# GCS-Q: Quantum Graph Coalition Structure Generation

Supreeth Mysore Venkatesh
Saarland University
Saarbruecken, Germany
<EMAIL_ADDRESS>
& Antonio Macaluso
German Research Center for Artificial Intelligence (DFKI)
Saarbruecken, Germany
<EMAIL_ADDRESS>
&Matthias Klusch
German Research Center for Artificial Intelligence (DFKI)
Saarbruecken, Germany
<EMAIL_ADDRESS>
corresponding author

###### Abstract

The problem of generating an optimal coalition structure for a given coalition
game of rational agents is to find a partition that maximizes their social
welfare and is known to be NP-hard. This paper proposes GCS-Q, a novel
quantum-supported solution for Induced Subgraph Games (ISGs) in coalition
structure generation. GCS-Q starts by considering the grand coalition as
initial coalition structure and proceeds by iteratively splitting the
coalitions into two nonempty subsets to obtain a coalition structure with a
higher coalition value. In particular, given an $n$-agent ISG, the GCS-Q
solves the optimal split problem $\mathcal{O}(n)$ times using a quantum
annealing device, exploring $\mathcal{O}(2^{n})$ partitions at each step. We
show that GCS-Q outperforms the currently best classical solvers with its
runtime in the order of $n^{2}$ and an expected worst-case approximation ratio
of $93\%$ on standard benchmark datasets.

_Keywords_ Multi-Agent Systems $\cdot$ Coalition Formation $\cdot$ Quantum
Computing $\cdot$ Quantum Artificial Intelligence

##  1 Problem Formulation

The problem of coalition structure generation (CSG) is to find a partition of
$n$ rational agents into mutually disjoint coalitions such that the sum of the
resulting coalition values for this coalition structure is maximized. Given a
coalition game $(A,v)$ with a set $A$ of $n$ agents and a characteristic
function $v:\mathcal{P}(A)\rightarrow\mathbb{R}$ for coalition values $v(C)$
for all non-empty coalitions $C$ in $A$, the problem is to find a coalition
structure $CS^{*}$ of $A$ that maximizes the social welfare $\sum_{C\in
CS^{*}}v(C)$. An induced subgraph game (ISG) $(A,v$) is a coalition game
defined by a connected, undirected, weighted graph $G(V,w)$: The set
$V=\\{i\\}_{i\in\\{1..n\\}}$ of nodes in $G$ represents the set $A$ of $n$
agents, and the real-valued weights $w_{i,j}$ of edges $(i,j)\ in\ w$ denote
the synergies of cooperation or joint utilities of agents $i,j\in A$ in
feasible coalitions $C\subseteq A$. A coalition $C$ is feasible if and only if
it induces a connected subgraph of $G$. The coalition values $v(C)$ of a
feasible coalition $C$ is

$\displaystyle v(C)=\sum_{(i,j)\in w,~{}i,j\in C}w_{i,j}.$ (1)

For a given coalition game $(A,v)$, the coalition structures $CS$ are
partitions of $A$ into mutually disjoint, feasible coalitions $C$. The
corresponding ISG is to find the optimal coalition structure $CS^{*}$ with
maximal coalition value (or social welfare) for $(A,v)$:

$CS^{*}=\arg\max_{CS}\sum_{C\in CS}v(C).$ (2)

In ISGs, the coalition values depend only on the pairwise interactions between
nodes/agents. Nevertheless, the set of possible solutions is not restricted,
and the problem remains NP-complete [1].

##  2 Methods

We propose GCS-Q (Quantum-supported solution for Graph Coalition Structure
generation), a novel quantum-supported anytime approximate algorithm for
solving ISG problems. GCS-Q starts from the grand coalition as the initial
coalition structure. Then, based on the graph induced by the coalition game,
it recursively performs a min-cut to find the optimal split of the agents
within each coalition. This approach’s primary complexity source is given by
finding the optimal split which is NP-hard [2]. However, properly formulating
this problem and delegating it to a quantum annealer allows for obtaining a
quantum-supported algorithm capable of outperforming state-of-the-art
classical and quantum solutions.

###  2.1 QUBO formulation for Min-Cut Problem

Operating the min-cut to a given graph $G(V,w)$, corresponds to finding the
bipartition of $G$ with the highest possible coalition value. Here we
reformulate the min-cut problem as a QUBO (Quadratic Unconstrained Binary
Optimization) in order to execute it using quantum annealing.

Consider a coalition game $(A,v)$ with underlying graph $G(V,w)$. Finding the
optimal bipartition of $G$ can be formulated as a quadratic objective function
of binary variables $\\{x_{i}\\}_{i=1,\dots,n}$, as follows:

$\begin{split}\arg\min_{x}\ \sum_{x_{i}\in
C,x_{j}\in\overline{C}}w_{i,j}x_{i}(1-x_{j})+{\sum w_{i}x_{i}}=x^{t}Wx\\\
\end{split}$ (3)

where $w_{i,j}$ is the weight connecting the nodes $i$ and $j$ and $w_{i}$ is
a self-loop on node $i$. Associated to the solution of the objective function
in Eq. (3), we differentiate the vertices belonging to the subsets $C$ or
$\overline{C}$ caused by the cut, as follows:

$\forall x_{i}\in A,x_{i}=\begin{cases}1&\text{if }x_{i}\in C\\\ 0&\text{if
}x_{i}\in\overline{C}\\\ \end{cases}$ (4)

Given the QUBO formulation in Eq. (3), we can easily define the corresponding
optimization problem of an Ising Hamiltonian $\mathcal{H}$ with the assignment
$x_{i}\rightarrow(1-Z_{i})/2$, where $Z_{i}$ is the Pauli-$Z$ operator with
eigenvalues $\pm 1$. Importantly, splitting the original graph into disjoint
sets guarantees always obtaining a valid coalition structure for the coalition
game.

###  2.2 GCS-Q Algorithm

The GCS-Q algorithm follows the strategy of hierarchical divisive clustering
methods and applies it in the context of ISGs. It starts by considering the
grand coalition $g_{c}$ as the initial coalition structure. Then, the first
step consists of splitting $g_{c}$ into two independent subsets
$\\{C,\overline{C}\\}$ solving the optimal split problem on a quantum
annealer. If the coalition function of the grand coalition $g_{c}$ (i.e.,
$v(g_{c})$) is greater than $v(\\{C,\overline{C}\\})$ (i.e., the coalition
structure comprising $\\{C\\}$ and {$\overline{C}\\}$ separately), then the
algorithm stops, and the $g_{c}$ is returned as the best coalition structure.
Otherwise, the algorithm finds the optimal split within each coalition for the
current coalition structure $\\{C,\overline{C}\\}$ using the quantum
annealing. This process continues for each coalition until a lower coalition
value is generated by performing the split. Notice that the additive nature of
the coalition value function in Eq. (1) for ISG problems allows acting
independently on the coalitions to maximize the overall coalition value of the
coalition structure.

In other terms, GCS-Q tries to split up all the coalitions of the current
coalition structure into two smaller ones. The split is performed if and only
if the generated coalitions have higher coalition values than the original. In
the worst-case scenario, the hierarchy is built in $n-1$ steps when the
coalition game contains $n$ agents. A key difference between a standard
divisive clustering algorithm and GCS-Q is how the partitions are determined.
In the case of divisive clustering, the splitting is usually based on a
distance matrix between the clusters (i.e., coalitions). This leads to a
worst-case complexity (for every single step) of $n^{2}$. Meanwhile, the GCS-Q
considers at each step all possible partitions which amounts to $2^{n-1}$.
This latter number grows exponentially fast, making this approach intractable
using classical resources. However, as it will be shown in the next section, a
quantum annealer can solve the optimal split problem with a runtime linear to
$n$, which leads to an overall complexity in the order of $n^{2}$.

The pseudocode of GCS-Q is shown in Algorithm 1.

