[title=Index of Notation]

# On polynomial solutions to the minimal surface equation

Yifan Guo  Department of Mathematics, Rowland Hall, University of California,
Irvine, Irvine, CA 92697, USA<EMAIL_ADDRESS>
###### Abstract.

We are interested in finding nonlinear polynomials $P$ on $\mathbb{R}^{n}$
that solves the minimal surface equation.

We first prove a structure theorem on such polynomials. We show that the
highest degree term $P_{m}$ must factor as $P_{0}^{k}Q_{m}$ where $k$ is odd,
$Q_{m}\geq 0$ on $\mathbb{R}^{n}$ and $P_{0}$ is irreducible, and that the
level sets of $P_{m}$ are all area minimizing. Moreover, the unique tangent
cone of $\operatorname{graph}P$ at $\infty$ is
$\\{P_{0}=0\\}\times\mathbb{R}$. If $k\geq 3$, we know further that lower
order terms down to some degree are divisible by $P_{0}$.

We also show that $P$ must contain terms of both high and low degree. In
particular, it cannot be homogeneous. As a corollary, we prove that there is
no quadratic or cubic polynomial solution.

Using a general decay lower bound of Green’s functions on minimal
hypersurfaces we are able to show that $\deg P_{0}+k^{-1}\deg Q_{m}\leq n-2$.

Finally, we prove that a polynomial minimal graph cannot have
$C\times\mathbb{R}^{l}$ as its tangent cone at $\infty$ where $l\geq 1$ and
$C$ is any isoparametric cone. We also show that the existence of a nonlinear
polynomial solutions on $\mathbb{R}^{8}$ will imply the existence of an area
minimizing but not strictly minimizing cubic cone in $\mathbb{R}^{8}$. These
results indicate that finding an explicit polynomial solution can be hard.

##  1\. Introduction

###  1.1. Entire solutions to the minimal surface equation

Let $P$ be a function on $\mathbb{R}^{n}$. The graph of $P$ has mean curvature
$0$ in $\mathbb{R}^{n+1}$ is equivalent to $P$ satisfying the minimal surface
equation:

(1) $\displaystyle(1+|\nabla P|)^{2}\Delta
P-\sum_{i,j=1}^{n}P_{x_{i}}P_{x_{j}}P_{x_{i}x_{j}}=0.$

Such function $P$, defined on all of $\mathbb{R}^{n}$, is called an entire
solution to the minimal surface equation.

A well-known theorem of Bernstein [3] says that the only entire solutions to
the minimal surface equation on $\mathbb{R}^{2}$ are affine functions. The
theorem was generalized to $\mathbb{R}^{7}$ by the successive works of Fleming
[13], De Giorgi [9], Almgren [2] and Simons [23].

On the other hand, Bombieri, De Giorgi, and Giusti [5] constructed nonlinear
entire solutions to the minimal surface equation on $\mathbb{R}^{n}$ for
$n\geq 8$ modeling on Simons’ cone. Thus the restriction on $n\leq 7$ in
Bernstein theorem is sharp. Many other entire solutions have been constructed
by Simon [21] modeling on isoparametric cones cf. subsection 1.5.

We are interested in finding entire solutions to the minimal surface equation
which are polynomials. This is a problem raised in [22]. While graphs of
complex polynomials are always minimal by a calibration argument, there is no
known example of a real polynomial whose graph is minimal except affine
functions. In fact, even though nonlinear entire minimal graphs are known to
exist, none of them is explicit. In order to find an explicit example, we
should try some special ansatz and polynomial seem to be a natural choice. The
other motivation is the long standing conjecture that all entire solutions to
the minimal surface equation have polynomial growth cf. [6, 22, 29]. This
conjecture is verified by Simon [21] under some technical assumptions.
Therefore, it is reasonable to believe that polynomials can be suitable
candidates for solutions to the minimal surface equation.

###  1.2. Structure of polynomial solutions

We start by considering suitable blow-downs of $\operatorname{graph}P$ which
will imply that a polynomial solution has a special structure. Neglecting the
constant term, we write $P$ as the sum of homogeneous polynomials

(2) $\displaystyle P=P_{m}+P_{m-1}+\cdots+P_{2}+P_{1}$

where $P_{i}$ is homogeneous of degree $i$. We consider the graph obtained by
scaling down $\operatorname{graph}P$ by $\lambda$ and translating vertically
by $t\lambda^{-m+1}$ for some fixed $t\in\mathbb{R}$. We are able to show the
following currents convergence

$\lim_{\lambda\to
0}\eta_{\lambda\\#}\operatorname{graph}P-t\lambda^{-m+1}e_{n+1}=\partial[P_{m}<t]\times\mathbb{R}$

where $\eta_{\lambda}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the map
$x\mapsto\lambda x$. By Federer-Fleming compactness, this means that for all
$t\in\mathbb{R}$ the current $\partial[P_{m}<t]$ is area minimizing i.e. the
level sets of $P_{m}$ are area minimizing. Using irreduciblily of area
minimizing currents as well as real Nullstellensatz, we obtain the following
structure theorem for polynomial solutions.

######  Theorem 1.1.

Let $P=P_{m}+P_{m-1}+\cdots+P_{2}+P_{1}$ with $m\geq 2$ solve the minimal
surface equation where $P_{i}$ is homogeneous of degree $i$. Then we have:

  1. (i)

For every $t\in\mathbb{R}$, the current $\partial[P_{m}<t]$ is area
minimizing.

  2. (ii)

There exist $k\geq 1$ odd, $P_{0}$ a nonlinear irreducible homogeneous
polynomial that changes sign and $Q_{m}$ an even degree homogeneous polynomial
which is non-negative on $\mathbb{R}^{n}$ and coprime with $P_{0}$, with
$\\{Q_{m}=0\\}$ of dimension $\leq n-2$ so that

(3) $\displaystyle P_{m}=P_{0}^{k}Q_{m}.$

  3. (iii)

The unique tangent cone of $\operatorname{graph}P$ at $\infty$ is
$\partial[P_{0}<0]\times\mathbb{R}$.

  4. (iv)

If $k\geq 3$ in (3), then there exists $2\leq s\leq m-1$ so that

$P=P_{0}^{k}Q_{m}+P_{0}^{k_{m-1}}Q_{m-1}+\cdots+P_{0}^{k_{s+1}}Q_{s+1}+P_{0}Q_{s}+P_{s-1}+\cdots+P_{1}$

where $P_{0}$ and $Q_{i}$ are coprime for $l\leq i\leq m$ and $k_{i}\geq 2$
for $s+1\leq i\leq m-1$.

From (i) and (ii), we see that the zero locus of $P_{0}$ defines an area-
minimizing cone. Moreover, $\\{P_{m}=t\\}$ for $t\neq 0$ are smooth area-
minimizing hypersurfaces lying on one side of $\\{P_{0}=0\\}$ and thus they
are the Hardt-Simon foliation of the cone $\\{P_{0}=0\\}$. The isoparametric
cones introduced by Cartan [8] are perfect examples of such cones. However, we
are going to show in Theorem 1.6 that all isoparametric cones cannot appear as
$\\{P_{0}=0\\}$ for any polynomial solution.

###  1.3. No homogeneous solution to the minimal surface equation

One convenience when we consider polynomial solutions to PDEs is that we can
consider each degree terms separately. Notice that we write the minimal
surface equation (1) in such a way that it maps a polynomial to a polynomial.
To proceed, let $L$ denote the 1-Laplacian:

$LP:=|\nabla P|^{2}\Delta P-\sum_{i,j=1}^{n}P_{x_{i}}P_{x_{j}}P_{x_{i}x_{j}}.$

Then the minimal surface equation becomes

$\Delta P+LP=0.$

If $P$ is homogeneous of degree $m$, then $LP$ is homogeneous of degree $3m-4$
and $\Delta P$ is homogeneous of degree $m-2$. Thus equation (1) is equivalent
to $3m-3$ equations for $P_{m},\dots,P_{1}$ which is obtained by expanding (1)
according to (2). The first 2 equations are:

(4) $\displaystyle LP_{m}$ $\displaystyle=0\quad(\text{degree }3m-4)$ (5)
$\displaystyle DL(P_{m})P_{m-1}$ $\displaystyle=0\quad(\text{degree }3m-5)$

where (5) means that $P_{m-1}$ satisfies the linearized equation of $L$ at
$P_{m}$ or explicitly

$|\nabla P_{m}|^{2}\Delta P_{m-1}+2\Delta P_{m}\nabla P_{m}\cdot\nabla
P_{m-1}-\sum_{i,j=1}^{n}\left(P_{m,x_{i}}P_{m,x_{i}}P_{m-1,x_{i}x_{j}}+2P_{m,x_{j}}P_{m,x_{i}x_{j}}P_{m-1,x_{i}}\right)=0.$

Essentially we will only use (4) explicitly in this article cf. Remark 3.2.
Note that $P_{m}$ solving $LP_{m}=0$ is equivalent to the fact that every
level set of $P_{m}$ has mean curvature 0 at each regular point. This is not
surprising in view of Theorem 1.1 (i). Solving $LP_{m}=0$ is the first step in
solving the full minimal surface equation and it is unknown whether there is
any nonlinear homogeneous polynomial solution except powers of linear
functions. One nontrivial solution to $Lu=0$ is the function $F$ constructed
in section III of [5] which gives the first proof of area minimality of
Simons’ cone. However, by Theorem 1.6, this function is not a polynomial.

It is natural to ask whether there is any homogeneous polynomial $P$ solving
the full minimal surface equation, not just the 1-Laplace equation. From
equations (1) and (4), we see that this is equivalent to $\Delta P=0$ and
$LP=0$. With an elementary result Proposition 4.1 of [26] (see also Lemma
3.4), we show that such polynomials cannot be a solution. Actually we have a
stronger theorem.

