Product Inequalities for $\mathbb{T}^{\rtimes}$-Stabilized Scalar Curvature

Product Inequalities for $\boldsymbol{\mathbb{T}^{\rtimes}}$-Stabilized Scalar
Curvature††This paper is a contribution to the Special Issue on Differential
Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon
for his 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Bourguignon.html

Misha GROMOV ab

M. Gromov

a) Courant Institute of Mathematical Sciences, New York University,
a) New York, NY 10012-1185, USA

b) Institut des Hautes Études Scientifiques, 91893 Bures-sur-Yvette, France
<EMAIL_ADDRESS>https://www.ihes.fr/~gromov/

Received June 26, 2023, in final form April 26, 2024; Published online May 08,
2024

We study metric invariants of Riemannian manifolds $X$ defined via the
$\mathbb{T}^{\rtimes}$-stabilized scalar curvatures of manifolds $Y$ mapped to
$X$ and prove in some cases additivity of these invariants under Riemannian
products $X_{1}\times X_{2}$.

scalar curvature; Riemannian manifold

53C21

To the 75th birthday
of Jean-Pierre Bourguignon

##  1 $\boldsymbol{\mathbb{T}^{\rtimes}}$-stabilization

A “warped” $\mathbb{T}^{N}$-extension, $N=0,1,\dots$, of a Riemannian manifold
$X=(X,g)$, possibly with a boundary, is

$X_{N}^{\rtimes}=X\rtimes\mathbb{T}^{N}=\bigl{(}X\times\mathbb{T}^{N},g^{\rtimes}\bigr{)},$

where $\mathbb{T}^{N}$ is the (flat split) $N$-torus and where
$X_{N}^{\rtimes}$ is endowed with a warped metric,

$g^{\rtimes}=g_{N}^{\rtimes}=g_{N,\\{\varphi_{i}\\}}^{\rtimes}=g+\sum_{i=1}^{N}\varphi_{i}^{2}\,{\rm
d}t_{i}^{2},$

where $\varphi_{i}(x)\geq 0$ are smooth positive functions, which are strictly
positive $(>0)$ in the interior $X\setminus\partial X$ of $X$.

Assume $g$ is smooth and let
$\operatorname{Sc}^{\rtimes}_{\\{\varphi_{i}\\}}(X)$ be the scalar curvature
of $g^{\rtimes}$, that is,

$\operatorname{Sc}^{\rtimes}_{\\{\varphi_{i}\\}}(X)=\operatorname{Sc}(g_{N,\\{\varphi_{i}\\}}^{\rtimes})=\operatorname{Sc}\left(g+\sum_{i=1}^{N}\varphi_{i}^{2}\,{\rm
d}t_{i}^{2}\right),$

where
$\operatorname{Sc}^{\rtimes}_{\\{\varphi_{i}\\}}(X)=\operatorname{Sc}^{\rtimes}_{\\{\varphi_{i}\\}}(X,x)$
is a function on $X$, since $g_{N}^{\rtimes}$ is invariant under the obvious
action of $\mathbb{T}^{N}$ on $X_{N}^{\rtimes}=X\times\mathbb{T}^{N}$.

Let $\operatorname{Sc}^{\rtimes}(X)$, $X=(X,g)$, be the supremum of the
numbers $\sigma$ such that
$\operatorname{Sc}_{N,\\{\varphi_{i}\\}}^{\rtimes}(X)>\sigma$ for some $N$ and
$\varphi_{i}$.

$\boldsymbol{\supset}$-Monotonicity. Clearly,

$\operatorname{Sc}^{\rtimes}(Y)\geq\operatorname{Sc}^{\rtimes}(X)$

for all smooth domains $Y\subset X$.

1.A. Formulas. A computation shows111See [10], [21, formulas (7.33) and
(12.5)], also [46], and [17, Section 2.4.1]. that

$\operatorname{Sc}\left(g+\sum_{i=1}^{N}\varphi_{i}^{2}\,{\rm
d}t_{i}^{2}\right)=\operatorname{Sc}(g)-2\sum^{N}_{i=1}\frac{\Delta\varphi_{i}}{\varphi_{i}}-2\sum_{i<j}\langle\nabla_{g}\log\varphi_{i},\nabla_{g}\log\varphi_{j}\rangle.$

For instance, if $N=1$ and $\varphi_{1}$ is a positive eigenfunction of the
operator $-\Delta+\frac{1}{2}\sigma$, for

$\sigma(x)=\operatorname{Sc}(g,x),$

that is,

$\varphi(x)\mapsto-\Delta\varphi(x)+\frac{1}{2}\sigma(x)\varphi(x)$

and

$-\Delta\varphi_{1}+\frac{1}{2}\sigma(x)\varphi_{1}=\lambda_{1}\varphi_{1},$

then

$\operatorname{Sc}\bigl{(}g+\varphi_{1}^{2}\,{\rm
d}t^{2}\bigr{)}=\sigma-2\frac{\Delta\varphi_{1}}{\varphi_{1}}=\sigma-2\frac{\frac{1}{2}\sigma\varphi_{1}-\lambda\varphi_{1}}{\varphi_{1}}=2\lambda_{1}.$

Thus,

$\operatorname{Sc}^{\rtimes}(X)\geq
2\lambda_{1}=2\lambda_{1}\left(\Delta+\frac{1}{2}\operatorname{Sc}(x)\right).$

Recall at this point that if $X$ is a compact connected manifold without
boundary, then, for all functions $\sigma(x)$,

$\displaystyle\lambda_{1}$ $\displaystyle=\inf_{\varphi\neq
0}\left(\frac{-\int_{X}\varphi(x)\Delta\varphi(x){\rm
d}y}{\int_{X}\varphi(x)^{2}{\rm
d}y}+\frac{1}{2}\frac{\int_{X}\sigma(x)\varphi^{2}(x){\rm
d}y}{\int_{X}\varphi(x)^{2}{\rm d}y}\right)$
$\displaystyle\geq\inf_{\varphi\neq
0}\left(\frac{-\int_{X}\varphi(x)\Delta\varphi(x){\rm
d}y}{\int_{X}\varphi(x)^{2}{\rm d}y}+\frac{1}{2}\inf_{x\in
X}\sigma(x)\right)\geq\frac{1}{2}\inf_{x\in X}\sigma(x),$

where the latter inequality follows from positivity of $-\Delta$ and this
inequality is strict $(>)$ with $\varphi=\varphi_{1}$, unless $\sigma(x)$ is
constant.

Also, the strict inequality $\lambda_{1}>\inf_{x\in X}\operatorname{Sc}(X,x)$
holds for the Dirichlet eigenvalue $\lambda_{1}$ on compact connected
manifolds with boundaries, since the above relations are satisfied for
functions $\varphi(x)$, which vanish on the boundary.

Next, now for all $N$ and all $\varphi_{i}$, rewrite the above expression for
$\operatorname{Sc}\bigl{(}g+\sum_{i=1}^{N}\varphi_{i}^{2}{\rm
d}t_{i}^{2}\bigr{)}$ with the function
$\Phi(x)=\log(\varphi_{1}(x)\cdots\varphi(x)_{N})$ as follows:

$\operatorname{Sc}\left(g+\sum_{i=1}^{N}\varphi_{i}^{2}\,{\rm
d}t_{i}^{2}\right)=\operatorname{Sc}(g)-2\Delta\Phi-||\nabla\Phi||^{2}-\sum_{i}{||\nabla\log\varphi_{i}||^{2}}.$

This shows that
$\operatorname{Sc}\bigl{(}g+\sum_{i=1}^{N}\varphi_{i}^{2}\,{\rm
d}t_{i}^{2}\bigr{)}$ increases under replacing all $\varphi_{i}$ by their
geometric mean, $\varphi_{i}\leadsto\phi=\sqrt[N]{\prod_{i}\varphi_{i}}$,
i.e.,

$\operatorname{Sc}(g)-2\Delta\Psi-\frac{N+1}{N}||\nabla\Psi||^{2}\geq\operatorname{Sc}(g)-2\Delta\Phi-||\nabla\Phi||^{2}-\sum_{i}||\nabla\log\varphi_{i}||^{2}$

for $\Psi=\log\phi^{N}$, where the equality holds only if all
$\nabla\log\varphi_{i}$ are mutually equal. Hence,

$\sup_{\varphi_{i}>0}\operatorname{Sc}\left(g+\sum_{i=1}^{N}\varphi_{i}^{2}\,{\rm
d}t_{i}^{2}\right)=\sup_{\Psi}\operatorname{Sc}(g)-2\Delta\Psi-\frac{N+1}{N}||\nabla\Psi||^{2},$

where this “$\sup$” increases with $N$; thus, by letting $N\to\infty$, we see
that

$\displaystyle\operatorname{Sc}^{\rtimes}(X)$
$\displaystyle=\sup_{\Psi(x)}\inf_{x\in
X}\operatorname{Sc}(X,x)-2\Delta\Psi(x)-||\nabla\Psi(x)||^{2}$
$\displaystyle=\sup_{\psi(x)>0}\inf_{x\in
X}\operatorname{Sc}(X,x)-2\frac{\Delta\psi(x)}{\psi(x)}+\frac{||\nabla\psi(x)||^{2}}{\psi(x)^{2}}.$

Rewrite this equation with $\Psi=2\Theta$ as follows:

$\displaystyle\operatorname{Sc}^{\rtimes}(X)$
$\displaystyle=\sup_{\Theta}\inf_{x\in
X}\operatorname{Sc}(X,x)-4\bigl{(}\Delta\Theta(x)+||\nabla\Theta(x)||^{2}\bigr{)}$
$\displaystyle=\sup_{\theta}\inf_{x\in
X}\operatorname{Sc}(X,x)-4\frac{\Delta\theta(x)}{\theta(x)}\qquad\text{for}\quad\theta=\exp\Theta.$

Therefore, if $X$ is compact, then

$\operatorname{Sc}^{\rtimes}(X)\geq 4\lambda_{1}^{\rtimes}(X),$

where $\lambda^{\rtimes}_{1}(X)$ is the lowest eigenvalue of the operator
$-\Delta+\frac{1}{4}\operatorname{Sc}$ on $X$ with the Dirichlet (vanishing on
the boundary) condition.

(If a connected manifold $X$ has no boundary and the scalar curvature of $X$
is constant, then $\operatorname{Sc}^{\rtimes}(X)=\operatorname{Sc}(X)$;
otherwise

$\operatorname{Sc}^{\rtimes}(X)>\inf_{x\in X}\operatorname{Sc}(X,x).$

Moreover,

$\operatorname{Sc}^{\rtimes}(X)>\beta^{-1}\lambda_{1}\left(-\Delta+\beta\operatorname{Sc}(X)\right)$

for all $\beta>1/4$, since the operator $-\Delta$ is strictly positive on non
constant functions on $X$.)

1.B. $\boldsymbol{\frac{1}{4}}$-Proposition. Let $X=(X,g)$ be a compact
Riemannian manifold with a boundary. Then

$\operatorname{Sc}^{\rtimes}(X)=4\lambda_{1}^{\rtimes}(X).$

In fact, let $\theta_{1}(x)=\theta^{\rtimes}_{1}(x)\geq 0$, be the first
Dirichlet eigenfunction of the operator $-\Delta+\frac{1}{4}\operatorname{Sc}$
and let $\theta(x)>0$ be an arbitrary smooth strictly positive function on
$X$.

Since $\theta_{1}$ is strictly positive in the interior of $X$, the ratio
$\frac{\theta_{1}(x)}{\theta(x)}$ assumes its maximum, call it $a$, at an
interior point $x_{0}\in X$, where

$\theta_{1}(x_{0})=a\theta(x),\qquad
a\nabla\theta_{1}(x_{0})=\nabla\theta(x)\qquad\text{and}\qquad\Delta\theta_{1}(x)\leq
a\Delta\theta(x),$

and consequently

$-\frac{\Delta\theta_{1}(x_{0})}{\theta_{1}(x_{0})}+\frac{1}{4}\operatorname{Sc}(X,x_{0})\geq-\frac{\Delta\theta(x_{0})}{\theta(x_{0})}+\frac{1}{4}\operatorname{Sc}(X,x_{0}).$

Since the sum
$-\frac{\Delta\theta_{1}(x)}{\theta_{1}(x)}+\frac{1}{4}\operatorname{Sc}(X,x)$
is constant ($=\lambda_{1}^{\rtimes}$), this inequality holds at the minimum
point $x_{o}\in X$ of the function
$-\frac{\Delta\theta(x)}{\theta(x)}+\frac{1}{4}\operatorname{Sc}(X,x)$; hence,
$\inf_{x\in
X}\operatorname{Sc}(X,x)-4\frac{\Delta\theta\rtimes_{1}(x)}{\theta^{\rtimes}(x)}$
majorizes $\inf_{x\in
X}\operatorname{Sc}(X,x)-4\frac{\Delta\theta_{1}(x)}{\theta(x)}$ for all
smooth strictly positive functions on $X$.

Finally, by $\supset$-monotonicity of $\operatorname{Sc}^{\rtimes}$ and the
continuity of the first eigenvalue
$\lambda_{1}(-\Delta_{g}+\operatorname{Sc}(g)/4)$ in (the space of
$C^{2}$-metrics) $g$, this majorization holds for functions $\theta$ which is
strictly positive only in the interior of $X$.222One needs to be slightly
careful here, since $\Delta\theta/\theta$ may, a priori, blow up at the
boundary of $X$. Then the proof of the $\frac{1}{4}$-proposition follows.