Algorithm 1 GCS-Q Algorithm

Input: Coalition game $(A,v)$, with underlying graph $G(A,w)$ where $|A|=n$
and ${w}:A\crossproduct A\to\mathbb{R}$
Output: Optimal coalition structure $CS^{*}$

1:$CS^{*}\leftarrow\\{\\}$ $\triangleright$ initialize $CS^{*}$ with empty
list

2:$queue\leftarrow g_{c}$ $\triangleright$ initialize $queue$ with grand
coalition $g_{c}$

3:while $queue\neq\emptyset$ do

4: $S\leftarrow queue.pop$ $\triangleright$ Fetch a coalition from $queue$

5: $\triangleright$ Optimal Split problem

6: Create a weight matrix $W$ from edges in $S$ $\triangleright$ Eq. (3)

7: Define the Ising Hamiltonian $\mathcal{H}$ for $W$

8: Solve Ising Hamiltonian $\mathcal{H}$ on a quantum annealer

9: Decode binary string to get $C$ and $\overline{C}$

10: $\triangleright$ End Optimal Split problem

11: if $\overline{C}=\emptyset$ then $\triangleright$ no splits of $S$ is
better than $S$

12: add $C$ to $CS^{*}$

13: else$\triangleright$ try further splitting $C$ and $\overline{C}$

14: add $C$ to $queue$

15: add $\overline{C}$ to $queue$

16: end if

17:end while

18:Return $CS^{*}$

To the best of our knowledge, only one approximate solver can find a solution
for an ISG problem with a fully connected graph in polynomial time, the C-Link
algorithm [3]. This approach is based on agglomerative clustering and makes
decisions by considering the local patterns without ever considering the
global patterns in the coalition game. On the contrary, the top-down approach
proposed by GCS-Q examines an exponentially large number of partitions at each
step. Thus, GCS-Q explores a larger portion of the solution space, potentially
leading to a better quality of solutions.

##  3 Evaluation

In this section, we evaluate the GCS-Q in terms of runtime and approximation
ratio on synthetic datasets, running experiments on a real quantum annealer,
the D-Wave 2000Q. In order to estimate the runtime of the quantum annealing in
solving the optimal split problem (sec. 2.1), two sets of ISGs are generated
by sampling the interaction score associated with the pairs of agents using
Laplace and Normal distributions111 $\mathcal{L}(\mu=0,b=5)$,
$\mathcal{N}(\mu=0,\sigma=5)$. The runtime for both distributions is reported
in Figure 1.

Figure 1:  The runtime of the D-Wave 2000Q when solving the optimal split
problem. For each number of agents, the same QUBO problem is solved $5$ times.
The blue line is the average runtime of the $5$ experiments. The yellow shaded
area represents the maximum and minimum runtimes. The green shaded area is
calculated considering the mean and the standard deviation of the runtimes for
each problem instance. The runtimes are reported in microseconds ($\mu s)$.

The order of growth of the runtime when increasing the number of agents is
linear. To confirm this hypothesis, we estimate a linear regression and
calculate the coefficient of determination $R^{2}$, which is equal to $1$ in
the case of a perfect deterministic linear function between the number of
agents $n$ and the runtime. A value of $97\%$ for both distributions indicates
that the relationship increases almost perfectly linearly in the order of
$\beta_{1}$. Thus, we can conclude that the average-case complexity of the
quantum annealing solution for solving the optimal split problem $\Theta(n)$.

Two sets of coalition games are generated based on the two distributions
mentioned above to assess the quality of GCS-Q. Thus, we implement the
Algorithm 1 into two variants: GCS-Q(c) solves the QUBO at each step using a
classical QUBO solver, which always returns the exact solution. While GCS-Q(q)
uses a D-Wave 2000Q. We compare the solutions of these two approaches for
coalition games up to $20$ agents with the optimal solution obtained by
running the IDP [4] algorithm on the same problem instances. Results are shown
in Figure 2.

Figure 2: Assessment of GSC-Q quality in terms of approximation error
calculated on the two benchmark datasets. The experiments are limited up to
$20$ agents since IDP has space complexity of $\mathcal{O}(2^{n})$ and time
complexity of $\mathcal{O}(3^{n})$, which makes most machines to go out of
resources.

For both distributions, the GCS-Q(c) has the largest error observed equally to
$7\%$. While the GCS-Q(q) produces the expected solution of the Algorithm 1
only for coalition games of size up to $10$. The approximation error suddenly
increases for problems with more than $10$ agents. The decrease in performance
is due to the poor quality of the solution provided by the quantum annealer.
In fact, with $n\geq 11$, the error is cascaded through further executions of
the algorithm, and the final solution attained for the ISG problem is far from
optimality.

###  3.1 Performance analysis

In this section, we compare the runtime of GCS-Q with state-of-the-art
classical and quantum solutions when considering an ISG problem with an
underlying fully connected graph. Since ISGs can be viewed as a particular
case of standard CSG, we include general solvers in the comparative analysis.

The state-of-the-art classical exact solvers for CSG problems are represented
by methods based on IDP [4], such as BOSS[5] and DyCE[6], with a worst-case
time complexity of $\mathcal{O}(3^{n})$. For ISGs, the CFSS [7] and KGC [8]
algorithms have shown good results with sparse graphs, but the worst-case
complexity for a complete graph still remains $\mathcal{O}(n^{n})$. The best
approximate solution in terms of the runtime is C-link [3], with scales as
$\mathcal{O}(n^{3})$.

BILP-Q[9] is the first quantum-supported solution with an average runtime in
the order of $2^{n}$ on quantum annealing. A graphical comparison of classical
and quantum solutions is provided in Figure 3.

Figure 3: Cost complexity as a function of the number of agents $n$. Classical
solutions are indicated with blue lines, while quantum solutions in green.

The ability of GCS-Q to be executed with a runtime quadratic in the number of
agents makes it the best solver for ISG problems.

##  4 Discussion

The proposed GCS-Q represents the best solver in terms of runtime for ISG
problems. However, the performance of the D-Wave 2000Q causes obtaining poor
quality solutions for problems with more than ten agents. The limits of the
D-Wave 2000Q have already been emphasized for specific optimization tasks[10].
Nonetheless, the latest generation of D-Wave, named Advantage, outperformed
D-Wave 2000Q for any problem size while solving larger problem instances. In
some cases, not only the Advantage system can find better-quality solutions
than the D-Wave 2000Q, but also obtains same quality solutions faster [11]222A
comparison between Advantage with the D-Wave 2000Q is reported [12] .
Nevertheless, several optimization strategies are still possible on the D-Wave
2000Q. For instance, the adoption of different embedding strategies or using
hybrid quantum annealing that can deliver better quality solutions at the cost
of worsening the runtime (for more details, see D-Wave
documentation333https://docs.dwavesys.com/docs/latest/index.html).

Compared with other approximate solvers, such as C-Link [3], GCS-Q examines a
larger portion of the solution space, potentially leading to a better
approximation ratio. In fact, C-link relies on hierarchical agglomerative
clustering where only local interactions are adopted for merging coalitions.
In other terms, given a set of $n$ coalitions, C-link explores the distance
matrix between coalitions and merges only the pair of coalitions with the
minimum distance ($n^{2}$ possibilities). On the contrary, GCS-Q examines all
possible bipartition of a given coalition exploring $\mathcal{O}(2^{n})$
possibilities at each step. This implies the total number of solutions
analyzed to be (worst-case) in the order of $\mathcal{O}(n2^{n})$ and thus
significantly higher than the number of solutions investigated by C-Link,
which is $\mathcal{O}(n^{3})$.

With respect to other quantum solutions, such as BILP-Q [9], GCS-Q delivers a
better qubit complexity. Although being an exact solver, BILP-Q requires
$2^{n}$ logical qubits to be executed, which is a substantial limitation
considering near-term quantum computers. In comparison, GCS-Q can solve an
$n$-agent ISG using $n$ logical qubits.

