# Energy-minimizing Mappings of Real Projective Spaces

Joseph Ansel Hoisington  Max Planck Institute for Mathematics
<EMAIL_ADDRESS>
###### Abstract.

We establish upper and lower bounds for the infimum of the energy in homotopy
classes of mappings from real projective spaces to Riemannian manifolds. We
also show, as a corollary of the work of White in [Wh86], that the infimum of
the energy in a homotopy class of mappings of real projective $n$-space is
determined by an associated class of mappings of the real projective plane.

###### Key words and phrases:

Energy-minimizing maps, infima of energy functionals in homotopy classes

###### 2020 Mathematics Subject Classification:

Primary 53C43 Secondary 53C35

##  1\. Introduction

A basic observation in the study of harmonic maps is that the identity
mappings of spheres of dimensions three and greater do not minimize energy in
their homotopy classes. Eells and Sampson illustrate this, with an example
which they attribute to Morrey, in their foundational work on the subject in
[ES64]. It is now known to be a special case of several more general results.
In particular, the work of White in [Wh86] shows that in all homotopy class of
mappings of quaternionic projective spaces, the Cayley projective plane, or
the sphere of dimension three or greater, the infimum of the energy is zero.

The results in [H23] show that, in all homotopy classes of mappings from
complex projective spaces to Riemannian manifolds, the infimum of the energy
is proportional to the infimal area in the class of mappings of the $2$-sphere
representing the induced action on the second homotopy group. Together with
the work in [Wh86] cited above, these results determine the infimum of the
energy in all homotopy classes of mappings of compact, rank-one Riemannian
symmetric spaces other than real projective spaces, and in particular for all
simply-connected spaces in this family. The purpose of this note is to address
this problem for real projective spaces. Our first result is the following
two-sided estimate for the infimum of the energy in a homotopy class:

######  Theorem 1.1.

Let $(\mathbb{R}P^{n},g_{0})$ be the real projective space with its Riemannian
metric of constant sectional curvature $1$. Let $\Psi$ be a homotopy class of
mappings from $(\mathbb{R}P^{n},g_{0})$ to a Riemannian manifold $(M,g)$ and
$\psi$ the homotopy class of mappings of $(\mathbb{R}P^{2},g_{0})$ represented
by composing the inclusion $\mathbb{R}P^{2}\subseteq\mathbb{R}P^{n}$ with
$F\in\Psi$. Let $B^{\star}$ be the infimal area of mappings $f\in\psi$; that
is:

$\displaystyle
B^{\star}=\inf\limits_{f\in\psi}\int\limits_{\mathbb{R}P^{2}}\sqrt{det(df^{T}\circ
df)}dA.$ (1.1)

Then, letting $E_{2}(F)$ be the energy of a mapping $F$, and letting
$\sigma(n)$ be the volume of the unit $n$-sphere and
$D_{n}=\frac{n\sigma(n)}{8\pi}$,

$\displaystyle D_{n}B^{\star}\leq\inf\limits_{F\in\Psi}E_{2}(F)\leq
2\left(\frac{n-1}{n}\right)D_{n}B^{\star}.$ (1.2)

If $n\geq 3$ and $F\in\Psi$ has $E_{2}(F)=D_{n}B^{\star}$, then $F$ is a
homothety onto a minimal submanifold.

In the homotopy class of the identity mapping of $(\mathbb{R}P^{n},g_{0})$,
the lower bound $D_{n}B^{\star}$ in (1.2) is equal to the energy of the
identity mapping, so this part of Theorem 1.1 is optimal. Croke first
established that the identity mapping of $(\mathbb{R}P^{n},g_{0})$ minimizes
energy in its homotopy class in [Cr87], and we will discuss his work in more
detail below. Although Theorem 1.1 generally does not give an exact value for
the infimum, the work of White cited above implies that, in a sense which we
make precise in our next result, the infimum in a class of mappings of real
projective $n$-space depends only on the induced class of mappings of the real
projective plane:

######  Theorem 1.2.

([H23], see also [Wh86, Corollary 1]) Let $\Psi$ be a homotopy class of
mappings from $(\mathbb{R}P^{n},g_{0})$ to a Riemannian manifold $(M,g)$, and
let $\widehat{\Psi}$ be the union of all homotopy classes of mappings from
$(\mathbb{R}P^{n},g_{0})$ to $(M,g)$ which give the same homotopy class of
mappings of $\mathbb{R}P^{2}$ as $\Psi$ when composed with the inclusion
$\mathbb{R}P^{2}\subseteq\mathbb{R}P^{n}$. Then:

$\displaystyle\inf\limits_{F\in\Psi}E_{2}(F)=\inf\limits_{F\in\widehat{\Psi}}E_{2}(F).$
(1.3)

Theorem 1.2 implies that the infimum of the energy is zero in any homotopy
class of mappings of $\mathbb{R}P^{n}$ which induces the trivial class of
mappings of $\mathbb{R}P^{2}$. Theorem 1.1 sharpens this by showing that the
infimum in a class of mappings of $\mathbb{R}P^{n}$ is zero if and only if it
is zero in the associated class of mappings of $\mathbb{R}P^{2}$.

We will finish this introduction by giving an outline of the rest of the paper
and explaining some of our notation: In Section 2, we will review some
properties of the energy functional and harmonic mappings and establish some
results needed in the proof of Theorem 1.1. In Section 3, we will prove
Theorem 1.1 and discuss some corollaries of the proof. We will write
$\sigma(n)$ for the volume of the unit $n$-sphere and $g_{0}$ for the
canonical Riemannian metrics on $\mathbb{R}P^{n}$ and its covering $S^{n}$.

### Acknolwedgements:

I am happy to thank Werner Ballmann, Christopher Croke, and Sergio Zamora for
their input and feedback about this work, and the Max Planck Institute for
Mathematics for support and hospitality.

##  2\. Preliminary Results

For Riemannian manifolds $(N^{n},h)$ and $(M^{m},g)$ and a Lipschitz map
$F:(N^{n},h)\rightarrow(M^{m},g)$, the energy of $F$ is:

$\displaystyle E_{2}(F)=\frac{1}{2}\int\limits_{N}|dF_{x}|^{2}dVol_{h},\vskip
6.0pt plus 2.0pt minus 2.0pt$ (2.1)

where $|dF_{x}|$ is the Euclidean norm of $dF:T_{x}N\rightarrow T_{F(x)}M$ at
a point $x\in N$ at which $F$ is differentiable. In [ES64], Eells and Sampson
initiated the study of mappings which are critical for the energy, known as
harmonic mappings. We will discuss some background related to harmonic
mappings below. The starting point for our proof of Theorem 1.1 is a formula
for the energy of a mapping due to Croke:

######  Proposition 2.1.

(Croke [Cr87, Proposition 1]) Let $F:(N^{n},h)\rightarrow(M^{m},g)$ be a
Lipschitz map of Riemannian manifolds, $U(N,h)$ the unit tangent bundle of
$(N,h)$, and $d\vec{u}$ the canonical measure on $U(N,h)$. The energy of $F$
is given by:

$\displaystyle
E_{2}(F)=\frac{n}{2\sigma(n-1)}\int\limits_{U(N,h)}|dF(\vec{u})|^{2}d\vec{u}.$
(2.2)

For mappings of real projective space, Proposition 2.1 implies:

######  Lemma 2.2.

Let $F:(\mathbb{R}P^{n},g_{0})\rightarrow(M,g)$ be a Lipschitz map to a
Riemannian manifold. Let $Gr_{k}$ be the family of totally geodesically
embedded $\mathbb{R}P^{k}$ in $(\mathbb{R}P^{n},g_{0})$, and for
$\mathcal{Q}\in Gr_{k}$, let $F|_{\mathcal{Q}}$ be the map given by composing
the inclusion $\mathcal{Q}\subseteq\mathbb{R}P^{n}$ with $F$. Let
$d\mathcal{Q}$ be the measure on $Gr_{k}$ which is invariant under the action
of the isometry group $Isom(\mathbb{R}P^{n},g_{0})$ of
$(\mathbb{R}P^{n},g_{0})$, normalized to have total volume
$\frac{n\sigma(n)}{k\sigma(k)}$. Then:

$\displaystyle
E_{2}(F)=\int\limits_{Gr_{k}}E_{2}(F|_{\mathcal{Q}})d\mathcal{Q}.$ (2.3)

###### Proof.

Let $\mathcal{V}_{k}$ denote the family of pairs $(\vec{u},\mathcal{Q})$,
where $\vec{u}$ belongs to the unit tangent bundle $U(\mathbb{R}P^{n},g_{0})$
of $\mathbb{R}P^{n}$, $\mathcal{Q}\in Gr_{k}$, and $\vec{u}$ is tangent to
$\mathcal{Q}$. Let $\mu_{k}$ be an $Isom(\mathbb{R}P^{n},g_{0})$-invariant
measure on $\mathcal{V}_{k}$. By Proposition 2.1, calculating via the
$Isom(\mathbb{R}P^{n},g_{0})$-equivariant fibration
$(\vec{u},\mathcal{Q})\mapsto\vec{u}$ of $\mathcal{V}_{k}$ over
$U(\mathbb{R}P^{n},g_{0})$ and choosing $\mu_{k}$ to have the correct value
for its total mass, we have:

$\displaystyle
E_{2}(F)=\int\limits_{\mathcal{V}_{k}}|dF(\vec{u})|^{2}d\mu_{k}.\vskip 6.0pt
plus 2.0pt minus 2.0pt$ (2.4)

By Proposition 2.1 as it applies to each $\mathcal{Q}\in Gr_{k}$, noting that
$F|_{\mathcal{Q}}$ is a Lipschitz map with well-defined energy, and
calculating via the $Isom(\mathbb{R}P^{n},g_{0})$-equivariant fibration
$(\vec{u},\mathcal{Q})\mapsto\mathcal{Q}$ of $\mathcal{V}_{k}$ over $Gr_{k}$,
we then have:

$\displaystyle\int\limits_{\mathcal{V}_{k}}|dF(\vec{u})|^{2}d\mu_{k}=\int\limits_{Gr_{k}}E_{2}(F|_{\mathcal{Q}})d\mathcal{Q}.$
(2.5)

∎

The case $k=1$ of Lemma 2.2, in which $Gr_{1}$ is the space of oriented
geodesics in $(\mathbb{R}P^{n},g_{0})$, is simpler than the general result:
each $\vec{u}\in U(\mathbb{R}P^{n},g_{0})$ is tangent to a unique geodesic, so
$U(\mathbb{R}P^{n},g_{0})$ admits an $Isom(\mathbb{R}P^{n},g_{0})$-equivariant
fibration over $Gr_{1}$ and the calculation via the “incidence variety”
$\mathcal{V}_{k}$ is unnecessary. In the work of Croke in [Cr87], this
observation leads to a proof that the identity mapping of
$(\mathbb{R}P^{n},g_{0})$ minimizes energy in its homotopy class. We will
discuss his results in more depth after the proof of Theorem 1.1. Our next
lemma shows that the infimum of the energy in a class of mappings of real
projective $n$-space can be bounded from above in terms of its action on
hypersurfaces:

######  Lemma 2.3.

Let $\Psi$ be a homotopy class of mappings from $(\mathbb{R}P^{n},g_{0})$ to a
Riemannian manifold $(M,g)$, and let $\Psi^{\prime}$ be the homotopy class of
mappings of $(\mathbb{R}P^{n-1},g_{0})$ which is represented by composing the
inclusion of a totally geodesic $\mathbb{R}P^{n-1}\subseteq\mathbb{R}P^{n}$
with $F\in\Psi$. Let $\beta$ be the infimum of the energy of mappings in
$\Psi$ and $\beta^{\prime}$ the infimum of the energy of mappings in
$\Psi^{\prime}$.

