# Calibrations and Energy-Minimizing Mappings of Rank-1 Symmetric Spaces

Joseph Ansel Hoisington  Department of Mathematics, University of Georgia,
Athens, GA 30602 USA<EMAIL_ADDRESS>
###### Abstract.

We prove lower bounds for energy functionals of mappings from real, complex
and quaternionic projective spaces with their canonical (symmetric space)
metrics to Riemannian manifolds. We show that these results for complex
projective space extend to all Kähler manifolds biholomorphic to
$\mathbb{C}P^{N}$. For real and complex projective spaces, these lower bounds
are sharp, and we characterize the family of energy-minimizing maps which
arise in these results. We discuss the connections between these results and
several theorems and questions in systolic geometry.

###### Key words and phrases:

Energy-minimizing mappings, harmonic maps, calibrated geometries, symmetric
spaces, systolic inequalities

###### 2010 Mathematics Subject Classification:

Primary 53C43, 53C35, 53C38 Secondary 53C26, 53C55, 58E20

##  1\. Introduction

The main results of this work are lower bounds for energy functionals of
mappings from real, complex and quaternionic projective spaces to Riemannian
manifolds and characterizations of mappings which minimize energy in these
results. For real projective space, we will prove:

######  Theorem 1.1.

Let $(\mathbb{R}P^{n},g_{0})$ be the $n$-dimensional real projective space
with its canonical (constant curvature) metric $g_{0}$, with $n\geq 2$. Let
$F:(\mathbb{R}P^{n},g_{0})\rightarrow(M^{m},g)$ be a Lipschitz mapping to a
Riemannian manifold $(M,g)$. Let $L^{\star}$ be the infimum of the lengths of
paths in the free homotopy class of $F_{*}(\gamma)$, where $\gamma$ represents
the non-trivial class in $\pi_{1}(\mathbb{R}P^{n})$, and let $E_{p}(F)$ be the
$p$-energy of $F$, as in Definition 2.1 below. For all $p\geq 1$,

$\displaystyle
E_{p}(F)\geq\frac{\sigma(n)}{2}\left(\frac{2\sqrt{n}}{\pi}L^{\star}\right)^{p},\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.1)

where $\sigma(n)$ is the volume of the unit $n$-sphere.

If $p>1$ and equality holds for $p$, then $F$ is a homothety onto a totally
geodesic submanifold of $(M,g)$ and equality holds for all $p\geq 1$. If $F$
is a smooth immersion, equality for $p=1$ also implies these conditions.

This implies in particular that the identity mapping of
$(\mathbb{R}P^{n},g_{0})$ minimizes $p$-energy in its homotopy class for all
$p\geq 1$.

The case $p=2$ of Theorem 1.1 was proven by Croke in [Cr87]. The $2$-energy,
referred to simply as the energy of a mapping, is the classical energy
functional of mappings of Riemannian manifolds and a fundamental invariant in
the theory of harmonic maps. It is a generalization of the energy of a path in
a Riemannian manifold and the Dirichlet integral of a real-valued function.
The fact that the identity mapping of $(\mathbb{R}P^{n},g_{0})$ minimizes
energy in its homotopy class, which was first established by Croke in this
result, is different from the round sphere $(S^{n},g_{0})$, $n\geq 3$: in that
case, conformal dilations give energy-decreasing deformations of the identity
mapping. In fact, they give 1-parameter families of mappings for which the
infimum of the energy is $0$. More generally, work of White [Wh86] implies
that in any closed Riemannian manifold $(M,g)$ with $\pi_{1}(M)=0$,
$\pi_{2}(M)=0$, the identity mapping is homotopic to mappings with arbitrarily
small energy. Croke also noted in [Cr87] that the results of Smith [Sm75]
imply there are metrics arbitrarily close to the canonical metric on
$\mathbb{R}P^{n}$, obtained by conformal deformations, for which the identity
mapping is not even a stable critical point of the energy functional.

Croke observed in [Cr87] that his argument could be adapted to prove that the
identity mapping of complex projective space $\mathbb{C}P^{N}$ with its
canonical metric $g_{0}$ minimizes $2$-energy in its homotopy class, but that
this was already known because $(\mathbb{C}P^{N},g_{0})$ is a Kähler manifold
– in fact, it is a result of Lichnerowicz [Li70] that any holomorphic mapping
of compact Kähler manifolds minimizes energy in its homotopy class.

A natural extension of Theorem 1.1 to complex projective space would give
lower bounds for the energy of mappings
$F:(\mathbb{C}P^{N},g_{0})\rightarrow(M^{m},g)$ to Riemannian manifolds in
terms of the infimum $A^{\star}$ of the areas of mappings $f:S^{2}\rightarrow
M$ which represent the homotopy or homology class of $F_{*}(\mathbb{C}P^{1})$.
However unlike the strong characterization of equality in Theorem 1.1, basic
properties of the energy functional and Kähler manifolds imply that in any
such optimal result, the equality case must be broad enough to include any
holomorphic mapping from $\mathbb{C}P^{N}$ to a compact Kähler manifold
$(X,h)$ – we will discuss this at the beginning of Section 4 below. Also,
although conformal deformations of the canonical metric give Riemannian
metrics on $\mathbb{C}P^{N}$ for which the identity mapping is not a stable
critical point of the energy functional, as with $\mathbb{R}P^{n}$, on
$\mathbb{C}P^{N}$ the result of Lichnerowicz cited above also gives an
infinite-dimensional family of metrics, obtained by Kähler deformations of the
canonical metric, for which the identity is energy-minimizing in its homotopy
class.

In Theorem 4.1 we will state and prove lower bounds for the $p$-energy of
Lipschitz mappings $F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(M,g)$, $p\geq
2$, where $(M,g)$ is a Riemannian manifold and $\widetilde{g}$ is any Kähler
metric on $\mathbb{C}P^{N}$, which are analogous to Theorem 1.1 for
$\mathbb{R}P^{n}$. This result generalizes an inequality for mappings of
$\mathbb{C}P^{N}$ with its canonical metric in [Cr87, Theorem 3]. The full
characterization of equality in Theorem 4.1 is somewhat technical and involves
several partial results under weaker assumptions, but for the complex
projective plane, this result implies that holomorphic mappings are
essentially the only energy-minimizing maps of this type. We record this in
the following:

######  Theorem 1.2.

Let $\widetilde{g}$ be a Kähler metric on $\mathbb{C}P^{2}$ and
$F:(\mathbb{C}P^{2},\widetilde{g})\rightarrow(M^{m},g)$ a Lipschitz mapping to
a Riemannian manifold $(M,g)$. Let $A^{\star}$ be the infimum of the areas of
mappings $f:S^{2}\rightarrow(M,g)$ in the free homotopy class of
$F_{*}(\mathbb{C}P^{1})$, and let $E_{p}(F)$ be the $p$-energy of $F$. Then
for all $p\geq 2$,

$\displaystyle
E_{p}(F)\geq\frac{\pi^{2}}{2}\left(\frac{4}{\pi}A^{\star}\right)^{\frac{p}{2}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.2)

Equality for $p=2$ implies that $F$ is smooth and, for any neighborhood
$\mathcal{V}$ of $\mathbb{C}P^{2}$ on which $F$ is an immersion,
$F^{*}g|_{\mathcal{V}}$ is a Kähler metric, $F(\mathcal{V})$ is minimal in
$(M,g)$ and the second fundamental form of $F(\mathcal{V})$ in $(M,g)$ can be
diagonalized by a unitary basis. If $p>2$ and equality holds for $p$, then $F$
has constant energy density and equality holds for all $p\geq 2$. If $F$ is an
immersion and equality holds for $p>2$, then $F$ is a homothety onto its
image.

It would be interesting to determine whether Theorem 1.2 is true for mappings
of $(\mathbb{C}P^{N},\widetilde{g})$, $N\geq 3$. The conclusion of Theorem 1.2
is similar to a result of Burns, Burstall, de Bartolomeis and Rawnsley
[BBdeBR89, Theorem 3] that if $\phi:(M^{4},g)\rightarrow(\mathcal{Z},h)$ is
stable harmonic map from a closed, real-analytic Riemannian $4$-manifold
$(M^{4},g)$ to a Hermitian symmetric space $(\mathcal{Z},h)$ and there is a
point of $M$ at which the rank of $d\phi$ is at least $3$, then there is a
unique Kähler structure on $M$ with respect to which $\phi$ is holomorphic.

When the mapping of $\mathbb{C}P^{N}$ in Theorem 4.1 is to a simply connected,
compact Kähler manifold, we will prove a stronger characterization of equality
– we record this equality case along with the general inequality in Theorem
4.1 in the following:

######  Theorem 1.3.

Let $\widetilde{g}$ be a Kähler metric on $\mathbb{C}P^{N}$,
$F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(M,g)$ a Lipschitz mapping to a
Riemannian manifold and $A^{\star}$ the infimum of the areas of mappings
$f:S^{2}\rightarrow M$ in the free homotopy class of $F_{*}(\mathbb{C}P^{1})$.
Then for $p\geq 2$,

$\displaystyle
E_{p}(F)\geq\frac{\pi^{N}}{N!}\left(\frac{2N}{\pi}A^{\star}\right)^{\frac{p}{2}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.3)

Suppose in addition that $(M,g)$ is a compact, simply connected Kähler
manifold and the class of $F_{*}(\mathbb{C}P^{1})$ in $H_{2}(X;\mathbb{Z})$
can be represented by a rational curve. Then if equality holds for $p=2$, $F$
is holomorphic or antiholomorphic. If $p>2$ and equality holds for $p$, then
$F$ is a homothety onto its image and equality holds for all $p\geq 2$.

Ohnita [Oh87] has proven that holomorphic and antiholomorphic mappings are the
only stable harmonic maps between $(\mathbb{C}P^{N_{1}},g_{0})$ and
$(\mathbb{C}P^{N_{2}},g_{0})$ with the canonical metric $g_{0}$. The equality
conditions in Theorem 1.3 and the more general results in Theorem 4.1 are
similar to those in several results of Ohnita [Oh87] and Burns, Burstall, de
Bartolomeis and Rawnsley [BBdeBR89], which we will discuss in Section 4.

White’s results [Wh86] cited above imply that for any Riemannian metric $g$ on
$\mathbb{C}P^{N}$ the identity mapping of $(\mathbb{C}P^{N},g)$ is homotopic
to maps with arbitrarily small $p$-energy for all $1\leq p<2$. They likewise
imply that the identity mapping of quaternionic projective space
$\mathbb{H}P^{N}$ is homotopic to maps with arbirarily small $p$-energy for
all $1\leq p<4$ and the identity mapping of the Cayley projective plane
$\mathcal{C}aP^{2}$ is homotopic to maps with arbitrarily small $p$-energy for
all $1\leq p<8$, for any Riemannian metrics on these spaces. In this sense,
Theorems 1.2 and 1.3 are optimal, and the strongest results one can hope to
establish in $\mathbb{H}P^{N}$ are lower bounds for the $p$-energy of mappings
for $p\geq 4$. Some aspects of the results for $\mathbb{R}P^{n}$ and
$\mathbb{C}P^{N}$ above make such lower bounds for $\mathbb{H}P^{N}$ a natural
extension of Theorems 1.1, 1.2 and 1.3 – we will discuss this below – but the
optimal result for $\mathbb{H}P^{N}$ cannot be as strong as our results for
$\mathbb{R}P^{n}$ and $\mathbb{C}P^{N}$: the work of Wei [We98] implies that
for $N\geq 2$, the identity mapping of $(\mathbb{H}P^{N},g_{0})$ is not a
stable critical point of the $4$-energy, and in particular does not minimize
$4$-energy in its homotopy class. We will show that for $p\geq 4$, the
$p$-energies of mappings of $(\mathbb{H}P^{N},g_{0})$ nonetheless do satisfy
lower bounds like our results for mappings of $(\mathbb{R}P^{n},g_{0})$ and
$(\mathbb{C}P^{N},\widetilde{g})$:

######  Theorem 1.4.

Let $(\mathbb{H}P^{N},g_{0})$ be the quaternionic projective space with its
canonical metric, $N\geq 2$. Let
$F:(\mathbb{H}P^{N},g_{0})\rightarrow(M^{m},g)$ be a non-constant Lipschitz
map to a closed Riemannian manifold and $B^{\star}$ the minimum mass of an
integral 4-current $T$ in $M$ which represents the class of
$F_{*}(\mathbb{H}P^{1})$ in $H_{4}(M;\mathbb{Z})$. Then for all $p\geq 4$,

$\displaystyle
E_{p}(F)>\frac{\pi^{2N}}{(2N+1)!}\left(K_{N}B^{\star}\right)^{\frac{p}{4}},\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.4)

where $K_{N}$ is a positive constant which depends only on $N$ and is given by
(5.15) below.

The minimum $B^{\star}$ in Theorem 1.4 is over all integral currents
homologous to $F_{*}(\mathbb{H}P^{1})$, which may be larger than the set of
mappings $f:S^{4}\rightarrow M$ homotopic to $F_{*}(\mathbb{H}P^{1})$ as in
Theorems 1.1, 1.2 and 1.3. However a theorem of White [Wh84] shows that if $M$
is simply connected, $B^{\star}$ is equal to the infimum of the areas of
mappings $f:S^{4}\rightarrow M$ in the free homotopy class of
$F_{*}(\mathbb{H}P^{1})$. We will discuss this after the proof of Theorem 1.4,
in Section 5.

The proof of Theorem 1.4 does show that
$Id:(\mathbb{H}P^{N},g_{0})\rightarrow(\mathbb{H}P^{N},g_{0})$ minimizes
energy in its homotopy class among maps which satisfy an additional hypothesis
– we will also discuss after the proof of Theorem 1.4. The strongest
conjecture which is consistent with Wei’s result [We98, Theorem 5.1] in all
dimensions is that the identity mapping of $(\mathbb{H}P^{N},g_{0})$ minimizes
$p$-energy in its homotopy class for $p\geq 6$. More precisely, Wei’s results
imply that the identity mapping of $(\mathbb{H}P^{N},g_{0})$ is an unstable
critical point of the $p$-energy for $1\leq p<2+4(\frac{N}{N+1})$ and a stable
critical point for $p\geq 2+4(\frac{N}{N+1})$. It would be interesting to find
optimal lower bounds for the $p$-energy of mappings of
$(\mathbb{H}P^{N},g_{0})$ and $(\mathcal{C}aP^{2},g_{0})$ and determine the
$p$-energies for which the identity mappings are energy-minimizing in their
homotopy classes – for the Cayley plane with its canonical metric, Wei’s
results imply that the identity mapping is an unstable critical point of the
$p$-energy for $1\leq p<10$ and a stable critical point for $p\geq 10$. At the
end of Section 5, we will sketch one possible approach to this problem for
$\mathbb{H}P^{N}$.