##  5 Conclusions

We proposed GCS-Q, a novel quantum-supported solution for induced subgraph
games in coalition structure generation. The key idea is to partition the
graph underlying the coalition game into two subsets to obtain a coalition
structure with a better coalition value at each step. By delegating the task
of finding the optimal split to a quantum annealer, we obtain a solver capable
of running faster than the state-of-the-art solutions (quantum and classical).

In addition, we provided a practical implementation of GCS-Q and evaluated its
performance on standard benchmark datasets. Specifically, we generated two
sets of ISGs with fully connected graphs by sampling the edge weights from two
distributions (Laplace and Normal).

When calculating the quality of the solutions, the performance of GCS-Q
deteriorates due to the limitations of the quantum hardware in use. For this
reason, the main challenge to tackle in the near future is the investigation
of alternative embedding strategies or the adoption of hybrid quantum-
classical solvers proposed by D-Wave.

Another natural follow-up is the execution of GCS-Q on better quantum
hardware, such as the D-Wave Advantage. This latest generation of quantum
devices has outperformed the D-Wave 2000Q in terms of both quality of
solutions and runtime in several combinatorial optimization problems.

In conclusion, we showed the feasibility and the benefit of adopting quantum
computation in multi-agent systems. Thanks to this work, we believe that GCS-Q
can represent the first quantum-supported solutions suitable for solving real-
world AI problems on near-term quantum technology.

## Acknowledgments

This work has been funded by the German Ministry for Education and Research
(BMB+F) in the project QAI2-QAICO under grant 13N15586.

## References

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Defining Pointwise Lower Scalar Curvature Bounds for $C^{0}$ Metrics

Defining Pointwise Lower Scalar Curvature Bounds
for $\boldsymbol{C^{0}}$ Metrics with Regularization by Ricci Flow††This paper
is a contribution to the Special Issue on Scalar and Ricci Curvature in honor
of Misha Gromov on his 75th Birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Gromov.html

Paula BURKHARDT-GUIM

P. Burkhardt-Guim

Department of Mathematics, University of California, Berkeley, USA
<EMAIL_ADDRESS>https://math.berkeley.edu/~paulab/

Received July 30, 2020, in final form November 19, 2020; Published online
December 04, 2020

We survey some recent work using Ricci flow to create a class of local
definitions of weak lower scalar curvature bounds that is well defined for
$C^{0}$ metrics. We discuss several properties of these definitions and
explain some applications of this approach to questions regarding uniform
convergence of metrics with scalar curvature bounded below. Finally, we
consider the relationship between this approach and some other generalized
notions of lower scalar curvature bounds.

Ricci flow; scalar curvature; synthetic lower curvature bounds

53E20; 53C21

##  1 Introduction

In [5] Gromov presented the following theorem on the uniform limit of metrics
satisfying a pointwise lower bound on the scalar curvature; cf. [5, p. 1118]
and [1, Theorem 1]:

######  Theorem 1.1.

Let $M$ be a smooth manifold and $\kappa\colon M\to R$ a continuous function
on $M$. Suppose $g_{i}$ is a sequence of $C^{2}$ metrics on $M$ that converges
locally uniformly to a $C^{2}$ metric $g$ on $M$. If $R(g_{i})\geq\kappa$
everywhere on $M$ for $i=1,2,\dots$, then $R(g)\geq\kappa$ everywhere on $M$
as well.

The proof described by Gromov involves formulating positive scalar curvature
as a $C^{0}$ quantity, by considering the mean convexity and dihedral angles
of small cubes. Bamler provided an alternative proof of Theorem 1.1 in [1],
which used the evolution of the scalar curvature under Ricci flow and some
results of Koch and Lamm [7] concerning the Ricci–DeTurck flow for a class of
possibly nonsmooth initial data on Euclidean space.

In light of Bamler’s approach to Theorem 1.1, it is natural to ask whether it
is possible to use Ricci flow to formulate a generalized definition of lower
scalar curvature bounds for $C^{0}$ metrics. In [3] the author proposed a
class of such definitions, and proved some related results. Generally
speaking, the lower scalar curvature bounds in [3] are determined by using the
Ricci flow to “regularize” the singular metric and then observing the scalar
curvature of the flow at small positive times. The purpose of this survey is
to describe more precisely how to formulate such a definition, to discuss
several natural properties of this definition, and to explain how to prove
some stability and rigidity results concerning the uniform convergence of
metrics satisfying certain lower scalar curvature bounds, such as the
following:

######  Theorem 1.2.

Let $g$ be a $C^{0}$ metric on a closed manifold $M$ which admits a uniform
approximation by $C^{2}$ metrics $g^{i}$ that have $R(g^{i})\geq\kappa_{i}$,
where $\kappa_{i}$ is some sequence of numbers such that
$\kappa_{i}\xrightarrow[i\to\infty]{}\kappa$ for some number $\kappa$. Then
$g$ admits a uniform approximation by smooth metrics with scalar curvature
bounded below by $\kappa$.

######  Corollary 1.3.

Let $g$ be a smooth metric on the torus which admits a uniform approximation
by smooth metrics $g^{i}$ such that
$R(g^{i})\geq-\varepsilon_{i}\xrightarrow[i\to\infty]{}0$. Then $g$ is flat.

##  2 Preliminaries: Ricci and Ricci–DeTurck flow

If $M$ is a smooth manifold and $g_{0}$ is a smooth Riemannian metric on $M$,
the Ricci flow starting from $g_{0}$ is a solution to

$\displaystyle\partial_{t}\bar{g}=-2\operatorname{Ric}(\bar{g})\qquad\text{in}\quad
M\times(0,T),$ $\displaystyle\bar{g}(0)=g_{0},$

where $\bar{g}(t)$ is a smooth, time-dependent family of Riemannian metrics on
a $M$ for $t\in(0,T)$. If $M$ is closed it is known that a short-time solution
to the Ricci flow always exists and is unique; see [12, Theorems 5.2.1 and
5.2.2].

We will also find it useful to consider the Ricci–DeTurck flow, introduced by
DeTurck in [4], a strongly parabolic flow that is related to the Ricci flow by
pullback via a family of diffeomorphisms. More specifically, we define the
following operator, which maps symmetric $2$-forms on $M$ to vector fields:

$X_{\bar{g}}(g):=\sum_{i=1}^{n}\big{(}\nabla^{\bar{g}}_{e_{i}}e_{i}-\nabla^{g}_{e_{i}}e_{i}\big{)},$

where $\\{e_{i}\\}_{i=1}^{n}$ is any local orthonormal frame with respect to
$g$. Then the Ricci–DeTurck equation is

$\partial_{t}g(t)=-2\operatorname{Ric}(g(t))-\mathcal{L}_{X_{\bar{g}(t)}(g(t))}g(t),$
(2.1)

where $\bar{g}(t)$ is a background Ricci flow. As mentioned, if $g(t)$ solves
(2.1) then it is related to a Ricci flow via pullback by diffeomorphisms. More
precisely, if $g(t)$ solves (2.1) and $\chi_{t}\colon M\to M$ is a family of
diffemorphisms satisfying

$\displaystyle X_{\bar{g}(t)}(g(t))f=\frac{\partial}{\partial
t}(f\circ\chi_{t})\text{ for all }f\in C^{\infty}(M),$
$\displaystyle\chi_{0}=\operatorname{id},$ (2.2)

then $\chi_{t}^{*}g(t)$ solves the Ricci flow equation with initial condition
$g(0)$.