$\boldsymbol{\frac{1}{4}}$-Remark. This $\frac{1}{4}$ agrees with that in the
Schrödinger–Lichnerowicz formula
$\mathcal{D}^{2}=\nabla^{\ast}\nabla+\frac{1}{4}\operatorname{Sc}$ via the
Kato inequality for the squared Dirac operator on
$X^{\rtimes}=\bigl{(}X\times\mathbb{T}^{N},g^{\rtimes}\bigr{)}$, which, to
make it index-wise more interesting, may be twisted with the canonical
$N$-parametric family of flat unitary complex line bundles over $X^{\rtimes}$.
(Probably, there is much of what we do not understand about the relations
between the two $\frac{1}{4}$.)

### Corollaries/Examples

1.B1. If $X$ is non-compact, then

$\operatorname{Sc}^{\rtimes}(X)=\lim_{i\to\infty}\operatorname{Sc}^{\rtimes}(X_{i})$

for compact equidimensional submanifolds $X_{1}\subset\cdots\subset
X_{i}\subset\cdots\subset X,$ which exhaust $X$.

1.B2. $\operatorname{Sc}^{\rtimes}$ is additive under Riemannian products:

$\operatorname{Sc}^{\rtimes}(X_{1}\times
X_{2})=\operatorname{Sc}^{\rtimes}(X_{1})+\operatorname{Sc}^{\rtimes}(X_{2}).$

For instance, the rectangular solids satisfy

$\operatorname{Sc}^{\rtimes}\left(\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{n}[-a_{i},b_{i}]\right)=4\sum_{1}^{n}\lambda_{1}[a_{i},b_{i}]=\sum_{1}^{n}\frac{4\pi^{2}}{(b_{i}-a_{i})^{2}}.$

1.B3. Manifolds $X$ with constant scalar curvature $\sigma$ satisfy

$\operatorname{Sc}^{\rtimes}(X)=4\lambda_{1}(X)+\sigma$

for the first eigenvalue $\lambda_{1}$ of the Laplace operator on $X$.

For instance, unit hemispheres satisfy

$\operatorname{Sc}^{\rtimes}(S^{n}_{+})=n(n-1)+4n=n(n+3)$

and the unit balls $B^{n}=B^{n}(1)\subset\mathbb{R}^{n}$ satisfy

$\operatorname{Sc}^{\rtimes}(B^{n})=4j_{\nu}^{2},$

for the first zero of the Bessel function $J_{\nu}$, $\nu=\frac{n}{2}-1$,
where $j_{-1/2}=\frac{\pi}{2}$, $j_{0}=2.4042\dots$, $j_{1/2}=\pi$ and if
$\nu>1/2$, then

$\nu+\frac{a\nu^{\frac{1}{3}}}{2^{\frac{1}{3}}}<j_{\nu}<\nu+\frac{a\nu^{\frac{1}{3}}}{2^{\frac{1}{3}}}+\frac{3}{20}\frac{2^{\frac{2}{3}}a^{2}}{\nu^{\frac{1}{2}}},$

where $a=\bigl{(}\frac{9\pi}{8}\bigr{)}^{\frac{2}{3}}(1+\varepsilon)\approx
2.32$ with $\varepsilon<0.13\bigl{(}\frac{8}{2.847\pi}\bigr{)}^{2}$ [36].

Specifically,

$\displaystyle\operatorname{Sc}^{\rtimes}\bigl{(}B^{2}\bigr{)}=4(2.404\dots)^{2}=23.116\ldots>10=\operatorname{Sc}^{\rtimes}\bigl{(}S^{2}_{+}\bigr{)},$
$\displaystyle\operatorname{Sc}^{\rtimes}\bigl{(}B^{3}\bigr{)}>36>18=\operatorname{Sc}^{\rtimes}\bigl{(}S^{3}_{+}\bigr{)},$
$\displaystyle\operatorname{Sc}^{\rtimes}\bigl{(}B^{4}\bigr{)}=4(3,817\dots)^{2}=52.727\ldots>28=\operatorname{Sc}^{\rtimes}\bigl{(}S^{4}_{+}\bigr{)},$
$\displaystyle\operatorname{Sc}^{\rtimes}\bigl{(}B^{8}\bigr{)}=4(6.380\dots)^{2}=162.827\ldots>88=\operatorname{Sc}^{\rtimes}\bigl{(}S^{8}_{+}\bigr{)}.$

1.B4. Ricci comparison inequality. Let $X$ be a (metrically) complete
Riemannian manifold with a boundary such that
$\operatorname{Ricci}(X)\geq(n-1)\kappa$ and ${\rm mean.curv}(\partial
x)\geq\mu$. Then

$\operatorname{Sc}^{\rtimes}(X)\geq\operatorname{Sc}^{\rtimes}(\underline{B}^{n}_{\kappa,\mu}),$

where $\underline{B}^{n}_{\kappa,\mu}$ is the ball in the complete simply
connected $n$-space $\underline{S}^{n}_{\kappa}$ with sectional curvature
$\kappa$, and where the mean curvature of the boundary
$\partial\underline{B}^{n}_{\kappa,\mu}$ is equal to $\mu$.333The
corresponding, comparison inequality for the Dirichlet (Schrödinger)
$\lambda_{1}(\Delta)$ (compare with [7] and [1]) has, undoubtedly, been known
for at least 45 years and the relation
$\lambda_{1}=-\sup_{\varphi>0}\inf_{x\in
X}\frac{\Delta\varphi(x)}{\varphi(x)}$ must be dated to the 19th century. My
apologies to the author(s), whose paper(s) I failed to find on the web.

For instance, if $\operatorname{Ricci}(X)\geq 0$ and
$\mathrm{mean.curv}(\partial X)\geq n-1$, then

$\operatorname{Sc}^{\rtimes}(X)\geq\operatorname{Sc}^{\rtimes}(B^{n}=\underline{B}^{n}_{0,n-1})=4j_{1}^{2}.$

In fact, let $\varphi(b)=\phi(d(b))=\phi_{\kappa,\mu}(d)$ be the first
Dirichlet eigenfunction in $B^{n}_{\rho,\mu}$ written as a function of
$d=d(b)=\operatorname{dist}(b,\partial(\underline{B}^{n}_{\kappa,\mu}))$ and
let $\varphi(x)=\phi(\operatorname{dist}(X,\partial x))$. Then, by the Bishop
comparison inequality,

$\frac{\Delta_{X}\varphi(x)}{\varphi(x)}\geq\frac{\Delta_{\underline{S}^{n}_{\kappa}}\phi(d(b))}{\phi(d(b))}=\lambda_{1}(\Delta_{\underline{B}^{n}_{\kappa}}),\qquad
d(b)=\operatorname{dist}(X,\partial x),$

and the proof follows.

1.B5. If $B_{-1}^{n}(r)$ is the hyperbolic $r$-ball, then, clearly,
$\operatorname{Sc}^{\rtimes}(B_{-1}^{n}(r))$ is monotonically decreasing in
$r$, asymptotically to $4\frac{j_{\nu}^{2}}{r^{2}}$ for $r\to 0$ and
$\operatorname{Sc}^{\rtimes}(B_{-1}^{n}(r))\to-(n-1)$ for $r\to\infty$.

In fact, it follows from [1, Theorem 3.3] that

$\operatorname{Sc}^{\rtimes}(B_{-1}^{n}(r))=-n(n-1)+(n-1)^{2}\left(\frac{1}{r^{2}}+c(r)\right)$

for a bounded positive function $c(r)$ such that $c(r)\to 1$ for $r\to\infty$,
and

$\frac{1}{6}\leq c(r)\leq 1\qquad\text{for $r\geq 1$ and $n\geq 2$}.$

Thus,

$\operatorname{Sc}^{\rtimes}(B_{-1}^{n}(r))>0\qquad\text{for
$r\leq\sqrt{\frac{6(n-1)}{5n+1}}$},\qquad\operatorname{Sc}^{\rtimes}(B_{-1}^{n}(r))<0\qquad\text{for
$r\geq 3$ and $n\geq 2$}$

roughly.

1.C. General torical stabilizations. The most permissive torical “extension”
of a Riemannian manifold $X$ is a Riemannian manifold $X^{\natural}$ with an
isometric $\mathbb{T}^{N}$-action and an isometry
$X^{\natural}/\mathbb{T}^{N}\leftrightarrow X$. Here, as earlier, one defines
the number $\operatorname{Sc}^{\natural}(X)$, which is clearly
$\geq\operatorname{Sc}^{\rtimes}(X)$.

It seems, however – I did not honestly checked this – that the curvature
formulas for Riemannian submersions [35] imply that
$\operatorname{Sc}^{\natural}(X)\leq\operatorname{Sc}^{\rtimes}(X)$.

Alternatively, if the fibration $X^{\natural}\to
X^{\natural}/\mathbb{T}^{N}=X$ admits a section, then the $\rtimes$-rendition
of the Schoen–Yau argument444See [18] and references therein. implies the
equality

$\operatorname{Sc}^{\natural}(X)=\operatorname{Sc}^{\rtimes}(X)$

for $\dim(X)=n\leq 8$,555The case $n=8$ depends on [40]. while for all $n$
this may follow from [43], where both arguments apply not only to Riemannian
submersions but to all distance non-increasing maps
$\bigl{(}X\times\mathbb{T}^{N},G\bigr{)}\to(X,g)$.

1.D. $\boldsymbol{\lambda_{1}(\beta)}$-Remarks. The formal properties of the
operators $-\Delta+\beta\operatorname{Sc}$ are similar for all $\beta$, e.g.,
if a Riemannian manifold $X_{0}$ admits a locally isometric equidimensional
map to $X$, then

$\lambda_{1}(-\Delta_{X_{0}}+\beta\operatorname{Sc}(X_{0}))\geq\lambda_{1}(-\Delta_{X}+\beta\operatorname{Sc}(X));$

the spectra $\\{\lambda_{1},\lambda_{2},\dots\\}$ of the operators
$-\Delta+\beta\operatorname{Sc}$ are additive under Riemannian products,

$\displaystyle\operatorname{spec}(-\Delta_{X_{1}\times
X_{2}}+\beta\operatorname{Sc}(X_{1}\times
X_{2}))=\operatorname{spec}(-\Delta_{X_{1}}+\beta\operatorname{Sc}(X_{1}))+\operatorname{spec}(-\Delta_{X_{2}}+\beta\operatorname{Sc}(X_{2})).$

There are several special values of $\beta$:

  * •

if $\beta=(N+1)/4N$ $(>1/4)$, then the corresponding $\lambda_{1}$ is equal to
the maximal constant scalar curvature of the warped metrics on
$X\times\mathbb{T}^{N}$;

  * •

if $\beta=(n-1)/4n$ $(<1/4)$, $n=\dim(X)$, this implies positivity of the
square of the Dirac operator by refined Kato’s inequality [24].

  * •

if $\beta=(n-2)(4(n-1))$ $(<(n-1)/4n)$, then the inequality $\lambda_{1}>0$
implies the existence of a conformal metric on $X$ with $\operatorname{Sc}>0$
by the Kazdan–Warner theorem.

The geometric meaning of other $\beta$, as well as of the higher eigenvalues
$\lambda_{i}(X,\beta)$ of $-\Delta+\beta\operatorname{Sc}$ is unclear.666The
second eigenvalue of $\lambda_{2}(-\Delta_{+}\frac{1}{2}\operatorname{Sc})$ is
used by Marques and Neves in the proof of the $S^{3}$-min-max theorem, [33],
but the role of $\lambda_{i}(X,\beta)$ remains problematic for $\dim(X)\geq
4$, $i\geq 2$ and all $\beta$.

##  2
$\boldsymbol{\operatorname{Sc}^{\rtimes_{\downarrow}},\operatorname{Sc}_{\rm
sp}^{\rtimes_{\downarrow}},\dots,\operatorname{Sc}^{\rtimes_{\downarrow}}_{\ast}}$
on homology

Let $X$ be a metric space, e.g., a Riemannin manifold
$\operatorname{Sc}^{\rtimes_{\downarrow}}(h)=\operatorname{Sc}_{\rm
dist}^{\rtimes_{\downarrow}}(h)$, $h\in H_{m}(X,\partial X)$, denote the
supremum of the numbers $\sigma$ such that the homology class $h$ is
representable by a distance decreasing map $f$ from an oriented Riemannian
$m$-manifold $Y$ with $\operatorname{Sc}^{\rtimes}(Y)\geq\sigma$,

$f\colon\ (Y,\partial Y)\to(X,\partial X),\qquad f_{\ast}[Y,\partial Y]=h.$

Smoothness remark. If $X$ is a smooth Riemannian manifold, then an obvious
approximation argument shows that requiring maps $f$ to be smooth does not
change the value of $\operatorname{Sc}_{\rm dist}^{\rtimes_{\downarrow}}(h).$
(However, smoothness of a distance non-increasing map $f$ in the extremal
case, where
$\operatorname{Sc}^{\rtimes}(Y)=\operatorname{Sc}^{\rtimes_{\downarrow}}(h)$
is a delicate matter, see [5].)

Below the are several versions of this definition with the generic notation
$\operatorname{Sc}^{\rtimes_{\downarrow}}_{\ast}$.

I. Restrict/relax the topology of $Y$, e.g., by requiring that

$\bullet_{\rm sp}$

$Y$ is spin;777A referee suggested to constrain maps rather then manifolds
$Y$, e.g., by allowing spin maps only (in the case, where $X$ is a manifold),
but I could not figure out what to do with it.

$\bullet_{\widetilde{\rm sp}}$

the universal covering of $Y$ is spin;888This condition is satisfied in
several interesting examples of non-spin manifolds $X$, e.g., where
$\pi_{2}=0$. At the same time, much of the Dirac theoretic scalar curvature
results apply to these $X$, see [13, Section $9\frac{1}{2}$].