Then for $n\geq 3$,

$\displaystyle\beta\leq\frac{\sigma(n-2)}{\sigma(n-3)}\beta^{\prime}.$ (2.6)

###### Proof.

Let $\mathcal{Q}_{0}$ be a fixed totally geodesic
$\mathbb{R}P^{n-1}\subseteq\mathbb{R}P^{n}$ and $p_{0}\in\mathbb{R}P^{n}$ the
unique point at distance $\frac{\pi}{2}$ from $\mathcal{Q}_{0}$. Let
$\tau:S^{n}\rightarrow\mathbb{R}P^{n}$ be the covering map,
$\widetilde{p}_{0}$ a point in $\tau^{-1}(p_{0})$, and
$\xi:S^{n}\setminus\\{-\widetilde{p}_{0}\\}\rightarrow\mathbb{R}^{n}$ the
stereographic coordinate chart which takes $\widetilde{p}_{0}$ to the origin.
Letting $\theta_{t}$ be the conformal diffeomorphism of $S^{n}$ which
corresponds to multiplication by $t$ in the coordinates given by $\xi$, a
calculation via polar coordinates in $\mathbb{R}^{n}$ shows that for $n\geq
3$, we have $\lim\limits_{t\rightarrow\infty}E_{2}(\theta_{t})=0$.

Let $\Omega\subseteq S^{n}$ be the open hemisphere centered at
$\widetilde{p}_{0}$, which we regard as a fundamental domain for the covering
$\tau$. Define the mapping
$\Theta_{t}:\mathbb{R}P^{n}\rightarrow\mathbb{R}P^{n}$ as follows: if
$p\in\mathbb{R}P^{n}$ has a preimage $\widetilde{p}$ via $\tau$ in
$\theta_{t}^{-1}(\Omega)$, then
$\Theta_{t}(p)=\tau\circ\theta_{t}(\widetilde{p})$; otherwise, $\Theta_{t}(p)$
is the image of $p$ under the nearest-point retraction of
$\mathbb{R}P^{n}\setminus\\{p_{0}\\}$ onto $\mathcal{Q}_{0}$.

Because $\theta_{t}$ is conformal and
$\lim\limits_{t\rightarrow\infty}E_{2}(\theta_{t})=0$, for any Lipschitz
mapping $F$ from $(\mathbb{R}P^{n},g_{0})$ to a Riemannian manifold $(M,g)$,
the energy of $F\circ\Theta_{t}$ on the domain $\tau(\theta_{t}^{-1}(\Omega))$
in $\mathbb{R}P^{n}$ also goes to $0$ as $t\rightarrow\infty$. Let $F$ be any
such mapping and $F^{\prime}$ the mapping of $\mathcal{Q}_{0}$ given by
composing $F$ with the inclusion $\mathcal{Q}_{0}\subseteq\mathbb{R}P^{n}$.
Calculating in Fermi coordinates about $\mathcal{Q}_{0}$ then shows:

$\displaystyle\lim\limits_{t\rightarrow\infty}E_{2}(F\circ\Theta_{t})=\lim\limits_{t\rightarrow\infty}\frac{1}{2}\int\limits_{\mathbb{R}P^{n}}|d(F\circ\Theta_{t})|^{2}dp\vskip
6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle=\int\limits_{\mathcal{Q}_{0}}\int\limits_{0}^{\frac{\pi}{2}}\cos^{n-3}(r)|dF^{\prime}|^{2}drdp=\frac{\sigma(n-2)}{\sigma(n-3)}E_{2}(F^{\prime}).\vskip
6.0pt plus 2.0pt minus 2.0pt$ (2.7)