Let $g_{t}$ be a solution to the Ricci–DeTurck equation, and write
$g_{t}=\bar{g}_{t}+h_{t}$, where $\bar{g}_{t}$ is the smooth background Ricci
flow from (2.1). Then the evolution equation (see [2, Appendix A] for more
details, and a more general setting) for $h_{t}$ is

$\partial_{t}h_{t}+Lh_{t}=Q[h_{t}],$ (2.3)

where the linear part, $L$, is

$Lh_{t}:=\Delta^{\bar{g}_{t}}h_{t}+2\operatorname{Rm}^{\bar{g}_{t}}(h_{t}):=\Delta^{\bar{g}_{t}}h_{t}+2{\bar{g}}^{pq}R_{pij}^{m}h_{qm}{\rm
d}x^{i}\otimes{\rm d}x^{j},$

and the quadratic term $Q$ may be written as

$Q[h_{t}]=Q^{0}_{t}+\nabla^{*}Q^{1}_{t},$

where

$Q^{0}_{t}=(\bar{g}+h)^{-1}\star(\bar{g}+h)^{-1}\star\nabla h\star\nabla
h+\big{(}(\bar{g}+h)^{-1}-\bar{g}^{-1}\big{)}\star\operatorname{Rm}^{\bar{g}_{t}}\star
h$

and

$\nabla^{*}Q^{1}_{t}:=\nabla_{p}\big{(}\big{(}(\bar{g}+h)^{pq}-\bar{g}^{pq}\big{)}\nabla_{q}h_{ij}\big{)}=\nabla\big{(}\big{(}(\bar{g}+h)^{-1}-\bar{g}^{-1}\big{)}\star\nabla
h\big{)}.$

Here we use the notation $A\star B$ for two tensor fields $A$ and $B$ to mean
a linear combination of products of the coefficients of $A$ and $B$, and
$(\bar{g}+h)^{-1}$ and $\bar{g}^{-1}$ denote tensor fields with coefficients
$(\bar{g}+h)^{ij}$ and $\bar{g}^{ij}$ respectively. We sometimes also write
$h_{t}$ using the integral representation

$\displaystyle h_{t}(x)=\int_{M}\bar{K}(x,t;y,0)h_{0}(y){\rm d}\bar{g}_{0}(y)$
$\displaystyle\hphantom{h_{t}(x)=}{}+\int_{M\times[0,t]}\big{(}\bar{K}(x,t;y,s)Q^{0}_{s}(y)+\nabla^{*}\bar{K}(x,t;y,s)Q^{1}_{s}(y)\big{)}{\rm
d}\bar{g}_{s}(y),$ (2.4)

where $\bar{K}$ denotes the heat kernel for the background Ricci flow
$\bar{g}(t)$. We refer to (2.3) as the Ricci–DeTurck perturbation equation.

There are several results on the existence of Ricci–DeTurck flows starting
from $C^{0}$ initial data, such as [11, Theorem $1.1$], [7, Theorem $4.3$],
and [8, Theorem $5.3$]. Because we will consider flows from initial metrics
that are close to one another in the $C^{0}$ sense, the result that is most
relevant to this survey is (cf. [3, Lemma $3.2$ and Corollary $3.3$]):

######  Proposition 2.1.

Let $M^{n}$ be a smooth, closed manifold. There exist constants
$\varepsilon=\varepsilon(n)$ and $C=C(n)$ such that the following is true:

For every metric $g_{0}\in C^{0}(M)$ and every smooth background metric
$\bar{g}_{0}$, if $||g_{0}-\bar{g}_{0}||_{L^{\infty}(M)}<\varepsilon$ and
$\bar{g}_{t}$ is the Ricci flow starting from $\bar{g}_{0}$, then there exists
$T=T(\bar{g}(t))$ sufficiently small so that there is a solution $g_{t}$ to
the integral equation (2.4) such that

$||g_{t}-\bar{g}_{t}||_{L^{\infty}(M\times(0,T))}\leq
C||g_{0}-\bar{g}_{0}||_{L^{\infty}(M)}.$

Moreover, by taking $\varepsilon$ and $T$ smaller, we may find constants
$c_{k}$ depending only on $k$, the dimension, and bounds for the derivatives
of $\operatorname{Rm}(\bar{g})$, such that the solution $g_{t}$ is smooth on
$M\times(0,T]$, continuous on $M\times[0,T]$, and satisfies

$\big{|}\nabla^{k}(g_{t}-\bar{g}_{t})\big{|}\leq\frac{c_{k}}{t^{k/2}}||g_{0}-\bar{g}_{0}||_{L^{\infty}(M)}$

for all $t\in(0,T^{\prime}]$, where $\nabla$ denotes the covariant derivative
with respect to $\bar{g}(t)$.

The existence of a solution to (2.4) is proven in a similar fashion to [7]: in
[7] Koch and Lamm construct suitable Banach spaces so that a solution to (2.4)
arises from an application of the Banach fixed point theorem. In [3] one
performs a similar construction, working on a Ricci flow background rather
than a stationary background. Once the existence of a solution to the integral
equation (2.4) has been established, smoothness of the solution and bounds on
the derivatives may be proven by iterative application of parabolic interior
estimates.

Under the Ricci flow, the scalar curvature evolves by

$\partial_{t}R=\Delta^{\bar{g}(t)}R+2|\operatorname{Ric}|^{2};$ (2.5)

see [12, Proposition 2.5.4]. Making an orthogonal decomposition, we may
conclude that

$\partial_{t}R\geq\Delta^{\bar{g}(t)}R+\frac{2}{n}R^{2};$ (2.6)

this is [12, Corollary 2.5.5]. For smooth Ricci flows on a closed manifold, a
general bound that follows from the maximum principle is (see [12, Corollary
3.2.5])

$R(x,t)\geq-\frac{n}{2t},$

for all $(x,t)\in M\times(0,T]$.

If, instead, $g(t)$ is a Ricci–DeTurck flow, then recall that
$g(t)=(\chi_{t}^{-1})^{*}\bar{g}(t)$ for some Ricci flow $\bar{g}(t)$, where
the family $(\chi_{t})$ satisfies (2.2) and $X$ is the corresponding vector
field. Therefore, pushing forward (2.6) by $(\chi_{t})$ we find that, under
the Ricci–DeTurck flow,

$\partial_{t}R\geq\Delta^{g(t)}R-\langle X,\nabla R\rangle+\frac{2}{n}R^{2};$

see also [1, p. 6]. It follows from the maximum principle that, if $g(t)$ is a
Ricci or Ricci–DeTurck flow on a closed manifold, starting from a smooth
initial metric, and defined on the interval $[0,T]$, then if
$R^{g_{0}}\geq\kappa_{0}\in\mathbb{R}$, we have (cf. [12, Theorem 3.2.1])

$R^{g(t)}\geq\frac{\kappa_{0}}{1-\big{(}\frac{2\kappa_{0}}{n}t\big{)}}\geq\kappa_{0},$
(2.7)

for all $t\in[0,T]$.

##  3 Regularizing Ricci flow
and pointwise lower scalar curvature bounds

As mentioned in Section 1, in order to determine the generalized lower scalar
curvature bound for a $C^{0}$ metric we first regularize the metric by the
Ricci flow and then study the scalar curvature of the flow for small positive
times. It is not obvious what is meant by a Ricci flow starting from $C^{0}$
initial data; however, we have [3, Theorem 1.1]:

######  Theorem 3.1.

Let $M$ be a closed manifold and $g_{0}$ a $C^{0}$ metric on $M$. Then there
exists a time-dependent family of smooth metrics $(\tilde{g}_{t})_{t\in(0,T]}$
and a continuous surjection $\chi\colon M\to M$ such that the following are
true:

  1. $(a)$

The family $(\tilde{g}_{t})_{t\in(0,T]}$ is a Ricci flow, and

  2. $(b)$

There exists a smooth family of diffeomorphisms $(\chi_{t})_{t\in(0,T]}:M\to
M$ such that

$\chi_{t}\xrightarrow[t\to
0]{C^{0}}\chi\qquad\text{and}\qquad||(\chi_{t})_{*}\tilde{g}_{t}-g_{0}||_{C^{0}(M)}\xrightarrow[t\to
0]{}0.$

Moreover, for any $x\in M$, $\operatorname{diam}\\{\chi_{s}(x)\colon
s\in(0,t]\\}\leq C\sqrt{t}$ for some constant $C>0$ independent of $x$, where
the diameter is measured with respect to a fixed smooth background metric, and
any two such families are isometric, in the sense that if
$\tilde{g}_{t}^{\prime}$ is another such family with corresponding continuous
surjection $\chi^{\prime}$, then there exists a stationary diffeomorphism
$\alpha\colon M\to M$ such that
$\alpha^{*}\tilde{g}_{t}=\tilde{g}_{t}^{\prime}$ and
$\chi\circ\alpha=\chi^{\prime}$.