$\bullet_{\pi_{2}=0}$

the second homotopy group of $Y$ is zero;

$\bullet_{\rm st.par}$

$Y$ is stably parallelizable;

$\bullet_{\odot}$

allow representation of $h$ by quasi-proper maps from complete manifolds $Y$
to $X$, where “quasi-proper” means locally constant at infinity.

Clearly,

$\displaystyle\operatorname{Sc}^{\rtimes_{\downarrow}}_{\odot}\geq\operatorname{Sc}^{\rtimes_{\downarrow}}\geq\operatorname{Sc}^{\rtimes_{\downarrow}}_{\widetilde{\rm
sp}}\geq\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm
sp}\geq\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm
st.par}\qquad\text{and}\qquad\operatorname{Sc}^{\rtimes_{\downarrow}}_{\widetilde{\rm
sp}}\geq\operatorname{Sc}^{\rtimes_{\downarrow}}_{\pi_{2}=0}.$

II. Assuming $X$ is a Riemannian manifold, relax the distance decreasing
condition on $f$ by the following

$\bullet_{\rm area}$ the map $f$ decreases the areas of all surfaces in
$Y$.999This makes sense for general metric spaces $X$ with the “Hilbertian
area” defined in [15].

Clearly, $\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}\geq\operatorname{Sc}^{\rtimes_{\downarrow}}$ and
we show in §2.E below that the ratio $\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}\over\operatorname{Sc}^{\rtimes_{\downarrow}}$ can
be arbitrarily large.

III. Replace the integer homology by the rational homology
$H_{\ast}(X;\mathbb{Q})$, which is essentially (but not quite) the same as
allowing maps $f\colon Y\to X$, where $f_{\ast}$ sends the fundamental class
of $Y$ to a non-zero multiple of $h$.

IV. Instead of homology, use a bordism group of $X$, e.g., the spin bordism
group, which is well adapted to $\operatorname{Sc}>0$.

V. Remarks on singular $Y$.

(a) If $h_{m}\in H_{m}(X)$ is not representable as $f_{\ast}[Y]$ for a smooth
manifold $Y$, it may be interesting to try pseudomanifolds $Y$ with suitably
defined singular Riemannian metrics $g$ with
${\operatorname{Sc}(g)\geq\sigma}$.

(b) Conical example. Here $Y$ has an isolated singularity at a point $y_{0}\in
Y$, where $g$ is a smooth Riemannian metric on the complement to $y_{0}$ such
that $\operatorname{Sc}(g)\geq\sigma$ and such that $g$ is (approximately)
conical at $y_{0}$.

This means that there exists an $\varepsilon$-neighbourhood (ball)
$U=U_{\varepsilon}\subset Y$ of $y_{0}$, which topologically splits away from
$y_{0}$,

$U\setminus\\{y_{0}\\}=Z\times(0,\varepsilon],$

where $Z=(Z,g_{Z})$ is a compact smooth Riemannian manifold such that the
metric $g$ restricted to $U\setminus\\{y_{0}\\}$ is related to $g_{Z}$ as
follows:

$g=a^{2}(t)t^{2}g_{Z}+{\rm d}t^{2},$

where $a^{2}(t)>0$ is a smooth positive function on the (now closed) interval
$[0,\varepsilon]$. (One may assume, if one wishes, that $a(t)$ is constant
near $t=0$.)

(c) One may additionally assume that
$\operatorname{Sc}(g_{Z})\geq\operatorname{Sc}\bigl{(}S^{m-1}\bigr{)}=(m-1)(m-2)$
for $m=\dim(Y)$ and, to make the metric truly conical, to require $a(t)$ to be
constant near $t=0$.

But this is not truly needed, since it can always be achieved by a small
deformation of our $g$ near $y_{0}$.

(d) Iterated conical singularities. Next, following [8], define
$m$-dimensional (roughly) cone-singular spaces $Y$ with
$\operatorname{Sc}(Y)\geq\sigma$ by induction on $m$, where (as in the above
conical case) the metric (i.e., the distance function) on $Y$ is defined by a
smooth Riemannian metric $g$ on the non-singular part $Y_{0}\subset Y$, where
the following conditions are satisfied:

  1. (i)

The singular locus $\Sigma=Y\setminus Y_{0}$ is a closed subset in $Y$ with
codimension two in $Y$.

  2. (ii)

$\operatorname{Sc}(g,y)\geq\sigma$ for all $y\in Y_{0}$.

  3. (iii)

Each $y_{0}\in Y$ admits a neighbourhood $U_{0}$, which is topologically (but
not metrically) cylindrically splits away from $y_{0}$,

$U_{0}\setminus\\{y_{0}\\}=Z_{0}\times(0,\varepsilon_{0}],$

where $Z_{0}=(Z_{0},g_{Z_{0}})$, is a compact $(m-1)$-dimensional cone-
singular space and where the Riemannian metric $g$ on the non-singular part of
$U_{0}\setminus\\{y_{0}\\}$, denoted $U_{00}\subset
U_{0}\setminus\\{y_{0}\\}$, is

$g=a_{0}^{2}(t)t^{2}g_{Z_{0}}+{\rm d}t^{2}+\delta_{0},$

where $a_{0}^{2}(t)>0$ is a smooth positive function on $[0,\varepsilon_{0}]$
and where $\delta_{0}=\delta_{0}(u)$ is a (small) smooth quadratic
differential form on $U_{0}$, which converges to zero for $u\to y_{0}$.

(e) Due to $h_{0}$, the above “conical” is slightly more general than how it
is defined in (b) for an isolated singularity $y_{0}$.

(f) Similarly to the isolated singularity case, the requirement
$\operatorname{Sc}(Z_{0})\geq m-1)(m-2)$ does not significantly change the
definition of $\operatorname{Sc}(Y)\geq\sigma$.

(g) One may also insist on the split-conical geometry at all points $y_{0}$:
if $y_{0}$ is contained in the interior of an $l$-dimensional strata
$S\subset\Sigma$, then a small neighbourhood $U_{0}\subset Y$ of $y_{0}$
metrically splits: $U_{0}=S_{0}\times N_{0}$, where $S_{0}=U_{0}\cap S$, and
where $N_{0}$ is a con-singular manifold with an $(m-l-1)$-dimensional base.

(h) Probably, as in the isolated singularity case, this additional condition
can be achieved by a small deformation of $g$ near $\Sigma$.

Question. How does the resulting
$\operatorname{Sc}^{\rtimes_{\downarrow}}(h)$, $h\in H_{m}(X)$, depend on the
topology of the singular locus $\Sigma\subset Y$?

For instance, $Y$ may be iterated conical space with initial cones based on
products of complex projective spaces and/or other generators of the oriented
bordisms groups with $\operatorname{Sc}>0$ metrics (e.g., as in [21]), where
we allow cones over $l$-dimensional $Y$ only if they admit metrics $g$ with
“suitably defined” $\operatorname{Sc}(g)>0$ and/or, which is probably
equivalent with $\operatorname{Sc}^{\rtimes}(g)>0$.

(Probably, stable minimal hypersurfaces and $\mu$-bubbles in such $Y$,
similarly to how it is in a smooth $Y$, enjoy necessary properties required
for the study of the scalar curvature and this is also conceivable for the
Dirac theoretic approach (compare with [2]).)

VI. If $X$ is non-compact, allow classes $h$ with infinite
supports101010Sometimes referred to as “locally finite homology classes” as
was pointed out to me by a referee. and use proper (and quasi-proper) maps
$f\colon Y\to X$.

Remark on Completeness of $Y$. Regardless of $X$ being complete or not, the
value $\operatorname{Sc}_{\rm area}^{\rtimes_{\downarrow}}[X]$ defined with
complete $Y$ mapped to $X$ may be very different without this completeness.

For instance, $\operatorname{Sc}_{\rm
area,sp}^{\rtimes_{\downarrow}}[\mathbb{R}^{n}]$ defined with non-complete $Y$
is infinite.

In fact, if $g_{0}$ is a metric on $\mathbb{R}^{2}$ such that
$\operatorname{Sc}(g_{0})=1$ and $\operatorname{area}(Y_{0},g_{0})=\infty$ (as
in §2.A below), then $Y=\bigl{(}\mathbb{R}^{n-2}\times\mathbb{R}^{2},g_{\rm
Eucl}+g_{0}\bigr{)}$ admits an obvious area contracting diffeomorphism onto
$\bigl{(}\mathbb{R}^{N},g_{\rm Eucl}\bigr{)}$.

But if we limit to complete spin manifolds $Y$, then $\operatorname{Sc}_{\rm
area,sp,compl}^{\rtimes_{\downarrow}}[\mathbb{R}^{n}]=0$.

(It is unknown for $n\geq 4$ whether $\operatorname{Sc}_{\rm
area,compl}^{\rtimes_{\downarrow}}[\mathbb{R}^{n}]$ is zero or infinity
without the spin assumption on $Y$.)

2.A. Surface examples. Closed connected simply connected, i.e., spherical
Riemann surfaces $X$ satisfy

$\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}[X]=\frac{8\pi}{\operatorname{area}(X)}.$

Indeed, the inequality $\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}[X]\geq\frac{8\pi}{\operatorname{area}(X)}$
follows from the existence of a measure preserving diffeomorphism from the
2-sphere with constant scalar curvature
$\sigma=\frac{8\pi}{\operatorname{area}(X)}$ onto $X$; the opposite inequality
follows from Zhu’s lemma (see [46] and [17, Section 2.8]).

Similarly, one shows that $\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}[X]$ for closed surfaces $X$ of positive genera.

On the opposite end of the spectrum, non-compact connected surfaces $X$
satisfy $\operatorname{Sc}_{\rm area}^{\rtimes_{\downarrow}}[X]=\infty$, since
all the surfaces $X$ admit Riemannian metrics with $\operatorname{Sc}=1$, and
with the given areas (including ${\rm area}=\infty$ for non-compact $X$) and
since connected mutually diffeomorphic Riemann surfaces $X_{1}$ and $X_{2}$ of
equal areas admit area preserving diffeomorphisms $X_{1}\leftrightarrow
X_{2}$.

Problem with
$\boldsymbol{\operatorname{Sc}^{\rtimes_{\downarrow}}[X]=\operatorname{Sc}_{\rm
dist}^{\rtimes_{\downarrow}}[X]}$. Unlike $\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}$, the geometric meaning of
$\operatorname{Sc}^{\rtimes_{\downarrow}}[X]$ for spherical surfaces $X$
remains obscure. All one knows besides Zhu’s lemma for general $X$ (see [17,
Section 2.8]) is that

$\operatorname{Sc}^{\rtimes_{\downarrow}}[X]<{4\pi^{2}\over\operatorname{diam}(X)^{2}}.$

2.B. $\boldsymbol{H_{2m}(\mathbb{C}P^{n})}$-example. Let the complex
projective space $\mathbb{C}P^{n}$ be endowed with the $U(n+1)$ invariant
(Fubini–Study)-metric such that the projective lines have scalar curvatures
equal 2 and let $\mathbb{C}P^{m}\hookrightarrow\mathbb{C}P^{n}$ be an
$m$-plane.

If $m$ is odd, then the manifold $\mathbb{C}P^{m}$ is spin, and both homology
and the spin-bordism class of $\mathbb{C}P^{m}\hookrightarrow\mathbb{C}P^{n}$
satisfy

$\operatorname{Sc}^{\rtimes_{\downarrow}}[\mathbb{C}P^{m}]\geq\operatorname{Sc}_{\rm
sp}^{\rtimes_{\downarrow}}(k[\mathbb{C}P^{m}])\geq\operatorname{Sc}^{\rtimes_{\downarrow}}[\mathbb{C}P^{m}]_{\rm
sp.brd}\geq\operatorname{Sc}(\mathbb{C}P^{m})=m(m+1)$

and the same holds for the multiples $k[\mathbb{C}P^{m}]\in
H_{2m}(\mathbb{C}P^{n})$, $k=\dots,-1,0,1,2,\dots$.

If $k$ is even, then if $m$ is even, then the
$\operatorname{Sc}^{\rtimes_{\downarrow}}k[\mathbb{C}P^{m}]\geq m(m+1)$
remains valid for all $k$. But if $k$ is odd, then $\operatorname{Sc}_{\rm
sp}^{\rtimes_{\downarrow}}(k[\mathbb{C}P^{m}])=0$, since the classes
$k[\mathbb{C}P^{m}]\in H_{2m}(\mathbb{C}P^{n})$ are not representable by the
maps from spin manifolds $Y\to\mathbb{C}P^{n}$.

In fact, if an oriented manifold $Y^{2m}$ contains a smooth hypersurface $H$
such that the $m$-fold self-intersection index
$\underset{m}{\underbrace{H\frown\cdots\frown H}}$ is odd, then the
$(m-1)$-fold intersection is an orientable surface $\Sigma\subset Y$, which
for even $m$ has non-trivial normal bundle; hence
$w_{2}[\Sigma]_{\mathbb{Z}_{2}}\neq 0$.

And if $k$ is even, then

$\operatorname{Sc}_{\rm sp}^{\rtimes_{\downarrow}}(k[\mathbb{C}P^{m}])\geq
m^{2}$

since the class $2[\mathbb{C}P^{m}])$ is represented by the quadric
$Q^{m}\subset\mathbb{C}P^{m+1}\subset\mathbb{C}P^{n}$ given by the equation
$z_{0}^{2}+z_{1}^{2}+\dots+z_{m}^{2}=0$, where this $Q^{m}$ is spin and has
scalar curvature $m^{2}$.