For any mapping $F\in\Psi$ and $\varepsilon>0$, the associated mapping
$F^{\prime}$ of $\mathcal{Q}_{0}$ is homotopic to a mapping
$\underline{F^{\prime}}$ with
$E_{2}(\underline{F^{\prime}})<\beta^{\prime}+\varepsilon$. By the homotopy
extension property for the inclusion
$\mathbb{R}P^{n-1}\subseteq\mathbb{R}P^{n}$, any homotopy
$F^{\prime}\simeq\underline{F^{\prime}}$ can be extended to a homotopy from
$F$ to a mapping $\underline{F}$ of $\mathbb{R}P^{n}$ which coincides with
$\underline{F^{\prime}}$ along $\mathcal{Q}_{0}$. By composing such a mapping
$\underline{F}$ with the deformation $\Theta_{t}$ described above, (2.7) shows
that $\beta\leq\frac{\sigma(n-2)}{\sigma(n-3)}\beta^{\prime}$. ∎

The proof of Theorem 1.1 depends on a basic fact about mappings of surfaces,
which we record here for later reference:

######  Lemma 2.4.

Let $f$ be a mapping from a Riemannian surface $(\Sigma,h)$ to a manifold
$(M,g)$.

Then the integrand $\frac{1}{2}|df|^{2}$ for the energy of $f$ is bounded
below pointwise by the integrand $\sqrt{det(df^{T}\circ df)}$ for the area of
the image of $f$, with equality precisely at points where $f$ is
semiconformal, i.e. where $f^{*}g$ is a nonnegative scalar multiple of $h$.

Lemma 2.4 follows from the arithmetic-geometric mean inequality for the
eigenvalues of $f^{*}g$ relative to $h$. One consequence is that conformal
mappings between Riemannian surfaces minimize energy in their homotopy class
and are therefore harmonic. For $(\mathbb{R}P^{2},g_{0})$, conversely, all
harmonic mappings are conformal:

######  Theorem 2.5.

(Lemaire [Lem78, Theorem 2.8]) A harmonic map from $(S^{2},g_{0})$ or
$(\mathbb{R}P^{2},g_{0})$ to a Riemannian manifold is a conformal branched
immersion.

Lemma 2.4 also implies that, in the notation of Theorem 1.1, the infimal area
$B^{\star}$ is a lower bound for the energy of mappings $f\in\psi$. The proof
of Theorem 1.1 depends on the fact that it is actually the infimum, i.e. the
case $n=2$ of the result. We will show this via the following lemma, adapted
from the solution of Plateau’s problem as in [Law80]:

######  Lemma 2.6.

Let $f:(S^{2},g_{0})\rightarrow(M,g)$ be a smooth mapping from the $2$-sphere
to a Riemannian manifold and $A(f)$ the area of its image.

Then for any $\delta>0$ there is a diffeomorphism $\phi:S^{2}\rightarrow
S^{2}$, which we can assume is homotopic to the identity, such that
$E_{2}(f\circ\phi)<A(f)+\delta$. In particular, $f$ is homotopic to a map with
energy less than $A(f)+\delta$. If $f$ is antipodally invariant, we can choose
$\phi$ to be antipodally invariant, so that the same conclusions hold for
mappings of $(\mathbb{R}P^{2},g_{0})$.

###### Proof.

Let $(S^{2},rg_{0})$ be the $2-$sphere of constant curvature $\frac{1}{r}$,
and for a mapping $f$ as above, let
$f_{r}:(S^{2},g_{0})\rightarrow(M,g)\times(S^{2},rg_{0})$ be the product of
$f$ with a homothety from $(S^{2},g_{0})$ to $(S^{2},rg_{0})$. The map $f_{r}$
is then a smooth immersion of $S^{2}$ into $M\times S^{2}$. By the
uniformization theorem, there is a diffeomorphism $\phi_{r}:S^{2}\rightarrow
S^{2}$, which we can take to be homotopic to the identity, such that
$f_{r}\circ\phi_{r}$ is conformal. Letting $A(f_{r})$ be the area of the image
of $f_{r}$, Lemma 2.4 implies:

$\displaystyle E_{2}(f_{r}\circ\phi_{r})=A(f_{r}).\vskip 6.0pt plus 2.0pt
minus 2.0pt$ (2.8)