######  Theorem 1.2.

Polynomials of the form

$P=P_{m}+P_{m-1}+\cdots+P_{\lceil\frac{m}{3}\rceil+1}$

with $m\geq 2$ and $P_{m}\neq 0$ cannot solve the minimal surface equation.
Here $\lceil x\rceil$ is the least integer greater than or equal to $x$.

In particular, there is no homogeneous polynomial of degree $m\geq 2$ solving
the minimal surface equation.

We also have a slight variant of the theorem above which will be an important
ingredient in Theorem 1.8.

######  Theorem 1.3.

We have the following:

  1. (i)

Polynomials of the form $P=P_{m}+P_{1}$ for $m\geq 2$ cannot solve the minimal
surface equation. In particular, there is no quadratic polynomial solution to
the minimal surface equation.

  2. (ii)

There is no cubic polynomial solution to the minimal surface equation.

###  1.4. Jacobi fields on level sets and lower bound of Green’s function

Since the level sets of $P_{m}$ are all minimal, the function
$u=\frac{1}{|\nabla(P_{0}Q_{m}^{1/k})|}$ satisfies the Jacobi field equation
on regular part of $\\{P_{m}=t\\}$ for all $t\in\mathbb{R}$ cf. Lemma 3.1.
Since a positive Jacobi field is bounded below by some multiple of Green’s
function, we get estimates on $\deg P_{0}+k^{-1}\deg Q_{m}$ if we have a lower
bound of the Green’s function. This is obtained in the following result proved
in [14].

######  Theorem 1.4 ([14]).

Let $n\geq 4$ and $\Sigma\subset\mathbb{R}^{n}$ be an area-minimizing boundary
i.e. $\Sigma=\partial[U]$ for some $U\subset\mathbb{R}^{n}$ open and is area
minimizing as a current. Then there exists $C_{1}(n),C_{2}(n)>0$ so that for
the Green’s function $G(x,y)$ on $\Sigma$, $x,y\in\operatorname{spt}\Sigma$

$C_{1}(n)|x-y|^{3-n}\geq G(x,y)\geq C_{2}(n)|x-y|^{3-n}$

where $|x|$ denotes the length of the vector $x\in\mathbb{R}^{n}$.

Using the above theorem, we have

######  Theorem 1.5.

If $P_{m}=P_{0}^{k}Q_{m}$ satisfies Theorem 1.1 (i) and (ii), then $\deg
P_{0}+k^{-1}\deg Q_{m}\leq n-2$.

###  1.5. “Non-existence” theorems

We have two “non-existence” theorems about polynomial solutions. We call them
“non-existence” since they show that the existence of polynomial solutions
will imply the existence of some algebraic area minimizing cones that are not
known to exist.

To state the first theorem, we recall that a minimal hypercone
$C\subset\mathbb{R}^{n}$ is called isoparametric if the link $C\cap S^{n-1}$
has constant principal curvature in $S^{n-1}$ cf. [7]. Every isoparametric
cone is the zero locus of a homogeneous polynomial called Cartan-Münzner
polynomial [8], [17]. Our first “non-existence” theorem is as follows.

######  Theorem 1.6.

If $P_{m}=P_{0}^{k}Q_{m}$ satisfies Theorem 1.1 (i) and (ii), then $P_{0}$ is
not a Cartan-Münzner polynomial.

In particular we must have $\deg P_{0}\geq 3$ since all quadratic minimal
cones are isoparametric.

In view of Theorem 1.1, this implies the following:

######  Corollary 1.7.

Polynomial minimal graph cannot have $C\times\mathbb{R}^{l}$ as tangent cone
at $\infty$ where $C$ is an isoparametric minimal cone and $l\geq 1$.

To the author’s best knowledge, isoparametric cones and their products with
$\mathbb{R}^{l}$ are the only known examples of area minimizing hypercones.
Therefore, we need to construct new algebraic area-minimizing cones in order
to find nonlinear polynomial solutions.

For the second “non-existence” theorem, we recall the Bernstein’s theorem
which says that the smallest dimension in which nonlinear entire solution
could exists is $n=8$. For polynomial solutions on $\mathbb{R}^{8}$, we have
the following “non-existence” theorem.

######  Theorem 1.8.

If there is a nonlinear polynomial solution to the minimal surface equation on
$\mathbb{R}^{8}$, then there exists a cubic, strictly stable, area minimizing
cone in $\mathbb{R}^{8}$ which is not strictly minimizing.

We can rephrase it as a Bernstein type theorem.

######  Corollary 1.9.

If $P$ is a polynomial on $\mathbb{R}^{8}$ solving the minimal surface
equation and the tangent cone of $\operatorname{graph}P$ at $\infty$ is
strictly minimizing, then $P$ is an affine function.

The idea is to combine the growth estimates of Simon [21] and no cubic
solutions from Theorem 1.3. The strictly minimizing property comes from the
assumption of [21]. Such assumption appears in many results of minimal
hypersurfaces. Note that there is no known example of area minimizing but not
strictly minimizing hypercones except for
$\mathbb{R}^{2}\subset\mathbb{R}^{3}$ cf. [16]. Moreover, in $\mathbb{R}^{8}$,
the only known area minimizing cones are cones over $S^{3}\times S^{3}$ and
$S^{2}\times S^{4}$, both of which are isoparametric and strictly minimizing.

The outline of the rest of the paper is as follows. In section 2, we prove the
structure theorem Theorem 1.1. In section 3, we prove Theorem 1.2 and 1.3. In
section 4, we use lower bound for Green’s function to prove Theorem 1.5. In
section 5, we prove the two “nonexistence” theorems: Theorem 1.6 and 1.8.

## Acknowledgement

The author would like to express his sincere gratitude to his advisor, Richard
Schoen, for constant support and guidance. He would also like to thank Connor
Mooney for helpful discussions on this problem.

##  2\. Structure of polynomial solutions

We introduce the following notations and definitions, mostly about geometric
measure theory cf. [11], [19]:

$|x|$: the length of the vector $x\in\mathbb{R}^{n}$;

$B_{r}(x)$: Euclidean ball radius $r$ in $\mathbb{R}^{n}$ centered at $x$
(write $B_{r}$ if $x=0$);

$U$: open set in $\mathbb{R}^{n}$;

$R_{k}^{loc}(U)$: integer multiplicity $k$-rectifiable currents in $U$;

$I_{k}^{loc}(U)$: $k$-integral currents in $U$;

$[M]$: the current associated to the smooth manifold $M$;

$[f<0]$: the current defined by the open set $\\{f<0\\}$;

$\|S\|$: Radon measure associated to $S\in R_{k}^{loc}(U)$;

$S\llcorner V$, $\mu\llcorner V$: restriction of the current $S\in
R_{k}^{loc}(U)$ or a Radon measure $\mu$ to a set $V\subset U$;

$\nabla u$: Euclidean gradient of a function $u$ at $x$

$\nabla_{S}u(x)$: tangential component of $\nabla u$ on $T_{x}S$;

$\Delta_{S}u$: Laplace-Beltrami operator of a smooth hypersurface $S$;

For later reference, we quote here the Nullstellensatz over $\mathbb{R}$ which
can be found in [4, Theorem 4.5.1].

######  Lemma 2.1 (Real Nullstellensatz).

Let $P\in\mathbb{R}[x_{1},\dots,x_{n}]$ be an irreducible polynomial. The
following are equivalent.

  1. (i)

The set $Z(P):=\\{P=0\\}$ contains a regular point, i.e. there exists
$x\in\\{P=0\\}$ so that $\nabla P(x)\neq 0$.

  2. (ii)

The polynomial $P$ changes sign.

  3. (iii)

$(P)=I(Z(P))$ where $(P)$ is the ideal generated by $P$ in
$\mathbb{R}[x_{1},\dots,x_{n}]$ and $I(Z(P))$ is the real radical i.e. the
ideal generated by polynomials that vanishes on $Z(P)$.

  4. (iv)

$\dim Z(P)=n-1$.

In particular, if $P$ satisfies any of the conditions above and $Q$ is a
polynomial that vanishes in a neighborhood of a regular point in $\\{P=0\\}$,
then $P|Q$.

######  Remark 2.2.

Throughout the article, the dimension of a zero locus $V$ of some polynomials
means one of the following, the algebraic dimension, the largest dimension of
the manifolds in the decomposition of $V$ into finite union of smooth
manifolds, or the Hausdorff dimension. They are all equivalent in this
setting.

######  Remark 2.3.

If $P$ is a polynomial that does not satisfy the conditions in Theorem 2.1,
then it cannot change sign, $Z(P)$ is of high codimension and $\nabla P\equiv
0$ on $\\{P=0\\}$. One such example is the function $x_{1}^{2}+x_{2}^{2}$ on
$\mathbb{R}^{3}$.

We first investigate the structure of the level sets of a homogeneous
polynomial.

Let $P_{m}$ be a homogeneous polynomial of degree $m$ in $\mathbb{R}^{n}$. We
write the irreducible factorization of $P_{m}$ as

(6) $\displaystyle P_{m}=c_{0}p_{1}^{k_{1}}\cdots p_{a}^{k_{a}}\cdots
p_{a+b}^{k_{a+b}}\cdots p_{a+b+c}^{k_{a+b+c}}$

where

(7) $\displaystyle c_{0}\neq 0\text{ is}$ $\displaystyle\text{ some
constant},$ $\displaystyle k_{i},a,b,c\geq 0$ are integers $\displaystyle
k_{i}\text{ is odd for }$ $\displaystyle 1\leq i\leq a,$ $\displaystyle
k_{i}\text{ is even for }$ $\displaystyle a+1\leq i\leq a+b$ $\displaystyle
p_{i}\text{ are irreducible polynomials for }$ $\displaystyle 1\leq i\leq
a+b+c,$ $\displaystyle\gcd(p_{i},p_{j})=1,$ $\displaystyle 1\leq i\neq j\leq
a+b+c,$ $\displaystyle p_{i}\text{ changes sign for }$ $\displaystyle 1\leq
i\leq a+b,$ $\displaystyle p_{i}\geq 0\text{ for }$ $\displaystyle a+b+1\leq
i\leq a+b+c.$

######  Lemma 2.4.

We have

  1. (i)

For any $t\neq 0$, the set $\\{P_{m}=t\\}$ is a smooth hypersurface in
$\mathbb{R}^{n}$ with $\nabla P_{m}$ nonvanishing on $\\{P_{m}=t\\}$.