The pair $((\tilde{g}_{t})_{t\in(0,T]},\chi)$ is called a _regularizing Ricci
flow_ for $g_{0}$.

######  Remark 3.2.

It has not been shown that there are metrics for which $\chi$ cannot be taken
to be a homeomorphism, though we suspect that such metrics do exist.

######  Remark 3.3.

Moreover, the regularizing Ricci flow is invariant under $C^{0}$ isometry in a
more general sense: if
$\varphi\colon\big{(}M_{2},g^{2}\big{)}\to\big{(}M_{1},g^{1}\big{)}$ is a
metric space isometry, then, for any two regularizing Ricci flows
$\big{(}\big{(}\tilde{g}^{1}(t)\big{)}_{t\in(0,T^{1}]},\chi^{1}\big{)}$ and
$\big{(}\big{(}\tilde{g}^{2}(t)\big{)}_{t\in(0,T^{2}]},\chi^{2}\big{)}$ for
$g^{1}$ and $g^{2}$ respectively, there is a stationary diffeomorphism
$\alpha\colon M_{2}\to M_{1}$ such that
$\alpha^{*}\tilde{g}^{1}(t)=\tilde{g}^{2}(t)$ for all
$t\in\big{(}0,\min\big{\\{}T^{1},T^{2}\big{\\}}\big{]}$ and
$\chi^{1}\circ\alpha=\varphi\circ\chi^{2}$; this is [3, Corollary 5.5].

In the smooth case, we may take $(\tilde{g}_{t})$ to be the classical Ricci
flow by choosing $(\chi_{t})$ to be the family of diffeomorphisms which solves
(2.2) and is defined for $t\in[0,T]$, and then pulling back the Ricci–DeTurck
flow by $(\chi_{t})_{t\in[0,T]}$. In the case of $C^{0}$ initial data, it is
still possible to solve the differential equation in (2.2) subject to the
“initial condition” $\chi_{t_{0}}=\operatorname{id}$ for some $t_{0}>0$, since
$X$ is nonsingular for $t_{0}>0$, but it is not possible to prescribe
$\chi_{0}=\operatorname{id}$, since $X$ may be singular at time $0$. Instead,
we take $\chi$ to be the uniform limit as $t\searrow 0$ of the solution
$(\chi_{t})_{t\in(0,T]}$ to the differential equation in (2.2) with
$\chi_{t_{0}}=\operatorname{id}$. The family $(\tilde{g}_{t})$ is then
constructed by pulling back the Ricci–DeTurck flow in Proposition 2.1 by
$(\chi_{t})_{t\in(0,T]}$.

The non-uniqueness of the regularizing Ricci flow is an artifact of the non-
unique choice of “initial data” $\chi_{t_{0}}$. That the regularizing Ricci
flow is unique up to isometry is due to the uniqueness of Ricci flows from
smooth initial data on closed manifolds, and compactness of the isometry group
for smooth Riemannian metrics on closed manifolds.

We now explain how to use the regularizing Ricci flow to define pointwise
lower scalar curvature bounds for $C^{0}$ metrics. Henceforth we will refer to
this definition as the “weak sense”. A satisfactory definition of the weak
sense should satisfy the following requirements: For any constant $\kappa$, we
should have

  1. 1.

Stability under greater-than-second-order perturbation: If $g^{\prime}$ and
$g^{\prime\prime}$ are two $C^{0}$ metrics that agree to greater than second
order around a point $x_{0}$, i.e., if, for some fixed smooth background
metric, we have $|g^{\prime}(x)-g^{\prime\prime}(x)|\leq cd^{2+\eta}(x,x_{0})$
for some $c,\eta>0$ and all $x$ in a neighborhood of $x_{0}$, then
$g^{\prime}$ should have scalar curvature bounded below by $\kappa$ in the
weak sense at $x_{0}$ if and only if $g^{\prime\prime}$ does. Moreover, if
$g^{\prime}$ and $g^{\prime\prime}$ are $C^{0}$ metrics on different manifolds
which merely agree to greater than second order under pullback by a locally
defined diffeomorphism, the conclusion should still hold.

  2. 2.

Preservation of global lower bounds under the Ricci flow: If $g$ is a $C^{0}$
metric on a closed manifold that has scalar curvature bounded below by
$\kappa$ in the weak sense at every point, and $\tilde{g}_{t}$ is a
regularizing Ricci flow for $g$, then $\tilde{g}_{t}$ should have scalar
curvature bounded below by $\kappa$ at every point for all $t>0$ for which the
flow is defined. This is true for Ricci flows starting from smooth initial
data.

  3. 3.

Agreement with the classical notion for $C^{2}$ metrics: If $g$ is a $C^{2}$
metric with scalar curvature bounded below by $\kappa$ at $x_{0}$ in the weak
sense for $C^{0}$ metrics, then $g$ should have scalar curvature bounded below
by $\kappa$ at $x_{0}$ in the classical sense. Conversely, if $g$ has scalar
curvature bounded below by $\kappa$ at $x_{0}$ in the classical sense, then
the same should hold in the weak sense.

Perhaps the most natural approach would be to declare that
$R(g_{0})\geq\kappa$ at $x$ if $\lim\limits_{t\searrow 0}R(g_{t},y)\geq\kappa$
for some $y\in\chi^{-1}(x)$ (we consider the scalar curvature at
$y\in\chi^{-1}(x)$ rather than at $x$ itself because $g_{t}$ only converges to
$g_{0}$ modulo pushforward by the family $(\chi_{t})$). As we shall discuss
below, we wish to leverage exponential heat kernel estimates in conjunction
with (2.5) to show that ($2$) holds, so we modify this approach by instead
requiring that the lower bound holds in the limit, in a small time-dependent
neighborhood of $y$, where the size of the neighborhood tends to $0$ as
$t\searrow 0$:

######  Definition 3.4.

Let $M^{n}$ be a closed manifold and $g_{0}$ a $C^{0}$ metric on $M$. For
$0<\beta<1/2$ we say that $g_{0}$ has scalar curvature bounded below by
$\kappa$ at $x$ in the _$\beta$ -weak sense_ if there exists a regularizing
Ricci flow $((\tilde{g}_{t})_{t\in(0,T]},\chi)$ for $g_{0}$ such that, for
some point $y\in M$ with $\chi(y)=x$, we have

$\inf_{C>0}\Big{(}\liminf_{t\searrow
0}\Big{(}\inf_{B_{\tilde{g}(t)}(y,Ct^{\beta})}R^{\tilde{g}}(\cdot,t)\Big{)}\Big{)}\geq\kappa,$
(3.1)

where $B_{\tilde{g}(t)}\big{(}y,Ct^{\beta}\big{)}$ denotes the ball of radius
$Ct^{\beta}$ about $y$, measured with respect to the metric $\tilde{g}(t)$,
and $R^{\tilde{g}}(\cdot,t)$ denotes the scalar curvature of $\tilde{g}_{t}$ .

######  Remark 3.5.

In fact, Definition 3.4 is independent of choice of $y$, so it is equivalent
to require that (3.1) hold at $y$ for _all_ $y$ with $\chi(y)=x$. Moreover,
the fact that any two regularizing Ricci flows for $g_{0}$ are isometric (see
Theorem 3.1) implies that Definition 3.4 holds for some regularizing Ricci
flow if and only if it holds for all regularizing Ricci flows for $g_{0}$.