An elementary calculation shows that
$E_{2}(f\circ\phi_{r})<E_{2}(f_{r}\circ\phi_{r})$, and that by choosing $r$
small enough we can ensure $A(f_{r})<A(f)+\delta$ for any $\delta>0$. Because
$\phi_{r}$ is homotopic to the identity, $f$ is homotopic to $f\circ\phi_{r}$.
If $f$ is antipodally invariant, $f_{r}$ is also antipodally invariant, and
the uniformization theorem implies we can choose an antipodally invariant
$\phi_{r}$ as well. ∎

If a map of the real projective plane minimizes energy in its homotopy class,
Theorem 2.5 and Lemma 2.6 imply that it has energy equal to the infimal area
$B^{\star}$ in (1.1), however a homotopy class of mappings need not contain a
continuous, energy-minimizing map. Sacks and Uhlenbeck studied the existence
of harmonic mappings of $2$-spheres in [SU81] and discuss this issue. Although
conformal mappings between Riemannian surfaces are always harmonic, mappings
of Riemannian manifolds of dimension $n\geq 3$ which are both harmonic and
conformal must be homotheties. We will show this using the following
characterization of harmonic maps:

######  Theorem 2.7.

(Eells, Sampson [ES64]) Let $F:(N,h)\rightarrow(M,g)$ be a smooth mapping of
Riemannian manifolds, with connections $\nabla^{h}$ and $\nabla^{g}$, and
$\alpha_{F}$ the second fundamental form of $F$; that is, $\alpha_{F}$ is the
$F^{*}TM$-valued symmetric bilinear form which, for vector fields $V,W$ on
$N$, satisfies
$\alpha_{F}(V,W)=F^{*}\nabla^{g}_{V}F_{*}W-F_{*}\nabla^{h}_{V}W$.

Then $F$ is harmonic if and only if $Tr(\alpha_{F})=0$.

For conformal maps, Theorem 2.7 implies:

######  Lemma 2.8.

Let $F:(N^{n},h)\rightarrow(M^{m},g)$ be an immersion of Riemannian manifolds,
and suppose that $F^{*}g=e^{2\eta}h$, where $\eta$ is a smooth function on
$N$.

If $n=2$, $F$ is harmonic if and only if the image of $F$ is a minimal
submanifold of $(M,g)$. If $n\geq 3$, $F$ is harmonic if and only if $F$ is a
homothety and the image of $F$ is a minimal submanifold of $(M,g)$.

###### Proof.

Let $\nabla^{h}$ be the connection of the metric $h$, and $\nabla^{h}\eta$ the
gradient of $\eta$ as calculated via the metric $h$, and let $\nabla^{*}$ be
the connection of the metric $F^{*}g$ on $N$. An elementary calculation shows
that for vector fields $V,W$ on $N$,

$\displaystyle\nabla^{*}_{V}W=\nabla^{h}_{V}W+V(\eta)W+W(\eta)V-h(V,W)\nabla^{h}\eta.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (2.9)

Letting $\nabla^{g}$ be the connection of the metric $g$ on $M$ and
$\alpha_{F}$ the second fundamental form of $F$ as in Theorem 2.7, we then
have:

$\displaystyle\alpha_{F}(V,W)=\nabla^{g}_{F_{*}V}F_{*}W-F_{*}\nabla^{h}_{V}W\vskip
6.0pt plus 2.0pt minus
2.0pt=(\nabla^{g}_{F_{*}V}F_{*}W)^{\perp}+F_{*}\left(\nabla^{*}_{V}W-\nabla^{h}_{V}W\right),\vskip
6.0pt plus 2.0pt minus 2.0pt$ (2.10)

where $(\nabla^{g}_{F_{*}V}F_{*}W)^{\perp}$ is the component of
$\nabla^{g}_{F_{*}V}F_{*}W$ normal to the image $F(N)$ of $F$ in $(M,g)$.
Taking the trace of (2.10) using the identity (2.9), and writing $\vec{H}$ for
the mean curvature vector of $F(N)$ in $(M,g)$, gives:

$\displaystyle Tr(\alpha_{F})=(2-n)\nabla^{h}\eta+e^{2\eta}\vec{H}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (2.11)

The characterization of conformal harmonic maps then follows from Theorem 2.7.
∎

##  3\. Upper and Lower Bounds for the Infimum

After proving Theorem 1.1, we will discuss some corollaries of the proof,
alternate approaches to this result, and the connections between this result
and the work of Croke in [Cr87].