  2. (ii)

For $1\leq i\leq a+b$, the set $\\{p_{i}=0\\}$ is $n-1$-dimensional and for
$a+b+1\leq j\leq a+b+c$ the set $\\{p_{j}=0\\}$ is of dimension $\leq n-2$.

  3. (iii)

For $1\leq i\neq j\leq a+b$, the set $\\{p_{i}=0\\}$ intersects
$\\{p_{j}=0\\}$ along a set of dimension $\leq n-2$.

###### Proof.

(i) Since $P_{m}$ is homogeneous of degree $m$, by Euler’s identity, we have,
for any $x\in\\{P_{m}=t\\}$

$x\cdot\nabla P_{m}(x)=mP_{m}(x)=mt\neq 0.$

Thus $\nabla P_{m}(x)\neq 0$ and $\\{P_{m}=t\\}$ is a smooth hypersurface.

(ii) This is direct consequence of Lemma 2.1 and (7).

(iii) By Lemma 2.1, $\dim\\{p_{i}=0\\}=n-1$. If for $1\leq j\leq a+b$, $j\neq
i$, $p_{j}$ vanishes on an $n-1$-dimensional subset of $\\{p_{i}=0\\}$, then
$p_{i}|p_{j}$. This is not possible since they are coprime by assumption (7).
Thus for $i\neq j$, the set $\\{p_{i}=p_{j}=0\\}$ have dimension $\leq n-2$. ∎

Now let $P=P_{m}+\cdots+P_{1}$ be a polynomial on $\mathbb{R}^{n}$ and
$\operatorname{graph}P$ denote the graph of $P$ with orientation given by the
unit normal

$\frac{(\nabla P(x),-1)}{\sqrt{1+|\nabla P(x)|^{2}}}.$

We consider the limit of $\operatorname{graph}P$ under appropriate blow down
and translation. Let $t\in\mathbb{R}$ be a number and $\lambda$ be a parameter
that tends to 0. We denote the map $x\mapsto\lambda x$ by $\eta_{\lambda}$. We
consider the current $\eta_{\lambda\\#}\operatorname{graph}P-t\lambda^{-m+1}$
obtained by scaling down $\operatorname{graph}P$ by $\lambda$ and translating
vertically by $t\lambda^{-m+1}$. Then we have
$\eta_{\lambda\\#}\operatorname{graph}P-t\lambda^{-m+1}e_{n+1}=\operatorname{graph}G_{t}^{(\lambda)}$
where

(8) $\displaystyle G_{t}^{(\lambda)}(x)=$ $\displaystyle\lambda
P(x/\lambda)-t\lambda^{-m+1}$ $\displaystyle=$
$\displaystyle\lambda^{-m+1}(P_{m}(x)-t)+\lambda^{-m+2}P_{m-1}(x)+\cdots+P_{1}(x)$

with the unit normal

(9) $\nu_{\operatorname{graph}G_{t}^{(\lambda)}}=\frac{(\nabla
G_{t}^{(\lambda)}(x),-1)}{\sqrt{1+|\nabla G_{t}^{(\lambda)}(x)|^{2}}}.$

We also denote $G_{0}^{(\lambda)}$ by $P^{(\lambda)}$. We have the following
lemma.

######  Lemma 2.5.

Let $T\in R_{n}^{loc}(\mathbb{R}^{n+1})$ be the limit of
$\operatorname{graph}G_{t}^{(\lambda_{j})}$ for some $\lambda_{j}\to 0$. Then
we have $\operatorname{spt}T\subset\\{P_{m}=t\\}\times\mathbb{R}$.

###### Proof.

A simple calculation shows

$\lim_{\lambda\to 0}|G_{t}^{(\lambda)}(x)|=\left\\{\begin{aligned}
&|P_{1}(x)|&\quad&\text{if }P_{m}(x)-t=P_{m-1}(x)=\cdots=P_{2}(x)=0\\\
&\infty&\quad&\text{otherwise}.\end{aligned}\right.$

Thus for any $x$ with $P_{m}(x)\neq t$, there is a neighborhood $U$ of $x$ in
$\mathbb{R}^{n}$ so that $|G_{t}^{(\lambda)}|\to\infty$ uniformly in $U$ as
$\lambda\to 0$. Thus $(U\times\mathbb{R})\cap\operatorname{spt}T=\emptyset$
and we have $\operatorname{spt}T\subset\\{P_{m}=t\\}\times\mathbb{R}$. ∎

Now let $t\neq 0$ and $z_{0}\in\\{P_{m}=t\\}$. We consider the convergence of
$\operatorname{graph}G_{t}^{(\lambda)}$ in a neighborhood of
$\\{z_{0}\\}\times\mathbb{R}$.

######  Lemma 2.6.

Let $P=P_{m}+\cdots+P_{1}$ be a polynomial. Let $t\neq 0$ and
$G_{t}^{(\lambda)}$ as in (8). Then for any $z_{0}\in\\{P_{m}=t\\}$, there
exists a neighborhood $U$ of $z_{0}$ in $\mathbb{R}^{n}$ so that

$\lim_{\lambda\to
0}\left[\operatorname{graph}G_{t}^{(\lambda)}\right]\llcorner(U\times\mathbb{R})=(\partial[P_{m}<t]\llcorner
U)\times\mathbb{R}$

where the convergence is both in $I_{n}^{loc}(U\times\mathbb{R})$ as currents
and in $C^{1}$ normal graphs over $\\{P_{m}=t\\}\times\mathbb{R}$.

###### Proof.

Since $t\neq 0$, by Lemma 2.4 (i), we define $\nu(z)=\frac{\nabla
P_{m}(z)}{|\nabla P_{m}(z)|}$ for all $z\in\\{P_{m}=t\\}$. By implicit
function theorem, we can choose a neighborhood $U$ of $z_{0}$ so that the
following are satisfied:

(i) We can write $U$ as $U=\psi(W\times(-\sigma,\sigma))$ where
$W=U\cap\\{P_{m}=t\\}$ and $\psi:W\times(-\sigma,\sigma)\to U$ is a
diffeomorphism given by $\psi(z,r)=z+r\nu(z)$.

(ii) There exists $\delta>0$ so that for all $(z,r)\in
W\times(-\sigma,\sigma)$ we have

$\nabla P_{m}(z+r\nu(z))\cdot\nu(z)\geq 2\delta>0.$

Using (8), we compute for $\lambda$ sufficiently small

$\displaystyle\frac{d}{dr}G_{t}^{(\lambda)}(z+r\nu(z))=$ $\displaystyle\nabla
G_{t}^{(\lambda)}(z+r\nu(z))\cdot\nu(z)$ $\displaystyle\geq$
$\displaystyle\lambda^{-m+1}\nabla
P_{m}(z+r\nu(z))\cdot\nu(z)-O(\lambda^{-m+2})$ $\displaystyle\geq$
$\displaystyle\delta\lambda^{-m+1}>0.$

Thus $r\mapsto P^{(\lambda)}(z+r\nu(z))$ is increasing for all
$-\sigma<r<\sigma$.

On $W$, we have

$|G_{t}^{(\lambda)}(z)|=|\lambda^{-m+2}P_{m-1}(z)+\cdots+\lambda^{-1}P_{2}(z)+P_{1}(z)|\leq
C\lambda^{-m+2}.$

Therefore, the interval
$[-\sigma\delta\lambda^{-m+1}+C\lambda^{-m+2},\sigma\delta\lambda^{-m+1}-C\lambda^{-m+2}]$
is in the range of $G_{t}^{(\lambda)}$ on $U$ by fundamental theorem of
calculus. Note that this interval converge to $\mathbb{R}$ as $\lambda\to 0$.
Let $R>0$ be fixed and $(z,y)\in W\times(-R,R)$. By mean value theorem, there
exists $\lambda_{0}(R)>0$ so that for all $0<\lambda<\lambda_{0}$, there is a
unique $-\sigma<\varphi^{(\lambda)}(z,y)<\sigma$ so that

(10)
$\displaystyle(x,G_{t}^{(\lambda)}(x))=(z+\varphi^{(\lambda)}(z,y)\nu(z),y).$

This means that we can write
$\left[\operatorname{graph}G_{t}^{(\lambda)}\right]\llcorner(U\times\mathbb{R})$
as a normal graph of $\varphi^{(\lambda)}$ over $W\times(-R,R)$. Using (10),
we compute the gradient of $\varphi^{(\lambda)}$ as

(11) $\displaystyle\frac{\partial\varphi^{(\lambda)}}{\partial y}(z,y)=$
$\displaystyle\frac{1}{\nabla
G_{t}^{(\lambda)}(x)\cdot\nu(z)}=O(\lambda^{m-1})$ (12)
$\displaystyle\frac{\partial\varphi^{(\lambda)}}{\partial z_{j}}(z,y)=$
$\displaystyle\frac{-1}{\nabla G_{t}^{(\lambda)}(x)\cdot\nu(z)}\frac{\partial
G_{t}^{(\lambda)}}{\partial
x_{i}}(x)(e_{ij}+\varphi^{(\lambda)}(z,y)\frac{\partial\nu^{i}}{\partial
z_{j}}(z))$ (13) $\displaystyle=$ $\displaystyle\frac{-1}{\nabla
P_{m}(x)\cdot\nu(z)}\frac{\partial P_{m}}{\partial
x_{i}}(x)(e_{ij}+\varphi^{(\lambda)}(z,y)\frac{\partial\nu^{i}}{\partial
z_{j}}(z))+O(\lambda)$

where $x=z+\varphi^{(\lambda)}(z,y)\nu(z)$ and
$(e_{ij})_{i=1,\dots,n,j=1,\dots,n-1}$ is the projection matrix from
$\mathbb{R}^{n}$ to $T_{z}\\{P_{m}=t\\}=\nu(z)^{\perp}$.