For mappings homotopic to the identity of $\mathbb{C}P^{N}$, the lower bound
in Theorem 1.3 follows from a lower bound for the area of surfaces homologous
to $\mathbb{C}P^{1}$ in $\mathbb{C}P^{N}$, which follows from the calibrated
structure of the Kähler $2$-form of $(\mathbb{C}P^{N},\widetilde{g})$. The
characterization of energy-minimizing maps as holomorphic or antiholomorphic
in Theorem 1.3 is related to the fact that a map which realizes the lower
bound in (1.3) must map all linearly embedded
$\mathbb{C}P^{1}\subseteq\mathbb{C}P^{N}$ to cycles which are calibrated by
the Kähler form of the image of $F$. We will explain at the end of Section 3
how the canonical $1$-form on the unit tangent bundle of
$(\mathbb{R}P^{n},g_{0})$ gives a calibration-like structure, and how the
energy-minimizing property of
$Id:(\mathbb{R}P^{n},g_{0})\rightarrow(\mathbb{R}P^{n},g_{0})$ can be derived
using this structure.

Quaternionic projective space also carries a calibration, by a parallel
$4$-form described in [Be07, Kr65, Kr66], and this plays a part in the proof
of Theorem 1.4. Despite these similarities, however, the fact that the
identity mapping of $(\mathbb{H}P^{N},g_{0})$ does not minimize $4$-energy in
its homotopy class, contrary to our results for the $1$-energy of mappings of
$(\mathbb{R}P^{n},g_{0})$ and the $2$-energy of mappings of
$(\mathbb{C}P^{N},g_{0})$, mirrors a surprising difference between the
systolic geometry of the projective planes $\mathbb{R}P^{2}$,
$\mathbb{C}P^{2}$ and $\mathbb{H}P^{2}$. These systolic results have several
connections to the results in this paper. Our results also have some
connections to the Blaschke conjecture and its generalizations to
$\mathbb{C}P^{N}$, $\mathbb{H}P^{N}$ and $\mathcal{C}aP^{2}$. We will finish
this introduction by briefly discussing the relationships between these
results and giving an outline of the rest of the paper.

Pao Ming Pu’s inequality, the first published result in systolic geometry,
gives a lower bound for the area of a Riemannian metric on the real projective
plane in terms of the minimum length of its non-contractible curves:

######  Theorem 1.5 (Pu’s Inequality, [Pu52]).

Let $g$ be a Riemannian metric on $\mathbb{R}P^{2}$. Let
$A(\mathbb{R}P^{2},g)$ be its area and $L^{\star}(g)$ its systole, that is,
the minimum length of a non-contractible closed curve in
$(\mathbb{R}P^{2},g)$. Then:

$\displaystyle
A(\mathbb{R}P^{2},g)\geq\left(\frac{2}{\pi}\right)L^{\star}(g)^{2}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.5)

Equality holds if and only if $(\mathbb{R}P^{2},g)$ has constant curvature.

Pu’s theorem follows from Croke’s proof of the $p=2$ case of Theorem 1.1 in
[Cr87] – we will discuss this in Section 2. Croke’s reasoning in the last
section of [Cr87] can also be used to show that the canonical metric on
$\mathbb{R}P^{n}$ is infinitesimally optimal for an inequality of the form
$Vol(\mathbb{R}P^{n},g)\geq G_{n}L^{\star}(g)^{n}$ for all $n\geq 3$. The
results of Gromov [Gr83] imply that such an inequality holds with a positive
constant $G_{n}$ for all $n\geq 2$, but for $n\geq 3$ the optimal constant
$G_{n}$ is currently not known.

Gromov has also proven an inequality for complex projective space which is
analogous to Pu’s inequality, in terms of an invariant known as the stable
$2$-systole. For a Riemannian metric $g$ on $\mathbb{C}P^{N}$, this can be
defined as follows: let $\mu_{k}(g)$ be the minimum area of a $2$-dimensional
current representing $k\in H_{2}(\mathbb{C}P^{N};\mathbb{Z})\cong\mathbb{Z}$.
Then the stable $2$-systole $A^{\star}(g)$ of $g$ is:

$\displaystyle
A^{\star}(g)=\lim\limits_{k\to\infty}\left(\frac{1}{k}\right)\displaystyle\mu_{k}(g).\vskip
12.0pt plus 4.0pt minus 4.0pt$ (1.6)

######  Theorem 1.6 (Gromov’s Stable Systolic Inequality for
$\mathbb{C}P^{N}$, [Gr81], see also [Gr07, BKSW09]).

Let $g$ be a Riemannian metric on $\mathbb{C}P^{N}$, let
$Vol(\mathbb{C}P^{N},g)$ be its volume and $A^{\star}(g)$ its stable
$2$-systole as above. Then:

$\displaystyle Vol(\mathbb{C}P^{N},g)\geq\frac{A^{\star}(g)^{N}}{N!}.$ (1.7)

As in Theorem 1.5, equality holds for the canonical metric $g_{0}$ on
$\mathbb{C}P^{N}$ in Theorem 1.6. Unlike the rigidity of the equality case in
Pu’s inequalty (1.5), however, equality also holds for all Kähler metrics on
$\mathbb{C}P^{N}$. As with the broader characterization of equality in Theorem
1.3 for complex projective space, compared to Theorem 1.1 for real projective
space, this is related to the fact that for any Kähler metric $\widetilde{g}$
on $\mathbb{C}P^{N}$, complex curves are calibrated and thus area minimizing
in their homology classes.

A result analogous to Pu’s and Gromov’s inequalities (1.5) and (1.7) holds for
the quaternionic projective plane. Like the result for $\mathbb{C}P^{N}$ in
Theorem 1.6, this inequality is in terms of the stable $4$-systole, which is
defined by a limit for $H_{4}(\mathbb{H}P^{2};\mathbb{Z})\cong\mathbb{Z}$ as
in (1.6). However unlike the results for $\mathbb{R}P^{2}$ and
$\mathbb{C}P^{2}$ in Pu’s and Gromov’s inequalities, Bangert, Katz, Shnider
and Weinberger have shown that the canonical metric on $\mathbb{H}P^{2}$ is
not optimal for this inequality:

######  Theorem 1.7 ([BKSW09]).

Let $g$ be a Riemannian metric on $\mathbb{H}P^{2}$ and $B^{\star}(g)$ the
stable $4$-systole of $g$ (as defined in [Gr81, BKSW09]). There is a positive
constant $D_{2}$, independent of $g$, such that:

$\displaystyle Vol(\mathbb{H}P^{2},g)\geq D_{2}B^{\star}(g)^{2}.\vskip 6.0pt
plus 2.0pt minus 2.0pt$ (1.8)

The optimal constant in (1.8) satisfies $\frac{1}{6}\geq
D_{2}\geq\frac{1}{14}$, which excludes the value $\frac{3}{10}$ of the
canonical metric.

The proof of the characterization of equality in Theorem 1.1 for $p=1$ is
different from the proof for $p>1$ and uses the characterization of the
canonical metric as the only Riemannian metric on $\mathbb{R}P^{n}$ for which
the first conjugate locus of each point $x_{0}$ consists of a single point (in
fact $x_{0}$ itself). Blaschke conjectured that this was the case and it was
proven by the combined work of Berger, Green, Kazdan and Yang, cf. [Gn62,
Be12]. The Blaschke conjecture therefore implies that, among mappings which
are immersions, the equality case in Theorem 1.1 for $p=1$ is the same as for
$p>1$. There are conjectures similar to the Blaschke conjecture for
$\mathbb{C}P^{N}$, $\mathbb{H}P^{N}$ and $\mathcal{C}aP^{2}$ which are open.
However, the statement which follows from the Blaschke conjecture for
$\mathbb{R}P^{n}$ in Theorem 1.1 is false for $\mathbb{C}P^{2}$: there are
holomorphic mappings of $\mathbb{C}P^{2}$ which are not isometries of the
canonical metric. These mappings minimize $2$-energy in their homotopy class
but by Theorem 1.2, they do not minimize $p$-energy for $p>2$. Therefore, the
equality case for $p=2$ in our results for $\mathbb{C}P^{2}$ is strictly
larger than for $p>2$.

An outline of this paper is as follows:

In Section 2 we will define the $p$-energy of a mapping of a Riemannian
manifold and establish some of its basic properties. In Section 3 we will
prove Theorem 1.1. In Section 4 we will prove Theorem 4.1, of which Theorem
1.2 is a special case, and Theorem 1.3. In Section 5, we will prove Theorem
1.4. The proof of Theorem 1.4 uses the twistor fibration
$\Psi:\mathbb{C}P^{2N+1}\rightarrow\mathbb{H}P^{N}$, and we will discuss some
background related to the twistor fibration at the beginning of Section 5.

Throughout, we will write $\sigma(m)$ for the volume of the unit sphere in
$\mathbb{R}^{m+1}$.

### Acknolwedgements:

I am very happy to thank Christopher Croke, Joseph H.G. Fu, Mikhail Katz,
Frank Morgan and Michael Usher for their input and feedback about this work.

##  2\. Energy Functionals of Mappings

In this section, we will define and study the $p$-energy of a mapping. We will
prove two elementary results, Lemmas 2.2 and 2.4, which we will use in the
proof of our main theorems below.

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

$\displaystyle E_{2}(F)=\int\limits_{M}|dF_{x}|^{2}dVol_{g},\vskip 12.0pt plus
4.0pt minus 4.0pt$ (2.1)

where $|dF_{x}|$ is the norm of $dF:T_{x}M\rightarrow T_{F(x)}N$ at a point
$x$ where $F$ is differentiable. We note that many authors define the energy
to be one half the expression in (2.1).

There are many equivalent ways to define the energy of a mapping – these are
discussed by Eells and Sampson in [ES64]. In this work, they initiated the
study of the critical points of the energy as a functional on mappings from
$(M,g)$ to $(N,h)$. These critical mappings are known as harmonic maps and
have many important connections to minimal submanifold theory, Kähler geometry
and several other topics in differential geometry and analysis.

The energy in (2.1) fits naturally into a $1$-parameter family of functionals:

######  Definition 2.1.

Let $F:(M^{m},g)\rightarrow(N^{n},h)$ be a Lipschitz mapping of closed
Riemannian manifolds. For $p\geq 1$, the $p$-energy of $F$ is:

$\displaystyle\int\limits_{M}|dF_{x}|^{p}dVol_{g}.$ (2.2)

The pointwise quantity $|dF_{x}|^{p}$, where defined, is called the $p$-energy
density. We will denote this $e_{p}(F)_{x}$. It will be helpful to keep in
mind that wherever this can be defined, i.e. at all $x\in M$ at which $F$ is
differentiable, $F^{*}h$ is a positive semidefinite, symmetric bilinear form
on $T_{x}M$ which can be diagonalized relative to $g$. Letting
$e_{1},e_{2},\cdots,e_{m}$ be an orthonormal basis for $T_{x}M$ (relative to
$g$) of eigenvectors for $F^{*}h$, we then have:

$\displaystyle|dF_{x}|^{2}=\sum\limits_{i=1}^{m}|dF(e_{i})|^{2}.\vskip 12.0pt
plus 4.0pt minus 4.0pt$ (2.3)

This gives the following elementary lower bound for the $p$ energy of
$F:(M^{m},g)\rightarrow(N^{n},h)$ for $p\geq dim(M)$:

######  Lemma 2.2.

Let $F:(M^{m},g)\rightarrow(N^{n},h)$ be a Lipschitz mapping, and let
$Vol_{g}(M,F^{*}h)$ be:

$\displaystyle\int\limits_{M}|\det(dF_{x})|dVol_{g},\vskip 6.0pt plus 2.0pt
minus 2.0pt$ (2.4)

where $det(dF_{x})$ is $0$ if $rk(dF_{x})<m$ and is the determinant of
$dF:T_{x}M\rightarrow dF(T_{x})\subseteq T_{F(x)}N$ if $rk(dF_{x})=m<n$.

Then for $p\geq m$,

$\displaystyle E_{p}(F)\geq
m^{\frac{p}{2}}\frac{Vol_{g}(M,F^{*}h)^{\frac{p}{m}}}{Vol(M,g)^{\frac{p-m}{m}}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (2.5)

For $p=m$, equality holds if and only if $dF_{x}$ is a homothety at almost all
$x\in M$ at which $F$ is differentiable. For $p>m$, equality holds if and only
if $dF_{x}$ is a homothety, by a constant factor $C_{F}$, at almost all $x\in
M$ at which $F$ is differentiable.

###### Proof.

For $x$ in $M$ at which $F$ is differentiable, let $e_{1},e_{2},\cdots e_{m}$
be an orthonormal basis of eigenvectors for $F^{*}h$ as in (2.3). By (2.3) and
the arithmetic-geometric mean inequality,

$\displaystyle
e_{m}(F)_{x}=\left(\sum\limits_{i=1}^{m}|dF(e_{i})|^{2}\right)^{\frac{p}{2}}\geq\left(m\sqrt[m]{\prod\limits_{i=1}^{m}|dF(e_{i})|^{2}}\right)^{\frac{p}{2}}\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=m^{\frac{p}{2}}|\det(dF_{x})|^{\frac{p}{m}}.\vskip 12.0pt plus
4.0pt minus 4.0pt$ (2.6)

For $p=m$, this immediately implies (2.5). Equality holds if and only if it
holds a.e. in (2.6), and equality in (2.6) holds at $x$ precisely if $dF_{x}$
is a homothety.

For $p>m$, by (2.6) and Hölder’s inequality,

$\displaystyle E_{p}(F)\geq
m^{\frac{p}{2}}\int\limits_{M}|\det(dF_{x})|^{\frac{p}{m}}dVol_{g}\geq
m^{\frac{p}{2}}\frac{Vol_{g}(M,F^{*}h)^{\frac{p}{m}}}{Vol(M,g)^{\frac{p-m}{m}}}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (2.7)

Equality holds if and only if it holds a.e. in (2.6), as in the case $p=m$,
and in Hölder’s inequality in (2.7). We have equality in Hölder’s inequality
precisely if $|\det(dF_{x})|$ is a.e. equal to a constant $C_{F}$. ∎

Note that in Lemma 2.2, equality for any $p>m$ implies equality for all $p\geq
m$.

If $F$ is a smooth mapping, the equality condition for $p=m$ in Lemma 2.2 says
that $F$ is a semiconformal mapping, that is $F^{*}h=\varphi(x)g$ for a
nonnegative function $\varphi(x)$ on $M$, and the equality condition for $p>m$
says that $F$ is a homothety, i.e. $F^{*}h$ is a rescaling of $g$. This
generalizes the well-known fact that for mappings of surfaces, the energy is
pointwise bounded below by the area of the image, with equality precisely
where the mapping is conformal. The uniformization theorem implies that every
Riemannian metric $g$ on $\mathbb{R}P^{2}$ is conformally equivalent to a
constant curvature metric $g_{0}$, which is unique up to scale. Letting
$g=\varphi(x)g_{0}$, Lemma 2.2 implies that
$Id:(\mathbb{R}P^{2},g_{0})\rightarrow(\mathbb{R}P^{2},g)$ is energy-
minimizing in its homotopy class and has energy equal to
$2A(\mathbb{R}P^{2},g)$ (with the energy defined with our normalization in
(2.1)). Pu’s Theorem 1.5 is then a special case of Croke’s lower bound for the
$2$-energy of mappings $F:(\mathbb{R}P^{n},g_{0})\rightarrow(M,g)$.