Finally,

$\operatorname{Sc}_{\rm st.par}^{\rtimes_{\downarrow}}(m!h_{2m})\geq{\rm
const}_{m}\cdot\operatorname{Sc}\bigl{(}\underset{m}{\underbrace{S^{2}(1)\times\dots\times
S^{2}(1)\bigr{)}}}=2m\qquad\mbox{for all $m$ and $n\geq m$},$

since the quotient space

$\bigl{(}S^{2}\bigr{)}^{m}/\Pi(m)\qquad\text{of}\quad\bigl{(}S^{2}\bigr{)}^{m}=\underset{m}{\underbrace{S^{2}(1)\times\dots\times
S^{2}(1)}}$

by the permutation group $\Pi(m)$ admits a natural biholomorphic map
$\psi\colon\bigl{(}S^{2}\bigr{)}^{m}\to\mathbb{C}P^{m}$, where ${\rm
const}_{m}>0$ is the squared reciprocal to the minimal Lipschitz constant of
maps in the homotopy class of this $\psi$.

Question. What are
$\operatorname{Sc}^{\rtimes_{\downarrow}}\bigl{[}\bigl{(}S^{2}\bigr{)}^{m}/\Pi(m)\bigr{]}$
and of the symmetric powers $[(X)^{m}/\Pi(m)]$ for more general manifolds
$X$?111111These are among most attractive singular quasi-conical spaces
discussed earlier.

2.C. Upper bounds and equalities. The ($\mathbb{T}^{\rtimes}$-stabilized and
$\widetilde{\rm sp}$-generalized) rigidity theorem by Min-Oo [34] and (the
spin cobordims version of) Goette–Semmelmann’s theorem from [11] imply that
the class $h_{2m}=[\mathbb{C}P^{m}]\in H_{2m}(\mathbb{C}P^{m})$ satisfy the
following relations:

$\displaystyle\operatorname{Sc}_{{\rm
area,sp.brd}\mathbb{C}P^{m}}^{\rtimes_{\downarrow}}(h_{2m})=m(m+1)\qquad\text{for
all $m$ and $n\geq m$},$

where “${\rm sp.brd}$” indicates that this $\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}$ defined with smooth maps $Y\to X$ which are
spin-bordant to the embedding
$\mathbb{C}P^{m}\hookrightarrow\mathbb{C}P^{n}$,121212It is unclear what
happens for $m\leq n\leq 2m-2$.

$\displaystyle\operatorname{Sc}_{{\rm area},\widetilde{\rm
sp}}^{\rtimes_{\downarrow}}(h_{2m})=m(m+1)\qquad\text{for odd $m$},$
$\displaystyle\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
area},\widetilde{\rm sp}}(h_{2m})=m^{2}\qquad\text{for even $m$},$
$\displaystyle\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
area,st.par}}(h_{2m})_{\rm area}=2m\qquad\text{for all $m$ and $n\geq 2m-1$}.$

2.D. Homological homogeneity conjecture. Let $X$ be a compact symmetric space
and $H\subset H_{m}(X,\mathbb{Q})$ be the linear subspace generated by the
fundamental classes $[Y_{i}]\in H_{m}(X)$ of homogeneous (not necessarily
totally geodesic) $m$-submanifolds $Y_{i}\subset X$.131313$Y\subset X$ is
homogeneous if an isometry group of $X$ preserves $Y$ and is transitive on
$Y$. Then all classes ${h_{m}\in H}$ can be represented by linear combinations
of homogeneous $Y_{i}$ such that
$\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm
area}[Y_{i}]\geq\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm area}(h)$. (This
maybe overoptimistic in general, but the $\widetilde{\rm sp}$-version of this
can be, probably, proved with available means for products of spheres, complex
and quaternionic projective spaces.)

2.E. Equivalence conjecture. All rational $h\in H_{m}(X)$ for compact
Riemannian manifolds $X$ without boundaries satisfy:141414The dimension $m=4$
may be special.

$\displaystyle\operatorname{Sc}^{\rtimes_{\downarrow}}_{\odot}(h)=\operatorname{Sc}^{\rtimes_{\downarrow}}(h)\qquad\text{and}\qquad\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
sp},\odot}(h)=\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm sp}(h),$
$\displaystyle\operatorname{Sc}^{\rtimes_{\downarrow}}(h)\leq
A\cdot\operatorname{Sc}^{\rtimes_{\downarrow}}_{\widetilde{\rm sp}}(h),$
$\displaystyle\operatorname{Sc}^{\rtimes_{\downarrow}}_{{{\widetilde{s}p}}}(h)\leq
B\cdot\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm sp}(h),$
$\displaystyle\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm sp}(h)\leq
C\cdot\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm st.par}(h),$

where $A=A_{n}$, $B=B_{n}$ and $C=C_{n}$ are universal constants, and where
the same relations are expected for the area version of these five
$\operatorname{Sc}^{\rtimes_{\downarrow}}$.

2.F. Positivity. Unlike $\operatorname{Sc}^{\rtimes}$, the values of all
$\operatorname{Sc}^{\rtimes_{\downarrow}}=\operatorname{Sc}_{\rm
dist}^{\rtimes_{\downarrow}}$-invariants are non-negative, since all (compact
or not) manifolds $Y$ admit arbitrarily large Riemannian metrics with
$\operatorname{Sc}>\varepsilon$.

Moreover, the fundamental classes of compact connected manifolds $X$ with non-
empty boundaries are strictly positive since such manifolds admit metrics with
$\operatorname{Sc}>0$. (For instance, the $r$-balls with hyperbolic metrics
$g$, ${\rm sect.curv}(g)=-1$, admit (obvious radial) metrics $g_{+}\geq g$
with $\operatorname{Sc}(g_{+})\geq\exp-4r$.)

2.G. $\boldsymbol{[\exists\operatorname{Sc}>0]}$-Conjecture. If a rational
homology class $h\in H_{m}(X;\mathbb{Q})$ vanishes under the classifying map
$\beta\colon X\to$B$(\pi_{1}(X))$

$\beta_{\ast}(h)=0,$

then $\operatorname{Sc}^{\rtimes_{\downarrow}}(h)>0$.151515This seems more
realistic if $\beta$ can be homotoped to the $(m-2)$-skeleton of (some cell
decomposition of) B$(\pi_{1}(X))$.

2.H. Finiteness. $\operatorname{Sc}^{\rtimes_{\downarrow}}(h)$ may be, a
priori, infinite. However, the finiteness of
$\operatorname{Sc}^{\rtimes_{\downarrow}}=\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm
dist}$ easily follows from the $\square^{m}$-inequality (3.8) in [17], where
the proof for $m\geq 9$ relies on Theorem 4.6 in [39] and where the finiteness
of $\operatorname{Sc}_{\rm sp}^{\rtimes_{\downarrow}}(h)$ for all $m$ follows
from [43]. (Probably, the arguments used in [43] generalize to
$\operatorname{Sc}_{\widetilde{\rm sp}}^{\rtimes_{\downarrow}}$.)

2.I. $\boldsymbol{\nexists\operatorname{Sc}>0}$-Problem. Does non-vanishing of
a rational $h\in H_{m}(X;\mathbb{Q})$ under the above classifying map
B$(\pi_{1}(X))$ imply that $\operatorname{Sc}^{\rtimes_{\downarrow}}=0$? This
is known for $m=3$, and also for
$\operatorname{Sc}^{\rtimes_{\downarrow}}_{\widetilde{\rm sp}}(h)$ and all $m$
if the spinorial curvature $\mathbb{S}$p$.{\rm
curv}^{\downarrow}(\beta_{\ast}(h)\in H_{m}($B$(\pi_{1}(X))$ defined in
Section 7 vanishes, e.g., if our B$(\pi_{1})$ admits a complete metric with
${\rm sect.curv}\leq 0$, see [17] and references therein. (I am not certain if
there are examples of non-zero rational homology classes $\underline{h}$ in
aspherical, say, compact finite-dimensional spaces such that
$\mathbb{S}$p$.{\rm curv}^{\downarrow}(\underline{h})\neq 0$.)

2.J. $\boldsymbol{\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}}$-finiteness question. Is $\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}[X]<\infty$ for all compact Riemannian manifolds
$X$ without boundaries?

Remarks. (a) All metrics $g_{+}$ on a compact Riemannian manifold $(X,g)$ such
that $\operatorname{area}_{g_{+}}(\Sigma)\geq\operatorname{area}_{g}(\Sigma)$
for all surfaces $\Sigma\subset X$ satisfy

$\operatorname{Sc}^{\rtimes}(g_{+})\leq\mathrm{const}\cdot(X,g)<\infty.$

In fact, this inequality holds for all $(Y,g_{+})$ what admit area decreasing
spin maps161616A continuous map between orientable manifolds, $f\colon Y\to
X$, is _spin_ if $f^{\ast}(w_{2}(X))=w_{2}(Y)$, where $w_{2}$ is the second
Stiefel–Whitney class. $f\colon Y\to X$ with non-zero degrees.

(b) Let $X=X_{0}\times Y$, where $Y$ is enlargeable171717A compact Riemannian
$n$-manifold $X$ is _enlargeable_ if there exists a sequence of oriented
coverings $\tilde{X}_{i}\to X$ and distance decreasing maps
$f_{i}\colon\tilde{X}_{i}\to S^{n}(R_{i})$, $R_{i}\to\infty$, which are
constant at infinity and which have non-zero degrees, compare with [4, 20, 23,
38], in [17, §4.7], [18, §2.A]. and $\dim(X_{0})=2$. Then the finiteness of
$\operatorname{Sc}_{\rm area}^{\rtimes_{\downarrow}}(X)$ for $X$ follows from
[47], where for $\dim(X)\geq 8$ one needs a version of Theorem 4.6 from [39].

Also the $\rtimes$-stabilized version of the area slicing theorem from [28]
(this stabilization is likely to be true) delivers an effective finite bound
on $\operatorname{Sc}_{\rm area}^{\rtimes_{\downarrow}}(X_{0}\times Y)$. For
$\dim(X_{0})=3$, provided $Y$ is enlargeable and $\dim(X)\leq 8$.

But the principal case, where $X=S^{n}$ remains problematic for all $n\geq 4$
and neither can one prove or disprove the existence of (necessarily non-spin)
complete orientable $n$-manifolds $Y$, $n\geq 4$, with
$\operatorname{Sc}\geq\sigma>0$, which admit smooth proper area decreasing
maps to $\mathbb{R}^{n}$, $n\geq 4$, with non-zero degrees.

2.K. Outline of construction for
$\boldsymbol{\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm
area}/\operatorname{Sc}_{\rm dist}^{\rtimes_{\downarrow}}\to\infty}$. Let
$g_{0}$ be a metric on a manifold $Y$ such that $\operatorname{Sc}(g_{0})>0$,
then there exists metrics $g$ on $Y$ with arbitrarily large ratios
$\operatorname{Sc}^{\rtimes\downarrow}_{\rm area}(g)/\operatorname{Sc}_{\rm
dist}^{\rtimes\downarrow}(g)$. In fact, let $Y=(Y,g_{0})$ be an arbitrary
Riemannian manifold and $U\subset Y$ an open subset. Then for all
$\varepsilon>0$ and $\delta>0$, there exists a Riemannian metric
$g_{\varepsilon,\delta}$ on $Y$ such that
$\operatorname{Sc}(g_{\varepsilon,\delta})\geq\operatorname{Sc}(g_{0})-\varepsilon$
and

$\bullet_{Y\setminus U}$

the metric $g_{\varepsilon,\delta}$ is equal to $g_{0}$ outside $U$;

$\bullet_{\rm area}$

the metric $g_{\varepsilon,\delta}$ is area-wise smaller than $g_{0}$,

$\operatorname{area}_{g_{\varepsilon,\delta}}(S)\leq\operatorname{area}_{g_{0}}(S)\qquad\text{for
all smooth surfaces $S\subset Y$};$

$\bullet_{\rm dist}$

all Riemannian manifolds $X$, which 1-Lipschitz dominate181818“Domination” is
a map with non-zero degree, see [17, §1.5]. $(Y,g_{\varepsilon,\delta})$, have
$\operatorname{Sc}^{\rtimes}(X)\leq\delta$. The only non-trivial condition
here is $\bullet_{\rm dist}$, which is achieved with the follow ing one.

$\bullet_{D}$

There is an open subset $U_{D}\subset U$ with $D=10\sqrt{\frac{1}{\delta}}$
such that $(U_{D},g_{\varepsilon,\delta})$ is isometric to the product

$\mathbb{T}^{1}(\varepsilon)\times\mathbb{T}^{n-2}(2\pi)\times[-D,D],$

where $T^{1}(\varepsilon)$ is the circle of length $\varepsilon$ and
$\mathbb{T}^{n-2}(2\pi)$ is the standard flat torus.

The construction of $g_{\varepsilon,\delta}$, which satisfies
$\bullet_{Y\setminus U}$, $\bullet_{\rm area}$ and $\bullet_{D}$ is elementary
(left to the reader191919To get an insight, start with $Y=S^{2}$, then look at
$Y=S^{2}\times\mathbb{T}^{n-2}$.) while the implication
$\bullet_{D}\Rightarrow\bullet_{\rm dist}$ follows the
$\frac{2\pi}{n}$-inequality (see [17, §3.6] and references therein).202020The
proof of $\frac{2\pi}{n}$-inequality for $n\geq 9$ relies on Theorem 4.6 in
[39], and if one is satisfied with $\operatorname{Sc}^{\rtimes\downarrow}_{\rm
area}(g)/\operatorname{Sc}_{{\rm dist,sp}}^{\rtimes\downarrow}(g)\to\infty$,
then one can use the spinorial version of $\frac{2\pi}{n}$ from [45].