From (8), we have

$P_{m}(x)-t=\lambda^{m-1}y-\lambda P_{m-1}(x)-\cdots-\lambda^{m-1}P_{1}(x).$

Hence with $x=z+\varphi^{(\lambda)}(z,y)\nu(z)$, we have $P_{m}(x)\to t$ as
$\lambda\to 0$. Thus $d(x,\\{P_{m}=t\\})\to 0$ and hence

$\lim_{\lambda\to 0}\varphi^{(\lambda)}(z,y)=0.$

Since $(e_{ij})$ is the projection onto $T_{z}\\{P_{m}=t\\}$ we have
$\sum_{i=1}^{n}e_{ij}\partial_{x_{i}}P_{m}(z)=0$. Thus the first term in (13)
converges to 0. Hence we see that $\operatorname{graph}G_{t}^{(\lambda)}$
converge to $(U\cap\\{P_{m}=t\\})\times\mathbb{R}$ in $C^{1}$. We observe that
the unit normal of $\operatorname{graph}G_{t}^{(t)}$ as in (9) converges to
$\nu$ as $\lambda\to 0$. Thus
$\left[\operatorname{graph}G_{t}^{(\lambda)}\right]\llcorner(U\times\mathbb{R})\to(\partial[P_{m}<t]\llcorner
U)\times\mathbb{R}$ as currents. ∎

Now we consider the convergence of $\partial[P_{m}<t]$ as $t\to 0$ in a
neighborhood of $z_{0}$ where $z_{0}$ is a regular point of $\\{P_{m}=0\\}$
i.e. for some $1\leq i\leq a+b$

(14) $\displaystyle p_{i}(z_{0})=0,\ \nabla p_{i}(z_{0})\neq 0,\text{ and}$
$\displaystyle p_{j}(z_{0})\neq 0\text{ for all }j\neq i.$

In view of the factorization (6), we denote the polynomial
$c_{0}p_{1}^{k_{1}}\cdots\widehat{p_{i}^{k_{i}}}\cdots p_{a+b+c}^{k_{a+b+c}}$
by $P_{m}/p_{i}^{k_{i}}$ for $1\leq i\leq a+b$.

######  Lemma 2.7.

Let $z_{0}$ be as in (14). Then there exists a neighborhood $U$ of $z_{0}$ in
$\mathbb{R}^{n}$ so that the following hold.

  1. (i)

If $1\leq i\leq a$, then

$\lim_{t\to
0}\left(\partial[P_{m}<t]\right)\llcorner{U}=\operatorname{sgn}\left(\left(P_{m}/p_{i}^{k_{i}}\right)(z_{0})\right)\left(\partial[p_{i}<0]\right)\llcorner{U}$

where the convergence is both in $I_{n-1}^{loc}(U)$ and in $C^{\infty}$ as
smooth manifolds.

In particular

$\lim_{t\to 0}\|\partial[P_{m}<t]\|\llcorner
U=\mathcal{H}^{n-1}\llcorner(U\cap\\{p_{i}=0\\})$

  2. (ii)

If $a+1\leq i\leq a+b$ then as currents in $I_{n-1}^{loc}(U)$

$\lim_{t\to 0}\partial[P_{m}<t]\llcorner U=0.$

(a) If $\operatorname{sgn}((P_{m}/p_{i}^{k_{i}})(z_{0}))>0$, then

$\displaystyle\lim_{t\to 0^{+}}\|\partial[P_{m}<t]\|\llcorner U=$
$\displaystyle 2\mathcal{H}^{n-1}\llcorner(U\cap\\{p_{i}=0\\}),$
$\displaystyle\lim_{t\to 0^{-}}\|\partial[P_{m}<t]\|\llcorner U=$
$\displaystyle 0.$

(b) If $\operatorname{sgn}((P_{m}/p_{i}^{k_{i}})(z_{0}))<0$, then

$\displaystyle\lim_{t\to 0^{+}}\|\partial[P_{m}<t]\|\llcorner U=$
$\displaystyle 0,$ $\displaystyle\lim_{t\to
0^{-}}\|\partial[P_{m}<t]\|\llcorner U=$ $\displaystyle
2\mathcal{H}^{n-1}\llcorner(U\cap\\{p_{i}=0\\}).$

###### Proof.

(i) By (7) $k_{i}$ is odd for $1\leq i\leq a+b$. We have
$\\{P_{m}=t\\}=\\{w=t^{1/k_{i}}\\}$ where

(15) $w=\left(P_{m}/p_{i}^{k_{i}}\right)^{1/k_{i}}p_{i}.$

Since $\nabla p_{i}(z_{0})\neq 0$ and $p_{j}(z_{0})\neq 0$ for $j\neq i$, we
have $\nabla w(z_{0})\neq 0$. By implicit function theorem, there exist a
small neighborhood $U$ of $z_{0}$, $\delta>0$ and
$f:U\cap\\{w=0\\}\times(-\delta,\delta)\to U$ a diffeomorphism so that
$w(f(z,t))=t$ and $f(z,0)=z$ for all $z\in U\cap\\{w=0\\}$,
$t\in(-\delta,\delta)$. Therefore,
$U\cap\\{w=t^{1/k_{i}}\\}=f(U\cap\\{w=0\\},t^{1/k_{i}})$. Since
$f(\cdot,t)\to\operatorname{id}_{U\cap\\{w=0\\}}$ in $C^{\infty}$ as $t\to 0$,
we have

$U\cap\\{P_{m}=t\\}=U\cap\\{w=t^{1/k_{i}}\\}\to
U\cap\\{w=0\\}=U\cap\\{p_{i}=0\\}\quad\text{in }C^{\infty}\text{ as }t\to 0.$

Note that on $\\{P_{m}=t\\}$ for $t\neq 0$, we have $\frac{\nabla
P_{m}}{|\nabla P_{m}|}=\frac{\nabla w}{|\nabla w|}$. Thus we have

$\frac{\nabla P_{m}}{|\nabla P_{m}|}=\frac{\nabla w}{|\nabla
w|}\to\operatorname{sgn}\left(\left(P_{m}/p_{i}^{k_{i}}\right)(z_{0})\right)\frac{\nabla
p_{i}}{|\nabla p_{i}|}\text{ as }t\to 0.$

This is the desired convergence.

(ii) We only prove (a). Item (b) follows from applying (a) to $-P_{m}$.

Since $\operatorname{sgn}((P_{m}/p_{i}^{k_{i}})(z_{0}))>0$ and $k_{i}$ is
even, we have $U\cap\\{P_{m}=t\\}=\emptyset$ for $t<0$. Thus the convergence
for $t\to 0^{-}$ is obtained.

For $t\to 0^{+}$, we consider
$w=\left(P_{m}/p_{i}^{k_{i}}\right)^{1/k_{i}}p_{i}$ on $U$. This is well
defined in $U$ since $\operatorname{sgn}((P_{m}/p_{i}^{k_{i}})(z_{0}))>0$. Now
we have

$U\cap\\{P_{m}=t\\}=(U\cap\\{w=t^{1/k_{i}}\\})\cup(U\cap\\{w=-t^{1/k_{i}}\\}).$

By the same argument as in (i), we have

$U\cap\\{w=\pm c^{1/k_{i}}\\}\to\pm U\cap\\{p_{i}=0\\}$

both in $I_{n-1}^{loc}(U)$ and in $C^{\infty}$ as manifolds. Thus we have
$\partial[P_{m}<t]\llcorner U\to 0$ and $\|\partial[P_{m}<t]\|\llcorner U\to
2\mathcal{H}^{n}|_{U\cap\\{p_{i}=0\\}}$ as $t\to 0^{+}$. ∎

######  Corollary 2.8.

We have the following convergence

$\displaystyle\lim_{t\to 0}\partial[P_{m}<t]=$
$\displaystyle\sum_{i=1}^{a}\operatorname{sgn}((P_{m}/p_{i}^{k_{i}})(z_{0}))\partial[p_{i}<0]=\partial[P_{m}<0]$
$\displaystyle\lim_{t\to 0+}\|\partial[P_{m}<t]\|=$
$\displaystyle\sum_{i=1}^{a}\mathcal{H}^{n-1}\llcorner{\\{p_{i}=0\\}}+\sum_{i=a+1}^{a+b}2\mathcal{H}^{n-1}\llcorner{\\{p_{i}=0,P_{m}/p_{i}^{k_{i}}>0\\}}$
$\displaystyle\lim_{t\to 0-}\|\partial[P_{m}<t]\|=$
$\displaystyle\sum_{i=1}^{a}\mathcal{H}^{n-1}\llcorner{\\{p_{i}=0\\}}+\sum_{i=a+1}^{a+b}2\mathcal{H}^{n-1}\llcorner{\\{p_{i}=0,P_{m}/p_{i}^{k_{i}}<0\\}}.$

###### Proof.

For any $z\in\\{P_{m}\neq 0\\}$, the set $\\{P_{m}=t\\}$ have positive
distance to $z$ for small $t$. Thus if $\partial[P_{m}<t]\to T$ for some $T\in
I_{n-1}^{loc}(\mathbb{R}^{n})$ then $\operatorname{spt}T\subset\\{P_{m}=0\\}$.
Let $z_{0}\in\\{P_{m}=0\\}$. By Lemma 2.4, the set of singular points of
$\\{P_{m}=0\\}$ has dimension $\leq n-2$. Thus we can consider the limits near
a regular point of $\\{P_{m}=0\\}$. All 3 identities follows from summing over
$1\leq i\leq a+b$ in Lemma 2.7.