The following formula for the energy density is used in Croke’s results in
[Cr87] and will also be used throughout the proofs of our results below:

######  Lemma 2.3 (See [Cr87]).

Let $F:(M^{m},g)\rightarrow(N^{n},h)$ be a Lipschitz mapping of Riemannian
manifolds and $x\in M$ with $F$ differentiable at $x$.

$\displaystyle|dF_{x}|^{2}=\frac{m}{\sigma(m-1)}\int\limits_{U_{x}M}|dF(\vec{u})|^{2}d\vec{u}.$
(2.8)

The identity in (2.8) is the basis for the following formula for the
$2$-energy of a mapping $F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(M,g)$,
where $\widetilde{g}$ is a Kähler metric on $\mathbb{C}P^{N}$. This result is
an elementary example of the type of arguments and calculations we will employ
below and is also an important lemma in several of them:

######  Lemma 2.4.

Let $g_{0}$ be the canonical metric on $\mathbb{C}P^{N}$, normalized so that
its sectional curvature $K$ satisfies $1\leq K\leq 4$, and let $\widetilde{g}$
be a Kähler metric cohomologous to $g_{0}$. Let $\mathcal{L}(\mathbb{C}P^{N})$
be the family of linearly embedded $1$-dimensional complex projective
subspaces $\mathbb{C}P^{1}\subseteq\mathbb{C}P^{N}$, let
$U(\mathbb{C}P^{N},\widetilde{g})$ be the unit tangent bundle of
$(\mathbb{C}P^{N},\widetilde{g})$ and let
$\mathsf{T}:U(\mathbb{C}P^{N},\widetilde{g})\rightarrow\mathcal{L}(\mathbb{C}P^{N})$
be the mapping which sends a unit tangent vector $\vec{u}$ to
$(\mathbb{C}P^{N},\widetilde{g})$ to the unique element of
$\mathcal{L}(\mathbb{C}P^{N})$ to which it is tangent. Let
$d\widetilde{\mathcal{M}}$ be the measure on $\mathcal{L}(\mathbb{C}P^{N})$
which is pushed forward from the Riemannian volume on
$U(\mathbb{C}P^{N},\widetilde{g})$ via $\mathsf{T}$.

Let $F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(M,g)$ be a Lipschitz mapping
to a Riemannian manifold $(M,g)$, and for
$\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$, let $F|_{\mathcal{P}}$ be the
mapping restricted to $\mathcal{P}$. Note that
$F|_{\mathcal{P}}:(\mathcal{P},\widetilde{g})\rightarrow(M,g)$ is also
Lipschitz. Then:

$\displaystyle E_{2}(F)=\frac{2\pi
N}{\sigma(2N-1)}\int\limits_{\mathcal{L}(\mathbb{C}P^{N})}E_{2}(F|_{\mathcal{P}})d\widetilde{\mathcal{M}}.$
(2.9)

###### Proof.

By (2.8), we have:

$\displaystyle
E_{2}(F)=\int\limits_{\mathbb{C}P^{N}}|dF_{x}|^{2}dVol_{\widetilde{g}}=\frac{2N}{\sigma(2N-1)}\int\limits_{U(\mathbb{C}P^{N},\widetilde{g})}|dF(\vec{u})|^{2}d\vec{u}\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=\frac{2N}{\sigma(2N-1)}\int\limits_{\mathcal{L}(\mathbb{C}P^{N})}\int\limits_{U(\mathcal{P},\widetilde{g})}|dF(\vec{u})|^{2}\
d\vec{u}\ d\widetilde{\mathcal{M}},\vskip 12.0pt plus 4.0pt minus 4.0pt$

where $U(\mathcal{P},\widetilde{g})$ is the unit tangent bundle of a complex
projective line $\mathcal{P}$ in the metric $\widetilde{g}|_{\mathcal{P}}$.
Using (2.8) again,

$\displaystyle\frac{2N}{\sigma(2N-1)}\int\limits_{\mathcal{L}(\mathbb{C}P^{N})}\int\limits_{U(\mathcal{P},\widetilde{g})}|dF(\vec{u})|^{2}\
d\vec{u}\ d\widetilde{\mathcal{M}}=\frac{2\pi
N}{\sigma(2N-1)}\int\limits_{\mathcal{L}(\mathbb{C}P^{N})}\int\limits_{\mathcal{P}}|dF|_{\mathcal{P}_{x}}|^{2}\
dx\ d\widetilde{\mathcal{M}}\vskip 12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=\frac{2\pi
N}{\sigma(2N-1)}\int\limits_{\mathcal{L}(\mathbb{C}P^{N})}E_{2}(F|_{\mathcal{P}})d\widetilde{\mathcal{M}}.$

∎

One can derive similar formulas for the energy of mappings of
$\mathbb{R}P^{n}$, $\mathbb{H}P^{N}$ and the Cayley plane $\mathcal{C}aP^{2}$
as integrals over the spaces of linearly embedded $\mathbb{R}P^{1}$,
$\mathbb{H}P^{1}$ and $\mathcal{C}aP^{1}\cong S^{8}$, and over the space of
geodesics in the the sphere $(S^{n},g_{0})$. The formula of this type for the
energy of mappings $F:(\mathbb{R}P^{n},g_{0})\rightarrow(M,g)$ plays a key
part in Croke’s proof of the $p=2$ case of Theorem 1.1 in [Cr87]. Even in the
cases $\mathbb{H}P^{N}$, $\mathcal{C}aP^{2}$ and $S^{n},n\geq 3$, where one
knows that the identity mapping is homotopic to maps with arbitrarily small
energy, one can use such a formula to show that a family of mappings whose
energies decay to $0$ must also have energies decaying to $0$ when restricted
to almost all linear subpsaces $\mathbb{H}P^{k}\subseteq\mathbb{H}P^{N}$,
$\mathcal{C}aP^{1}\subseteq\mathcal{C}aP^{2}$, and almost all great subspheres
$S^{k}\subseteq S^{n}$.

We end this section with a few comments about the regularity of harmonic and
energy-minimizing maps:

Continuous, weakly harmonic maps are smooth [Aub13, Ch.10]. In particular, any
continuous map which minimizes energy in its homotopy class is smooth.
Assuming only Lipschitz regularity, mappings which realize equality for $p=2$
in Theorems 1.1, 1.2, 1.3 and 4.1 are therefore $C^{\infty}$. However, unless
one has established that equality holds for $p=2$, one cannot assume
smoothness because for $p\neq 2$ there are $p$-energy minimizing maps which
are $C^{1,\alpha}$ for $\alpha<1$ but are not $C^{2}$ [EL88, Section 3]. In
dimensions $3$ and greater, $p$-energy minimizing maps also need not be
continuous [EL88, Section 3]. White has shown [Wh88] that $p$-energy
minimizing sequences of maps of compact Riemannian manifolds converge, in an
appropriate topology, to mappings which belong to a Sobolev space of mappings
and have well-defined homotopy classes when restricted to lower-dimensional
skeleta of their domain. However a $p$-energy minimizing sequence of maps in
one homotopy class can converge, in a weak sense, to a map in another homotopy
class – for example, on $(S^{n},g_{0})$, a family of conformal dilations with
energy decaying to $0$, in the homotopy class of the identity, converges
weakly to a constant map. In light of these results, Lipschitz regularity is a
stronger assumption than is natural for $p$-energy minimizing maps in general.
However, the Lipschitz condition works well in our setting because it is
inherited by the restriction of maps $F:(M,g)\rightarrow(N,h)$ to any
submanifold of $M$. In our case, this implies that a Lipshitz mapping
$F:(\mathbb{K}P^{N},g_{0})\rightarrow(M,g)$ will be Lipshitz when restricted
to all $\mathbb{K}P^{1}\subseteq\mathbb{K}P^{N}$, where $\mathbb{K}$ is
$\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. For $\mathbb{H}P^{N}$, we will
also consider the restriction of $F$ to totally geodesic submanifolds
isometric to $(\mathbb{C}P^{2},g_{0})$. This will allow us to draw conclusions
about the mapping of $\mathbb{K}P^{N}$ from its behavior along lower-
dimensional subspaces.

##  3\. Real Projective Space

In this section, we will prove Theorem 1.1. We will write $g_{0}$ for the
canonical metric on $\mathbb{R}P^{n}$, normalized to have constant sectional
curvature $1$. We define the space of oriented geodesics in
$(\mathbb{R}P^{n},g_{0})$ to be the quotient of the unit tangent bundle
$U(\mathbb{R}P^{n},g_{0})$ by the geodesic flow. We denote this
$\mathcal{G}(\mathbb{R}P^{n},g_{0})$ and we equip
$\mathcal{G}(\mathbb{R}P^{n},g_{0})$ with the measure $d\gamma$ pushed forward
from the measure on $U(\mathbb{R}P^{n},g_{0})$, so that:

$\displaystyle
Vol(\mathcal{G}(\mathbb{R}P^{n},g_{0}))=\frac{Vol(U(\mathbb{R}P^{n},g_{0}))}{\pi}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (3.1)

Because $Vol(U(\mathbb{R}P^{n},g_{0}))=\frac{\sigma(n)\sigma(n-1)}{2}$, we
have:

$\displaystyle
Vol(\mathcal{G}(\mathbb{R}P^{n},g_{0}))=\frac{\sigma(n)\sigma(n-1)}{2\pi}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (3.2)

###### Proof of Theorem 1.1.

By the formula (2.8) for the energy density and the Cauchy-Schwarz inequality,

$\displaystyle
E_{p}(F)=\int\limits_{\mathbb{R}P^{n}}|dF_{x}|^{p}dVol_{g_{0}}=\int\limits_{\mathbb{R}P^{n}}\left(\frac{n}{\sigma(n-1)}\int\limits_{U_{x}\mathbb{R}P^{n}}|dF(\vec{u})|^{2}d\vec{u}\right)^{(\frac{p}{2})}dVol_{g_{0}}\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle\geq\frac{n^{\frac{p}{2}}}{\sigma(n-1)^{p}}\int\limits_{\mathbb{R}P^{n}}\left(\int\limits_{U_{x}\mathbb{R}P^{n}}|dF(\vec{u})|d\vec{u}\right)^{p}dVol_{g_{0}}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (3.3)

For $p=1$, this says:

$\displaystyle
E_{1}(F)\geq\frac{\sqrt{n}}{\sigma(n-1)}\int\limits_{U(\mathbb{R}P^{n},g_{0})}|dF(\vec{u})|d\vec{u}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (3.4)

For $p>1$, by (3.3) and Hölder’s inequality,

$\displaystyle
E_{p}(F)\geq\frac{2^{p-1}n^{\frac{p}{2}}}{\sigma(n-1)^{p}\sigma(n)^{p-1}}\left(\int\limits_{U(\mathbb{R}P^{n},g_{0})}|dF(\vec{u})|d\vec{u}\right)^{p}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (3.5)

For each $\gamma\in\mathcal{G}(\mathbb{R}P^{n},g_{0})$, $F\circ\gamma$ is an
oriented Lipschitz $1$-cycle in $(M,g)$. Letting $|F\circ\gamma|$ be its mass,
and writing $\gamma:[0,\pi]\rightarrow\mathbb{R}P^{n}$ for a unit-speed
parametrization of $\gamma$ and $F\circ\gamma:[0,\pi]\rightarrow M$ for the
associated parametrization of $F\circ\gamma$, we then have:

$\displaystyle|F\circ\gamma|=\int\limits_{0}^{\pi}|(F\circ\gamma)^{\prime}(t)|dt,\vskip
12.0pt plus 4.0pt minus 4.0pt$

where we have used that the Lipschitz mapping $F\circ\gamma:[0,\pi]\rightarrow
M$ is differentiable almost everywhere. For all $t$ such that $F$ is
differentiable at $\gamma(t)$,
$(F\circ\gamma)^{\prime}(t)=dF(\gamma^{\prime}(t))$. By Fubini’s theorem, the
right-hand sides of (3.4) and (3.5) can therefore be rewritten as an integral
over $\mathcal{G}(\mathbb{R}P^{n},g_{0})$, which implies:

$\displaystyle
E_{p}(F)\geq\frac{2^{p-1}n^{\frac{p}{2}}}{\sigma(n-1)^{p}\sigma(n)^{p-1}}\left(\int\limits_{\mathcal{G}(\mathbb{R}P^{n},g_{0})}|F\circ\gamma|d\gamma\right)^{p}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (3.6)

Because each geodesic $\gamma$ represents a generator of
$\pi_{1}(\mathbb{R}P^{n})$, $|F\circ\gamma|\geq L^{\star}$, and therefore,

$\displaystyle
E_{p}(F)\geq\frac{2^{p-1}n^{\frac{p}{2}}}{\sigma(n-1)^{p}\sigma(n)^{p-1}}\left(Vol(\mathcal{G}(\mathbb{R}P^{n},g_{0}))L^{\star}\right)^{p},\vskip
12.0pt plus 4.0pt minus 4.0pt$

which is (1.1).

Suppose equality holds for $p=1$.

This implies that equality holds in the Cauchy-Schwarz inequality in (3.3) for
a.e. $x\in\mathbb{R}P^{n}$. For all $x$ at which $F$ is differentiable and for
which this equality holds, $|dF_{x}(\vec{u})|$ depends only on $x$. This
implies that $F^{*}g$ is a.e. equal to $\varphi(x)g_{0}$, where $\varphi(x)$
is a nonnegative function on $\mathbb{R}P^{n}$. Because all
$\gamma\in\mathcal{G}(\mathbb{R}P^{n},g_{0})$ map to rectifiable currents
$F\circ\gamma$ with well-defined lengths in $(M,g)$, equality also implies
that for almost all $\gamma$, $|F\circ\gamma|=L^{\star}$. Because
$|F\circ\gamma|$ is lower semicontinuous on
$\mathcal{G}(\mathbb{R}P^{n},g_{0})$, we in fact have
$|F\circ\gamma|=L^{\star}$ for all $\gamma$. The image via $F$ of each
geodesic $\gamma$ in $(\mathbb{R}P^{n},g_{0})$ is therefore a closed geodesic
in $(M,g)$, of minimal length $L^{\star}$ in its free homotopy class, although
a priori $F\circ\gamma$ may not be parametrized by arc length.