Thus, we see that if a non-torsion homology class $h$ in a compact manifold
$X$ satisfies $\operatorname{Sc}_{\rm dist}^{\rtimes_{\downarrow}}(h)>0$, then
the ratio $\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
area,sp}}(h)/\operatorname{Sc}_{{\rm dist,sp}}(h)^{\rtimes_{\downarrow}}$ can
be made arbitrarily large with some Riemannian metric on $X$.

2.L. Question on $\boldsymbol{\lambda^{\downarrow}_{1}(h,\beta)}$. The
definition of $\operatorname{Sc}^{\rtimes_{\downarrow}}(h)$, which depends on
$\lambda_{1}(Y,\beta=1/4)$ (see $(N+1)/4N$-Remark 1.D) makes sense for all
$\beta$ and the arguments which depends on stable minimal hypersurfaces and
$\mu$-bubbles generalize to all $\beta$, e.g., as the
$\square^{\exists\exists}(n,m,N)$-inequality, which is stated in [18, §2.B]
for $\beta=N/(N+1)$. However, the geometric significance of this for
$\beta\neq N/(N+1)$ is unclear.

Probably, if $X$ is simply connected, $\beta\leq\beta_{m}>0$ and $m\geq 3$,
then an integer multiples $lh$ for some $l\neq 0$ and all $h\in H_{m}(X)$ are
representable by a distance decreasing maps $Y\to X$, where
$\lambda_{1}(Y,\beta)\geq C$ for a given $C>0$.

### Exercises

2.M. Let $g$ be a Riemannian metric on an open manifold212121A manifold $X$ is
_open_ if it contains no closed manifold connected component. $X$ of dimension
$\dim(X)=n\geq 2$. Show that there exists a Riemannian metric $g_{+}$ on $X$
such that $\operatorname{Sc}(g_{+})=1$ and
$\operatorname{area}_{g_{+}}(\Sigma)\geq\operatorname{area}_{g}(\Sigma)$ for
all smooth surfaces $\Sigma\subset X$.

Hint: Observe that $[0,1]\times\mathbb{R}^{n-1}$ admits an area decreasing
diffeomorphism onto $\mathbb{R}^{n}$ and use products of surfaces with
constant curvatures by $\mathbb{R}^{n-2}$ as building blocks for $(X,g_{+})$.

Remark. If $Y$ is a complete spin $n$-manifold with
$\operatorname{Sc}^{\rtimes}(Y)\geq\sigma>0$, then it admits no proper area
decreasing map to $\mathbb{R}^{n}$ with non-zero degree [21].

2.N. Show that non-zero multiples of homology classes $h$ in simply connected
manifolds $X$ have $\operatorname{Sc}_{{\rm
st.par}}^{\rtimes_{\downarrow}}(ih)>0$, for some $i\neq 0$.

Hint. Recall the Serre–Thom theorem on framed bordisms and apply Stolz’
theorem on spin manifolds [41].

##  3 $\boldsymbol{\rtimes^{\downarrow}}$-extremality and
$\boldsymbol{\rtimes^{\downarrow}}$-rigidity

3.A. Homological
$\boldsymbol{\operatorname{Sc}_{\ast}^{\rtimes_{\downarrow}}}$-problems. Let
$X$ be a Riemannian manifold and $h\in H_{m}(X)$ a homology class, e.g.,
$m=n=\dim(X)$, and let $h$ be the fundamental class $[X]$ of $X$, where $X$ is
assumed oriented.

Evaluate $\operatorname{Sc}_{\ast}^{\rtimes_{\downarrow}}$ and/or find
relations between $\operatorname{Sc}_{\ast}^{\rtimes_{\downarrow}}$ and more
accessible metric invariants of $X$.

Decide if $\operatorname{Sc}_{\ast}^{\rtimes_{\downarrow}}(h)$ is represented
by an $\operatorname{Sc}_{\ast}^{\rtimes_{\downarrow}}$-extremal, or, for
brevity, $\rtimes^{\downarrow}_{\ast}$-extremal, oriented Riemannian
$m$-manifold mapped to $X$,

$Y\overset{f}{\rightarrow}X\quad\text{such that $f_{\ast}(Y)=h$ and
$\operatorname{Sc}^{\rtimes}(Y)=\operatorname{Sc}_{\ast}^{\rtimes_{\downarrow}}(h)$},$

where $f$ is the distance or the area decreasing depending on “$\ast$” and
where, ideally, $f$ is an isometric immersion.

For instance, given a submanifold $Y\hookrightarrow X$, e.g., $Y=X$ decide if
it is $\operatorname{Sc}_{\ast}^{\rtimes_{\downarrow}}$-extremal, or,
moreover, if it is rigid, that is unique extremal (compare with §3.D below).

Find examples of $h$, where there is no extremal manifold $Y\to X$ with
$f_{\ast}[Y]=h$, but such a generalized $Y$, e.g., a singular extremal one
does exist. (We saw some potential examples of such singular $Y$, and stable
minimal singular hypersurfaces suggest further examples.)

Determine which closed manifolds $X$ admit $\rtimes^{\downarrow}$-extremal
Riemannian metrics.

(Possibly, metrics $g_{0}$ with $\operatorname{Ricci}(g_{0})>0$ can be
deformed to $g\geq g_{0}$ with
$\operatorname{Sc}^{\rtimes}(g)=\operatorname{Sc}^{\rtimes_{\downarrow}}(g)$.
But, for instance, metrics $g_{0}=g_{1}+g_{2}$ on $X=X_{1}\times X_{2}$, where
${\rm sect.curv}(g_{1})=1$ and ${\rm sect.curv}(g_{2})<0$ admit no such
deformations. However, the pointed Hausdorff limit manifolds
$\lim_{\lambda\to\infty}(X,g_{+}\lambda g_{2})$, which are isometric to
$X_{1}\times\mathbb{R}^{\dim(X)}$, are extremal.

In general, the existence of a metric $g_{0}$ on $X$ with
$\operatorname{Ricci}(g_{0})\geq 0$ might be necessary for the existence of an
extremal metric $g$ on $X$.

### Examples

3.B$\boldsymbol{{}_{S^{n}}}$. Complete manifolds with constant sectional
curvatures, e.g., unit spheres, flat tori and Euclidean spaces are
$\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm area},\widetilde{\rm
sp}}$-extremal.

This follows from the $\rtimes$-stabilized Llarull’s theorem (see [17] and
references therein).

3.BR>0. A compact spin manifold $X$ with non-negative curvature operator,
$\mathcal{R}(X)\geq 0$, e.g., a compact symmetric space is
$\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm area},\widetilde{\rm
sp}}$-extremal, provided scalar curvature $\operatorname{Sc}(X)$ is
constant222222There are lots of metrics with $\mathcal{R}>0$ on spheres
$S^{n}$ and if $n\geq 3$ many of these have constant scalar curvatures. On the
other hand, it is possible that a closer look at the curvature term in the
twisted Schrödinger–Lichnerowicz formula (see Section 5) would allow one to
drop the constancy of the scalar curvature condition. and the Euler
characteristic of the universal covering $\tilde{X}$ does not vanish.

This follows by an elaboration on the proof of the Goette–Semmelmann
extremality theorem [12]. (We say a few words about it in Section 5.)

Probably, the corresponding rigidity arguments (see [29] and references
therein) also admit $\rtimes$-stabilization, but I did not check this
carefully.

Also the condition $\chi\bigl{(}\tilde{X}\bigr{)}\neq 0$ seems redundant and
$\rtimes^{\downarrow}_{{\rm area},\widetilde{\rm sp}}$-extremality can be
strengthened, also conjecturally, to the $\rtimes^{\downarrow}_{\rm
area}$-extremality.

3.B$\boldsymbol{{}_{\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}[a_{1},b_{i}]}}$.
The rectangular solids
$\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{n}[-a_{i},b_{i}]$
are $\rtimes^{\downarrow}_{\rm sp}$-extremal and, if $n\leq 8$, they are
${\rtimes^{\downarrow}}$-extremal.

In fact, $\rtimes^{\downarrow}_{\rm sp}$-extremality follows by a slight
generalization of the argument from [43], which, probably, can be adapted for
the proof of the $\rtimes^{\downarrow}_{\widetilde{\rm sp}}$-extremality.

As for $\rtimes^{\downarrow}$-extremality for $n\leq 8$, this follows from the
$\square^{\exists\exists}(n,m,N)$-inequality [18, §2.B].

Furthermore, the generic regularity theorem from [9] extended to $\mu$-bubbles
(I have not check this extension) yields the
$\square^{\exists\exists}(n,m,N)$-inequality and thus
$\rtimes^{\downarrow}$-extremality of solids for $n\leq 10$.

Moreover, granted a $\mu$-bubble generalization of Theorem 4.6 from [39], the
$\rtimes^{\downarrow}$-extremality (but not the
$\square^{\exists\exists}(n,m,N)$-inequality) would follow for all $n$.

3.B$\boldsymbol{{}_{\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}}}$.
Riemannian products of the manifolds from the above examples, e.g.,

$X=\bigl{(}\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{n-k}[-a_{i},b_{i}]\bigr{)}\times
S^{k},$

are $\rtimes^{\downarrow}_{\rm sp}$-extremal.

As above, this follows by a simple generalization of argument from [43]
combined with the basic (algebraic) inequality in [12] for twisted Dirac
operators on manifolds with $\mathcal{R}\geq 0$.

But the $\rtimes^{\downarrow}_{\rm sp}$-extremality remains problematic even
for $n\leq 8$.

For instance, if $k\leq 4$ and $n\leq 8$ (probably $n\leq 10$ will do), then
the $\square^{\exists\exists}(n,m,N)$-inequality combined with the warped
product splitting argument in [17, §5.5] yield $\rtimes^{\downarrow}_{\rm
sp}$-extremality of
$X=\bigl{(}\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{n-k}[-a_{i},b_{i}]\bigr{)}\times
S^{k}$.

Yet, there is no approach so far to non-spin extremality of the spheres
$S^{k}$ for $k\geq 5$.232323The warped product splitting argument (combined
with a stable version of [12]) applies to $S^{4}$, because 3-manifolds are
spin.

3.Bwarp. There are several classes of log-concave warped product manifolds,
e.g., $S^{n}$ minus a point, where the $\rtimes_{\rm sp}$-extremality (and
$\rtimes$-extremality for $n=4$) follow by $\rtimes$-stabilization of the
arguments in [17, §§5.5–5.7] and [6]. In fact, the $\rtimes$-extremality for
warped manifolds is more common then non-stabilized extremality.

For instance, geodesic balls in spheres and in $\mathbb{R}^{n}$ are not non-
stably extremal: one can increase their metrics without diminishing the scalar
curvatures. But, probably, they are $\rtimes_{\downarrow}$-extremal.

3.C. Questions.

  1. (i)

Which convex subsets in $\mathbb{R}^{n}$ are $\rtimes^{\downarrow}$-extremal?

  2. (ii)

Which surfaces are $\rtimes^{\downarrow}$-extremal?

3.D. About rigidity. The proofs of extremality of the manifolds $X$ in the
above examples can be upgraded to rigidity that says in the present case that
if a smooth distance non-increasing positive degree map $f\colon Y\to X$
satisfies
$\operatorname{Sc}_{\ast}^{\rtimes}(Y)\geq\operatorname{Sc}_{\ast}^{\rtimes\downarrow}(X)$
$($where
$\operatorname{Sc}_{\ast}^{\rtimes\downarrow}(X)=\operatorname{Sc}_{\ast}^{\rtimes}(X)$
by extremality$)$, then $f$ is homotopic to a local isometry, where one can
drop “homotopic to” if $X$ has no local scalar flat factors.

This follows by combining the $\rtimes$-stabilized rigidity arguments in [12]
and [29] with those in [17, §5.7] but to be honest, I did not check this in
full generality.

##  4 $\boldsymbol{\operatorname{Sc}^{\rtimes\downarrow}}$-product
inequalities, conjectures and problems

4.A. Additivity for cylinders. Since, obviously,

$\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{1}\times
X_{2}]\geq\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{1}]+\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{2}],$

then, for all Riemannian manifolds $X_{1}$ and $X_{2}$, the inequality

$\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{1}\times
X_{2}]\leq\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{1}]+\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{2}],$

is equivalent to the equality

$\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{1}\times
X_{2}]=\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{1}]+\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{2}].$

Thus, in particular, the $\square^{\exists\exists}(n,m,N)$-inequality from
[18] and/or equivariant separation theorem for stable $\mu$-bubbles242424See
[17, §5.4] and compare with [22, 37] and with [18, the proof of §2.B]. along
with the equality

$\operatorname{Sc}^{\rtimes}[a,b]=4\lambda_{1}[a,b]=\frac{4\pi^{2}}{(b-a)^{2}}.$

imply the following.

Proposition. The fundamental homology classes of oriented Riemannian
cylindrical manifolds $X=Y\times[a,b]$ of dimensions $\leq 8$ satisfy

$\operatorname{Sc}^{\rtimes_{\downarrow}}[X]=\operatorname{Sc}^{\rtimes_{\downarrow}}[Y]+\operatorname{Sc}^{\rtimes_{\downarrow}}[a,b].$

(This generalizes
$\operatorname{Sc}^{\rtimes_{\downarrow}}\left(\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{n}[a_{i},b_{i}]\right)=\sum_{i}\operatorname{Sc}^{\rtimes\downarrow}[a_{i},b_{i}]$,
that is
§3.B${}_{\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}[a_{1},b_{i}]}$
from the previous section.)