######  Remark 3.6.

Definition 3.4 may also be formulated in terms of Ricci–DeTurck flow (see [3,
Lemma 6.4]). The equivalent statement using Ricci–DeTurck flow is that there
is a Ricci–DeTurck flow $g_{t}$ starting from $g_{0}$ in the sense of
Proposition 2.1 and $\bar{g}_{0}$ a stationary metric on $M$ that is uniformly
bilipschitz to $(g_{t})_{t\in(0,T]}$ such that

$\inf_{C>0}\Big{(}\liminf_{t\searrow
0}\Big{(}\inf_{B_{\bar{g}_{0}}(x,Ct^{\beta})}R^{g_{t}}(\cdot)\Big{)}\Big{)}\geq\kappa.$

We take the infimum over all $C>0$ to ensure that the condition that
$R(g_{0})\geq 0$ at $x$ in the $\beta$-weak sense is scaling invariant, since
the condition $0<\beta<1/2$ disrupts the parabolic scaling of the expression.
The significance of $\beta$ in this range is that away from
$B_{\tilde{g}(t)}\big{(}y,Ct^{\beta}\big{)}$ the heat kernel for the flow
decays exponentially as $t\searrow 0$; we will discuss how this is used below.
It is unknown whether $\beta$ could instead be replaced by $1/2$, or even to
replace (3.1) by $\liminf\limits_{t\searrow 0}R(g_{t},y)$. It is clear that
Definition 3.4 satisfies item ($3$), since if $g_{0}$ is $C^{2}$, then by
Remark 3.5 we may take the regularizing Ricci flow as the usual Ricci flow
with $\chi=\operatorname{id}$, and

$\inf_{C>0}\Big{(}\liminf_{t\searrow
0}\Big{(}\inf_{B_{\tilde{g}(t)}(y,Ct^{\beta})}R^{\tilde{g}}(\cdot,t)\Big{)}\Big{)}=\lim_{t\to
0}R^{\tilde{g}}(x,t)=R^{g}(x).$

Item ($1$) holds due to the following stability result for the scalar
curvatures of regularizing Ricci flows from metrics which agree to greater
than second order at a point; this is [3, Theorem 1.4]:

######  Theorem 3.7.

Suppose $g^{\prime}$ and $g^{\prime\prime}$ are two $C^{0}$ metrics on closed
manifolds $M^{\prime}$ and $M^{\prime\prime}$ respectively, and that there is
a locally defined diffeomorphism $\phi\colon U\to V$ where $U$ is a
neighborhood of $x_{0}^{\prime}$ in $M^{\prime}$ and $V$ is a neighborhood of
$x_{0}^{\prime\prime}$ in $M^{\prime\prime}$ with
$\phi(x_{0}^{\prime})=x_{0}^{\prime\prime}$. Suppose furthermore that
$g^{\prime}$ and $\phi^{*}g^{\prime\prime}$ agree to greater than second order
around $x_{0}^{\prime}$, i.e., with respect to some fixed smooth background
metric, $|g^{\prime}(x)-\phi^{*}g^{\prime\prime}(x)|\leq cd^{2+\eta}(x,x_{0})$
for some $c,\eta>0$ and all $x$ in a neighborhood of $x_{0}^{\prime}$. Then
there exist regularizing Ricci flows $(\tilde{g}_{t}^{\prime},\chi^{\prime})$
and $(\tilde{g}_{t}^{\prime\prime},\chi^{\prime\prime})$ for $g^{\prime}$ and
$g^{\prime\prime}$ respectively such that, for $1/(2+\eta)<\beta<1/2$, $C>0$,
and $t$ sufficiently small depending on $C$, $\beta$, and $\eta$, we have

$\sup_{B(x_{0}^{\prime},Ct^{\beta})}\big{|}R^{(\chi_{t}^{\prime})_{*}\tilde{g}^{\prime}_{t}}-\phi^{*}R^{(\chi_{t}^{\prime\prime})_{*}\tilde{g}^{\prime\prime}_{t}}\big{|}\leq
ct^{\omega},$

where $\omega$ is some positive exponent, $c$ is a constant that does not
depend on $t$ or $C$, $R^{(\chi_{t}^{\prime})_{*}\tilde{g}^{\prime}_{t}}$ and
$R^{(\chi_{t}^{\prime\prime})_{*}\tilde{g}^{\prime\prime}_{t}}$ denote the
scalar curvatures with respect to
$(\chi_{t}^{\prime})_{*}\tilde{g}_{t}^{\prime}$ and
$(\chi_{t}^{\prime\prime})_{*}\tilde{g}_{t}^{\prime\prime}$ respectively, and
$(\chi_{t}^{\prime})$ and $(\chi_{t}^{\prime\prime})$ are the smooth families
of diffeomorphisms for $\tilde{g}_{t}^{\prime}$ and
$\tilde{g}_{t}^{\prime\prime}$ respectively, whose existence is given by (b)
in Theorem 3.1.

In particular, Definition 3.4 holds for $g^{\prime}$ at $x_{0}^{\prime}$ if
and only if it holds for $g^{\prime\prime}$ and $x_{0}^{\prime\prime}$.

######  Remark 3.8.

By Theorem 3.7, Definition 3.4 descends to the space of germs of metrics at a
point, and further descends to the quotient space induced on the space of
germs of metrics at $x_{0}$ by the relation $[g]\sim[g^{\prime}]$ if $g$ and
$g^{\prime}$ agree to greater than second order at $x_{0}$. One may also use
this fact to define weak pointwise lower scalar curvature bounds for $C^{0}$
metrics on open manifolds.

The strategy for proving Theorem 3.7 is essentially to endow the Banach spaces
defined by Koch and Lamm in [7] with the weight
$\omega(x,t)=\max\big{\\{}\big{(}d(x_{0},x)+\sqrt{t}\big{)}^{-2-\eta},1\big{\\}}$
to offset the evolution of two metrics that initially agree to greater than
second order about $x_{0}$, and then to show that
$||\omega(\cdot,t)(g_{t}^{\prime}-\phi^{*}g_{t}^{\prime\prime})||_{C^{0}(M)}\leq
C(||\omega(\cdot,0)g_{0}^{\prime}-\phi^{*}g_{0}^{\prime\prime}||)<\infty$.
Closeness of the scalar curvatures is obtained from the $C^{0}$ estimate by
interpolating between higher and lower order derivatives.

Moreover, we have that item ($2$) holds; this is [3, Theorem 1.5]:

######  Theorem 3.9.

Suppose that $g_{0}$ is a $C^{0}$ metric on a closed manifold $M$, and suppose
there is some $\beta\in(0,1/2)$ such that $g_{0}$ has scalar curvature bounded
below by $\kappa$ in the $\beta$-weak sense at all points in $M$. Suppose also
that $(\tilde{g}(t))_{t\in(0,T]}$ is a Ricci flow starting from $g_{0}$ in the
sense of Theorem 3.1. Then the scalar curvature of $\tilde{g}(t)$,
$R(\tilde{g}(t))$, satisfies $R(\tilde{g}(t))\geq\kappa$ everywhere on $M$,
for all $t\in(0,T]$.

To prove Theorem 3.9 one supposes that the theorem is false and then
iteratively uses the heat kernel estimates and (2.5) to show that there must
be a sequence $(x_{i},t_{i})$ with $t_{i}\searrow 0$ and all $x_{i}\in
B_{\tilde{g}(t_{i})}\big{(}x_{0},t_{i}^{\beta}\big{)}$ for some $x_{0}$, such
that $R(x_{i},t_{i})$ is bounded above away from $\kappa$ for all $i$, so that
Definition 3.4 must fail at $x_{0}$.

##  4 Some applications: global lower bounds

Recall that the purpose of Definition 3.4 was to formulate a _pointwise_ lower
scalar curvature bound for a $C^{0}$ metric $g$. Nevertheless, if $g$
satisfies a (constant) global lower bound on the scalar curvature in the sense
of Definition 3.4, Theorem 3.9 leads to several alternate characterizations of
$g$, which we will discuss in this section.

