near the boundary. Then we can glue this metric
onto $g_{M}$ to obtain a metric on the closed manifold $M$ which has non-
negative and non-vanishing scalar curvature. But this would mean that $M$ also
has a metric of positive scalar curvature (for instance, run the Ricci flow
for a short time), a contradiction to the choice of $M$. ∎

Theorem B affirmatively answers [5, Question 3]. In _loc. cit._ it is shown
that an $\mathrm{L}^{\infty}$-metric $g$ admits a “smoothing family”
$(g_{t})_{t>0}$ such as in the statement of Theorem B, provided the metric $g$
is $\mathrm{L}^{\infty}$-close to the Euclidean one. [5, Question 3] then asks
if the assumption of $\mathrm{L}^{\infty}$-closeness is necessary. Indeed, our
Theorem B shows that this condition cannot be replaced by merely requiring the
metric to be uniformly bilipschitz to the Euclidean metric.

##  4\. Relative Dirac obstructions

The proof of Theorem C is based on the following proposition.

######  Proposition 4.1.

Let $M$ be a closed $n$-dimensional spin manifold with fundamental group
$\Gamma$. Let $S$ be a finite subset of $M$, and let $g$ be a uniformly
Euclidean metric on $M$ which is smooth on $M\setminus S$. If
$\operatorname{scal}_{g}>0$ on $M\setminus S$, then
$j(\alpha(M;1))=\alpha(M;\Gamma)$.

###### Proof.

To simplify the notation, set $\hat{M}=M\setminus S$ and
$\hat{g}=g|_{\hat{M}}$. We will first construct a complete psc metric on
$\hat{M}$ following [28, Proof of Proposition 6.2] and [23, Proof of Theorem
A]. Consider the modified conformal Laplacian

$\mathcal{L}_{\sigma}\coloneqq-4\frac{n-1}{n-2}\Delta+\sigma,$

where $\Delta$ is the Laplace-Beltrami operator for $(\hat{M},\hat{g})$, and
$\sigma$ is a positive function on $M\setminus S$ such that
$0<\sigma\leq\min(1,\operatorname{scal}_{\hat{g}})$. Let $G$ be the positive
distributional solution of the elliptic PDE

$\mathcal{L}_{\sigma}G=\delta_{S}\qquad\textrm{on }M,$

where $\delta_{S}$ is the Dirac delta measure on $S$. By [29, Theorem 7.1]
(see also [9, Appendix A] for more details), there exists $c>0$ such that

(4.1) $c^{-1}\operatorname{dist}_{g}(S,x)^{2-n}\leq G(x)\leq
c\operatorname{dist}_{g}(S,x)^{2-n}\qquad\textrm{on }M\setminus S.$

For $\varepsilon>0$, consider the metric
$g_{\varepsilon}\coloneqq\left(1+\varepsilon\mathop{G}\right)^{\frac{4}{n-2}}\hat{g}$
on $\hat{M}$. By (4.1), $g_{\varepsilon}$ is complete. By direct computation,

$\operatorname{scal}_{g_{\varepsilon}}=\left(1+\varepsilon\mathop{G}\right)^{-\frac{n+2}{n-2}}\left(\operatorname{scal}_{\hat{g}}+\varepsilon\left(\operatorname{scal}_{\hat{g}}-\sigma\right)\mathop{G}\right)\geq\operatorname{scal}_{\hat{g}}\left(1+\varepsilon\mathop{G}\right)^{-\frac{n+2}{n-2}}>0.$

In conclusion, we constructed a complete metric of positive scalar curvature
on $\hat{M}$. By a direct application of [6, Theorem C], we conclude that
$j(\alpha(M;1))=\alpha(M;\Gamma)$. ∎

###### Proof of Theorem C.

We proceed by contraposition. Suppose that $\operatorname{Ric}_{g}\not\equiv
0$ on $M\setminus S$. By [28, Section 7, Proof of Theorem 1.7], there exists a
uniformly Euclidean metric $g_{1}$ on $M$ which is smooth $M\setminus S$ and
such that $\operatorname{scal}_{g_{1}}>0$ on $M\setminus S$. Then Proposition
4.1 shows that $j(\alpha(M;1))=\alpha(M;\Gamma)$. ∎

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#  An approach to implement Reinforcement Learning for Heterogeneous Vehicular
Networks

Bhavya Peshavaria, Sagar Kavaiya, Dhaval K. Patel, Bhavya Peshavaria, Sagar
Kavaiya and Dhaval K Patel are with School of Engineering and Applied Science,
Ahmedabad University, India.
(Email:bhavya.p<EMAIL_ADDRESS>dhaval.patel@ahduni.edu.in).