###### Proof of Theorem 1.1.

Let $\Psi$ be a homotopy class of mappings from $(\mathbb{R}P^{n},g_{0})$ to a
Riemannian manifold $(M,g)$ and $\psi$ the induced class of mappings of
$(\mathbb{R}P^{2},g_{0})$. The lower bound $D_{n}B^{\star}$ for the energy of
mappings $F\in\Psi$ follows from the formula for $E_{2}(F)$ in Proposition
2.2, as it applies to the collection $Gr_{2}$ of totally geodesic
$\mathbb{R}P^{2}\subseteq\mathbb{R}P^{n}$, and the fact that $B^{\star}$ is a
lower bound for the energy of mappings $f\in\psi$ by Lemma 2.4. For $n\geq 3$,
successive application of Lemma 2.3, together with Lemma 2.6, shows that the
infimum of the energy of mappings $F\in\Psi$ is no greater than
$\frac{\sigma(n-2)}{2}B^{\star}=2(\frac{n-1}{n})D_{n}B^{\star}$.

Suppose $F\in\Psi$ has energy equal to $D_{n}B^{\star}$ and $n\geq 3$. Because
$F$ minimizes energy in its homotopy class, $F$ is harmonic, and therefore
smooth by elliptic regularity for harmonic mappings (cf. [Aub13, Ch.10]).
Proposition 2.2 implies that for all totally geodesic
$\mathcal{Q}\cong\mathbb{R}P^{2}$ in $(\mathbb{R}P^{n},g_{0})$,
$F|_{\mathcal{Q}}$ minimizes energy in the homotopy class $\psi$ and is
therefore also a smooth harmonic map, and therefore a conformal branched
immersion by Theorem 2.5. At any $x\in\mathbb{R}P^{n}$, any two unit tangent
vectors $\vec{u}_{1}$, $\vec{u}_{2}$ in $U_{x}(\mathbb{R}P^{n},g_{0})$ are
tangent to a unique totally geodesic
$\mathcal{Q}(\vec{u}_{1},\vec{u}_{2})\cong\mathbb{R}P^{2}$. Because
$F|_{\mathcal{Q}(\vec{u}_{1},\vec{u}_{2})}$ is a conformal branched immersion,
$|dF(\vec{u}_{1})|=|dF(\vec{u}_{2})|$. This implies that $|dF(\vec{u})|$ is
constant on the unit tangent space $U_{x}(\mathbb{R}P^{n},g_{0})$ at $x$, and
that $F$ is a semiconformal mapping of $(\mathbb{R}P^{n},g_{0})$. By Lemma
2.8, $F$ is a homothety onto a minimal submanifold of $(M,g)$. ∎

In [Cr87], via the case $k=1$ of Lemma 2.2, Croke establishes the following
lower bound for the energy of a mapping
$F:(\mathbb{R}P^{n},g_{0})\rightarrow(M,g)$: letting $\gamma$ denote the
nontrivial free homotopy class of loops in $\mathbb{R}P^{n}$, and letting
$L^{\star}$ denote the infimal length of curves in the class $F_{*}\gamma$ in
$M$,

$\displaystyle E_{2}(F)\geq\frac{n\sigma(n)}{4\pi^{2}}L^{\star^{2}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (3.1)

The inequality in (3.1) also follows from Theorem 1.1 and Pu’s systolic
inequality [Pu52, CK03], together with some mild technical observations,
however the equality case is stronger than in Theorem 1.1: equality in (3.1)
implies $F$ is a homothety onto a totally geodesic submanfiold. The lower
bound in (3.1) and the rigidity of the equality case imply that the identity
mapping of $(\mathbb{R}P^{n},g_{0})$ minimizes energy in its homotopy class,
uniquely up to isometries.