We have (cf. [3, Corollary 1.6]):

######  Theorem 4.1.

Let $g$ be a $C^{0}$ metric on a closed manifold $M$. Let $\kappa$ be some
constant. The following are equivalent:

  1. $1.$

There exists some $\beta<1/2$ such that $R(g)\geq\kappa$ in the $\beta$-weak
sense everywhere on $M$.

  2. $2.$

For any regularizing Ricci flow $(\tilde{g}(t))_{t\in(0,T]}$ for $g$,
$R(\tilde{g}(t))\geq\kappa$ everywhere on $M$, for all $t\in(0,T]$.

  3. $3.$

There exists a sequence of smooth metrics $g^{i}$ on $M$ such that
$R(g^{i})\geq\kappa$ everywhere on $M$ for $i=1,2,\dots$, and
$g^{i}\xrightarrow[i\to\infty]{C^{0}}g$.

######  Remark 4.2.

In particular, Theorem 4.1 implies that if the global lower bound
$R(g)\geq\kappa$ holds in the $\beta$-weak sense for some value of
$\beta<1/2$, then it holds for all $\beta<1/2$.

To see why Theorem 4.1 is true, first choose a sequence of times
$t_{i}\searrow 0$, and let $\tilde{g}(t)$ be a regularizing Ricci flow for
$g$. Then Theorem 3.1 says that $(\chi_{t_{i}})_{*}\tilde{g}(t_{i})$ converges
uniformly to $g$. Moreover, Theorem 3.9 guarantees that
$R(\tilde{g}(t_{i}))\geq\kappa$ for all $i$, so the same is true for
$R((\chi_{t_{i}})_{*}\tilde{g}(t_{i}))$, so $(1)\Rightarrow(3)$.

Furthermore, if there exists a sequence of smooth approximators $g^{i}$ as in
$(3)$, then for sufficiently large $i$ all $g^{i}$ are close enough to some
smooth background metric $\bar{g}_{0}$ so that there exists a uniform value
$T$ such that the Ricci–DeTurck flows $g^{i}(t)$ starting from $g^{i}$ in the
sense of Proposition 2.1 are defined for $t\in(0,T]$. Moreover,
$R(g^{i}(t))\geq\kappa$ for all $t\in(0,T]$ and $i$, according to (2.7).

By the derivative estimates given by Proposition 2.1, the flows $g^{i}(t)$
converge smoothly to the Ricci–DeTurck flow $g(t)$ for $g$ on $M$ for all
fixed $t>0$. In particular, for any fixed $t\in(0,T]$ and all $i$ sufficiently
large, we have $R(g(t))\geq\kappa$. Since $g(t)$ is related to a regularizing
Ricci flow $\tilde{g}(t)$ for $g$ by a family of diffeomorphisms, we find that
$R(\tilde{g}(t))\geq\kappa$ for all $t\in(0,T]$, so $(3)\Rightarrow(2)$. It is
clear that $(2)\Rightarrow(1)$ for any $\beta<1/2$. In particular, we have
shown that $(1)\Rightarrow(3)\Rightarrow(2)\Rightarrow(1)$, which proves
Theorem 4.1.

Observe that in our argument to show that $(3)\\!\Rightarrow\\!(2)$ it was
only necessary to have $R(g^{i}(t))\\!\geq\\!\kappa_{i}$ where
$\kappa_{i}\to\kappa$, since by smooth convergence of the time slices we would
then have $R(g(t))\geq\kappa_{i}$ for all $i$ sufficiently large, and hence
$R(g(t))\geq\kappa$. In particular, Theorems 4.1 and 3.9 imply that it is
sufficient to require that the approximating metrics $g^{i}$ be only $C^{0}$
with $R(g^{i})\geq\kappa_{i}$ in the sense of Definition 3.4. In particular,
we have the following “continuity property” for global lower bounds in the
sense of Definition 3.4 (cf. [3, Theorem 1.7]):

######  Theorem 4.3.

Let $g$ be a $C^{0}$ metric on a closed manifold $M$ which admits a uniform
approximation by $C^{0}$ metrics $g^{i}$ such that there is some $\beta<1/2$
so that $g^{i}$ has scalar curvature bounded below by $\kappa_{i}$ in the
$\beta$-weak sense everywhere on $M$, where $\kappa_{i}$ is some sequence of
numbers such that $\kappa_{i}\xrightarrow[i\to\infty]{}\kappa$ for some number
$\kappa$. Then $g$ has scalar curvature bounded below by $\kappa$ in the
$\beta$-weak sense. In particular, any regularizing Ricci flow
$(\tilde{g}(t))_{t\in(0,T]}$ for $g$ satisfies $R(\tilde{g}(t))\geq\kappa$ for
all $t\in(0,T]$, so $g$ admits a uniform approximation by smooth metrics with
scalar curvature bounded below by $\kappa$.

Theorem 1.2 follows from Theorem 4.3 in the case where the $g^{i}$ are
$C^{2}$. We remark that Theorem 1.2 may also be proved directly by using the
Ricci–DeTurck flow as in $(3)\Rightarrow(2)$ in the proof of Theorem 4.1,
using Proposition 2.1 or [11, Theorem 1.1]. By setting
$\kappa_{i}=-\varepsilon_{i}$, Theorem 1.2 answers in the affirmative the
following question, posed by Gromov in [5]:

######  Question 4.4 ([5, p. 1119]).

Let $g$ be a continuous Riemannian metric on a closed manifold $M$ which
admits a $C^{0}$-approximation by smooth Riemannian metrics $g_{i}$ with
$R(g_{i})\geq-\varepsilon_{i}\xrightarrow[i\to\infty]{}0$. Does $M$ admit a
smooth metric of nonnegative scalar curvature?

Moreover, Theorem 1.2 implies that $g$ admits a uniform approximation by
smooth metrics with nonnegative scalar curvature.

In light of the scalar torus rigidity theorem, which says that any Riemannian
manifold with nonnegative scalar curvature that is diffeomorphic to the torus
must be isometric to the flat torus (see [6, 10]), it is natural to ask
whether Definition 3.4 satisfies an analogous rigidity result for $C^{0}$
metrics on the torus. If $g$ is a $C^{0}$ metric on the torus with nonnegative
scalar curvature in the $\beta$-weak sense, for some value of $\beta<1/2$, and
$(\tilde{g}_{t},\chi_{t})$ is a regularizing Ricci flow for $g$, then Theorem
3.9 implies that $R(\tilde{g}(t))\geq 0$ for all $t>0$, and hence the usual
scalar torus rigidity theorem implies that $\tilde{g}(t)$ is flat for all
$t>0$. In particular, the Ricci flow equation implies that
$\tilde{g}(t)\equiv\tilde{g}$ for some stationary smooth flat metric
$\tilde{g}$. Then $(\chi_{t})_{\tilde{g}}\xrightarrow[t\to 0]{C^{0}}g$ implies
that $\tilde{g}$ is equivalent to $g$ in the Gromov–Hausdorff sense. We have
shown:

######  Corollary 4.5.

Suppose $g$ is a $C^{0}$ metric on the torus $\mathbb{T}$, and that there is
some $\beta\in(0,1/2)$ such that $g$ has nonnegative scalar curvature in the
$\beta$-weak sense everywhere. Then $(\mathbb{T},g)$ is isometric as a metric
space to the standard flat metric on $\mathbb{T}$.

Corollary 4.5 and Theorem 1.2 together imply Corollary 1.3.