4.B. The spin case. This additivity formula remains problematic for $n\geq
9$,252525The dimensions $n=9,10$, probably, can be taken care by the argument
in [9]. but the spin cube inequality from [43] (proved with an index theorem
for deformed Dirac operators on manifolds with boundaries) implies, as we
stated earlier, that

$\operatorname{Sc}_{\rm
sp}^{\rtimes\downarrow}(\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{n}[-a_{i},b_{i}])=\operatorname{Sc}^{\rtimes}(\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{n}[-a_{i},b_{i}])=\sum_{1}^{n}\frac{4\pi^{2}}{(b_{i}-a_{i})^{2}}=\sum_{1}^{n}\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm
sp}[a_{i},b_{i}]$

for all $n$.

Yet, as far as I can see, the present day Dirac theoretic argument does not
yield the general $\operatorname{Sc}_{\rm sp}$-inequality

$\operatorname{Sc}_{\rm
sp}^{\rtimes_{\downarrow}}[Y\times[a,b]]\leq\operatorname{Sc}_{\rm
sp}^{\rtimes_{\downarrow}}[Y]+\operatorname{Sc}_{\rm
sp}^{\rtimes_{\downarrow}}[a,b].$

However, this argument does apply, if $Y$ is a special (extremal) manifold as
in §3.B××, e.g., a product of spheres.

4.C. [$\bf sect.curv\leq 0$]-Remark. Let $Y$ and $Z$ be compact Riemannian
manifolds, where $Z$ has no boundary and the sectional curvature ${\rm
sect.curv}(Y)\leq 0$. Then, similarly as above, one can prove additivity in
the following two cases:

  1. (i)

If $\dim(Y\times Z)=n\leq 8$,262626In view of [9], the inequality $n\leq 10$
may suffice. then

$\operatorname{Sc}^{\rtimes_{\downarrow}}[Y\times
Z]=\operatorname{Sc}^{\rtimes_{\downarrow}}[Y]=\operatorname{Sc}^{\rtimes_{\downarrow}}[Y]+\operatorname{Sc}^{\rtimes_{\downarrow}}[Z]=\operatorname{Sc}^{\rtimes_{\downarrow}}[Y].$

  2. (ii)

If $Y$ is as in §3.B××, then

$\operatorname{Sc}_{\rm sp}^{\rtimes_{\downarrow}}[Y\times
Z]=\operatorname{Sc}_{\rm sp}^{\rtimes_{\downarrow}}[Y]=\operatorname{Sc}_{\rm
sp}^{\rtimes_{\downarrow}}[Y]+\operatorname{Sc}_{\rm
sp}^{\rtimes_{\downarrow}}[Z]=\operatorname{Sc}^{\rtimes_{\downarrow}}[Y]$

for all $n$.

4.D. Riemannian additivity conjecture. Riemannian products of all oriented
Riemannian manifolds satisfy

$\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{1}\times
X_{2}]=\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{1}]+\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{2}].$

In fact, the following stronger inequality might be true.

4.E. Sup-metric product conjecture. Let $X_{i}$, $i=1,\dots,k$, be metric
spaces (e.g., closed oriented Riemannian manifolds) and let

$X=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{\sup_{i}}X_{i}=(X_{1}\times\cdots\times
X_{k},\operatorname{dist}_{\sup})$

be their product endowed with the sup-metric

$\operatorname{dist}((x_{1},\dots,x_{k}),(x^{\prime}_{1},\dots,x^{\prime}_{k}))=\max_{i=1,\dots,k}\operatorname{dist}(x_{i},x^{\prime}_{i}).$

Then rational homology classes $h_{i}\in H_{m_{i}}(X_{i};\mathbb{Q})$ (e.g.,
the rational fundamental classes $[X_{i}]$272727“Rational” in the case of
compact locally contractible spaces means “a non-zero integer multiple of”,
that is,
$\operatorname{Sc}^{\rtimes_{\downarrow}}(h_{\mathbb{Q}})\overset{\rm
def}{=}\sup_{N\neq 0}\operatorname{Sc}^{\rtimes_{\downarrow}}(N\cdot
h_{\mathbb{Q}}).$ ) satisfy

$\operatorname{Sc}^{\rtimes_{\downarrow}}(\otimes_{i}h_{i})\leq\sum_{i=1,\dots,k}\operatorname{Sc}^{\rtimes_{\downarrow}}(h_{i}),\qquad\text{e.g.,}\qquad\operatorname{Sc}^{\rtimes_{\downarrow}}[\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{\sup_{i}}X_{i}]_{\mathbb{Q}}\leq\sum_{i=1,\dots,k}\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{i}]_{\mathbb{Q}},$
(4.1)

where the opposite inequality

$\operatorname{Sc}^{\rtimes_{\downarrow}}(\otimes_{i}h_{i})\geq\sum_{i=1,\dots,k}\operatorname{Sc}^{\rtimes_{\downarrow}}(h_{i})$

follows from additivity of the scalar curvature; hence, (4.1) implies the
equality

$\operatorname{Sc}^{\rtimes_{\downarrow}}(\otimes_{i}h_{i})=\sum_{i=1,\dots,k}\operatorname{Sc}^{\rtimes_{\downarrow}}(h_{i}).$

4.F.
$\boldsymbol{\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{\sup_{i}}[a_{i},b_{i}]}$-Example.
The above indicated proofs of §4.A and §4.B actually show that the rectangular
solids
$\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{n}[a_{i},b_{i}]$
with the Riemannian product and the sup-product metrics have the same
$\operatorname{Sc}^{\rtimes_{\downarrow}}$ for $n\leq 8$ and have the same
$\operatorname{Sc}^{\rtimes_{\downarrow}}_{\rm sp}$ for all $n$. This confirms
the validity of (4.1) for rectangular solids.

4.F′.
$\boldsymbol{\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}[0,d_{i}]}$-Sub-
Example. Let $Y\subset\mathbb{R}^{n}$ be a diffeomorphic image of the $n$-cube
and let $d_{i}$, $i=1,\dots,n$, be the distances between the images in $Y$ of
the pairs of the opposite $(n-1)$-faces of the cube. Then the first Dirichlet
eigenvalue of the Laplacian $-\Delta_{Y}$ is bounded by that of the solid
$\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{i}[0,d_{i}]$,

$\lambda_{1}(-\Delta_{Y})\leq\sum_{i}\frac{\pi^{2}}{d^{2}_{i}}.$

Exercise. Find a direct elementary proof of this inequality.282828To my shame,
I could not solve it.

Sup-distance, sup-area and $\boldsymbol{\operatorname{Sc}_{\rm
sup.area}^{\rtimes_{\downarrow}}}$. The Riemannian product metric, that is the
Pythagorean one

$\operatorname{dist}((x_{1},\dots,x_{k}),(x^{\prime}_{1},\dots,x^{\prime}_{k}))=\sqrt{\sum_{i}\operatorname{dist}(x_{i},x^{\prime}_{i})^{2}},$

is greater than the sup-metric but only by a factor $\sqrt{k}$,

$1\leq{\operatorname{dist}((x_{1},\dots,x_{k}),(x^{\prime}_{1},\dots,x^{\prime}_{k}))\over\operatorname{dist}_{\sup}((x_{1},\dots,x_{k}),(x^{\prime}_{1},\dots,x^{\prime}_{k}))}\leq\sqrt{k}.$

The situation is somewhat different with areas. Namely, let
$X=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{i}X_{i}$
be the product of Riemannian manifolds and let
$\sup_{i}$-$\operatorname{area}(\Sigma)$ for a smooth surface $\Sigma\subset
X$ be the maximum of the areas of the projections $\Sigma\to X_{i}$. Here
again

${\sup_{i}}\mbox{-}{\operatorname{area}}(\Sigma)\leq\operatorname{area}(\Sigma)$

but now, unlike to how it is with the distances, the ratio

$\frac{\operatorname{area}(\Sigma)}{{\sup_{i}}\mbox{-}{\operatorname{area}}(\Sigma)}$

may be infinite. Accordingly, the corresponding $\operatorname{Sc}_{{\rm
sup.area}}^{\rtimes_{\downarrow}}(h)$, $h\in
H_{m}(\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{i}X_{i})$,
defined with smooth maps $f\colon
Y^{m}\to\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{i}X_{i}$,
$f_{\ast}[Y]=h$ such that the corresponding $f_{i}\colon Y^{m}\to X_{i}$ are
area decreasing, can be significantly greater than $\operatorname{Sc}_{{\rm
sup.area}}^{\rtimes_{\downarrow}}(h)$, where the maps $f$ must be area
decreasing themselves.

Thus, the area version of (4.1),

$\operatorname{Sc}_{\rm
sup.area}^{\rtimes_{\downarrow}}(\otimes_{i}h_{i})\leq\sum_{i=1,\dots,k}\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}(h_{i}),$ (4.2)

e.g.,

$\operatorname{Sc}_{\rm
sup.area}^{\rtimes_{\downarrow}}[\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{\sup_{i}}X_{i}]_{\mathbb{Q}}\leq\sum_{i=1,\dots,k}\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}[X_{i}]_{\mathbb{Q}},$

is qualitatively stronger than corresponding inequality for
$\operatorname{Sc}_{\rm area}^{\rtimes_{\downarrow}}(\otimes_{i}h_{i}).$

Although we have no known means for bounding $\operatorname{Sc}_{\rm
area}^{\rtimes_{\downarrow}}$ and even less for $\operatorname{Sc}_{{\rm
sup.area}}^{\rtimes_{\downarrow}}$ in most cases, we shall do this in the next
section for $\operatorname{Sc}_{{\rm sup.area},\widetilde{\rm
sp}}^{\rtimes_{\downarrow}}$ and thus prove the $\widetilde{\rm sp}$-version
of (4.2) in some cases.

4.G. Semiadditivity problem. Let $X=X^{n}$ and $Z=Z^{k}$, $k\leq n-2$, be
compact Riemannian manifolds, possibly with boundaries, and let $f\colon X\to
Z$ be a smooth distance decreasing map such that $\partial
X\overset{f}{\to}\partial Z$, and let $h_{m}=\bigl{[}f^{-1}(z)\bigr{]}\in
H_{m}(X)$, $m=n-k$, be the homology class of the pullback of a generic $z\in
Z$.

Identify the cases, where

$\operatorname{Sc}^{\rtimes\downarrow}(h_{m})_{\mathbb{Q}}\geq\operatorname{Sc}^{\rtimes\downarrow}[X]_{\mathbb{Q}}-\operatorname{Sc}^{\rtimes\downarrow}[Y]_{\mathbb{Q}},$

at least for “simple” manifolds $Z$, e.g., compact convex domains in
$\mathbb{R}^{k}$ and in $S^{k}$ and, in general, evaluate the difference

$\operatorname{Sc}^{\rtimes\downarrow}[X]_{\mathbb{Q}}-\operatorname{Sc}^{\rtimes\downarrow}[Y]_{\mathbb{Q}}-\operatorname{Sc}^{\rtimes\downarrow}(h_{m})_{\mathbb{Q}}$

in terms of the geometry of $Z$, for instance, where $Z$ is the product of
balls
$Z=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{i}B^{k_{i}}(R_{i})$
or product of spheres $S^{k_{i}}(R_{i})$.

If $n=m+k\leq 8$, a satisfactory lower bound on
$\operatorname{Sc}^{\rtimes\downarrow}(h_{m})$ for rectangular solids $Z$
follows from §4.A. Also [18, §2.B] yields similar bounds for products of
2-discs and 2-spheres (compare [22]). But it is unclear, for instance, how
large the difference
$\operatorname{Sc}^{\rtimes\downarrow}[X]_{\mathbb{Q}}-\operatorname{Sc}^{\rtimes\downarrow}[Y]_{\mathbb{Q}}-\operatorname{Sc}^{\rtimes\downarrow}(h_{m})_{\mathbb{Q}}$
can be for the balls $B^{k}\subset\mathbb{R}^{k}$ and spheres $S^{k}$ for
large $k$.

##  5 Additivity of the twisted SLWB-formula and applications

Let $Y$ be a Riemannian spin $n$-manifold and $V\to X$ be a complex vector
bundle with a unitary connection $\nabla$ and let $\mathcal{D}_{\otimes V}$
denote the Dirac operator on spinors $\mathbb{S}$ on $Y$ tensored with $V$.
Then the square of $\mathcal{D}_{\otimes V}$ satisfies the following
Schrödinger–Lichnerowicz–Weitzenböck–Bochner formula (see [27])

${\cal D}_{\otimes V}^{2}=\nabla^{2}_{\otimes
V}+\frac{1}{4}\operatorname{Sc}(Y)+{\cal K}_{\otimes V},$

where $\mathcal{K}_{\otimes V}$ is an endomorphism $\mathbb{S}\otimes
V\to\mathbb{S}\otimes V$ such that

$\mathcal{K}_{\otimes V}(s\otimes v)=\frac{1}{2}\sum_{i,j}(e_{i}\circ
e_{j}\circ s)\otimes R^{V}_{e_{i}\wedge e_{j}}(v),$

where $e_{i}\in T_{y}(Y)$, $i=1,\dots,m=\dim(Y)$, are orthonormal tangent
vectors at $y\in Y$, where $\circ$ is the Clifford multiplication and
$R^{V}_{e_{i}\wedge e_{j}}\colon V\to V$ is the curvature operator of
$\nabla$.