It remains to show the second equality in the first equation. Let
$z_{0}\in\\{p_{i}=0\\}$ be a regular point and $U$ be a small neighborhood of
$x$. We define $w$ in $U$ as in (15).

For $1\leq i\leq a$, we have $[P_{m}<0]\llcorner
U=\operatorname{sgn}((P_{m}/p_{i}^{k_{i}})(z_{0}))[w<0]\llcorner U$. Thus
$\partial[P_{m}<0]\llcorner
U=\operatorname{sgn}((P_{m}/p_{i}^{k_{i}})(z_{0}))\partial[p_{i}<0]\llcorner
U$.

For $a+1\leq i\leq a+b$, we have $[P_{m}<0]\llcorner U=[w\neq 0]\llcorner
U=[U]$. Thus $\partial[P_{m}<0]=0$. This finishes the proof. ∎

######  Theorem 2.9.

Let $P=P_{m}+P_{m-1}+\cdots+P_{2}+P_{1}$ with $m\geq 2$ solve the minimal
surface equation where $P_{i}$ is homogeneous of degree $i$. Then we have:

  1. (i)

For every $t\in\mathbb{R}$, the current $\partial[P_{m}<t]$ is area
minimizing.

  2. (ii)

There exist $k\geq 1$ odd, $P_{0}$ a nonlinear irreducible homogeneous
polynomial that changes sign and $Q_{m}$ an even degree homogeneous polynomial
which is non-negative on $\mathbb{R}^{n}$ and coprime with $P_{0}$, with
$\\{Q_{m}=0\\}$ of dimension $\leq n-2$ so that

(16) $\displaystyle P_{m}=P_{0}^{k}Q_{m}.$

  3. (iii)

The unique tangent cone of $\operatorname{graph}P$ at $\infty$ is
$\partial[P_{0}<0]\times\mathbb{R}$.

  4. (iv)

If $k\geq 3$ in (16), then there exists $2\leq s\leq m-1$ so that

$P=P_{0}^{k}Q_{m}+P_{0}^{k_{m-1}}Q_{m-1}+\cdots+P_{0}^{k_{s+1}}Q_{s+1}+P_{0}Q_{s}+P_{s-1}+\cdots+P_{1}$

where $P_{0}$ and $Q_{i}$ are coprime for $l\leq i\leq m$ and $k_{i}\geq 2$
for $s+1\leq i\leq m-1$.

###### Proof.

(i) By Lemma 2.6 and Federer-Fleming compactness [19], we have for any $t\neq
0$, $\partial[P_{m}<t]$ is area minimizing. By Corollary 2.8, we have
$\partial[P_{m}<t]\to\partial[P_{m}<0]$ as $t\to 0$. Thus by Federer-Fleming
compactness [19] again, $\partial[P_{m}<0]$ is area minimizing.

(ii) By convergence of area-minimizing currents [19], we have

$\|\partial[P_{m}<t]\|\to\|\partial[P_{m}<0]\|=\sum_{i=1}^{a}\mathcal{H}^{n-1}\llcorner{\\{p_{i}=0\\}}.$

Thus $a\geq 1$ in (6). Moreover, in view of Corollary 2.8, we see that $b=0$
in (6). In other words, there is no factor $p_{i}^{k_{i}}$ where $p_{i}$
changes sign and $k_{i}$ is even.

We claim that $a=1$. Indeed, suppose $a\geq 2$ for contradiction. We can
assume $c_{0}>0$ in (6) by adjusting $p_{1}$ to be $-p_{1}$ if necessary. We
have

$\\{P_{m}<0\\}=\\{p_{1}<0,p_{2}\cdots p_{a}>0\\}\cup\\{p_{1}>0,p_{2}\cdots
p_{a}<0\\}.$

Observe that the two open sets on the right hand side are disjoint. This
contradicts Theorem 1 of [6], which asserts that $\\{P_{m}<0\\}$ is connected.

Thus $a=1$. We let $P_{0}=p_{1}$, $k=k_{1}$ and
$Q_{m}=c_{0}p_{a+b+1}^{k_{a+b+1}}\dots p_{a+b+c}^{k_{a+b+c}}$. Then the
conclusion follows.

(iii) Let $T$ be a tangent cone at $\infty$ of $\operatorname{graph}P$. Recall
that we wrote
$\eta_{\lambda\\#}\operatorname{graph}P=\operatorname{graph}P^{(\lambda)}$ in
(8). By Lemma 2.5, $\operatorname{spt}T\subset\\{P_{m}=0\\}\times\mathbb{R}$.
By [20, §2] we know that $T=\partial[U]\times\mathbb{R}$ for some connected
open set $U\subset\mathbb{R}^{n}$. Hence we have
$\operatorname{spt}\partial[U]\subset\\{P_{m}=0\\}$. By (6), we have

$\\{P_{m}=0\\}=\\{P_{0}=0\\}\bigcup\left(\bigcup_{i=2}^{c+1}\\{p_{i}=0\\}\right).$

By Lemma 2.1, the set $\\{p_{i}=0\\}$ have dimension $\leq n-2$ for $2\leq
i\leq c+1$. Thus $\partial[P_{m}<0]=\partial[P_{0}<0]$ and
$\operatorname{spt}\partial[U]\subset\\{P_{0}=0\\}$. If we take $\phi\in
C_{c}^{\infty}(\\{P_{0}<0\\})$ then $\partial[U](\phi)=0$. Thus
$\partial[U]=0$ in $I_{n-1}^{loc}(\\{P_{0}<0\\})$. Similarly $\partial[U]=0$
in $I_{n-1}^{loc}(\\{P_{0}>0\\})$. Thus by constancy theorem, $[U]$ is
constant on both $\\{P_{0}<0\\}$ and $\\{P_{0}>0\\}$. Hence either
$[U]=[P_{0}<0]$ or $[U]=[P_{0}>0]$. Since we take the normal vector of
$\operatorname{graph}P^{(\lambda)}$ as in (9), for $x\notin\\{P_{0}=0\\}$,
$\nu_{\operatorname{graph}P^{(\lambda)}}$ is close to $\frac{\nabla
P_{0}}{|\nabla P_{0}|}$ for $\lambda$ small. Thus we have
$U\cap\\{P_{0}<0\\}\neq\emptyset$. Therefore $[U]=[P_{0}<0]$.

(iv) Suppose $k\geq 3$ in (16). Then $\nabla P_{m}=0$ on $\\{P_{0}=0\\}$.
Since
$[\operatorname{graph}P^{(\lambda)}]\to\partial[P_{0}<0]\times\mathbb{R}$ as
$\lambda\to 0$, by Allard regularity, for any regular point
$(z_{0},y_{0})\in\\{P_{0}=0\\}\times\mathbb{R}$, there is a neighborhood $U$
of $(z,y)$ so that $\operatorname{graph}P^{(\lambda)}\cap U$ can be written as
normal graph of $\varphi^{(\lambda)}$ over $\\{P_{0}=0\\}\times\mathbb{R}\cap
U$ and $\varphi^{(\lambda)}\to 0$ smoothly. In particular, with
$x=z+\varphi^{(\lambda)}(z,y)\nu(z)$, we have

$\frac{(\nabla P^{(\lambda)}(x),-1)}{\sqrt{1+|\nabla
P^{(\lambda)}(x)|^{2}}}=\nu_{\operatorname{graph}P^{(\lambda)}}(x)\to\nu(z)=\frac{(\nabla
P_{0}(z),0)}{|\nabla P_{0}(z)|}\text{ as }\lambda\to 0.$

Since $\frac{1}{\sqrt{1+|\nabla P^{(\lambda)}|^{2}}}\to 0$ and $\nabla
P_{m}(z)=0$, there exists $2\leq s\leq m-1$ so that

(17) $\nabla P_{m}(z)=\nabla P_{m-1}(z)=\cdots=\nabla P_{s+1}(z)=0\text{ and
}\nabla P_{s}(z)\neq 0.$

Since for every regular point $z\in\\{P_{0}=0\\}$, there exists $s$ satisfying
(17), we may choose the maximal $s$ so that (17) holds on a open set $V$ of
$\\{P_{0}=0\\}$. By Lemma 2.1, we have $P_{0}|\nabla P_{i}$ for $s+1\leq i\leq
m-1$. By Euler’s formula $iP_{i}(x)=x\cdot\nabla P_{i}(x)$, we have
$P_{0}|P_{i}$. Thus we can write $P_{i}=P_{0}\tilde{Q}_{i}$. Since

$0=\nabla P_{i}=\tilde{Q}_{i}\nabla
P_{0}+P_{0}\nabla\tilde{Q}_{i}=\tilde{Q}_{i}\nabla P_{0}\quad\text{in }V,$

we see that $\tilde{Q}_{i}$ vanishes in $V$. Thus $P_{0}|\tilde{Q}_{i}$ and
hence $P_{0}^{2}|P_{i}$.