###### Abstract

This paper presents the extension of the idea of spectrum sharing in the
vehicular networks towards the Heterogeneous Vehicular Network(HetVNET) based
on multi-agent reinforcement learning. Here, the multiple vehicle-to-
vehicle(V2V) links reuse the spectrum of other vehicle-to-interface(V2I) and
also those of other networks. The fast-changing environment in vehicular
networks limits the idea of centralizing the CSI and allocate the channels.
So, the idea of implementing ML-based methods is used here so that it can be
implemented in a distributed manner in all vehicles. Here each On-Board
Unit(OBU) can sense the signals in the channel and based on that information
runs the RL to decide which channel to autonomously take up. Here, each V2V
link will be an agent in MARL. The idea is to train the RL model in such a way
that these agents will collaborate rather than compete.

###### Index Terms:

Vehicular networks, distributed spectrum access, multi-agent reinforcement
learning ,heterogeneous networks,

##  I Introduction

Vehicular communication had been introduced first time in 3GPP Release 12 for
communications with devices with mobility named D2D communication. Upon its
evolution towards release 14 and later such a communication link was termed as
vehicle-to-everything(V2X). Vehicular communication has achieved a high amount
of attention from research scholars and companies as it is a key enabler of
Intelligent Transportation Systems (ITS), smart cities, and autonomous
driving. In the recent 3GPP release 16, the concept of vehicular communication
is broadened to improve road safety by letting the On-Board Units(OBUs) of
cars communicate with each other. Such communications include disseminating
safety messages, traffic hazards ahead and other sensor-based information
picked up by the system. These communications are meant to improve road
safety, the comfort of transport, traffic congestion problems and enhance
situational awareness. However, due to the high mobility of the vehicles, it
is difficult to establish a reliable communication channel between two
vehicles to satisfy the necessity of lower latency.

Besides, the use cases of the V2X have been extended to incorporate autonomous
driving. According to the SAE International’s Standard J3016 [1], vehicles are
classified into six levels of autonomy. Of which, the vehicles with the
autonomy levels 0 and 1 are manually driven with no technological assistance.
However, the vehicles of levels 3 and 4 have partial and conditional autonomy;
here, vehicular communication could be implemented to let the drivers know of
the information about other vehicles. Those of levels 4 and 5 have a high
level of autonomy and no need for human interference. Such autonomous vehicles
need to constantly communicate among those in the vicinity to deliver messages
about their actions so that other vehicles can decide their actions. Such a
scenario needs highly reliable and low latency communication.

The focus of this paper will be on optimizing the communication in the
vehicle-to-vehicle(V2V) links. These links have one crucial necessity of
having low latency as they are responsible for disseminating the safety
messages for other vehicles and user equipment.

The paper is organized as follows. In Section I(A), the background of the
concepts involved in this paper is given. It includes a brief of different
works of literature in heterogeneous vehicular networks. In Section II, we
present the key takeaways and understanding of the work done throughout this
duration. Section III describes the future work which will be carried out in
the following months. It has plans on how to connect the dots of the concepts
discussed in this paper to bring out innovation.

###  I-A Background

Figure 1: A visual representation of the communication links in the 3GPP
Release 16 for C-V2X communication

According to 3GPP release 16, the following types of communication are
included in vehicular communication: Vehicle-to-vehicle (V2V) : This type of
communication will be utilized for the communication among vehicles. The main
use of it would be to transmit the sensor data, traffic hazards, vehicle
breakdown and other road safety data. Use cases of this can vary from
platooning, sending warning messages of vehicles out of site and event-driven
safety information. Vehicle-to-infrastructure (V2I) This type of link will
serve the purpose of providing a reliable and low-latency data connection to
the internet for the user devices inside the vehicle. Such links will mainly
be used for entertainment purposes in vehicles. Vehicle-to-pedestrian (V2P)
This link is very similar to V2V links. The idea behind this is to disseminate
the warning messages for the pedestrians of any possible accidents. Use cases
of this could be to warn pedestrians of any vehicular issuers like brake
failure so that they have enough time to react. Besides, it can be used to
warn pedestrians about the vehicles out of sight. In [3], the idea of spectrum
sharing in cellular networks has been discussed. The goal in [3] is to utilize
unused and available bands of the licensed cellular network for unlicensed V2V
communication so as to improve the throughput of dissemination of the safety
messages while not causing any disturbance in the currently occupied cellular
bands. The concept used there was to use a cellular network(Uu) for the V2I
links to provide a reliable and fast internet connection to the User
Equipment(UE), whereas use sidelink(PC5) for the V2V communication. This
sidelink was according to 3GPP Release 15, mode 4 where the UE can sense the
channels in the sensing window and then based on the received CSI,
autonomously select a band for the V2V link. A multi-agent reinforcement
learning(MARL) framework is used on the UE to autonomously pick the most
efficient selection scheme in the sensed conditions. This way, by implementing
spectrum sharing, the available spectrum can be better utilized without
needing to clear out the bands for such ad-hoc communications.