If $F$ is a smooth immersion, then because each geodesic $\gamma$ in
$(\mathbb{R}P^{n},g_{0})$ maps to a closed geodesic in $M$, the image of $F$
is a totally geodesic submanifold, and because $F\circ\gamma$ is of minimal
length in its free homotopy class in $(M,g)$, $F^{*}g$ is a Blaschke metric on
$\mathbb{R}P^{n}$, cf. Remark 3.1 below. By the Berger-Green-Kazdan-Yang proof
of the Blaschke conjecture [Gn62, Be12], $F^{*}g$ is therefore isometric to a
round metric. This does not yet imply that $F$ is an isometry or a homothety.
However, letting
$\psi:(\mathbb{R}P^{n},F^{*}g)\rightarrow(\mathbb{R}P^{n},g_{0})$ be an
isometry (or homothety), we then have $\psi^{*}g_{0}=\varphi(x)g_{0}$, where
$\varphi(x)$ is the semi-conformal factor as above and is in fact a conformal
factor, i.e. is everywhere-defined and positive, because $F$ is an immersion.
By the classification of conformal diffeomorphisms of the round sphere
$(S^{n},g_{0})$, the mapping $\psi$ is therefore an isometry of $g_{0}$, up to
rescaling, and $F$ is an isometry or homothety onto its image.

Now suppose $p>1$ and equality holds for $p$.

Supposing only that $F$ is Lipschitz, this implies all of the conditions which
follow for Lipshitz mappings which realize equality for $p=1$ and also implies
equality in Hölder’s inequality in (3.5). Equality in Hölder’s inequality
implies that the semiconformal factor $\varphi(x)$ is a.e. equal to a constant
$C_{F}$. This implies that equality holds for all $p\geq 1$. Because equality
holds for $p=2$, $F$ is smooth, so $\varphi(x)$ is an everywhere-defined
constant function and $F$ is a homothety onto its image. Because all
$\gamma\in\mathcal{G}(\mathbb{R}P^{n},g_{0})$ maps to a geodesic of $(M,g)$,
the image of $F$ is a totally geodesic submanifold. ∎

######  Remark 3.1.

To see that the conditions for equality when $p=1$ and $F$ is an immersion
imply that $F^{*}g$ is a Blaschke metric, i.e. that the first conjugate locus
of each point $x_{0}$ in $(\mathbb{R}P^{N},F^{*}g)$ is a single point, in fact
$x_{0}$, note that each unit-speed geodesic
$c:[0,\infty)\rightarrow(\mathbb{R}P^{N},F^{*}g)$ has a conjugate point at
$c(L^{\star})=c(0)$, where all geodesics based at $c(0)$ intersect, and that
this must be the first conjugate point to $c(0)$ along $c$ because
$c([0,L^{\star}])$ is length-minimizing in its homotopy class.

Also, curiously, this part of Theorem 1.1 is false without the stipulation
that $n\geq 2$: any diffeomorphism of
$\mathbb{R}P^{1}=\mathbb{R}/\pi\mathbb{Z}$ has $1$-energy equal to $\pi$. In
fact, any homeomorphism of $\mathbb{R}P^{1}\cong S^{1}$ is differentiable a.e.
and has $1$-energy equal to the identity mapping.

In the next section, we will prove a lower bound similar to Theorem 1.1 for
mappings of any Kähler manifold biholomorphic to $\mathbb{C}P^{N}$. In light
of this result, it might be natural to ask whether the result of Theorem 1.1
applies to a larger family of metrics on $\mathbb{R}P^{n}$. The proof of
Theorem 1.1 would give the same lower bounds for mappings of
$(\mathbb{R}P^{n},\widehat{g})$, where $\widehat{g}$ is any metric on
$\mathbb{R}P^{n}$ all of whose geodesics are closed and non-contractible, but
the results of Lin and Schmidt in [LS17] show that for $n\neq 3$, the only
metrics on $\mathbb{R}P^{n}$ all of whose geodesics are closed are those of
constant curvature. A proof of the Berger conjecture in dimension $n=3$, as in
[GG81] for $n=2$ and [RW17] for $n\geq 4$, would imply that this is the case
for $\mathbb{R}P^{3}$ as well.

We note that for $n\geq 2$, non-isometric projective linear transformations of
$\mathbb{R}P^{n}$ have the length-preserving property for
$\gamma\in\mathcal{G}(\mathbb{R}P^{n},g_{0})$ which is implied by equality for
$p=1$ in Theorem 1.1 but do not minimize $1$-energy in their homotopy class.
In complex projective space, however, non-isometric projective complex linear
transformations are holomorphic and minimize $2$-energy in the homotopy class
of the identity.

We end this section by re-interpreting the fact that the identity mapping of
$(\mathbb{R}P^{n},g_{0})$ minimizes $1$-energy in its homotopy class in terms
of a calibration-like property of the canonical $1$-form on the unit tangent
bundle of $(\mathbb{R}P^{n},g_{0})$:

For any oriented integral $1$-chain $\sum_{i}a_{i}\tau_{i}$ in
$\mathbb{R}P^{n}$, where $a_{i}\in\mathbb{Z}$ and
$\tau_{i}:\Delta^{1}\rightarrow\mathbb{R}P^{n}$ are oriented $1$-simplices,
one can define a $1$-current in the unit tangent bundle
$U(\mathbb{R}P^{n},g_{0})$ by associating each point $\tau_{i}(t)$ at which
$\tau_{i}$ is differentiable to the unit vector in
$T_{\tau_{i}(t)}\mathbb{R}P^{N}$ tangent to $\tau_{i}$ in the oriented
direction, with multiplicity $a_{i}$. Letting $\alpha$ be the canonical
$1$-form on the unit tangent bundle of $(\mathbb{R}P^{n},g_{0})$, the mass of
this current is its pairing with $\alpha$. The current of this type associated
to any non-contractible closed curve in $\mathbb{R}P^{n}$ has mass greater
than or equal to $\pi$, and any Lipshitz mapping
$F:(\mathbb{R}P^{n},g_{0})\rightarrow(\mathbb{R}P^{n},g_{0})$ homotopic to the
identity sends each $\gamma\in\mathcal{G}(\mathbb{R}P^{n},g_{0})$ to such a
closed curve. This implies the lower bound for the $1$-energy of mappings
homotopic to $Id:(\mathbb{R}P^{n},g_{0})\rightarrow(\mathbb{R}P^{n},g_{0})$ in
Theorem 1.1.

Unlike a calibration, however, $\alpha$ is not closed: $d\alpha$ is the
Liouville symplectic form on the tangent bundle.

##  4\. Complex Projective Space

In this section, we will prove lower bounds for the energy of mappings from
Kähler manifolds biholomorphic to $\mathbb{C}P^{N}$ to Riemannian manifolds,
similar to the estimate for mappings of real projective space in Theorem 1.1.
The equality case in these results is much broader than in Theorem 1.1, but
for mappings from $\mathbb{C}P^{2}$ it implies Theorem 1.2. For mappings from
$\mathbb{C}P^{N}$ to simply connected, compact Kähler manifolds, we will prove
a stronger characterization of equality in Theorem 1.3.

As in Lemma 2.4, we will let $\mathcal{L}(\mathbb{C}P^{N})$ be the space of
linearly embedded $\mathbb{C}P^{1}\subseteq\mathbb{C}P^{N}$. We will use the
fact that all Kähler metrics on $\mathbb{C}P^{N}$ are cohomologous, up to
rescaling, to the canonical metric $g_{0}$, and that for any Kähler metric
$\widetilde{g}$ cohomologous to $g_{0}$,
$Vol(\mathbb{C}P^{N},\widetilde{g})=\frac{\pi^{N}}{N!}$ and for all
$\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$,
$A(\mathcal{P},\widetilde{g})=\pi$. As in Lemma 2.4, we will let
$d\widetilde{\mathcal{M}}$ be the measure on $\mathcal{L}(\mathbb{C}P^{N})$
pushed forward from the measure on the unit tangent bundle
$U(\mathbb{C}P^{N},\widetilde{g})$ of $(\mathbb{C}P^{N},\widetilde{g})$ via
the mapping
$\mathsf{T}:U(\mathbb{C}P^{N},\widetilde{g})\rightarrow\mathcal{L}(\mathbb{C}P^{N})$
which sends non-zero tangent vectors to $\mathbb{C}P^{N}$ to the unique
element of $\mathcal{L}(\mathbb{C}P^{N})$ to which they are tangent. The total
volume of $\mathcal{L}(\mathbb{C}P^{N})$ in the measure
$d\widetilde{\mathcal{M}}$ is:

$\displaystyle
Vol(\mathcal{L}(\mathbb{C}P^{N}),d\widetilde{\mathcal{M}})=\frac{Vol(U(\mathbb{C}P^{N},\widetilde{g}))}{Vol(U(\mathbb{C}P^{1},\widetilde{g}))}=\frac{\sigma(2N-1)\pi^{N-2}}{2N!}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (4.1)

It is a fundamental fact of Kähler geometry that closed complex submanifolds
of a compact Kähler manifold $(X,h)$ are area-minimizing in their homology
classes – this follows from the calibrated structure given by the Kähler form
[HL82]. Moreover, for any compact complex curve $\Sigma$ and any holomorphic
mapping $F$ of $\Sigma$ to a compact Kähler manifold $(X,h)$, it follows from
Lemma 2.2 that:

$\displaystyle E_{2}(F)=2Area(\Sigma,F^{*}h),\vskip 12.0pt plus 4.0pt minus
4.0pt$ (4.2)

where $Area(\Sigma,F^{*}h)$ is equal to the area of $F(\Sigma)$ and is the
minimum area of any cycle representing the class of $F(\Sigma)$ in
$H_{2}(X;\mathbb{Z})$. For any holomorphic mapping
$F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(X,h)$, letting $A^{\star}$ this
minimum area for $F(\mathcal{P})$,
$\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$, Lemma 2.4 then implies:

$\displaystyle E_{2}(F)=\frac{2\pi
N}{\sigma(2N-1)}\int\limits_{\mathcal{L}(\mathbb{C}P^{N})}E_{2}(F|_{\mathcal{P}})d\widetilde{\mathcal{M}}$
$\displaystyle=\frac{2\pi N}{\sigma(2N-1)}\times
Vol(\mathcal{L}(\mathbb{C}P^{N}),d\widetilde{\mathcal{M}})\times
2A^{\star}=\frac{2\pi^{N-1}}{(N-1)!}A^{\star}.\vskip 12.0pt plus 4.0pt minus
4.0pt$ (4.3)

We will see that this coincides with a sharp lower bound for the energy of any
Lipschitz mapping from a Kähler manifold $(\mathbb{C}P^{N},\widetilde{g})$ to
a Riemannian manifold $(M,g)$:

######  Theorem 4.1.

Let $\widetilde{g}$ be a Kähler metric on $\mathbb{C}P^{N}$, which we suppose
without loss of generality is cohomologous to the canonical metric $g_{0}$
with sectional curvature $K$ satisfying $1\leq K\leq 4$. Let
$F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(M^{m},g)$ be a Lipschitz mapping
to a Riemannian manifold $(M,g)$ and $A^{\star}$ the infimum of the areas of
mappings $f:S^{2}\rightarrow(M,g)$ in the free homotopy class of
$F_{*}(\mathbb{C}P^{1})$.

For all $p\geq 2$,

$\displaystyle
E_{p}(F)\geq\frac{\pi^{N}}{N!}\left(\frac{2N}{\pi}A^{\star}\right)^{\frac{p}{2}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (4.4)

If equality holds for $p=2$, then $F^{*}g$ is a positive semidefinite
Hermitian bilinear form on $\mathbb{C}P^{N}$. In particular, on any
neighborhood $\mathcal{V}$ on which $F$ is an immersion, $F^{*}g$ is a
Hermitian metric. Moreover, for any neighborhood $\mathcal{V}$ on which $F$ is
an immersion, $F(\mathcal{V})$ is a minimal submanifold of $(M,g)$, the second
fundamental form of $F(\mathcal{V})$ in $(M,g)$ can be diagonalized by a
unitary basis of $F^{*}g$ and, letting $\omega^{*}$ denote the Kähler form of
$F^{*}g$, $d\omega^{*}$ vanishes on all subspaces of $T_{x}\mathcal{V}$ of
complex dimension $2$, for all $x\in\mathcal{V}$.

If $p>2$ and equality holds for $p$, then equality holds for all $p\geq 2$ and
$F$ has constant energy density.

If $p>2$, $\mathbb{C}P^{2}\subseteq\mathbb{C}P^{N}$ is a linearly embedded
subspace and $F|_{\mathbb{C}P^{2}}$ is an immersion and realizes equality for
$p$, then $F|_{\mathbb{C}P^{2}}$ is a homothety onto its image.

###### Proof.

Throughout the proof, we will let $I$ denote the complex structure of
$\mathbb{C}P^{N}$. By (2.8),

$\displaystyle
E_{p}(F)=\int\limits_{\mathbb{C}P^{N}}|dF_{x}|^{p}dVol_{\widetilde{g}}=\left(\frac{2N}{\sigma(2N-1)}\right)^{\frac{p}{2}}\int\limits_{\mathbb{C}P^{N}}\left(\int\limits_{U_{x}\mathbb{C}P^{N}}|dF(\vec{u})|^{2}d\vec{u}\right)^{\frac{p}{2}}dVol_{\widetilde{g}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (4.5)

For $x\in\mathbb{C}P^{N}$, let $G_{1}^{\mathbb{C}}(x)$ be the space of complex
lines in $T_{x}\mathbb{C}P^{N}$. $G_{1}^{\mathbb{C}}(x)$ is then a copy of
$\mathbb{C}P^{N-1}$, and the fibration
$\mathsf{T}:U(\mathbb{C}P^{N},\widetilde{g})\rightarrow\mathcal{L}(\mathbb{C}P^{N})$
factors through a quotient mapping which is given fibrewise by
$U_{x}(\mathbb{C}P^{N},\widetilde{g})\rightarrow G_{1}^{\mathbb{C}}(x)$. For
$\Lambda\in G_{1}^{\mathbb{C}}(x)$, let $U(\Lambda,\widetilde{g})$ be the unit
circle in $\Lambda$ in the metric $\widetilde{g}$. For each such $\Lambda$ at
each point $x$ where $F$ is differentiable, $F^{*}g|_{\Lambda}$ is a positive
semidefinite, symmetric $2$-form and can be diagonalized relative to the
metric $\widetilde{g}|_{\Lambda}$. Letting $\vec{u}_{1},\vec{u}_{2}$ be a set
of eigenvectors for $F^{*}g|_{\Lambda}$ which are orthonormal in
$\widetilde{g}$, by the arithmetic-geometric mean inequality,

$\displaystyle\int\limits_{U(\Lambda,\widetilde{g})}|dF(\vec{u})|^{2}d\vec{u}=\int\limits_{0}^{2\pi}\left(\cos^{2}(\theta)|dF(\vec{u}_{1})|^{2}+\sin^{2}(\theta)|dF(\vec{u}_{2})|^{2}\right)d\theta\vskip
6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle=\pi\left(|dF(\vec{u}_{1})|^{2}+|dF(\vec{u}_{2})|^{2}\right)\geq
2\pi|dF(\vec{u}_{1})||dF(\vec{u}_{2})|=2\pi|\det(dF|_{\Lambda})|.\vskip 12.0pt
plus 4.0pt minus 4.0pt$ (4.6)

For $p=2$, (4.5), (4.6) and Fubini’s theorem imply:

$\displaystyle
E_{2}(F)=\frac{2N}{\sigma(2N-1)}\int\limits_{\mathbb{C}P^{N}}\int\limits_{G_{1}^{\mathbb{C}}(x)}\int\limits_{U(\Lambda,\widetilde{g})}|dF(\vec{u})|^{2}\
d\vec{u}\ d\Lambda\ dVol_{\widetilde{g}}\vskip 12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle\geq\frac{4N\pi}{\sigma(2N-1)}\int\limits_{\mathbb{C}P^{N}}\int\limits_{G_{1}^{\mathbb{C}}(x)}|\det(dF|_{\Lambda})|d\Lambda
dVol_{\widetilde{g}}=\frac{4N\pi}{\sigma(2N-1)}\int\limits_{\mathcal{L}(\mathbb{C}P^{N})}|F(\mathcal{P})|d\widetilde{\mathcal{M}},\vskip
12.0pt plus 4.0pt minus 4.0pt$ (4.7)

where $|F(\mathcal{P})|$ is the area of the image via $F$ of a complex
projective line $\mathcal{P}$ in $\mathbb{C}P^{N}$ and where the measure on
$G_{1}^{\mathbb{C}}(x)$ is its canonical measure as a quotient of the unit
sphere $U_{x}(\mathbb{C}P^{N},\widetilde{g})$. As in the proof of Theorem 1.1,
we have used the fact that $F|_{\mathcal{P}}$ is Lipschitz, so that
$F(\mathcal{P})$ represents a rectifiable $2$-current with a well-defined mass
$|F(\mathcal{P})|$, for all $\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$.