Next, recall that the curvature of the tensor product of two bundles with
connections satisfies

$R^{V_{1}\otimes V_{2}}=1^{V_{1}}\otimes R^{V_{2}}+R^{V_{1}}\otimes
1^{V_{2}},$

where $1^{V}\colon L\to V$ is the identity operator, and observe that the
operators on $\mathbb{S}\otimes V_{1}\otimes V_{2}$ defined by

$s\otimes v_{1}\otimes v_{2}\mapsto\frac{1}{2}\sum_{i,j}(e_{i}\circ e_{j}\circ
s)\otimes R^{V_{1}}_{e_{i}\wedge e_{j}}(v_{1})\otimes v_{2}$

and by

$s\otimes v_{1}\otimes v_{2}\mapsto\frac{1}{2}\sum_{i,j}(e_{i}\circ e_{j}\circ
s)\otimes v_{1}\otimes R^{V_{2}}_{e_{i}\wedge e_{j}}(v_{2})$

have the same spectra up to multiplicity as

$s\otimes v_{1}\mapsto\frac{1}{2}\sum_{i,j}(e_{i}\circ e_{j}\circ s)\otimes
R^{V_{1}}_{e_{i}\wedge e_{j}}(v_{1})$

and

$s\otimes v_{2}\mapsto\frac{1}{2}\sum_{i,j}(e_{i}\circ e_{j}\circ s)\otimes
R^{V_{2}}_{e_{i}\wedge e_{j}}(v_{2})$

correspondingly. Therefore, the lowest eigenvalue $\lambda_{\otimes 1\otimes
2}$ (often negative) of the (self-adjoint) operator
$\mathcal{R}_{\otimes(V_{1}\otimes V_{2})}$ is bounded from below by the sum
of these for $\mathcal{R}_{\otimes V_{1}}$ and $\mathcal{R}_{\otimes
V_{2}}$,292929A referee pointed out to me that $\lambda_{\otimes 1\otimes
2}=\lambda_{\otimes 1}+\lambda_{\otimes 2}$.

$\lambda_{\otimes 1\otimes 2}\geq\lambda_{\otimes 1}+\lambda_{\otimes 2}.$

This yields the following.

5.A. Theorem.303030This is a refinement of the
Llarull–Goette–Semmelmann–Listing rigidity theorem. Let
$X=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{k}X_{k}$,
$k=1,\dots,l$, be an orientable Riemannian $n$-manifold split into Riemannian
product, where the factors $X_{k}=(X_{k},\underline{g}_{k})$ are either

  1. (a)

compact $n_{k}$-manifolds with non-negative curvature operators,
$\mathcal{R}^{X_{k}}\geq 0$ (e.g., closed convex hypersurfaces in
$\mathbb{R}^{n_{k}+1}$) and with non-vanishing Euler characteristics
$\chi(X_{k})\neq 0$ (hence of even dimensions $n_{k}$), or

  2. (b)

spheres $S^{n_{k}}$ with constant sectional curvatures $($possibly of odd
dimension $n_{k})$.

Let
$\underline{g}^{\natural}_{k}=\operatorname{Sc}(\underline{g}_{k})\cdot\underline{g}_{k}$.
Let $Y=(Y,g)$ be a smooth complete orientable Riemannian $(n+N)$-manifold with
$\operatorname{Sc}(g)>0$, and let $g^{\natural}=\operatorname{Sc}(g)\cdot g$.
Let $Z$ be an orientable enlargeable $N$-manifold, e.g., $Z=\mathbb{R}^{N}$,
and let $f\colon Y\to X\times Z$ be a smooth proper (quasi-proper will do) map
such that the corresponding maps $f_{k}\colon Y\to X_{k}$ are strictly sum-
wise area decreasing with respect to $g^{\natural}$ in $Y$ and
$\underline{g}^{\natural}_{k}$ in $X_{k}$.313131The quadratic forms
$\underline{g}^{\natural}_{k}$ on $X_{k}$ may vanish but “area” makes sense
anyway.

This means that the norms of the exterior squares of the differentials of
$f_{k}$ with respect the $\natural$-metrics satisfies

$\sum_{k}||\wedge^{2}{\rm d}f_{k}||<1.$ (5.1)

(Notice that $\sum_{k}||\wedge^{2}{\rm d}f_{k}||=1$ if $X=Y$ and $f$ is the
identity map.)

If either the map $f$ is spin or the universal covering of $Y$ is spin, then
the topological degree of $f$ is zero.

Proof. First, let $X$ and $Y$ be spin, let $f\colon Y\to X$ be a smooth map
and let $V\to Y$ be the $f$-pullback of the spin bundle $\mathcal{S}(X)\to X$
to $Y$. Then, if $\mathcal{R}^{X}\geq 0$ and $f\colon Y\to X$ is
$\natural$-area decreasing at point $y\in Y$, i.e., $\|{\wedge}^{2}{\rm
d}f(y)\|\leq 1$, then according to [12] (also see [29]) the lowest eigenvalue
of the operator $\mathcal{R}_{\otimes V}$ at $y\in Y$ satisfies

$\lambda_{\otimes V}\geq-{\operatorname{Sc}(Y,y)\over 4},$

where this inequality is strict if $f$ is strictly $\natural$-area decreasing
at $y$.

Next, let $X_{k}$ be spin, let
$X=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{k}X_{k}$
and let $V\to Y$ be the tensor product $V=\bigotimes_{k}V_{k}$ of the
pullbacks $V_{k}=f^{\ast}_{k}(\mathbb{S}_{k})\to Y$ of
$f^{\ast}_{k}(\mathbb{S}_{k})=\mathbb{S}(X_{k})\to X_{k}$ to $Y$ for
$f_{k}\colon Y\to X_{k}$.

Then, if the maps $f_{k}$ are $\natural$-area decreasing and at least one of
$f_{k}$ is strictly $\natural$-area decreasing at a point $y\in Y$ then,
assuming $Y$ is connected and spin, the Dirac operator $\mathcal{D}_{\otimes
V}$ on $Y$ has index zero.

On the other hand, if $\chi(X_{k})\neq 0$, if $\dim(Y)=\dim(X)$ and
$\deg(f)\neq 0$,323232If $\dim(Y)=\dim(X)+4m$, then instead of $\deg(f)\neq 0$
one assumes that the $f$-pullback of a generic point $x\in X$ has non-zero hat
A-genus. then $\operatorname{ind}(\mathcal{D}_{\otimes V})\neq 0$ by the
Atiyah–Singer theorem (compare with [12, 29, 30]).

This proves §5.A in the case where the manifold $X$ is spin and it contains
neither a $Z$-factor, nor an odd spherical factor.

To pass to the general case we argue as follows:

1\. Odd dimensional spheres are suspended to even dimensional ones
$S^{n_{k}}\leadsto S^{n_{k}+1}$, where these suspensions are accompanied by
multiplying $Y$ by a long circles and a suspending $[f_{k}\colon Y\to
S^{n_{k}}\leadsto[Y\times S^{1}\to S^{n_{k}+1}]$ as in [30], also see [17,
§3.4.1] and [18].

2\. If a $Z$, which may be assumed even-dimensional, is enlargeable, it
supports an almost flat bundle, say $W\to Z$ with non-zero top-dimensional
Chern class and the above $V\to Y$ is tensored by the pullback
$f_{Z}^{\ast}(W)\to Y$ of $W$ to $Y$.

3\. If neither $X$ nor $Y$ are spin but the map $f$ is spin, then the Dirac
operator $\mathcal{D}_{\otimes V}$ is defined (this is explained in the
present context in [34] and in [12]) and the above applies.

5.B. Spherical trace and symplectic remarks. The $||\wedge^{2}{\rm d}f_{k}||$
contribution of each spherical factor $X_{k}$ with constant sectional
curvature can be replaced in the formula (5.1) by an a priori smaller entity,
that is, $2||\wedge^{2}{\rm d}f||_{\rm trace}\over n_{k}(n_{k}-1)$, where

$||\wedge^{2}{\rm d}f_{k}(y)||_{\rm trace}=\sum_{1\leq i<j\leq
n+N}\lambda_{i,k}(y)\lambda_{j,k}(y),$

and where the numbers $\lambda_{j,k}(y)\geq 0$ are defined by diagonalizing
the differential ${\rm d}f_{k}\colon T_{y}(Y)\to T_{f_{k}}(X_{k})$ with an
orthonormal frame $e_{i,k}\in T(y)(Y)$, which is sent by ${\rm d}f_{k}$ to an
orthogonal frame in $T_{f_{k}}(X_{k})$ with the vectors of lengths
$\lambda_{j,k}(y)$.

In fact, this follows from [30, formula (4.6)] (also [29] and [17, §3.4]).

The $S^{2}$ factors in $X$ contribute to complex line bundles as
$\otimes$-factors in $V\to Y$. This, in view of Schrödinger–Hitchin (see [25])
formula for $\mathcal{D}_{\otimes L}$ allows one to replace the product of
these $S^{2}$ by a single (quasi)symplectic manifold (compare with [17, §2.7
and §3.4.4(4)]).

5.C. $\boldsymbol{\operatorname{Sc}_{{\rm area},\widetilde{\rm
sp}}^{\rtimes_{\downarrow}}}$-additivity corollary. Let $X_{k}$ be manifolds
as in $5.A$, where we additionally assume that they are spin and have constant
scalar curvatures. Then the fundamental classes $[X_{k}]$ satisfy the
$\widetilde{\rm sp}$-version of the $\operatorname{Sc}_{{\rm
sup.area}}^{\rtimes_{\downarrow}}$-additivity (4.2) in §4.F:

$\operatorname{Sc}_{{\rm sup.area},\widetilde{\rm
sp}}^{\rtimes_{\downarrow}}(\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{k}X_{k})=\sum_{k}\operatorname{Sc}_{\widetilde{\rm
sp}}^{\rtimes_{\downarrow}}(X_{k})=\sum_{k}\operatorname{Sc}^{\rtimes}(X_{k})=\sum_{k}\operatorname{Sc}(X_{k}).$

Consequently,

$\operatorname{Sc}_{\widetilde{\rm
sp}}^{\rtimes_{\downarrow}}(\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{\sup_{k}}X_{k})=\sum_{k}\operatorname{Sc}(X_{k}).$

5.D. Questions. (i) Does vanishing of
$\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{k}]_{\mathbb{Q}}$ for closed
manifolds $X_{k}$ (this is a homotopy invariant of $X$) implies vanishing of
$\operatorname{Sc}^{\rtimes_{\downarrow}}[\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{k}X_{k}]_{\mathbb{Q}}$?

There are examples of manifolds $X_{k}$, where
$\operatorname{Sc}^{\rtimes_{\downarrow}}[X_{i}]=0$ and where their products
admit metrics with $\operatorname{Sc}>0$; hence,
$\operatorname{Sc}^{\rtimes_{\downarrow}}[\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{i}X_{i}]>0$
for these $X_{i}$, see [19].

(ii) Do products of spheres $X=S^{n_{1}}\times S^{n_{2}}$, $n_{1},n_{2}\geq
2$, admit Riemannian metrics $g_{\varepsilon}$, for all $\varepsilon>0$, with
$\operatorname{Sc}(g_{\varepsilon})\geq 1$ and such that all non-zero homology
classes $h$ in $H_{n_{1}}(X)$ and in $H_{n_{2}}(X)$ satisfy
$\operatorname{Sc}_{\operatorname{area}_{g_{\varepsilon}}}^{\rtimes_{\downarrow}}(h)\leq\varepsilon$?

The existence of such a $g_{\varepsilon}$, for $n_{1}=n_{2}=2$, would imply
the absence of the lower bounds on the 2-systoles of manifolds $(X,g)$ in
terms of $\sigma(g)=\inf_{x\in X}\operatorname{Sc}(X,g,x)>0$,333333Such a
counter example would undermine (but not disprove) the conjectural bound ${\rm
waist}_{2}(X)\leq\frac{{\rm const}_{n}}{\sigma}$ for compact Riemannian
$n$-manifolds with $\operatorname{Sc}(X)\geq\sigma>0$. Thus, it may be safer
to assume $n_{1},n_{2}\geq 3$.

$\sup_{\sigma(g)\geq 1}{\rm syst}_{2}(X,g)=\infty.$

Recall, that the 2-systole is the infimum of the areas of all non-zero classes
$h\in H_{2}(X)$, for $\operatorname{area}(h)=\inf_{[c]\in
h}\operatorname{area}(c)$ for the 2-cycles $c\subset Y$ that represent
$h$.343434There are bounds on the 2-systoles of manifolds $X$ with
$\operatorname{Sc}^{\rtimes}(X)\geq\sigma$ in terms of their
$\tilde{\square}^{\perp}$-spreads (see [37, 44]) which are proved as
$\square^{\exists\exists}(n,m)$-inequality in §2.B with a use of minimal
hypersurfaces and $\mu$-bubbles. Also there are similar bounds on the stable
systoles of spin manifolds obtained with Dirac operators twisted with line
bundles, where, recall, ${\rm
st.syst}_{2}(X)=\liminf_{N\to\infty}\frac{\operatorname{area}(Nh)}{N}$.

(iii) Let $X$ be a compact symmetric space. What is the minimal seminorm on
the linear maps $\lambda^{2}{\rm
d}\colon\mathbb{\wedge}^{2}R^{n}\to\wedge^{2}T(X)$, say $||\lambda^{2}{\rm
d}||_{\min}$ such that the $\natural$-normalized inequality $||\wedge^{2}{\rm
d}f||_{\min}<1$ for smooth equidimensional spin maps $f\colon Y\to X$ would
imply that $\deg(f)=0$?353535This norm must be invariant under isometries of
$X$.

(If $X$ is the products of spheres, this seminorm is equal to the sum of the
mean trace norms (as in §5.B) for maps $\mathbb{R}^{n}\overset{{\rm
d}_{k}}{\to}X_{k}=S^{n_{k}}$ and for all symmetric spaces $X$ of dimension
$\geq 4$ with $\chi(X)\neq 0$ this norm is, probably, strictly smaller then
the sup-norm $||\wedge^{2}{\rm d}||$ from the Goette–Semmelmann theorem.)