By (17), we have as $\lambda\to 0$

$\frac{(\nabla P^{(\lambda)}(x),-1)}{\sqrt{1+|\nabla
P^{(\lambda)}(x)|^{2}}}\to\frac{(\nabla P_{s}(z),0)}{|\nabla P_{s}(z)|}.$

Hence for all $z\in V$, we have

$\frac{\nabla P_{s}(z)}{|\nabla P_{s}(z)|}=\frac{\nabla P_{0}(z)}{|\nabla
P_{0}(z)|}.$

By Euler’s identity,

$sP_{s}(z)=z\cdot\nabla P_{s}(z)=z\cdot\nabla P_{0}(z)\frac{|\nabla
P_{s}(z)|}{|\nabla P_{0}(z)|}=\deg P_{0}\cdot P_{0}(z)\frac{|\nabla
P_{s}(z)|}{|\nabla P_{0}(z)|}=0.$

Thus we have $P_{s}(z)=0$ in an open set $V$ of $C$. Thus by Lemma 2.1,
$P_{0}|P_{s}$. Since $\nabla P_{s}\neq 0$ on $V$, we have $P_{0}^{2}\nmid
P_{s}$. This finishes the proof. ∎

##  3\. No homogeneous polynomial solution to the minimal surface equation

In this section, we are going to show that polynomial solutions to the minimal
surface equation must contain terms of both high and low degree. First we need
a preparation lemma.

######  Lemma 3.1.

Let $P=P_{m}+\cdots+P_{1}$ be a polynomial solution to the minimal surface
equation. Then

(i) $P_{m}$ satisfies

$LP_{m}=|\nabla P|^{2}\Delta
P-\sum_{i,j=1}^{n}P_{x_{i}}P_{x_{j}}P_{x_{i}x_{j}}=0.$

(ii) Let $P_{m}=P_{0}^{k}Q_{m}$ be as in Theorem 2.9 (ii). Then
$u=\frac{1}{|\nabla(P_{0}Q_{m}^{1/k})|}$ is the velocity of the family of
minimal hypersurfaces $(\\{P_{0}Q_{m}^{1/k}=t\\})_{t\in\mathbb{R}}$. Hence $u$
satisfies the Jacobi field equation on regular part of $\\{P_{m}=t\\}$ for all
$t\in\mathbb{R}$.

(iii) $P_{m-1}$ satisfies

(18) $\displaystyle DL(P_{m})P_{m-1}=$ $\displaystyle|\nabla P_{m}|^{2}\Delta
P_{m-1}+2\Delta P_{m}\nabla P_{m}\cdot\nabla P_{m-1}$
$\displaystyle-\sum_{i,j=1}^{n}\left(P_{m,x_{i}}P_{m,x_{i}}P_{m-1,x_{i}x_{j}}+2P_{m,x_{j}}P_{m,x_{i}x_{j}}P_{m-1,x_{i}}\right)=0.$

(iv) The function $v=-\frac{P_{m-1}}{|\nabla P_{m}|}$ is a Jacobi field on
$\\{P_{m}=t\\}$ with $t\neq 0$. If $k=1$ in (16), then $v$ is also a Jacobi
field on regular part of $\\{P_{m}=0\\}$.

(v) Suppose $t\neq 0$. The function $v$ viewed as a function on
$\\{P_{m}=t\\}\times\mathbb{R}$ is the velocity of the family of minimal
hypersurfaces
$(\operatorname{graph}G_{t}^{(\lambda)})_{\lambda\in[0,\epsilon)}$ at
$\lambda=0$ where $G_{t}^{(\lambda)}$ is defined as in (8).

######  Remark 3.2.

If we have $P_{m}$ solving $LP_{m}=0$, then $\partial_{x_{i}}P_{m}$ for $1\leq
i\leq n$ are solutions to (18). This corresponds to translations of the level
sets in $\mathbb{R}^{n}$. This indicates that (18) may not give too much
information on the polynomial.

###### Proof.

(i) This is the degree $3m-4$ term in the expansion of (1) according to (2).

(ii) By (i), Lemma 2.4 and Theorem 2.9 (ii) for any $t\in\mathbb{R}$, the
level set $\\{P_{0}Q_{m}^{1/k}=t\\}=\\{P_{m}=t^{k}\\}$ is minimal. We take a
$\frac{1}{k}$ power of $P_{m}$ since
$\nabla(P_{0}Q_{m}^{1/k})=Q_{m}^{1/k}\nabla P_{0}\neq 0$ on regular part of
$\\{P_{0}=0\\}$. Now the velocity of the family
$(\\{P_{0}Q_{m}^{1/k}=t\\})_{t\in\mathbb{R}}$ is
$u=\frac{1}{|\nabla(P_{0}Q_{m}^{1/k})|}$ and hence it satisfies the Jacobi
equation on regular part of $\\{P_{m}=t\\}$.

(iii) This is the degree $3m-5$ term in the expansion of (1) according to (2).

(iv) For $t\neq 0$, Suppose $\gamma(s)$ is a curve in $\mathbb{R}^{n}$ so that
$\gamma(0)=x\in\\{P_{m}=t\\}$, $\dot{\gamma}(0)=v(x)\frac{\nabla
P_{m}(x)}{|\nabla P_{m}(x)|}$ and $(P_{m}+sP_{m-1})(\gamma(s))=0$ for all
$s\in(-\epsilon,\epsilon)$. Then

$0=\left.\frac{d}{ds}\right|_{s=0}(P_{m}+sP_{m-1})(\gamma(s))=P_{m-1}(x)+\nabla
P_{m}\cdot\left(v(x)\frac{\nabla P_{m}(x)}{|\nabla P_{m}(x)|}\right).$

Thus $v(x)=-\frac{P_{m-1}}{|\nabla P_{m}|}$ is the velocity field of the
hypersurfaces $(\\{P_{m}+sP_{m-1}=t\\})_{s\in(-\epsilon,\epsilon)}$ at $s=0$.
Equation (18) shows that the level set of $P_{m}+sP_{m-1}$ has 0 mean
curvature up to the first order in $s$, thus $v$ is a Jacobi field on
$\\{P_{m}=t\\}$ for $t\neq 0$.

If $k=1$, then $\nabla P_{m}\neq 0$ on regular part of $\\{P_{m}=0\\}$. The
same argument as above shows that $v$ is a Jacobi field on $\\{P_{m}=0\\}$.

(v) Let $t\neq 0$. We write $\operatorname{graph}G_{t}^{(\lambda)}$ as a
normal graph of $\varphi^{(\lambda)}$ over $\\{P_{m}=t\\}\times\mathbb{R}$ as
in (10). Using the expression of $G_{t}^{(\lambda)}$ in (8) and writing
$x=z+\varphi^{(\lambda)}(z,y)\nu(z)$, we have

$\displaystyle 0=$
$\displaystyle\frac{d}{d\lambda}G_{t}^{(\lambda)}\left(z+\varphi^{(\lambda)}(z,y)\nu(z)\right)=\frac{\partial
G_{t}^{(\lambda)}}{\partial\lambda}(x)+\nabla
G_{t}^{(\lambda)}(x)\cdot\nu(z)\frac{\partial\varphi^{{(\lambda)}}}{\partial\lambda}(z)$
$\displaystyle=$
$\displaystyle\lambda^{-m}(1-m)(P_{m}(x)-t)+\lambda^{1-m}\left((2-m)P_{m-1}(x)+\nabla
P_{m}(x)\cdot\nu(z)\frac{\partial\varphi^{(\lambda)}}{\partial\lambda}(z)\right)$
$\displaystyle+O(\lambda^{2-m})$ $\displaystyle=$
$\displaystyle\lambda^{1-m}(2-m)\left(P_{m-1}(x)+|\nabla
P_{m}(z)|\frac{\partial\varphi^{(\lambda)}}{\partial\lambda}(z)\right)+O(\lambda^{2-m})$

where in the last step, since $P_{m}(x)-t\to 0$ as $\lambda\to 0$, we expand
the first term as

$\lambda^{-m}(1-m)(P_{m}(x)-t)=\lambda^{1-m}(1-m)|\nabla
P_{m}(z)|\frac{\partial\varphi^{(\lambda)}}{\partial\lambda}(z)+O(\lambda^{2-m}).$

Taking $\lambda\to 0$, we get (v). ∎

######  Theorem 3.3.

Polynomials of the form

$P=P_{m}+P_{m-1}+\cdots+P_{\lceil\frac{m}{3}\rceil+1}$

with $m\geq 2$ and $P_{m}\neq 0$ cannot solve the minimal surface equation.
Here $\lceil x\rceil$ is the least integer greater than or equal to $x$.

In particular, there is no homogeneous polynomial of degree $m\geq 2$ solving
the minimal surface equation.

###### Proof.

Suppose there is such a polynomial solving the minimal surface equation. By
Lemma 3.1 (i), we have $LP_{m}=0$. The lowest degree term in $LP$ is
$LP_{\lceil\frac{m}{3}\rceil+1}=0$ which has degree
$3\lceil\frac{m}{3}\rceil-1>m-2=\deg\Delta P_{m}$. Thus the $m-2$ degree term
in $\Delta P+LP$ is $\Delta P_{m}$ and we have $\Delta P_{m}=0$. Combining it
with $LP_{m}=0$, we have a nonzero homogeneous polynomial $P_{m}$ satisfying
$\Delta P_{m}=0$ and the $\infty$-Laplacian equation

$\sum_{i,j=1}^{n}P_{x_{i}}P_{x_{j}}P_{x_{i}x_{j}}=0.$

This is a contradiction in view of the next lemma. ∎

######  Lemma 3.4 ([26, Proposition 4.1]).

Let $P$ be a smooth function on $\mathbb{R}^{n}\setminus\\{0\\}$ which is
homogeneous of degree $m\notin\\{0,1\\}$, and satisfies $\Delta P=0$ and
$\sum_{i,j=1}^{n}P_{x_{i}x_{j}}P_{x_{i}}P_{x_{j}}=0$. Then $P=0$.

######  Remark 3.5.

Note that $m$ is not required to be an integer. Hence the lemma actually shows
that there is no homogeneous solution to the minimal surface equation on
$\mathbb{R}^{n}\setminus\\{0\\}$.

We write down the proof of the lemma for the sake of convenience.