For $p>2$, (4.5), (4.6), Fubini’s theorem and Hölder’s inequality imply:

$\displaystyle
E_{p}(F)\geq\frac{(4N\pi)^{\frac{p}{2}}}{\sigma(2N-1)^{\frac{p}{2}}Vol(\mathbb{C}P^{N})^{\frac{p-2}{2}}}\left(\int\limits_{\mathcal{L}(\mathbb{C}P^{N})}|F(\mathcal{P})|d\widetilde{\mathcal{M}}\right)^{\frac{p}{2}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$ (4.8)

Since $|F(\mathcal{P})|\geq A^{\star}$, (4.7) and (4.8) imply:

$\displaystyle
E_{p}(F)\geq\frac{(4N\pi)^{\frac{p}{2}}}{\sigma(2N-1)^{\frac{p}{2}}Vol(\mathbb{C}P^{N})^{\frac{p-2}{2}}}\left(A^{\star}Vol(\mathcal{L}(\mathbb{C}P^{N},g_{0}))\right)^{\frac{p}{2}},\vskip
6.0pt plus 2.0pt minus 2.0pt$ (4.9)

which is (4.4).

Suppose equality holds for $p=2$.

$F$ is therefore smooth. Equality holds in (4.6) for all $\Lambda\in
G_{1}^{\mathbb{C}}(x)$, at all $x\in\mathbb{C}P^{N}$. Equality in (4.6)
implies that $|dF(\vec{u})|$ is $S^{1}$-invariant on $\Lambda$, where $S^{1}$
represents the unit complex numbers acting by multiplication on
$T_{x}\mathbb{C}P^{N}$. $F^{*}g$ is therefore a positive semidefinite
Hermitian bilinear form on $T_{x}\mathbb{C}P^{N}$. In particular, $F^{*}g$ is
a Hermitian metric on any neighborhood of $\mathbb{C}P^{N}$ on which $F$ is an
immersion. We also have that $|F(\mathcal{P})|=A^{\star}$ for all
$\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$ and, by Lemma 2.4, that
$F|_{\mathcal{P}}$ minimizes energy in its homotopy class of mappings
$F:\mathbb{C}P^{1}\rightarrow(M,g)$ for all $\mathcal{P}\in\mathcal{L}$.

Now let $\mathcal{V}$ be a neighborhood of $\mathbb{C}P^{N}$ such that
$F|_{\mathcal{V}}$ is an immersion. We will show that $F(\mathcal{V})$ is a
minimal submanifold of $(M,g)$, whose second fundamental form can be
diagonalized by a unitary basis of $F^{*}g$.

For $x_{0}\in\mathcal{V}$, let $\vec{n}$ be a unit normal vector in $(M,g)$ to
$F(\mathbb{C}P^{N})$ at $F(x_{0})$. Let $S_{\vec{n}}^{\mathbb{C}P^{N}}$ be the
shape operator of $F(\mathbb{C}P^{N})$ in the normal direction $\vec{n}$. Let
$\vec{u}_{0}$ be a principal vector for $S_{\vec{n}}^{\mathbb{C}P^{N}}$, with
$|S_{\vec{n}}^{\mathbb{C}P^{N}}(\vec{u}_{0})|$ maximal, and let
$\mathcal{P}_{0}=\mathsf{T}(\vec{u}_{0})$. Let $S_{\vec{n}}^{\mathcal{P}_{0}}$
be the shape operator of $\mathcal{P}_{0}$ in the normal direction $\vec{n}$.
Then $\vec{u}_{0}$ is also a principal vector for
$S_{\vec{n}}^{\mathcal{P}_{0}}$

$F(\mathcal{P}_{0}\cap\mathcal{V})$ is minimal in $(M,g)$ because
$F(\mathcal{P}_{0})$ minimizes area in its homotopy class of mappings
$\mathbb{C}P^{1}\rightarrow(M,g)$. By the minimality of
$F(\mathcal{P}_{0}\cap\mathcal{V})$ and the fact that the metric $F^{*}g$ on
$\mathcal{V}$ is Hermitian, $I(\vec{u}_{0})$ is also a principal vector for
$S_{\vec{n}}^{\mathcal{P}_{0}}$ and
$g\left(S_{\vec{n}}^{\mathcal{P}_{0}}(I(\vec{u}_{0})),I(\vec{u}_{0})\right)=-g\left(S_{\vec{n}}^{\mathcal{P}_{0}}(\vec{u}_{0}),\vec{u}_{0}\right)$.
In particular,
$|S_{\vec{n}}^{\mathcal{P}_{0}}(I(\vec{u}_{0}))|=|S_{\vec{n}}^{\mathcal{P}_{0}}(\vec{u}_{0})|$.
Because
$|S_{\vec{n}}^{\mathbb{C}P^{N}}(I(\vec{u}_{0}))|\geq|S_{\vec{n}}^{\mathcal{P}_{0}}(I(\vec{u}_{0}))|$
and $|S_{\vec{n}}^{\mathbb{C}P^{N}}(\vec{u}_{0})|$ is maximal, this implies
that $I(\vec{u}_{0})$ is also a principal vector for
$S_{\vec{n}}^{\mathbb{C}P^{N}}$, and that the principal curvature of
$S_{\vec{n}}^{\mathbb{C}P^{N}}$ along $I(\vec{u}_{0})$ is the negative of its
principal curvature along $\vec{u}_{0}$.

Now let $\vec{u}_{1}$ be a principal vector for
$S_{\vec{n}}^{\mathbb{C}P^{N}}$ which maximizes
$|S_{\vec{n}}^{\mathbb{C}P^{N}}(\vec{u}_{1})|$ in the subspace of
$T_{x_{0}}\mathbb{C}P^{N}$ which is orthogonal via $F^{*}g$ to
$span(\vec{u}_{0},I(\vec{u}_{0}))$ and let $\mathcal{P}_{1}$ be the linear
$\mathbb{C}P^{1}$ to which $\vec{u}_{1}$ is tangent. As above, $\vec{u}_{1}$
is a principal vector for $S_{\vec{n}}^{\mathcal{P}_{1}}$, which implies that
$I(\vec{u}_{1})$ is a principal vector for $S_{\vec{n}}^{\mathcal{P}_{1}}$
whose principal curvature is the negative of the principal curvature of
$\vec{u}_{1}$, and therefore that $\vec{u},I(\vec{u})$ are principal vectors
of $S_{\vec{n}}^{\mathbb{C}P^{N}}$ whose principal curvatures are equal in
magnitude and opposite in sign. Continuing in this way diagonalizes
$S_{\vec{n}}^{\mathbb{C}P^{N}}$ with a unitary basis for
$T_{x_{0}}\mathbb{C}P^{N}$.

Now let $\nabla^{*}$ and $\omega^{*}$ be the connection and Kähler form of
$F^{*}g$ on $\mathcal{V}$, that is, for $\vec{v},\vec{w}\in
T_{x_{0}}\mathbb{C}P^{N}$,
$\omega^{*}(\vec{v},\vec{w})=F^{*}g(I(\vec{v}),\vec{w})$. Let $\vec{v}$ be a
unit tangent vector to $(\mathbb{C}P^{N},F^{*}g)$ at a point $x_{0}$ of
$\mathcal{V}$ and let $\vec{w},I(\vec{w})$ be a pair of unit vectors which
span a complex $1$-dimensional subspace orthogonal to $\vec{v}$ in
$T_{x_{0}}\mathbb{C}P^{N}$, where the norm and orthogonality of $\vec{v}$,
$\vec{w}$, $I(\vec{w})$ are again with respect to $F^{*}g$. Let
$\mathsf{T}(\vec{w})=\mathsf{T}(I(\vec{w}))=\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$.
Choose a system of normal coordinates for $\mathcal{P}$ about $x_{0}$, in the
metric $F^{*}g$, based on the frame $\vec{w},I(\vec{w})$. Extend $\vec{v}$ to
an orthonormal frame for the normal space to $\mathcal{P}$ at $x_{0}$, then
extend this frame to an orthonormal frame field for the normal bundle of
$\mathcal{P}$ in $M$, on the normal coordinate neighborhood of $\mathcal{P}$
defined above. Use this local orthonormal frame field to define a system of
Fermi coordinates about $x_{0}$ in $(\mathcal{V},F^{*}g)$. Let $V$ be the
coordinate vector field which coincides with $\vec{v}$ and $W_{1},W_{2}$ the
coordinate vector fields which coincide with $\vec{w},I(\vec{w})$ at $x_{0}$.
We then have:

$\displaystyle
d\omega^{*}(\vec{v},\vec{w},I(\vec{w}))=V(\omega^{*}(W_{1},W_{2}))-W_{1}(\omega^{*}(V,W_{2}))+W_{2}(\omega^{*}(V,W_{1}))\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=V(F^{*}g(I(W_{1}),W_{2}))-W_{1}(F^{*}g(I(V),W_{2}))+W_{2}(F^{*}g(I(V),W_{1})).\vskip
12.0pt plus 4.0pt minus 4.0pt$ (4.10)

Because $W_{1},W_{2}$ are tangent and $V$ is normal to to the complex
submanifold $\mathcal{P}$ of $\mathbb{C}P^{N}$ and $F^{*}g$ is Hermitian, the
terms $W_{1}(\omega^{*}(V,W_{2}))$ and $W_{2}(\omega^{*}(V,W_{1}))$ vanish,
and we have:

$\displaystyle
d\omega^{*}(\vec{v},\vec{w},I(\vec{w}))=V(\omega^{*}(W_{1},W_{2}))=V(F^{*}g(I(W_{1}),W_{2})).\vskip
12.0pt plus 4.0pt minus 4.0pt$ (4.11)

Although $I(W_{1})$ may not be equal to $W_{2}$ at $x\neq x_{0}$, by (4.11),

$\displaystyle
d\omega^{*}(\vec{v},\vec{w},I(\vec{w}))=F^{*}g(\nabla^{*}_{V}I(W_{1}),I(W_{1}))+F^{*}g(W_{2},\nabla^{*}_{V}W_{2})\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=\frac{1}{2}V\left(F^{*}g(I(W_{1}),I(W_{1}))+F^{*}g(W_{2},W_{2})\right).\vskip
12.0pt plus 4.0pt minus 4.0pt$ (4.12)

Using the fact that $F^{*}g$ is Hermitian, so that
$F^{*}g(I(W_{1}),I(W_{1}))\equiv F^{*}g(W_{1},W_{1})$, and that
$\nabla^{*}_{V}W_{1}-\nabla^{*}_{W_{1}}V=[V,W_{1}]=0$,
$\nabla^{*}_{V}W_{2}-\nabla^{*}_{W_{2}}V=[V,W_{2}]=0$ because $V,W_{1},W_{2}$
are coordinate vector fields from the same Fermi coordinate system defined
above, we then have:

$\displaystyle
d\omega^{*}(\vec{v},\vec{w},I(\vec{w}))=F^{*}g(\nabla^{*}_{V}W_{1},W_{1})+F^{*}g(\nabla^{*}_{V}W_{2},W_{2})\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=F^{*}g(\nabla^{*}_{W_{1}}V,W_{1})+F^{*}g(\nabla^{*}_{W_{2}}V,W_{2}).\vskip
12.0pt plus 4.0pt minus 4.0pt$ (4.13)

This is the negative of the mean curvature of $\mathcal{P}$ at $x_{0}$ in the
normal direction determined by $\vec{v}$ in the metric $F^{*}g$. Because
$F(\mathcal{P}\cap\mathcal{V})$ is minimal, $d\omega^{*}$ vanishes on
$\vec{v},\vec{w},I(\vec{w})$, and therefore on any subspace of
$T_{x_{0}}\mathbb{C}P^{N}$ of complex dimension $2$.

Now suppose $p>2$ and equality holds for $p$.

Assuming only that $F$ is Lipschitz, equality then holds in (4.6) for almost
all $\Lambda\in G_{1}^{\mathbb{C}}(x)$, at almost all $x\in\mathbb{C}P^{N}$.
This implies that equality holds for $p=2$, and therefore that $F$ is smooth.
Equality must also hold in Hölder’s inequality in (4.8). This implies that
$\int_{U_{x}\mathbb{C}P^{N}}|dF(\vec{u})|^{2}d\vec{u}$ is a constant function
of $F$, which implies that $F$ has constant energy density.

Suppose $\mathbb{C}P^{2}\subseteq\mathbb{C}P^{N}$ is a linear subspace, $p>2$,
$F|_{\mathbb{C}P^{2}}$ is an immersion and realizes equality in (4.4) for $p$,
and therefore for all $p\in[2,\infty)$. The equality for $p=2$ implies that
$F^{*}g$ is a Kähler metric on $\mathbb{C}P^{2}$. Letting $\omega^{*}$ be the
Kähler form of $F^{*}g$ as above, for all
$\mathbb{C}P^{1}\subseteq\mathbb{C}P^{2}$,
$\int_{\mathbb{C}P^{1}}\omega^{*}=A^{\star}$. Letting $\widetilde{\omega}$ be
the Kähler form of $\widetilde{g}$ on $\mathbb{C}P^{2}$, this implies that
$\omega^{*}$ is cohomologous to $(\frac{A^{\star}}{\pi})\widetilde{\omega}$,
and therefore that $Vol(\mathbb{C}P^{2},F^{*}g)=\frac{A^{\star^{2}}}{2}$. For
$p>4$, the lower bound in Theorem 4.1 therefore coincides with the lower bound
in Lemma 2.2. Because equality in (4.4) holds for all $p\geq 2$, including
$p>4$, $F$ realizes equality in Lemma 2.2 for $p>4$ and is therefore a
homothety onto its image. ∎

######  Remark 4.2.

The proof Theorem 4.1 includes a proof of the following fact: if $h$ is a
Hermitian metric on a complex surface $X$ such that all complex curves in $X$
are minimal, then $h$ is a Kähler metric. In fact, one need only assume that
all complex lines $\Pi\subset T_{p}X$ are tangent to a complex curve which is
minimal in $(X,h)$. The author does not know of a reference for this fact.