In [5], a cascaded Hungarian channel allocation algorithm has been described
for non-orthogonal multiple access (NOMA) based heterogeneous vehicular
networks. A chance-constrained throughput optimization problem has been
formulated to combat the imperfect CSI which arises due to high mobility in
vehicular networks.

[6] has utilized the white space spectrum in TV networks as an approach to
implementing Heterogeneous vehicular networks. They chose a non-cooperative
game theory approach with correlated equilibrium for the allocation of
resources. Here, the idea is to allow the macro-cell base station to allow
sharing available unused resources to UE without causing much interference to
macro-cell performance.

[7] has presented QoS based resource allocation scheme for an SDN(Software-
defined network) by sharing the LTE and Wi-Fi networks. A heuristic solution
was used for RA to be used in central as well as hybrid SDN. The simulation
consisted of 20 vehicles, one remote server, an eNodeB and 3 Wi-Fi AP(Access
Points). The proposed schemes were then evaluated using NS-3.

An approach about HetVNET for LTE and DSRC has been proposed in [8]. The
approach first tries to allocate all the LTE resources optimally and that way
reduce the channel congestion in DSRC.

##  II Contribution

###  II-A Study of MARL

Figure 2: Logical flow of MARL as discussed in [3]

The fig. 2 is the representation of the flow of RL implemented in [1]. In this
scenario, the V2I links are already assigned, the task is to assign V2V links.
Here, each V2V link is an agent in the RL framework.

The simulation starts with all the k agents(V2V links) getting their
respective observations from the environment. The observation function Zt(k)
represents the observation of the kth agent at time t. Once all the agents
have received their observations, they take independent decisions on which
available links to use and then take corresponding actions. Here, all the
agents take action simultaneously. The action function At(k) represents the
action of the kth agent at time t. Once the actions are taken, the joint
action At goes to the environment to simulate the data transfer. Once the
simulations are finished, the rewards are given to all the agents according to
their actions. And then the simulation goes to the next iteration at time t+1.
However, it waits till all the agents have updated their Q network based on
the result at the given time t. Once all the agents are ready, they observe
the environment again from the observation function. This was one iteration.

###  II-B Reward Function

The reward function is a key element of the RL framework. This function takes
the simulated result of the actions of agents as input for processing. It then
rates this input according to the metric coded in it to rate the decisions of
the agents. So, if the actions of the agent lead it closer to the defined
goal, then it will give more reward and vice versa. This function basically
takes the simulation towards the necessary goal.

In [1] reward function was needed to be carefully designed. As all the agents
are working in the same environment, the actions of one will affect the other.
Say the reward function is kept simply as each agent gets the reward according
to their successful transmission. In this scenario, the agents will evolve to
compete rather than collaborate. Each agent will try to hog as many spectral
resources as it can to get more rewards for itself. Such a reward function
will not give us the model we need.

So, in order to establish collaborative efforts among the agents, the reward
function is designed in a different way. Here, the scheme is that all the
agents will get a reward only if each of the agents has successfully completed
its transmission; if one agent fails then none would get the reward. In this
way, the agents will now try to collaborate with each other to optimize the
resource allocation such that they collectively achieve the maximal efficiency
they can in the given situation.

In this way, the reward function is defined in [1] to achieve collaboration
instead of competition.

###  II-C Reproduced Results

The following results are reproduced from [1]. Note: Due to less computation
power, there is less accuracy. This leads to the graph differing from the
original paper.