###### Proof.

We can rewrite the $\infty$-Laplacian equation as

(19) $\frac{1}{2}\langle\nabla P,\nabla|\nabla P|^{2}\rangle=0.$

Let $x\in S^{n-1}$ so that $P(x)=\max_{y\in S^{n-1}}P(y)$. By the method of
Lagrange multiplier, we have $\nabla P(x)=\lambda x$. Evaluating (19) at $x$,
and using Euler’s formula for the homogeneous function $|\nabla P|^{2}$, we
have

$0=\frac{1}{2}\langle\lambda x,\nabla|\nabla
P|^{2}(x)\rangle=\lambda(m-1)|\nabla P(x)|^{2}=\lambda^{3}(m-1).$

Since $m\neq 1$, we have $\lambda=0$ and thus $\nabla P(x)=0$.

On the other hand, using Euler’s formula for the homogeneous function $P$, we
have

$mP(x)=\langle x,\nabla P(x)\rangle=0.$

Hence $\max_{y\in S^{n-1}}P(y)=0$. Similarly we have $\min_{y\in
S^{n-1}}P(y)=0$ and thus $P=0$. ∎

The following theorem is a slight variant of Theorem 3.3. This is important in
later applications cf. Theorem 5.6.

######  Theorem 3.6.

We have

  1. (i)

Polynomials of the form $P=P_{m}+P_{1}$ for $m\geq 2$ cannot solve the minimal
surface equation. In particular, there is no quadratic polynomial solution to
the minimal surface equation.

  2. (ii)

There is no cubic polynomial solution to the minimal surface equation.

###### Proof.

(i) After rotating the coordinate in $\mathbb{R}^{n}$, we may assume that
$P_{1}(x)=ax_{1}$ with $a\neq 0$. Plugging $P_{1}=ax_{1}$ in (1), and
considering degree $m-2$ and $2m-3$ terms, we obtain

(20) $\displaystyle(1+a^{2})\Delta P_{m}-a^{2}P_{m,x_{1}x_{1}}=0$ (21)
$\displaystyle P_{m,x_{1}}\Delta
P_{m}-\sum_{i=1}^{n}P_{m,x_{i}x_{1}}P_{m,x_{i}}=0.$

Eliminating $\Delta P_{m}$ in (20) and (21), we get

$\frac{a^{2}}{1+a^{2}}P_{m,x_{1}x_{1}}P_{m,x_{1}}-\sum_{i=1}^{n}P_{m,x_{i}x_{1}}P_{m,x_{i}}=0$

i.e.

$\left(\frac{1}{1+a^{2}}P_{m,x_{1}}^{2}+P_{m,x_{2}}^{2}+\cdots+P_{m,x_{n}}^{2}\right)_{x_{1}}=0.$

Thus we have $\frac{1}{1+a^{2}}P_{m,x_{1}}^{2}+\sum_{i=2}^{n}P_{m,x_{i}}^{2}$
is independent of $x_{1}$.

We arrange the terms in $P_{m}$ according to the degree in $x_{1}$:

$P_{m}(x)=c_{0}x_{1}^{m}+c_{1}(x^{\prime})x_{1}^{m-1}+\cdots+c_{m}(x^{\prime})$

where $x^{\prime}=(x_{2},\dots,x_{n})$ and $c_{i}$ is a polynomial in
$x^{\prime}$ of degree $i$.

The term in $\frac{1}{1+a^{2}}P_{m,x_{1}}^{2}+\sum_{i=2}^{n}P_{m,x_{i}}^{2}$
with highest $x_{1}$ degree is

$\frac{1}{1+a^{2}}c_{0}^{2}m^{2}x_{1}^{2m-2}+\sum_{i=2}^{n}c_{1,x_{i}}^{2}x_{1}^{2m-2}.$

Thus we must have $c_{0}=c_{1}=0$.

Assume that $j$ is the smallest index such that $c_{j}\neq 0$, then we have
$c_{0}=\dots=c_{j-1}=0$. The term in
$\frac{1}{1+a^{2}}P_{m,x_{1}}^{2}+\sum_{i=2}^{n}P_{m,x_{i}}^{2}$ with highest
$x_{1}$ degree is

$\sum_{i=2}^{n}c_{j,x_{i}}(x^{\prime})^{2}x_{1}^{2m-2j}.$

Since $c_{j}\neq 0$, in order for
$\frac{1}{1+a^{2}}P_{m,x_{1}}^{2}+\sum_{i=2}^{n}P_{m,x_{i}}^{2}$ to be
independent of $x_{1}$, we must have $m=j$. Hence $c_{1}=\dots=c_{m-1}=0$ and
$P_{m}$ is independent of $x_{1}$.

By equation (20), we have $\Delta P_{m}=0$. Since $LP_{m}=0$, Lemma 3.4
implies that $P_{m}=0$ i.e. $P=P_{1}$.

(ii) Let $P=P_{3}+P_{2}+P_{1}$ be a solution to the minimal surface equation
with $P_{3}\neq 0$.

By Theorem 2.9 (ii), we know that $P_{3}=P_{0}^{k}Q_{3}$ where $Q_{3}\geq 0$
has even degree. If $\deg Q_{3}=2$, then $\deg P_{0}=1$ and $k=1$. Thus by
Theorem 2.9 (ii), the tangent cone at infinity of $\operatorname{graph}P$ is
$\partial[P_{0}<0]\times\mathbb{R}$ whjch is a hyperplane. This implies that
$\operatorname{graph}P$ is a hyperplane, a contradiction. If $\deg Q_{3}=0$
i.e. $Q_{3}=1$ then $P_{3}=P_{0}^{k}$. If $k=3$, then $\deg P_{0}=1$ and
$\partial[P_{0}<0]\times\mathbb{R}$ is again a hyperplane which is not
possible.

Thus we know that $k=1$ and $P_{3}$ is irreducible. We have $LP_{3}=0$. By
Theorem 1 of [25], we know that any irreducible cubic polynomial solving
$LP_{3}=\lambda|x|^{2}P_{3}$ for some $\lambda\in\mathbb{R}$ is harmonic. Thus
$\Delta P_{3}=0$. Hence $P_{3}=0$ by Lemma 3.4, a contradiction. ∎

##  4\. Lower bounds for Green’s function and estimating the degree of $\deg
P_{0}$

In this section we prove the following theorem using Theorem 1.4 proved in
[14].

######  Theorem 4.1.

If $P_{m}=P_{0}^{k}Q_{m}$ satisfies Theorem 2.9 (i) and (ii), then $\deg
P_{0}+k^{-1}\deg Q_{m}\leq n-2$.

###### Proof.

Let $u$ be the function $\frac{1}{|\nabla(P_{0}Q_{m}^{1/k})|}$. By Lemma 3.1,
$u$ is a positive Jacobi field on $\Sigma=\\{P_{m}=1\\}$ i.e.

$\Delta_{\Sigma}u+|A_{\Sigma}|^{2}u=0.$

In particular, $\Delta_{\Sigma}u\leq 0$ on $\Sigma$.

Let $x\in\operatorname{reg}\Sigma$ and $r_{0}>0$ be fixed. By [14, Theorem
3.2], for any $R>r_{0}$, there exists $G_{R}(x,y)$ the Dirichlet Green’s
function on $\Sigma\cap B_{R}(x)$ satisfying $G_{R}(x,y)\leq
C_{1}(n)|x-y|^{3-n}$. Thus have

$\sup_{y\in\Sigma\cap\partial B_{r_{0}}(x)}G_{R}(x,y)\leq
C_{1}r_{0}^{3-n}\quad\forall R>r_{0}.$

Since $G_{R}(x,y)=0$ for any $y\in\partial B_{R}(x)$, by weak maximum
principle on $\Sigma\cap B_{R}(x)\backslash B_{r_{0}}(x)$, we have for any
$y\in\Sigma\cap B_{R}(x)\backslash B_{r_{0}}(x)$

(22) $u(y)\geq\left(\inf_{\Sigma\cap\partial
B_{r_{0}}(x)}u\right)C_{1}^{-1}r_{0}^{n-3}G_{R}(x,y).$

Letting $R\to\infty$ in (22) and applying Theorem 1.4, we have for all
$y\in\Sigma$, $|x-y|>r_{0}$

(23) $u(y)\geq\left(\inf_{\Sigma\cap\partial
B_{r_{0}}(x)}u\right)C_{1}^{-1}r_{0}^{n-3}G(x,y)\geq c_{1}|x-y|^{3-n}$

where $c_{1}=\left(\inf_{\Sigma\cap\partial
B_{r_{0}}(x)}u\right)C_{2}C_{1}^{-1}r_{0}^{n-3}>0$ is independent of $y$.

Now we want to transplant the lower bound of $u$ on $\Sigma$ to a lower bound
of $u$ on $\\{P_{0}=0\\}$. By Lemma 3.1, $u$ is the velocity of the family
$(\\{P_{0}Q_{m}^{1/k}=t\\})$. Then there exists $R_{0}>0$, a conical
neighborhood $V$ in $\mathbb{R}^{n}$ of a regular point of $\\{P_{0}=0\\}$ and
a function $\phi$ over $\\{P_{0}=0\\}\cap V\setminus B_{R_{0}}$ so that
$\operatorname{graph}\phi\subset\\{P_{m}=1\\}$ and $\\{P_{m}=1\\}\cap
V\setminus\operatorname{graph}\phi$ is compact. Moreover for some $\alpha>0$
and any $|z|\geq R_{0}$, we have

$\phi(z)=c_{2}|z|^{-\gamma}+O(|z|^{-\gamma-\alpha})$

where $\gamma=\deg P_{0}+k^{-1}\deg Q_{m}-1$ and $c_{2}>0$. Since $\deg
P_{0}\geq 2$, we have $\phi\to 0$ as $|z|\to\infty$. Then by (23), we have

$u(z+\phi(z)\nu(z))\geq c_{1}|z+\phi(z)\nu(z)-x|^{3-n}\geq c_{3}|z|^{3-n}$

for some $c_{3}>0$ and all $z\in\\{P_{0}=0\\}\cap V$ with $|z|$ large. On the
other hand, for $z\in\\{P_{0}=0\\}\cap V$ with $|z|$ large

$|u(z+\phi(z)\nu(z))-u(z)|\leq\sup_{t\in[0,1]}\|\nabla
u(z+t\phi(z)\nu(z))\||\phi(z)|\leq
c_{4}|z|^{\gamma}2c_{2}|z|^{-\gamma}=c_{5}.$

Hence combining the above, we have for $z\in\\{P_{0}=0\\}\cap V$ with $|z|$
large

$u(z)\geq c_{2}|z|^{3-n}-c_{5}.$

Letting $|z|\to\infty$, we obtain $\deg P_{0}+k^{-1}\deg Q_{m}\leq n-2$. ∎

##  5\. “Non-existence” theorems

We first introduce some concepts related to minimal cones cf. [15].