We note that Ohnita [Oh87] has proven that if
$\phi:(\mathbb{C}P^{N},g_{0})\rightarrow(M,g)$ is a stable harmonic map from
$\mathbb{C}P^{N}$ with its canonical metric $g_{0}$ to any Riemannian manifold
$(M,g)$, then $\phi$ is pluriharmonic – for continuous maps, this is
equivalent to the statement that $\phi|_{\Sigma}$ is harmonic for all complex
curves $\Sigma\subseteq\mathbb{C}P^{N}$.

For $\mathbb{C}P^{1}$, the lower bound in Theorem 4.1 coincides with the bound
implied by Lemma 2.2, so equality for $p>2$ for
$F:(\mathbb{C}P^{1},\widetilde{g})\rightarrow(M,g)$ in Theorem 4.1 implies
that $F$ is a homothety onto its image. For mappings of $\mathbb{C}P^{N}$,
$N\geq 2$, let $Vol_{\widetilde{g}}(\mathbb{C}P^{N},F^{*}g)$ be the invariant
of the mapping $F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(M,g)$ as defined
in Lemma 2.2. We can then ask whether
$Vol_{\widetilde{g}}(\mathbb{C}P^{N},F^{*}g)$ satisfies the following
inequality along the lines of Pu’s and Gromov’s inequalities in Theorems 1.5
and 1.6:

$\displaystyle
Vol_{\widetilde{g}}(\mathbb{C}P^{N},F^{*}g)\geq\frac{A^{\star^{N}}}{N!}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (4.14)

For mappings for which (4.14) holds, the lower bound for $E_{p}(F)$ in Lemma
2.2 is bounded below by the lower bound in Theorem 4.1. Mappings which realize
equality in Theorem 4.1 for $p>2$ and satisfy (4.14) must therefore be
homotheties. However, it is known that one cannot replace the limit (1.6) in
Theorem 1.6 by the minimum area of a current representing a generator of
$H_{2}(\mathbb{C}P^{N};\mathbb{Z})$, so (4.14) need not hold. In fact, there
are Riemannian metrics on $\mathbb{C}P^{2}$ of arbitrarily small volume for
which the minimum area of a cycle generating
$H_{2}(\mathbb{C}P^{N};\mathbb{Z})$ is $1$. This is an example of a phenomenon
known as systolic freedom and is discussed in [CK03].

Also, unlike in the proof of Theorem 1.1 for real projective space, where one
knows that all length-minimizing paths in a free homotopy class are smoothly
immersed closed geodesics, the infima of the area and energy in a homotopy
class of mappings $f:S^{2}\rightarrow(M,g)$, even if they are realized, may
not be realized by a smooth immersion. This is discussed by Sacks and
Uhlenbeck in [SU81], where they prove that if $(M,g)$ is a compact Riemannian
manifold with $\pi_{2}(M)\neq 0$, one can find a generating set for
$\pi_{2}(M)$ which consists of conformal, branched minimal immersions of
$S^{2}$ which minimize area and energy in their homotopy classes.

When one has more information about the regularity of mappings which minimize
area and energy in the homotopy class of $F(\mathbb{C}P^{1})$ in Theorem 4.1,
one may be able to draw stronger conclusions about the regularity of energy-
minimizing maps $F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(M,g)$. This is
the basis for our proof of Theorem 1.3. The conclusion of Theorem 1.3 for
mappings which realize equality for $p=2$, that they must be holomorphic or
antiholomorphic, will then allow us to conclude that mappings which realize
equality for $p>2$ in Theorem 1.3 must be homotheties, by an argument similar
to the argument sketched above for mappings which satisfy (4.14).

###### Proof of Theorem 1.3.

Let $F:(\mathbb{C}P^{N},\widetilde{g})\rightarrow(X^{d},h)$ be a Lipschitz
mapping to a compact, simply connected Kähler manifold, and suppose equality
holds in Theorem 4.1 for $p=2$. This implies $F$ is smooth. By the Hurewicz
theorem, the natural mapping $\pi_{2}(X)\rightarrow H_{2}(X;\mathbb{Z})$ is an
isomorphism, so the family of mappings $f:S^{2}\rightarrow X$ which is
homotopic to $F|_{\mathbb{C}P^{1}}$ coincides with the family which is
homologous. Because $F_{*}([\mathbb{C}P^{1}])\in H_{2}(X;\mathbb{Z})$ can be
represented by a rational curve, the infimum $A^{\star}$ of the areas in this
homotopy class is in fact a minimum and is realized exclusively by cycles
which are calibrated by the Kähler form of $(X,h)$. By Lemma 2.4, for all
$\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$, $F|_{\mathcal{P}}$ must be area-
minimizing and energy-minimizing in its homotopy class and must therefore be a
holomorphic or antiholomorphic map to a complex curve in $(X,h)$. Letting
$\omega_{h}$ be the Kähler form of $(X,h)$,
$\int_{\mathcal{P}}F^{*}\omega_{h}$ is constant on
$\mathcal{L}(\mathbb{C}P^{N})$. $F|_{\mathcal{P}}$ is either holomorphic for
all $\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$ or antiholomorphic for all
$\mathcal{P}\in\mathcal{L}(\mathbb{C}P^{N})$, depending on whether this
constant is positive or negative.

Reversing the complex structure of $(X,h)$ if necessary, we can suppose
$F|_{\mathcal{P}}$ is holomorphic. Choose affine coordinates
$(z^{1},z^{2},\cdots,z^{N})$ on a neighborhood $\mathcal{U}$ of a point
$x_{0}$ in $\mathbb{C}P^{N}$ and holomorphic coordinates on a neighborhood of
$F(x_{0})$ in $X$. For each $i=1,2,\dots,N$ and each fixed
$z_{0}^{1},z_{0}^{2},\dots,z_{0}^{i-1},z_{0}^{i+1},\dots,z_{0}^{N}$, the
affine line $(z_{0}^{1},\dots,z^{i},\dots,z_{0}^{N})$ in these coordinates on
$\mathcal{U}$ corresponds to a complex $2$-plane through the origin in
$\mathbb{C}^{N+1}$, via the association
$(z^{1},z^{2},\cdots,z^{N})\rightarrow[1:z^{1}:z^{2}:\cdots:z^{N}]$ of affine
and homogeneous coordinates on $\mathcal{U}$, and thus to a complex projective
line $\mathcal{P}$ in $\mathbb{C}P^{N}$. Because $F|_{\mathcal{P}}$ is
holomorphic, each of the functions $F_{1},F_{2},\dots,F_{d}$ representing $F$
in these coordinates on $X$ is holomorphic when restricted to any such affine
line in $\mathcal{U}$. Osgood’s Lemma [GR65] then implies that $F$ is
holomorphic on $\mathcal{U}$, and therefore that $F$ is a holomorphic mapping
of $\mathbb{C}P^{N}$.

If $p>2$ and equality holds for $p$, then by Theorem 4.1, equality holds for
all $p\in[2,\infty)$. Equality for $p=2$ implies that, reversing orientation
if necessary, $F$ is holomorphic. Because
$\int_{\mathbb{C}P^{1}}F^{*}\omega_{h}=A^{\star}$ for linear
$\mathbb{C}P^{1}\subseteq\mathbb{C}P^{N}$, $F^{*}\omega_{h}$ is cohomologous
to $(\frac{A^{\star}}{\pi})\widetilde{\omega}$, where $\widetilde{\omega}$
again represents the Kähler form of $\widetilde{g}$. This implies that, in the
notation of Lemma 2.2, $Vol(\mathbb{C}P^{N},F^{*}h)=\frac{A^{\star^{N}}}{N!}$,
and that for $p>2N$, the lower bound in Theorem 4.1 coincides with the lower
bound in Lemma 2.2. This implies that $F$ realizes equality in Lemma 2.2 for
$p>2N$, and therefore that $F$ is a homothety onto its image. ∎

We note that Burns, Burstall, de Bartolomeis and Rawnsley [BBdeBR89] have
proven that any stable harmonic map
$\phi:(\mathbb{C}P^{N},g_{0})\rightarrow(\mathcal{Z},h)$ to a compact, simple
Hermitian symmetric space $(\mathcal{Z},h)$ is holomorphic or antiholomorphic.
This generalizes the result of Ohnita [Oh87] for mappings
$F:(\mathbb{C}P^{N_{1}},g_{0})\rightarrow(\mathbb{C}P^{N_{2}},g_{0})$
mentioned in the introduction.

##  5\. Quaternionic Projective Space

In this section, we will prove Theorem 1.4. In the proof of Theorem 1.4, we
will make use of the twistor fibration
$\Psi:\mathbb{C}P^{2N+1}\rightarrow\mathbb{H}P^{N}$. This mapping gives a
parametrization by $\mathbb{C}P^{2N+1}$ of the complex structures on tangent
spaces to $\mathbb{H}P^{N}$ which satisfy a local compatibility condition with
the the canonical metric $g_{0}$. We will describe this below, emphasizing the
features which we will use, and refer to [Sa82] and [Be07, Ch. 14] for more
background. The results in Theorem 1.4 for $(\mathbb{H}P^{N},g_{0})$, $N\geq
2$ are different from the corresponding results for $(\mathbb{H}P^{1},g_{0})$,
which is isometric to a rescaling of $(S^{4},g_{0})$. These results for
$(\mathbb{H}P^{1},g_{0})$ therefore follow from Lemma 2.2, and throughout the
discussion below we will restrict to the case $N\geq 2$.

We fix a basis $1,I,J,K$ for $\mathbb{H}$ and a Euclidean inner product on
$\mathbb{H}^{N+1}$, which we view as a right $\mathbb{H}$-module. Throughout,
we will identify $\mathbb{C}$ with the subfield $\mathbb{R}+\mathbb{R}I$ of
$\mathbb{H}$. In this way we will view $\mathbb{H}^{N+1}$ and all of its
$\mathbb{H}$-submodules as complex vector spaces. Letting $U(1)$ denote the
unit complex numbers and $Sp(1)$ the group of unit quaternions, we identify
$\mathbb{H}P^{N}$ with the quotient
$\mathbb{H}^{N+1}/\mathbb{H}^{*}=S^{4N+3}/Sp(1)$, and we identify
$\mathbb{C}P^{2N+1}$ with $\mathbb{H}^{N+1}/\mathbb{C}^{*}=S^{4N+3}/U(1)$.
Given a $1$-dimensional $\mathbb{H}$-submodule of $\mathbb{H}^{N+1}$, we will
write $l$ to denote both the associated point in $\mathbb{H}P^{N}$ and the
subspace of $\mathbb{H}^{N+1}$. We will write $Z_{l}$ for the fibre of the
twistor fibration $\Psi:\mathbb{C}P^{2N+1}\rightarrow\mathbb{H}P^{N}$, which
can be defined as follows:

The choice of a point $p$ in the unit sphere $S^{3}$ in
$l\subseteq\mathbb{H}^{N+1}$ determines an $\mathbb{R}$-linear isomorphism
between the subspace $l^{\perp}$ orthogonal to $l$ in $\mathbb{H}^{N+1}$ and
the tangent space $T_{l}\mathbb{H}P^{N}$ to $\mathbb{H}P^{N}$ at $l$, via the
differential $d\Phi_{p}$ of the Hopf fibration
$\Phi:S^{4N+3}\rightarrow\mathbb{H}P^{N}$. This isomorphism can be used to
transfer the $\mathbb{C}$-vector space and $\mathbb{H}$-module structures of
$l^{\perp}\subseteq\mathbb{H}^{N+1}$ to $T_{l}\mathbb{H}P^{N}$, but the
structures induced on $T_{l}\mathbb{H}P^{N}$ in this way depend on the choice
of the point $p$. In particular, because the action of $Sp(1)$ on $l^{\perp}$
is not $U(1)$-equivariant, the complex structure on $T_{l}\mathbb{H}P^{N}$
induced in this way depends on $p$. The subgroup of $Sp(1)$ which acts
$U(1)$-equivariantly is $U(1)$ itself. The complex structures induced on
$T_{l}\mathbb{H}P^{N}$ in this way are therefore parametrized by $Sp(1)/U(1)$,
which is canonically identified with the projectivization of
$l\cong\mathbb{C}^{2}$ as a complex vector space. The collection of complex
structures induced on $T_{l}\mathbb{H}P^{N}$ as above, paramatrized by
$Sp(1)/U(1)$, is the fibre $Z_{l}$ at $l$ of the twistor fibration
$\Psi:\mathbb{C}P^{2N+1}\rightarrow\mathbb{H}P^{N}$. Viewing
$\mathbb{C}P^{2N+1}$ and $\mathbb{H}P^{N}$ as quotients of $S^{4N+3}$ by
$U(1)$, resp. $Sp(1)$, the twistor fibration corresponds to the mapping
$S^{4N+3}/U(1)\rightarrow S^{4N+3}/Sp(1)$. The canonical metrics on
$\mathbb{C}P^{2N+1}$ and $\mathbb{H}P^{N}$ are the base metrics of Riemannian
submersions from $(S^{4N+3},g_{0})$ and the twistor fibration is itself a
Riemannian submersion
$(\mathbb{C}P^{2N+1},g_{0})\rightarrow(\mathbb{H}P^{N},g_{0})$, with totally
geodesic fibres $\mathbb{C}P^{1}\subseteq\mathbb{C}P^{2N+1}$ isometric to
round $2$-spheres with curvature $4$ and area $\pi$.

Later, it will important that although the induced $\mathbb{C}$ vector space
and $\mathbb{H}$-module structures on $T_{l}\mathbb{H}P^{N}$ depend on $p$,
the $1$-dimensional $\mathbb{H}$-submodule of $T_{l}\mathbb{H}P^{N}$
determined by a tangent vector $v\in T_{l}\mathbb{H}P^{N}$ is well-defined
independent of $p$. Likewise, for a complex structure $\tau\in Z_{l}$ and a
complex $1$-dimensional subspace $\lambda$ of the complex vector space
$(T_{l}\mathbb{H}P^{N},\tau)$, the $1$-dimensional $\mathbb{H}$-submodule of
$T_{l}\mathbb{H}P^{N}$ containing $\lambda$ is independent of $p$. We will
denote the $1$-dimensional quaternionic subspace determined by $v\in
T_{l}\mathcal{H}P^{N}$ by $v\mathbb{H}$ and the subspace determined by a
complex line $\lambda$ in $(T_{l}\mathbb{H}P^{N},\tau)$ by
$\lambda\mathbb{H}$.