####  II-C1 Figure 3 of [3]

Figure 3: Sum capacity performance of V2I links with varying payload of V2V

In fig. 3 the blue line(marl_sum) shows the sum capacity of V2I when RA of V2V
was done with the help of the proposed MARL and the orange line (sarl_sum)
shows the sum of the sum capacity of V2I when RA of V2V was done with
SARL(Single agent reinforcement learning).

The fig. 3 shows the graph of sum capacity performance of V2I links with
varying V2V payload sizes. As we can observe from the graph that the general
trend of the sum of speeds of V2I links is decreasing with the increase in the
payload size of V2V links. This can be explained by the fact that V2V starts
occupying more bandwidth and hence affecting the performance of V2I. The main
goal in [1] was to allocate the resources optimally so that the sum of V2I
speed remains maximal while sharing the spectrum with V2V links.

####  II-C2 Figure 4 of [3]

Figure 4: V2V payload transmission success probability with varying payload of
V2V

In fig. 4 the blue line(marl_sum) shows the V2V payload transmission success
probability with varying payload sizes when done with the help of the proposed
MARL and the orange line (sarl_sum) shows the V2V payload transmission success
probability with varying payload sizes when RA of V2V was done with
SARL(Single agent reinforcement learning).

The fig. 4 shows the V2V payload transmission success probability with varying
payload sizes. It can be observed from the graph trend of all the lines that
with increasing transmission payload, the probability of successful
transmission decreases. This can be explained as the loss of data packets over
the transmission. The payload which is to be passed is broken down into
smaller chunks and then sent over the network. In this, if one packet gets
lost, then the whole payload is said to fail. So, the larger the payload, the
more the packets and hence the higher chances of transmission failure. This in
turn decreases the successful payload transmission as the payload size
increases.

##  III Future Work

With the increasing number of devices working on wireless communication
technology, it is possible to have a high UE density in some regions. In such
cases, the latency and packet drop may increase in that particular network.
Such conditions are not favourable for V2V transmission as they have time
crucial data which needs to be sent with high reliability and low latency. In
such cases, it is beneficial to switch towards other networks.

As discussed in the background section, there has been a lot of literature
around using heterogeneous vehicular networks. By combining the works in [5],
[6], [7] and [8] a HetVNET of the white TV spectrum, DSRC, Wi-Fi AP and LTE/NR
can be achieved. Such a large heterogeneous network has been discussed in [4].

To proceed further, the selection of which network to use at any given
instance needs to be decided in real-time. There have been some opportunistic
and greedy algorithms for such tasks but they cannot be malleable to all
situations and environments.

The concept of allocating the spectrum according to the environment had been
discussed in [3]. There, the concept was to allocate mobile network resources
to the V2V links dynamically. Now, that concept can be extended to work with
the HetVNET. In this model, the OBU will sense the channels of all the
networks enabled (i.e. white TV spectrum, DSRC, Wi-Fi AP and 5G LTE or NR)and
then based on the environment and the CSI will decide which band of which
network to choose for communication.

##  IV End Goal

As discussed in the previous section about future work, the idea is to extend
the concept of the implementation of MARL for efficient allocation of
resources to include all the other possible networks of HetVNET. This way
there will be a diverse range of spectral resources available for
communication and the agents can choose which are the most efficient links to
communicate over in the given scenario.

For example, if the communicating vehicles are going at similar speeds in the
same direction then they would choose DSRC over cellular as they are going to
stay in range for long enough for successful transmission and the latency of
DSRC is lower than Uu. Similarly, the agents can choose other networks based
on their pros and cons.

This implementation of HetVNET will also act as a load balancing mechanism for
the traffic in all the networks. If a network would get too crowded in terms
of resource requests, then the devices with this technology enabled could
easily switch over to the other network to offload the burden on the network
while making sure their transmission is not hindered by the network traffic.

##  V Conclusion

This paper has discussed the idea of combining HetVNET and MARL for resource
allocation on a superficial level as an in-depth study of this concept is out
of scope here. The main aim here is to impart this concept to other scholars
for the implementation and research of this in further studies. There are more
challenges to the implementation of such a technology at the present time.
Here each OBU is assumed to be having the necessary hardware resource to
process the whole MARL based framework in real-time. Such hardware could be
tough to fit in vehicles and consume more energy for performing the necessary
processes. However, when implemented, this concept would let the autonomous
and semi-autonomous vehicles communicate with each other in a more reliable
way.

##  VI Reference

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