##  6 $\boldsymbol{P}$-families of maps to product of spheres

Let $Y=(Y,g)$ be an $n$-dimensional Riemannian manifold with
$\operatorname{Sc}(X)>0$, where as earlier
$g^{\natural}=\operatorname{Sc}(Y)\cdot g$, let $h_{m}\in H_{m}(Y)$ be a
homology class and let $P$ be a locally contractible topological space, e.g.,
a manifold and $h_{K}\in H_{K}(P)$ be a homology class.

Let $X$ be a product of spheres of variable radii,

$X=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{k}S^{n_{k}}(R_{k}),$

where $\dim(X)=\sum_{k}n_{k}=m+K$, and where the spheres are endowed with the
usual metrics with sectional curvatures $1/R_{k}^{2}$.

Let $F\colon Y\times P\to X$ be a continuous map such that the maps
$F_{p}=F_{|Y\times p}\colon Y\to X$ are smooth and $C^{1}$-continuous in $p\in
P$.

Let the universal covering of $Y$ be spin and let $h_{m}$ be equal to the
homology class of the pullback of a genetic point under a smooth map
$\phi\colon Y\to Z$, where $Z$ is a smooth enlargeable manifold of dimension
$\dim(Y)-m$. For instance, $m=n$ and $h_{m}=[Y]$ or
$Y=Y_{0}^{m}\times\mathbb{T}^{n-m}$ and $h_{m}=[Y_{0}]$.

6.A. Theorem. Let the norms of the exterior squares of the differentials of
the maps $f_{k}\colon Y\to S^{n_{k}}(R_{k})$ with respect to the
$\natural$-metrics in $Y$ and in $S^{n_{k}}(R_{k})$ satisfy

$\sum_{k}||\wedge^{2}{\rm d}f_{k}||<1.$

Then, in the following two cases, the $F$-image $F_{\ast}(h_{m}\otimes
h_{K})\in H_{m-K}(X)=\mathbb{Z}$ vanishes:

  1. (1)

The ranks of the (differentials of the) maps $f_{p}\colon Y\to X$ are
everywhere $\leq m$, e.g., $\dim(Y)=m$ and $h_{m}=[Y]$.

  2. (2)

The dimension of $Y$ is bounded by $n\leq 8$.

Proof. Case 1. If $\operatorname{ranks}(f_{p})\leq m$, then the Llarull
(Listing) trace inequality (4.6) in [30] together withe the above
$\lambda_{\otimes V}$-additivity show that index of the family of the Dirac
operators on $Y$, twisted with the pullbacks of $\bigotimes_{k}\mathbb{S}_{k}$
as in §5.A, vanishes and the Atiyah–Singer theorem for families shows that
$F_{\ast}(h_{m}\otimes h_{K})=0$. (See in [17, §4] and references therein.)

Case 2. If $\dim(Y)\leq 8$, then, at last generically, the homology class
$h_{m}$ can be realized by an $m$-submanifold $Y_{0}\subset Y$ such that the
product $Y_{0}\times\mathbb{T}^{\dim(Y)-m}$ admits a warped product metric
$g^{\rtimes}$ such that
$\operatorname{Sc}(g^{\rtimes},y)\geq\operatorname{Sc}(Y,y)$ for all $y\in
Y_{0}$ (see [18, §3] and references therein). Now the case 1 applies to
$Y_{0}\times\mathbb{T}^{\dim(Y)-m}$ and the proof follows.

Remarks/Problems. (a) For all we know, the spin and $\dim(Y)\leq 8$ condition
are redundant and there is a fair chance that a further study of singularities
of minimal hypersurfaces in he spirit of [39] and/or [31, 32] will allow one
to remove the latter. But removing the spin condition needs a new idea.

The argument in Case 1 can be extended to maps of foliated manifolds to
$\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{k}S^{n_{k}}$
as in [42], but a foliated version of Case 2 is problematic.

##  7 Spinorial curvature

Given a closed orientable even dimensional Riemannian manifold $Y$ let
$\mathbb{S}{\sf p}.\mathrm{curv}(Y)$ be the infimum of the numbers $\kappa\geq
0$ such that there exist a complex vector bundle $V\to X$ with a unitary
connection such that

$\bigl{(}\mathrm{Ch}(V)\smile\hat{A}\bigr{)}[Y]\neq 0$

and the lowest eigenvalue of the operators $\mathcal{K}_{\otimes V}$ on
$(\mathcal{S}\otimes V)_{y}$ (see Section 5) satisfies

$\lambda_{\otimes V}\geq-\kappa$

at all points $y\in Y$.363636Although the spin bundle $\mathcal{S}\to Y$ is
defined only for spin manifolds, this definition, being local, makes sense for
all $Y$, since $\lambda_{\otimes V}$ does not depend on the spin structure.

Observe that

$\mathbb{S}\mbox{\sf p}.{\rm curv}(Y_{1}\times Y_{2})\leq\mathbb{S}\mbox{\sf
p}.{\rm curv}(Y_{1})+\mathbb{S}\mbox{\sf p}.{\rm curv}(Y_{2})$

by the inequality $\lambda_{\otimes 1\otimes 2}\geq\lambda_{\otimes
1}+\lambda_{\otimes 2}$ from Section 5, that

$\mathbb{S}\mbox{\sf p}.{\rm curv}(Y)=0$

for enlargeable manifolds $Y$ and that if the universal coverings of $Y$ is
spin, then

$\operatorname{Sc}^{\rtimes}(Y)\leq 4\kappa$

by the ($\mathbb{T}^{\rtimes}$-stabilized) index theorem, SLWB-formula and
Kato’s inequality.373737Instead of the $\mathbb{T}^{\rtimes}$-stabilization
and Kato’s inequality one may use Kazdan–Warner conformal change theorem [26]
and conformal invariance of harmonic spinors [25].

Remarks on $\lambda_{1}(X,\beta)$ for $\beta<1/4$. (a) The refined Kato
inequality strengthens the above to

$\lambda_{1}(X,\beta)\leq 4\kappa\qquad\text{for}\quad\beta=(n-1)/4n,$

where $\lambda_{1}(X,\beta)$ is the lowest eigenvalue of
$-\Delta+\beta\operatorname{Sc}(X)$ (see §1.D).

(b) The Kazdan–Warner conformal change theorem [26] and conformal invariance
of harmonic spinors [25] show that if $\lambda_{1}(X,\beta)>0$ for
$\beta=(n-2)(4(n-2)$, then $X$ supports no non-zero harmonic spinors.

However, it is unclear how to extract further geometric, rather than
topological information from the inequality $\lambda_{1}(X,\beta)>\sigma$ for
$\beta<(n-1)/4n$ and $\sigma>0$.

Next, let $X$ be a Riemannian manifold, let $h_{m}\in H_{m}(X)$ be a homology
class and let $\mathcal{Y}$ be a class of smooth closed orientable
$m$-manifolds $Y$ along with maps $f\colon Y\to X$.

Define $\mathbb{S}$p$.{\rm curv}^{\downarrow}_{\mathcal{Y}}(X)$ via smooth
maps $F\colon Y\times\mathbb{T}^{N}\to X\times\mathbb{T}^{N}$ and Riemannian
metrics $G$ on $Y\times\mathbb{T}^{N}$ as the infimum

$\mathbb{S}\mbox{\sf p}.{\rm curv}^{\downarrow}_{\mathcal{Y}}(X)\mbox{\sf
p}\inf_{Y,N,G,F}\mathbb{S}\mbox{\sf p}.{\rm
curv}\bigl{(}Y\times\mathbb{T}^{N},G\bigr{)},$

where the infimum is taken over $N$ such that $m+N$ is even, where $F$ is area
decreasing with respect to the metric $G$ and where

$F_{\ast}\bigl{[}Y\times\mathbb{T}^{N}\bigr{]}=h_{m}\otimes\bigl{[}\mathbb{T}^{N}\bigr{]}\in
H_{m+N}\bigl{(}X\times\mathbb{T}^{N}\bigr{)}\qquad\text{and}\qquad(Y,F_{|Y\times
0})\in\mathcal{Y}.$

Clearly, by the above, if the universal coverings of manifolds $Y\in\cal Y$
are spin,383838This condition is necessary but its $\mathbb{Q}$-version may be
redundant. then

$\mathbb{S}\mbox{\sf p}.{\rm
curv}^{\downarrow}_{\mathcal{Y}}(h_{m})\geq\frac{1}{4}\bigl{(}\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
area},\mathcal{Y}}(h_{m})\bigr{)}.$

Remark. If the universal coverings of the manifolds $Y\in\cal Y$ are spin,
then the fundamental classes $[X]$ of compact symmetric spaces $X$ with
$\chi(X)\neq 0$, satisfy the equally

$\mathbb{S}\mbox{\sf p}.{\rm
curv}^{\downarrow}_{\mathcal{Y}}(X)=\frac{1}{4}\bigl{(}\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
area},\mathcal{Y}}(h)\bigr{)}$

by the $\mathbb{T}^{\rtimes}$-stabilized Goette–Semmelmann theorem and this
equally applies to products
$X=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}X_{i}$,
where $X_{i}$ are as in §5.A.

Possibly, (a version of) this equality holds true for all symmetric spaces but
it seems unlikely in general, even for rational homology classes $h$, that the
Dirac operator is the only source of bounds on
$\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm area},\mathcal{Y}}(h).$

Power stabilization. Let

$\displaystyle X^{M}=\underset{M}{\underbrace{X\times X\times\cdots\times
X}},$ $\displaystyle\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
area},\mathcal{Y}}\bigl{(}\otimes^{[\infty/\infty]}h_{m}\bigr{)}=\sup_{M=1,2,\dots}\frac{1}{M}\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
area},\mathcal{Y}}(h_{Mm}),\qquad h_{Mm}=\otimes^{M}h_{m}\in
H_{Mm}\bigl{(}X^{M}\bigr{)}$

and

$\displaystyle\mathbb{S}\mbox{\sf c}.{\rm curv}^{\rtimes_{\downarrow}}_{{\rm
area},\mathcal{Y}}\bigl{(}\otimes^{[\infty/\infty]}h_{m}\bigr{)}=\inf_{M=1,2,\dots}\frac{1}{M}\mathbb{S}\mbox{\sf
c}.{\rm curv}^{\rtimes_{\downarrow}}_{{\rm area},\mathcal{Y}}(h_{Mm}),$
$\displaystyle h_{Mm}=\otimes^{M}h_{m}\in H_{Mm}\bigl{(}X^{M}\bigr{)}.$

Questions. I. What are further instances (besides the above $h=[X]$) of the
equality

$\mathbb{S}\mbox{\sf p}.{\rm
curv}^{\downarrow}_{\mathcal{Y}}(\otimes^{[\infty/\infty]}h_{m})=\frac{1}{4}\bigl{(}\operatorname{Sc}^{\rtimes_{\downarrow}}_{{\rm
area},\mathcal{Y}}\bigl{(}\otimes^{[\infty/\infty]}h_{m}\bigr{)}\bigr{)},$

and what are examples where this fails to be true?

II. Can one pass to the limit, set $M=\infty$ and prove scalar curvature
bounds for “Riemannian metrics” $G$ on infinite dimensional manifolds $X$,
e.g., where such a G differs from the infinite sum of Riemannian metrics,
$\sum_{1}^{\infty}g_{i}$, on
$X=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{\infty}(X_{i},g_{i})$
(and/or on $Y$ mapped to $X$) by a “fast decaying in $i$” error term $\Delta$?

Remarks. (a) If $\Delta=\Delta_{i,j}$ decays very fast, for $i$ and/or $j$
tending to infinity, then finite products
$X_{M}=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{M}X_{i}$
embed to
$X=\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{1.5}{$\SavedStyle\times$}}}}}_{1}^{\infty}(X_{i},g_{i})$
with small relative curvatures and a bound on “$\operatorname{Sc}(X)$” may be
derived in some cases from such a bound on $X_{M}$, but it would be more
interesting to develop a truly infinite dimensional argument for bounds on
“$\operatorname{Sc}(X)$” and/or to find applications of such bounds.

Test question. Let $X=\\{x_{i}\\}_{\sum_{i}x_{i}^{2}\leq\infty}$ be the
Hilbert space and $G=G_{ij}$ be a smooth Riemannian metric on $X$, which is
greater than the background Hilbertian metric,

$G(\tau,\tau)\geq||\tau||^{2}$

for all tangent vectors $\tau\in T(X)$ and let $M=3,4,\dots$ be an integer.

Can the $M$-scalar curvature of $G$ (defined below) be strictly positive, say
$\operatorname{Sc}_{M}(G)\geq 1$? Here $\operatorname{Sc}_{M}$ is the function
on the tangent $M$-planes $P^{M}\in T_{x}(X)$, $x\in X$, which is equal to the
scalar curvature at zero in $P^{M}$ $\bigl{(}{=}\,\mathbb{R}^{M}\bigr{)}$ of
the Riemannian metric induced by the exponential map $\exp\colon P^{M}\to X$
from $G$. (It may be worthwhile to compare $\operatorname{Sc}_{M}$ with with
the $m$-intermediate curvature from [3].)

(b) A natural approach to these problems is by a finite-dimensional
approximation as in (a) but this seems that uncomfortably restrictive
conditions on $G$ are needed (compare with [14]).

(c) Basic features of positive scalar curvature have their counterparts for
mean convex hypersurfaces (see [16]), where the infinite dimensional geometry
is a bit more transparent than that of the scalar curvature.

### Acknowledgements

I am thankful to anonymous referees for helpful suggestions and for pointing
out errors in the first version of this paper.

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