For $\tau\in Z_{l}$, we define the following family of bases of the complex
vector space $(T_{l}\mathbb{H}P^{N},\tau)$:

######  Definition 5.1.

Given a point $l$ in $\mathbb{H}P^{N}$ and $\tau\in Z_{l}$, let
$\mathcal{F}_{l}(\tau)$ be the set of ordered orthonormal frames
$e_{1},e_{2},e_{3},\cdots,e_{4N-1},e_{4N}$ for $T_{l}\mathbb{H}P^{N}$ such
that:

  1. (1)

$e_{2i}=\tau(e_{2i-1})$ for all $i=1,2,\cdots,2N$.

  2. (2)

$e_{4j-3},e_{4j-2},e_{4j-1},e_{4j}$ span a quaternionic line in
$T_{l}\mathbb{H}P^{N}$ for each $j=1,2,\cdots,N$.

We will write $\mathcal{F}(N)$ for the volume of the space of frames
$\mathcal{F}_{l}(\tau)$ in Definition 5.1, when viewed as a subset of the
Steifel manifold of frames for $\mathbb{R}^{4N}$. We then have:

$\displaystyle\mathcal{F}(1)=\sigma(3)\sigma(1)=4\pi^{3},\vskip 6.0pt plus
2.0pt minus 2.0pt$ (5.1)

and for $N\geq 2$,

$\displaystyle\mathcal{F}(N)=Vol(\mathbb{H}P^{N-1})\mathcal{F}(N-1)\mathcal{F}(1).\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.2)

By convention, we define $\mathcal{F}(0)=1$. Using that the volume of
$\mathbb{H}P^{N}$ in the metric $g_{0}$ is equal to
$\frac{\pi^{2N}}{(2N+1)!}$, we then have:

$\displaystyle\mathcal{F}(N)=\frac{\pi^{N(N+2)}4^{N}}{\prod_{j=1}^{N-1}(2j+1)!}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.3)

It will also be helpful to note that:

$\displaystyle\frac{\mathcal{F}(1)\mathcal{F}(N-1)}{\mathcal{F}(N)}=\frac{(2N-1)!}{\pi^{2N-2}},\vskip
6.0pt plus 2.0pt minus 2.0pt$ (5.4)
$\displaystyle\frac{\mathcal{F}(N-2)}{\mathcal{F}(N)}=\frac{(2N-2)!(2N-3)!}{16\pi^{4N}}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.5)

Given a mapping $F:(\mathbb{H}P^{N},g_{0})\rightarrow(M^{m},g)$, the following
formula for the $4$-energy of $F$ at a point $l\in\mathbb{H}P^{N}$ is an
elementary consequence of (2.3):

$\displaystyle
e_{4}(F)_{l}=\frac{1}{\mathcal{F}(N)}\int\limits_{\mathcal{F}_{l}(\tau)}\left(\sum\limits_{i=1}^{2N}|dF(e_{2i-1})|^{2}+|dF(e_{2i})|^{2}\right)^{2}de\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=\frac{1}{\pi\mathcal{F}(N)}\int\limits_{Z_{l}}\int\limits_{\mathcal{F}_{l}(\tau)}\left(\sum\limits_{i=1}^{2N}|dF(e_{2i-1})|^{2}+|dF(e_{2i})|^{2}\right)^{2}de\
d\tau.\vskip 12.0pt plus 4.0pt minus 4.0pt$ (5.6)

######  Definition 5.2.

Given $l\in\mathbb{H}P^{N}$ and $\tau\in Z_{l}$ as above, let
$\mathcal{C}_{2}(\tau)$ be the family of subspaces $V$ of
$T_{l}\mathbb{H}P^{N}$ which are complex subspaces of dimension $2$ with
respect to the complex structure $\tau$, with the following property: for any
complex $1$-dimensional subspace $\lambda$ of $V$, the orthogonal complement
of $\lambda$ in the quaternionic line $\lambda\mathbb{H}$ is also orthogonal
to $V$.

Each such subspace $V\in\mathcal{C}_{2}(\tau)$ again determines a well-defined
subspace of $T_{l}\mathbb{H}P^{N}$ of real dimension $8$ which is an
$\mathbb{H}$-submodule for any $\mathbb{H}$-module structure induced on
$T_{l}\mathbb{H}P^{N}$ as above. We will denote this $V\mathbb{H}$.
$\mathcal{C}_{2}(\tau)$ is the base of a fibration
$\mathcal{F}_{l}(\tau)\rightarrow\mathcal{C}_{2}(\tau)$, which sends a frame
$e_{1},e_{2},e_{3},\cdots,e_{4N-1},e_{4N}$ to $span(e_{1},e_{2},e_{5},e_{6})$.
We will equip $\mathcal{C}_{2}(\tau)$ with the measure pushed forward from
$\mathcal{F}_{l}(\tau)$ via this mapping – we then have:

$\displaystyle
Vol(\mathcal{C}_{2}(\tau))=\frac{\mathcal{F}(N)}{4\pi^{2}\mathcal{F}(1)\mathcal{F}(N-2)}=\frac{(2N-2)!(2N-3)!}{256\pi^{4(N+1)+1}},$
(5.7)

where we have used (5.4) and the fact that, for each frame
$e_{1},e_{2},e_{5},e_{6}$ for $V$, the volume of the space of frames
$e_{3},e_{4},e_{7},e_{8}$ for $V^{\perp}\subseteq V\mathbb{H}$ which give a
frame $e_{1},e_{2},\cdots,e_{7},e_{8}$ for $V\mathbb{H}$ as in Definition 5.1
is equal to $4\pi^{2}$.

We will let $Gr_{1}^{\mathbb{H}}(l)$ denote the family of quaternionic
$1$-dimensional subspaces of $T_{l}\mathbb{H}P^{N}$, that is, the family of
$1$-dimensional $\mathbb{H}$-modules determined by $v\in T_{l}\mathbb{H}P^{N}$
– this space is a copy of $\mathbb{H}P^{N-1}$. The fundamental pointwise
result which is the basis for Theorem 1.4 is:

######  Lemma 5.3.

Let $F:(\mathbb{H}P^{N},g_{0})\rightarrow(M^{m},g)$ be a Lipschitz mapping and
$l\in\mathbb{H}P^{N}$ a point at which $F$ is differentiable. Then:

$\displaystyle
e_{4}(F)_{l}\geq\scriptstyle\frac{16N^{2}(2N-2)!}{\pi^{2N-2}}\displaystyle\left(\int\limits_{Gr_{1}^{\mathbb{H}}(l)}|\det(dF|_{\Lambda})|d\Lambda+\scriptstyle\frac{(2N-2)!}{(2N-1)\pi^{2N-2}}\displaystyle\int\limits_{Z_{l}}\int\limits_{\mathcal{C}_{2}(\tau)}|\det(dF|_{V})|dVd\tau\right).$
(5.8)

Equality holds if and only $dF_{l}$ is a homothety.

###### Proof.

We begin with the identity for $e_{4}(F)_{l}$ from (5.6). By Newton’s
inequality for symmetric functions (cf. [ES64] p.113) and the arithmetic-
geometric mean inequality,

$=\displaystyle
e_{4}(F)_{l}\geq\frac{16N}{(2N-1)\pi\mathcal{F}(N)}\int\limits_{Z_{l}}\int\limits_{\mathcal{F}_{l}(\tau)}\sum\limits_{i=1}^{2N-1}\sum\limits_{j=i+1}^{2N}|dF(e_{2i-1})||dF(e_{2i})|dF(e_{2j-1})||dF(e_{2j})|ded\tau\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=\frac{16N}{(2N-1)\pi\mathcal{F}(N)}\text{\Huge{[}}\int\limits_{Z_{l}}\int\limits_{\mathcal{F}_{l}(\tau)}\sum\limits_{i=1}^{N}|dF(e_{4i-3})||dF(e_{4i-2})||dF(e_{4i-1})||dF(e_{4i})|ded\tau$
$\displaystyle+\int\limits_{Z_{l}}\int\limits_{\mathcal{F}_{l}(\tau)}\text{\Large{(}}\sum\limits_{j=1}^{N-1}\sum\limits_{k=j+1}^{N}|dF(e_{4j-3})||dF(e_{4j-2})||dF(e_{4k-3})||dF(e_{4k-2})|$
(5.9)
$\displaystyle\displaystyle+\sum\limits_{j=1}^{N-1}\sum\limits_{k=j+1}^{N}|dF(e_{4j-3})||dF(e_{4j-2})||dF(e_{4k-1})||dF(e_{4k})|$
$\displaystyle+\sum\limits_{j=1}^{N-1}\sum\limits_{k=j+1}^{N}|dF(e_{4j-1})||dF(e_{4j})||dF(e_{4k-3})||dF(e_{4k-2})|$
$\displaystyle+\sum\limits_{j=1}^{N-1}\sum\limits_{k=j+1}^{N}|dF(e_{4j-1})||dF(e_{4j})||dF(e_{4k-1})||dF(e_{4k})|\text{\Large{)}}ded\tau\text{\Huge{]}}.\vskip
12.0pt plus 4.0pt minus 4.0pt$

For each $\Lambda\in Gr_{1}^{\mathbb{H}}(l)$ and each orthonormal frame
$e_{4i-3},e_{4i-2},e_{4i-1},e_{4i}$ for $\Lambda$, we have:

$\displaystyle|dF(e_{4i-3})||dF(e_{4i-2})||dF(e_{4i-1})||dF(e_{4i})|\geq|\det(dF|_{\Lambda})|.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.10)

Therefore, for each $\tau\in Z_{l}$,

$\displaystyle\int\limits_{\mathcal{F}_{l}(\tau)}\sum\limits_{i=1}^{N}|dF(e_{4i-3})||dF(e_{4i-2})||dF(e_{4i-1})||dF(e_{4i})|de\vskip
12.0pt plus 4.0pt minus 4.0pt\geq\scriptstyle
N\mathcal{F}(1)\mathcal{F}(N-1)\displaystyle\int\limits_{Gr_{1}^{\mathbb{H}}(l)}|\det(dF|_{\Lambda})|d\Lambda,\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.11)

Note that the constant $N\mathcal{F}(1)\mathcal{F}(N)$ in (5.11) is the
product of the number of indices $i=1,2,\cdots,N$ in the summation, the volume
$\mathcal{F}(1)$ of the space of frames for $\Lambda$ and the volume
$\mathcal{F}(N-1)$ of the space of frames for $\Lambda^{\perp}$.

Similarly, for each $\tau\in Z_{l}$ and each $V\in\mathcal{C}_{2}(\tau)$, the
space of frames $e_{4j-3},e_{4j-2},e_{4k-3},e_{4k-2}$ for $V$ has volume
$\sigma(3)\sigma(1)=4\pi^{3}$. The volume of the space of frames
$e_{4j-1},e_{4j},e_{4k-1},e_{4k}$ for $V^{\perp}\subseteq V\mathbb{H}$ has
volume $\sigma(1)^{2}=4\pi^{2}$, the space of frames for $V\mathbb{H}^{\perp}$
has volume $\mathcal{F}(N-2)$ and there are $\binom{N}{2}$ terms in the
summation $\sum_{j=1}^{N-1}\sum_{k=j+1}^{N}$ at which the frame may occur. The
same is true for the other possible indexings of each frame, i.e.
$e_{4j-3},e_{4j-2},e_{4k-1},e_{4k}$; $e_{4j-1},e_{4j},e_{4k-3},e_{4k-2}$ and
$e_{4j-1},e_{4j},e_{4k-1},e_{4k}$. For each $\tau\in Z_{l}$, the second
integral term in (5.9) is therefore bounded below by:

$\displaystyle
32N(N-1)\pi^{5}\mathcal{F}(N-2)\int\limits_{\mathcal{C}_{2}(\tau)}|\det(dF|_{V})|dV.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.12)

Combining (5.9) with the lower bounds in (5.11) and (5.12), noting that the
integration over $Z_{l}$ in (5.9) adds a factor $\pi$ to the constant in
(5.11) and using the identities (5.4) gives (5.8).

Equality requires equality in Newton’s inequality and the arithmetic-geometric
mean inequality in (5.9). This implies that for all $\tau\in Z_{l}$,
$\\{e_{1},e_{2},\dots,e_{4N}\\}\in\mathcal{F}_{l}\tau$, for all
$i,j=1,2,\cdots 2N$,
$|dF(e_{2i-1})|^{2}+|dF(e_{2i})|^{2}=|dF(e_{2j-1})|^{2}+|dF(e_{2j})|^{2}$, and
that $|dF(e_{2i-1})|=|dF(e_{2i})|$. This implies $|dF(\vec{u})|$ is the same
for all unit vectors $\vec{u}$ tangent to $\mathbb{H}P^{N}$ at $l$, so that
$dF$ is a homothety on $T_{l}\mathbb{H}P^{N}$. ∎

For each $\tau\in Z_{l}$ and each $V\in\mathcal{C}_{2}(\tau)$, $V$ is tangent
to a totally geodesic submanifold of $\mathbb{H}P^{N}$ which is isometric to
$\mathbb{C}P^{2}$ with its canonical metric, cf. [Be12, Ch. 5]. In the
following, we describe these totally geodesic subspaces and collect several of
their properties which we will use in the proof of Theorem 1.4:

######  Lemma 5.4.

Let $\tau\in Z_{l},V\in\mathcal{C}_{2}(\tau)$ as above.

  1. (1)

Let $\lambda_{\tau}$ be the complex line in
$l\cong\mathbb{C}^{2}\subseteq\mathbb{H}^{N+1}$ such that $\lambda_{\tau}\cap
S^{3}\subseteq l$ corresponds to $\tau$ by the twistor fibration. Let
$p_{1},p_{2}\in\lambda_{\tau}\cap S^{3}$, and let $V_{1},V_{2}\subseteq
l^{\perp}$ such that $d\Phi_{p_{i}}(V_{i})=V$. Then $V_{1}=V_{2}$. We will
denote this subspace of $l^{\perp}$ by $V_{\tau}$.

  2. (2)

For each subspace $\lambda^{\prime}\cong\mathbb{C}$ in
$V_{\tau}\oplus\lambda_{\tau}\cong\mathbb{C}^{3}\subseteq\mathbb{H}^{N+1}$,
the orthogonal complement $\lambda^{\prime\perp}$ to $\lambda^{\prime}$ in
$\lambda^{\prime}\mathbb{H}$ is orthogonal to $V_{\tau}\oplus\lambda_{\tau}$
in $\mathbb{H}^{N+1}$.

  3. (3)

Let $\widetilde{X}_{\tau}$ be the image of $V_{\tau}\oplus\lambda_{\tau}$ in
$\mathbb{C}P^{2N+1}$ via the projectivization of
$\mathbb{C}^{2N+2}=\mathbb{H}^{N+1}$. Then $\widetilde{X}_{\tau}$ maps
injectively and isometrically to $\mathbb{H}P^{N}$ via the twistor fibration.
Its image is a totally geodesic submanifold of $\mathbb{H}P^{N}$ which we
denote $X_{\tau}$.