Theorem 1.1 and the inequality in (3.1) are the only results known to the
author which give quantitative lower bounds for the energy of a mapping of
real projective space, however for a homotopy class $\Psi$ of mappings from
$(\mathbb{R}P^{n},g_{0})$ to a manifold $(M,g)$, letting $\Psi^{(k)}$ be the
induced class of mappings of $(\mathbb{R}P^{n-k},g_{0})$, a stronger lower
bound for the energy of mappings in $\Psi^{(k)}$, together with Lemma 2.2 as
it applies to $Gr_{n-k}$, would give a stronger lower bound for the energy of
mappings in $\Psi$. Likewise, if $\Psi^{(k)}$ contains maps with energy less
than the upper bound $2(\frac{n-k-1}{n-k})D_{n-k}B^{\star}$ for the infimum in
Theorem 1.1, Lemma 2.3 leads to a stronger upper bound for the infimum of the
energy of mappings in $\Psi$.

We proved the upper bound for the infimum in Theorem 1.1 by repeated
application of the estimate in Lemma 2.3, which we derived by conformally
expanding a neighborhood $N_{0}$ of the point $p_{0}$ onto the complement of
the totally geodesic hypersurface $\mathcal{Q}_{0}$ and retracting
$\mathbb{R}P^{n}\setminus N_{0}$ onto $\mathcal{Q}_{0}$. An alternate approach
to this part of the proof of Theorem 1.1 would be to choose a totally geodesic
$\mathcal{P}_{0}\cong\mathbb{R}P^{2}$ in $(\mathbb{R}P^{n},g_{0})$, with
$\mathcal{R}_{0}$ the unique totally geodesic $\mathbb{R}P^{n-3}$ at maximal
distance from $\mathcal{P}_{0}$, and deform the identity mapping of
$(\mathbb{R}P^{n},g_{0})$ along geodesic segments from $\mathcal{R}_{0}$ to
$\mathcal{P}_{0}$, expanding a tubular neighborhood $T_{0}$ of
$\mathcal{R}_{0}$ onto $\mathbb{R}P^{n}\setminus\mathcal{P}_{0}$ and
retracting $\mathbb{R}P^{n}\setminus T_{0}$ onto $\mathcal{P}_{0}$. One can
give a proof along these lines, however the resulting upper bound is the same
as from our argument above.

## References

  * [Aub13] Thierry Aubin: Some Nonlinear Problems in Riemannian Geometry, Springer Science and Business Media, 2013.
  * [Cr87] Christopher B. Croke: Lower Bounds on the Energy of Maps, Duke Mathematical Journal 55.4 (1987), 901-908.
  * [CK03] Christopher B. Croke and Mikhail Katz: Universal Volume Bounds in Riemannian Manifolds, Surveys in Differential Geometry 8.1 (2003), 109-137.
  * [ES64] James Eells and Joseph H. Sampson: Harmonic Mappings of Riemannian Manifolds, American Journal of Mathematics 86.1 (1964), 109-160.
  * [H23] Joseph Hoisington: Energy-minimizing Mappings of Complex Projective Spaces, ArXiv preprint, arxiv:2311.08285 (2023).
  * [Law80] H. Blaine Lawson, Jr.: Lectures on Minimal Submanifolds, Vol. I, Publish or Perish. Inc., Berkeley (1980).
  * [Lem78] Luc Lemaire: Applications Harmoniques de Surfaces Riemanniennes, Journal of Differential Geometry 13.1 (1978), 51-78.
  * [Pu52] Pao Ming Pu: Some Inequalities in Certain Nonorientable Riemannian Manifolds, Pacific J. Math 2.1 (1952), 55-71.
  * [SU81] Jonathan Sacks and Karen Uhlenbeck: The Existence of Minimal Immersions of 2-spheres, Annals of Mathematics 113.1 (1981), 1-24.
  * [Wh86] Brian White: Infima of Energy Functionals in Homotopy Classes of Mappings, Journal of Differential Geometry 23.2 (1986), 127-142.