######  Remark 4.6.

Corollary 4.5 is in fact the optimal result, i.e., it is not possible to show
that there is a _Riemannian_ isometry between $g$ and the standard flat
metric. In the case where $g^{1}$ and $g^{2}$ are smooth metrics, a metric
space isometry is automatically a smooth Riemannian isometry. However, there
exist examples of $C^{0}$ isometries between $C^{0}$ metrics which are not
$C^{1}$. Moreover, by Remark 3.3, the regularizing Ricci flow is invariant
under $C^{0}$ isometry, so it is not possible to use regularizing Ricci flow
to distinguish Riemannian isometries from isometries which are merely metric
space isometries.

##  5 Relation to some notions of positive scalar curvature

We will now discuss the relationship between Definition 3.4 and some other
possible definitions of lower scalar curvature bounds for $C^{0}$ metrics,
both in the global and local setting. Throughout we take $M$ to be a closed
manifold. We first define the following classes of $C^{0}$ metrics on $M$,
each of which is a class of $C^{0}$ metrics that have scalar curvature
globally bounded below by a constant $\kappa$ in some reasonable generalized
sense:

$\displaystyle C^{0}_{\beta}(M,\kappa):=\big{\\{}g\in C^{0}(M)\colon
R(g)\geq\kappa\text{ in the $\beta$-weak sense}\big{\\}},$ $\displaystyle
C^{0}_{\operatorname{met}}(M,\kappa):=\Big{\\{}g\in C^{0}(M)\colon\text{ there
exists }g^{i}\in C^{2}\text{ with }R(g^{i})\geq\kappa\text{ and
}g^{i}\xrightarrow[i\to\infty]{C^{0}}g\Big{\\}},$ $\displaystyle
C^{0}_{\operatorname{Rf}}(M,\kappa):=\big{\\{}g\in
C^{0}(M)\colon\exists\,(\tilde{g}(t))_{t\in(0,T]}\text{ a regularizing Ricci
flow for }g$
$\displaystyle\hphantom{C^{0}_{\operatorname{Rf}}(M,\kappa):=\big{\\{}}{}\text{with
}R(\tilde{g}(t))\geq\kappa\,\forall\,t\in(0,T]\big{\\}},$ $\displaystyle
C^{0}_{\operatorname{G}}(M,\kappa):=\big{\\{}c\in C^{0}(M)\colon
R(g)\geq\kappa\text{ in the sense of Gromov
\cite[cite]{[\@@bibref{}{Gro}{}{}]}}\big{\\}}.$

It is natural to study $C^{0}_{\operatorname{Rf}}(M,\kappa)$ in light of [1].
A metric $g$ is in $C^{0}_{\operatorname{G}}(M,0)$ if essentially it does not
contain a cube with strictly mean convex faces, such that the dihedral angles
are acute, or more generally, it is in $C^{0}_{\operatorname{G}}(M,\kappa)$ if
the product metric of its product with an appropriate space form $S_{\kappa}$
is in $C^{0}_{\operatorname{G}}(M\times S_{-\kappa},0)$; see [5, p. 1119].
This is a natural definition because, if such a cube were to exist in a
manifold with (classical) nonnegative scalar curvature, then, as in [5, pp.
1144–1145], one could glue together $2n$ copies of the cube and obtain a non-
flat torus of nonnegative scalar curvature, contradicting the scalar torus
rigidity theorem ([10, Corollary 2] and [6, Corollary A]).

By Theorem 4.1 we have that for any $\beta<1/2$,
$C^{0}_{\beta}(M,\kappa)=C^{0}_{\operatorname{met}}(M,\kappa)=C^{0}_{\operatorname{Rf}}(M,\kappa)$.
The Ricci flow proof of Theorem 1.1 suggests a relationship between
$C^{0}_{\beta}(M,\kappa)=C^{0}_{\operatorname{Rf}}(M,\kappa)$ and
$C^{0}_{\operatorname{G}}(M,\kappa)$. We have that
$C^{0}_{\operatorname{met}}(M,\kappa)\subset
C^{0}_{\operatorname{G}}(M,\kappa)$, since
$C^{0}_{\operatorname{G}}(M,\kappa)$ contains all $C^{2}$ metrics $g$ with
$R(g)\geq\kappa$ and is closed in $C^{0}$. Thus,
$C^{0}_{\operatorname{Rf}}(M,\kappa)\subset
C^{0}_{\operatorname{G}}(M,\kappa)$. One key question is whether, if a $C^{0}$
metric on a closed manifold has scalar curvature bounded below in the sense of
[5], it necessarily has scalar curvature bounded below under the Ricci flow:

######  Question 5.1.

Suppose that $M$ is closed. Is $C^{0}_{\operatorname{G}}(M,\kappa)\subset
C^{0}_{\operatorname{Rf}}(M,\kappa)$?

If so, then this would imply that all $C^{0}$ notions of a global lower scalar
curvature bound that we have mentioned would agree.

We remark that, aside from the classes of $C^{0}$ metrics listed above, Lee
and LeFloch [9] have introduced a notion of positive distributional scalar
curvature for metrics in $C^{0}\cap W^{1,n}_{\text{loc}}$, which they have
used to prove a version of the Positive Mass Theorem. In this paper we work
with $C^{0}$ metrics without any assumptions regarding their distributional
derivatives, but in the case of metrics in $C^{0}\cap W^{1,n}_{\text{loc}}$,
it is an open question whether the definition in [9] is equivalent to
Definition 3.4.

We now discuss some open questions in the pointwise setting. One feature of
Gromov’s definition is that it may be localized around a point $x\in M$, by
requiring only that there exist a neighborhood of $x$ such that no cube within
the neighborhood that contains $x$ has strictly mean convex faces and acute
dihedral angles; see [5, p. 1144]. In light of this, for $x\in M$ define
$C^{0}_{\operatorname{G}}(x,\kappa)$ to be the space of germs of $C^{0}$
metrics on $M$ at $x$ that have scalar curvature bounded below by $\kappa$ at
$x$ in the sense of [5].

Define $C^{0}_{\beta}(x,\kappa)$ to be the space of germs of $C^{0}$ metrics
on $M$ at $x$ that have scalar curvature bounded below by $\kappa$ at $x$ in
the sense of Definition 3.4. By Theorem 3.9, this is a reasonable of
localization of $C^{0}_{\operatorname{Rf}}(M,\kappa)$, as was intended.

######  Question 5.2.

Let $M$ be a closed manifold and $x\in M$. Do we have
$C^{0}_{\operatorname{G}}(x,\kappa)=C^{0}_{\beta}(x,\kappa)$?

######  Remark 5.3.

In contrast with the case of global lower bounds, the results in this paper do
_not_ imply that $C^{0}_{\beta}(x,\kappa)\subset
C^{0}_{\operatorname{G}}(x,\kappa)$.

Theorem 3.7 says that the perturbation of a metric by greater than second
order does not affect $\beta$-weak lower bounds on the scalar curvature. Thus,
it is natural to ask whether one can characterize, up to higher order
perturbation, those metrics, on $\mathbb{R}^{n}$, say, that have nonnegative
scalar curvature in the sense of Definition 3.4:

######  Question 5.4.

Suppose $g=g_{ij}{\rm d}x^{i}\otimes{\rm d}x^{j}$ is a metric on a
neighborhood of the origin in $\mathbb{R}^{n}$, and that we may write

$g_{ij}(x)=\delta_{ij}+r^{2}G_{ij}\left(\frac{x}{r}\right)+O\big{(}|x|^{2+\eta}\big{)},$

where the $G_{ij}$ are functions on $\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$
satisfying $x^{i}x^{j}G_{ij}(x)=0$. Is there an explicit characterization of
metrics of this form that have nonnegative scalar curvature at the origin, in
the sense of Definition 3.4?

### Acknowledgements

This survey is dedicated to Misha Gromov on the occasion of his 75th birthday.
I would like to thank him for his interest in my work. I would also like to
thank my advisor, Richard Bamler, for his guidance during the writing process.
Finally, I would like to thank the referees for numerous helpful comments on a
previous draft of this paper. This material is based upon work supported by
the National Science Foundation Graduate Research Fellowship Program under
Grant No. DGE 1752814.

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