  4. (4)

$V$ is tangent to $X_{\tau}$ at $l$.

  5. (5)

For each complex $1$-dimensional subspace $\lambda^{\prime}$ of
$V_{\tau}\oplus\lambda_{\tau}$, let
$l^{\prime}=\lambda^{\prime}\mathbb{H}\subseteq\mathbb{H}^{N+1}$ and let
$\tau^{\prime}$ be the complex structure on $T_{l^{\prime}}\mathbb{H}P^{N}$
associated to $\lambda^{\prime}$ by the twistor fibration, by the
identification of the twistor fibre with the projectivization of $l^{\prime}$
as above. Then $T_{l^{\prime}}X_{\tau}$ is $\tau^{\prime}$-invariant. In
particular, $\widetilde{X}_{\tau}$ is a section of the twistor fibration over
$X_{\tau}$, and the complex structure on $T\mathbb{H}P^{N}|_{X_{\tau}}$
defined in this way preserves $TX_{\tau}$.

We will write $\mathcal{C}(\mathbb{H}P^{N},g_{0})$ for the space of all such
totally geodesic $X_{\tau}$ in $\mathbb{H}P^{N}$. We will equip
$\mathcal{C}(\mathbb{H}P^{N},g_{0})$ with the measure pushed forward from the
total space of the bundle over $\mathbb{C}P^{2N+1}$ whose fibre over $\tau$ is
$\mathcal{C}_{2}(\tau)$, via the natural fibration $V\mapsto X_{\tau}$ of this
space over $\mathcal{C}(\mathbb{H}P^{N},g_{0})$. We then have:

$\displaystyle
Vol\left(\mathcal{C}(\mathbb{H}P^{N},g_{0})\right)=\frac{Vol(\mathbb{C}P^{2N+1},g_{0})Vol(\mathcal{C}_{2}(\tau))}{Vol(\mathbb{C}P^{2})}=\frac{(2N-3)!}{256N(4N^{2}-1)\pi^{2N+6}}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.13)

We will let $\mathcal{H}(\mathbb{H}P^{N},g_{0})$ be the space of linearly
embedded $\mathbb{H}P^{1}\subseteq\mathbb{H}P^{N}$, with the measure pushed
forward from the unit tangent bundle $U(\mathbb{H}P^{N},g_{0})$. Then:

$\displaystyle
Vol\left(\mathcal{H}(\mathbb{H}P^{N},g_{0})\right)=\frac{Vol\left(U(\mathbb{H}P^{N},g_{0})\right)}{Vol\left(U(\mathbb{H}P^{1},g_{0})\right)}=\frac{6\sigma(4N-1)\pi^{2N-1}}{\sigma(3)(2N+1)!}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.14)

The constant $K_{N}$ in Theorem 1.4 can be defined in terms of
$\mathcal{H}(\mathbb{H}P^{N},g_{0})$ and $\mathcal{C}(\mathbb{H}P^{N},g_{0})$:

$\displaystyle
K_{N}=\frac{16N^{2}(2N+1)!(2N-2)!}{\pi^{4N-2}}\left(Vol\left(\mathcal{H}(\mathbb{H}P^{N},g_{0})\right)+\frac{2(N-1)(2N-3)!}{(2N-1)\pi^{2N-2}}Vol\left(\mathcal{C}(\mathbb{H}P^{N},g_{0})\right)\right)\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=\frac{32N^{2}(2N+1)!(2N-2)!}{\pi^{4N-2}}\left(\frac{3\sigma(4N-1)\pi^{2N-1}}{\sigma(3)(2N+1)!}+\frac{(N-1)(2N-3)!^{2}}{256N(2N-1)(4N^{2}-1)\pi^{4(N+1)}}\right).\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.15)

Given a triple of complex structures $I,J,K$ on a neighborhood of
$\mathbb{H}P^{N}$ which satisfy the quaternion relations
$I^{2}=J^{2}=K^{2}=IJK=-Id$, we can form their associated Kähler forms
$\omega_{I}$, $\omega_{J}$, $\omega_{K}$. The 4-form
$\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2}$ is independent of the choice of
$I,J,K$. This form is therefore a local representation of a canonical,
globally-defined closed 4-form on $\mathbb{H}P^{N}$, known as the fundamental
4-form or Kraines 4-form (cf. [Be12, Kr65, Kr66]), whose powers generate the
cohomology of $\mathbb{H}P^{N}$ and calibrate the linear subspaces
$\mathbb{H}P^{k}\subseteq\mathbb{H}P^{N}$.

More precisely, we define $\Omega$ to be the form which coincides with
$\frac{1}{\pi^{2}}\left(\omega_{I}^{2}+\omega_{J}^{2}+\omega_{K}^{2}\right)$
for any choice of $I,J,K$ as above. $\Omega$ is a closed, parallel form
satisfying $\langle\Omega,\mathbb{H}P^{1}\rangle=1$. In
$H^{4}(\mathbb{H}P^{N};\mathbb{R})$, the cohomology class of $\Omega$ is the
image of a generator of $H^{4}(\mathbb{H}P^{N};\mathbb{Z})$ via the natural
homomorphism $H^{4}(\mathbb{H}P^{N};\mathbb{Z})\rightarrow
H^{4}(\mathbb{H}P^{N};\mathbb{R})$. For any orthonormal frame
$e,I(e),J(e),K(e)$ for a quaternionic line in $T_{l}\mathbb{H}P^{N}$,
$\Omega(e,I(e),J(e),K(e))=\frac{6}{\pi^{2}}$. More generally,
$(\frac{\pi^{2}}{6})\Omega$ has comass $\equiv 1$ and gives a calibration of
$(\mathbb{H}P^{N},g_{0})$, whose calibrated submanifolds are precisely the
linearly embedded $\mathbb{H}P^{1}$ in $\mathbb{H}P^{N}$. The powers of
$\Omega$, approriately rescaled, likewise give calibrations whose calibrated
submanifolds are linear subspaces $\mathbb{H}P^{k}\subseteq\mathbb{H}P^{N}$.

If $X\in\mathcal{C}(\mathbb{H}P^{N},g_{0})$, then by choosing $I$ to coincide
with the complex structure $\tau$ on $T_{l}X$ along $X$ as in Lemma 5.4 (5),
we have that for any orthonormal frame $e_{1},I(e_{1}),e_{2},I(e_{2})$ for
$T_{l}X$, $\Omega(e_{1},I(e_{1}),e_{2},I(e_{2}))=\frac{2}{\pi^{2}}$. Since
$Vol(\mathbb{C}P^{2})=3Vol(\mathbb{H}P^{1})$, this implies that $X$ also
represents a generator of $H_{4}(\mathbb{H}P^{N};\mathbb{Z})$ and is “one
third calibrated” by $\Omega$.

###### Proof of Theorem 1.4.

Let $F:(\mathbb{H}P^{N},g_{0})\rightarrow(M^{m},g)$ be a mapping as above and
$p\geq 4$. By Lemma 5.3,

$\displaystyle
E_{p}(F)=\int\limits_{\mathbb{H}P^{N}}|dF|^{p}dVol_{g_{0}}\vskip 12.0pt plus
4.0pt minus 4.0pt$
$\displaystyle\geq\scriptstyle\left(\frac{16N^{2}(2N-2)!}{\pi^{2N-2}}\right)^{\frac{p}{4}}\displaystyle\int\limits_{\mathbb{H}P^{N}}\left(\int\limits_{Gr_{1}^{\mathbb{H}}(x)}|\det(dF|_{\Lambda})|d\Lambda+\scriptstyle\frac{(2N-2)!}{(2N-1)\pi^{2N-2}}\displaystyle\int\limits_{Z_{x}}\int\limits_{\mathcal{C}_{2}(\tau)}|\det(dF|_{V})|dV\right)^{\frac{p}{4}}dVol_{g_{0}}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.16)

For $p=4$ this immediately implies:

$\displaystyle
E_{4}(F)\geq\scriptstyle\left(\frac{16N^{2}(2N-2)!}{\pi^{2N-2}}\right)\displaystyle\left(\int\limits_{\mathcal{H}(\mathbb{H}P^{N},g_{0})}|F(\mathcal{Q})|d\mathcal{Q}+\scriptstyle\frac{(2N-2)!}{(2N-1)\pi^{2N-2}}\displaystyle\int\limits_{\mathcal{C}(\mathbb{H}P^{N},g_{0})}|F(X)|dX\right).\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.17)

For $p>4$, (5.16) and Hölder’s inequality imply:

$\displaystyle
E_{p}(F)\geq\scriptstyle\left(\frac{((2N+1)!(2N-2)!16N^{2})^{p}}{(2N+1)!^{4}\pi^{(4N-2)p-8N}}\right)^{\frac{1}{4}}\displaystyle\left(\int\limits_{\mathcal{H}(\mathbb{H}P^{N},g_{0})}|F(\mathcal{Q})|d\mathcal{Q}+\scriptstyle\frac{(2N-2)!}{(2N-1)\pi^{2N-2}}\displaystyle\int\limits_{\mathcal{C}(\mathbb{H}P^{N},g_{0})}|F(X)|dX\right)^{\frac{p}{4}}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.18)

Because all $\mathcal{Q}\in\mathcal{H}(\mathbb{H}P^{N},g_{0})$ and all
$X\in\mathcal{C}(\mathbb{H}P^{N},g_{0})$ represent generators of
$H_{4}(\mathbb{H}P^{N};\mathbb{Z})$ we have $|F(\mathcal{Q})|,|F(X)|\geq
B^{\star}$, which implies (1.4), as a non-strict inequality.

For (1.4) to be an equality, equality would have to hold a.e. in (5.16), and
therefore in Lemma 5.3. This would imply that $F^{*}g$ is equal a.e. to
$\varphi(x)g_{0}$, where $\varphi(x)$ is an a.e.-defined non-negative function
on $\mathbb{H}P^{N}$. Equality would also require that $F$ take almost all
$\mathcal{Q},X\subseteq\mathbb{H}P^{N}$ to area-minimizing currents in their
homology class in $H_{4}(M;\mathbb{Z})$. Together, these conditions would
imply that for almost all $\mathcal{Q},X$,

$\displaystyle\int\limits_{\mathcal{Q}}\varphi(x)^{2}dx=B^{\star},\int\limits_{X}\varphi(x)^{2}dx=B^{\star}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.19)

Letting $\zeta:U(\mathbb{H}P^{N},g_{0})\rightarrow\mathbb{H}P^{N}$ be the
bundle projection of the unit tangent bundle of $\mathbb{H}P^{N}$, we would
then have:

$\displaystyle\int\limits_{\mathbb{H}P^{N}}\varphi(x)^{2}dVol_{g_{0}}=\frac{1}{\sigma(4N-1)}\int\limits_{U(\mathbb{H}P^{N},g_{0})}\varphi(\zeta(\vec{u}))^{2}d\vec{u}\vskip
12.0pt plus 4.0pt minus
4.0pt=\frac{1}{\sigma(4N-1)}\int\limits_{\mathcal{H}(\mathbb{H}P^{N},g_{0})}\int\limits_{U(\mathcal{Q},g_{0})}\varphi(\zeta(\vec{u}))^{2}d\vec{u}d\mathcal{Q}\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=\frac{\sigma(3)}{\sigma(4N-1)}\int\limits_{\mathcal{H}(\mathbb{H}P^{N},g_{0})}\int\limits_{\mathcal{Q}}\varphi(x)^{2}dxd\mathcal{Q}=\left(\frac{6\pi^{2N-1}}{(2N+1)!}\right)B^{\star}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.20)

On the other hand, we would also have:

$\displaystyle\int\limits_{\mathbb{H}P^{N}}\varphi(x)^{2}dVol_{g_{0}}=\frac{1}{\pi}\int\limits_{\mathbb{C}P^{2N+1}}\varphi(\Psi(\tau))^{2}d\tau=\frac{256\pi^{4(N+1)}}{(2N-2)!(2N-3)!}\int\limits_{\mathbb{C}P^{2N+1}}\int\limits_{\mathcal{C}_{2}(\tau)}\varphi(\Psi(\tau))^{2}dVd\tau\vskip
12.0pt plus 4.0pt minus 4.0pt$
$\displaystyle=\frac{256\pi^{4(N+1)}}{(2N-2)!(2N-3)!}\int\limits_{\mathcal{C}(\mathbb{H}P^{N},g_{0})}\int\limits_{X}\varphi(x)^{2}dxdX=\left(\frac{\pi^{2N-2}}{(2N-2)!}\right)B^{\star}.\vskip
12.0pt plus 4.0pt minus 4.0pt$ (5.21)

The right-hand sides of (5.20) and (5.21) can only be equal if $B^{\star}=0$.
Therefore, equality is only possible in Theorem 1.4 if $F$ is a constant map.
∎

A theorem of White [Wh84] shows that if $F:(W^{d},g)\rightarrow(N^{n},h)$ is a
mapping of compact, connected, oriented Riemannian manifolds with
$F_{*}:\pi_{1}(W)\rightarrow\pi_{1}(N)$ surjective and $\dim(W)\geq 3$, then
the infimum of the areas of mappings homotopic to $F$ is equal to the minimum
mass of an integral current $T$ representing $F([W])$ in
$H_{d}(N;\mathbb{Z})$. If $M$ in Theorem 1.4 is simply connected, $B^{\star}$
is therefore equal to the infimum of the areas of mappings $f:S^{4}\rightarrow
M$ in the free homotopy class of $F_{*}(\mathbb{H}P^{1})$, as in Theorems 1.1
for $\mathbb{R}P^{n}$ and 4.1 for $\mathbb{C}P^{N}$.

The proof of Theorem 1.4 implies that the identity mapping of
$(\mathbb{H}P^{N},g_{0})$ minimizes energy among maps $F$ in its homotopy
class such that the average volume of $F(X)$ for
$X\in\mathcal{C}(\mathbb{H}P^{N},g_{0})$ is at least
$Vol(\mathbb{C}P^{2},g_{0})=\frac{\pi^{2}}{2}$. More generally, with more
precise information about the minima or averages of the areas of
$F(\mathbb{H}P^{1})$ and $F(\mathbb{C}P^{2})$, one can deduce stronger lower
bounds for energy functionals of the mapping $F$.

One possible approach to investigating the $p$-energy of mappings homotopic to
the identity of $\mathbb{H}P^{N}$ is to use the homotopy lifting property to
consider the family of mappings
$\widetilde{F}:\mathbb{C}P^{2N+1}\rightarrow\mathbb{C}P^{2N+1}$ which cover a
mapping $F:\mathbb{H}P^{N}\rightarrow\mathbb{H}P^{N}$ via the twistor
fibration. In particular, this may be helpful in finding optimal lower bound
for the $p$-energy of mappings homotopic to the identity mapping of
$\mathbb{H}P^{N}$ and determining when the identity is $p$-energy minimizing
in its homotopy class.

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