######  Definition 5.1.

A minimal hypercone $C\subset\mathbb{R}^{n}$ is called regular if
$\operatorname{sing}C\subset\\{0\\}$. Let $\Sigma=C\cap S^{n-1}$ and
$\lambda_{1}$ be the first eigenvalue of $\Delta_{\Sigma}+|A_{\Sigma}|^{2}$.
The minimal cone $C$ is called stable (strictly stable) if
$(n-3)^{2}+4\lambda_{1}\geq 0$ ($>0$). There are 2 independent positive Jacobi
fields on a stable cone $C$. If $C$ is strictly stable, they are
$|x|^{-\gamma_{\pm}}\phi_{1}$. If $C$ is not strictly stable, they are
$|x|^{-(n-3)/2}\phi_{1}$ and $|x|^{-(n-3)/2}(\log|x|)\phi_{1}$. Here
$\phi_{1}$ is the first eigenfunction of $\Delta_{\Sigma}+|A_{\Sigma}|^{2}$ on
$\Sigma$ and

(24) $\gamma_{\pm}=\frac{n-3\pm\sqrt{(n-3)^{2}+4\lambda_{1}}}{2}.$

By [15, Theorem 2.1] if $C$ is regular area minimizing cone, then
$\mathbb{R}^{n}\setminus C$ consists of two components $E_{\pm}$ and each
$E_{\pm}$ is foliated uniquely by homotheties of smooth minimal hypersurface
$S_{\pm}\subset E_{\pm}$. There exists $R(C)>0,\alpha>0$ so that
$S_{\pm}\setminus B_{R}(0)=\operatorname{graph}_{C}(v_{\pm})$ and as
$|x|\to\infty$

$v_{\pm}(x)=w_{\pm}(x)+O(r^{-\gamma_{+}-\alpha})$

where $\pm w_{\pm}>0$ and $\Delta_{C}w_{\pm}+|A_{C}|^{2}w_{\pm}=0$.

By [15, Theorem 3.1], a regular area minimizing cone $C$ is called strictly
minimizing if $w$ is the positive Jacobi field with slower decay rate near
infinity.

######  Definition 5.2.

A minimal hypercone $C\subset\mathbb{R}^{n}$ is called isoparametric if the
link $C\cap S^{n-1}$ has constant principal curvature in $S^{n-1}$. Every
isoparametric cone is the zero locus of a polynomial called Cartan-Münzner
polynomial [8], [17].

######  Lemma 5.3 ([1],[8],[17],[27]).

Let $P_{0}$ be a Cartan-Münzner polynomial on $\mathbb{R}^{l}$. The following
are all possibilities for the degree $\deg P_{0}$, number of variables $l$ of
$P_{0}$ and the norm squared of the second fundamental form $|A|^{2}$ of the
link of $\\{P_{0}=0\\}$.

(i) $\deg P_{0}=1$, $l\geq 2$, $|A|^{2}=0$,

(ii) $\deg P_{0}=2$, $l\geq 4$, $|A|^{2}=l-2$,

(iii) $\deg P_{0}=3$, $l=5,8,14,26$, $|A|^{2}=2(l-2)$,

(iv)$\deg P_{0}=4$, $l\geq 6$ is even, $|A|^{2}=3(l-2)$,

(v) $\deg P_{0}=6$, $l=8,14$, $|A|^{2}=5(l-2)$.

Moreover, $\\{P_{0}=0\\}$ is area minimizing if and only if $l\geq 4\deg
P_{0}$ and $\Sigma$ is not $S^{1}\times S^{5}$ or $SO(8)\times
SO(2)/SO(6)\times\mathbb{Z}_{2}$.

###### Proof.

By [17], $\deg P_{0}$ is equal to the number of distinct principal curvatures
of the link and it can only be $1,2,3,4,6$. By [8], for $\deg P_{0}\leq 3$,
the isoparametric hypersurfaces are homogeneous and $l$ can only take the
value we mention. By [17], if the $i$-th principal curvature has multiplicity
$m_{i}$, then $m_{i}=m_{i+2}$ with index understood mod $\deg P_{0}$. Thus in
case (iv), we have $l=2(m_{1}+m_{2})+2\geq 6$ is even. For $\deg P_{0}=6$, by
[1], $m_{1}=m_{2}$ can only be $1$ or $2$ and thus $l=3(m_{1}+m_{2})+2=8$ or
$14$. The value of $|A|^{2}$ follows from exact values of principal curvatures
of minimal isoparametric cones [17]. Calculation can be found in [24] in
equations (48) (49) (50) and the formula below (53). The area minimizing
property is a result of [27]. ∎

######  Theorem 5.4.

If $P_{m}=P_{0}^{k}Q_{m}$ satisfies Theorem 2.9 (i) and (ii), then $P_{0}$ is
not a Cartan-Münzner polynomial.

In particular $\deg P_{0}\geq 3$ since all quadratic minimal cones are
isoparametric.

In view of Theorem 2.9(iii) we have the following corollary.

######  Corollary 5.5.

If a polynomial $P$ solves the minimal surface equation, then the tangent cone
of $\operatorname{graph}P$ at $\infty$ is not $C\times\mathbb{R}^{l}$ where
$l\geq 1$ and $C$ is any isoparametric cone.

###### Proof.

Suppose $P_{0}$ is a Cartan-Münzner polynomial of degree $\geq 2$. Permuting
the $x_{i}$’s if necessary, we assume that $x_{1},\dots,x_{l}$ are all
variables involved in $P_{0}$ for some $1\leq l\leq n$ .

We claim that $P_{m}=P_{0}^{k}Q_{m}$ is independent of $x_{l+1},\dots,x_{n}$.
Indeed, by [27], the isoparametric cone $\\{P_{0}=0\\}$ is strictly stable and
strictly minimizing. By Lemma 2.4 and Theorem 2.9, for $t\neq 0$,
$\\{P_{m}=t\\}$ is a smooth minimal hypersurface lying on one side of the cone
$\\{P_{m}=0\\}=\\{P_{0}=0\\}\times\mathbb{R}^{n-l}$. Moreover, the blow-down
of $\\{P_{m}=t\\}$ is $\\{P_{0}=0\\}\times\mathbb{R}^{n-l}$ by Corollary 2.8.
By the Liouville theorem of [10], $\\{P_{m}=t\\}$ is of the form
$H\times\mathbb{R}^{n-l}$ where $H$ is a leaf of the Hardt-Simon foliation of
$\\{P_{0}=0\\}$ in $\mathbb{R}^{l}$. Thus for all $t\in\mathbb{R}$,
$\xi\in\mathbb{R}^{n-l}$, $\\{P_{m}=t\\}$ is invariant under $x\mapsto x+\xi$.
This show that $P_{m}$ is independent of $x_{l+1},\dots,x_{n}$.

Since $\\{P_{0}=0\\}$ is strictly minimizing, the Hardt-Simon foliation
$\\{P_{m}=t\\}$ is a graph of $v$ over $\\{P_{0}=0\\}$ with $v$ decays like
$|x|^{-\gamma_{-}}\phi_{1}$ near infinity. Since
$\frac{1}{|\nabla(P_{0}Q_{m}^{1/k})|}$ is the velocity of the family
$(\\{P_{m}=t^{k}\\})_{t\in(-\epsilon,\epsilon)}$ at $t=0$ near regular points
of $\\{P_{m}=0\\}$, we have

(25) $\displaystyle\deg P_{0}-1+\frac{\deg
Q_{m}}{k}=\frac{l-3-\sqrt{(l-3)^{2}+4\lambda_{1}}}{2}.$

Equation (25) implies

(26) $(l-3)^{2}+4\lambda_{1}\text{ is a perfect square integer.}$

To see this, since $|A|^{2}$ is an integer for isoparametric cones by Lemma
5.3, we have $\lambda_{1}=-|A|^{2}$ is also an integer. Since (25) implies
that the square root in (25) is rational, it follows that the integer
$(l-3)^{2}+4\lambda_{1}$ is a perfect square.

By Theorem 2.9 (ii), we also have

(27) $k\text{ is odd and }\deg Q_{m}\text{ is even.}$

We claim that (25), (26), (27) cannot hold simultaneously if $P_{0}$ is a
Cartan-Münzner polynomial. Indeed, we show this for each case in Lemma 5.3.

Case (i) $\deg P_{0}=1$. This is not what we want by our assumption.

Case (ii) $\deg P_{0}=2$, $\lambda_{1}=-(l-2)$.

In order for $\\{P_{0}=0\\}$ to be area minimizing, we have $l\geq 8$. By
(26), there is $a\in\mathbb{N}$ so that

$(l-3)^{2}-4(l-2)=a^{2}.$

This is equivalent to

$(l-5-a)(l-5+a)=8.$

The only nonegative integer solutions to this equation is

$(l,a)=(8,1).$

By (25), we have $k^{-1}\deg Q_{m}=1$. This contradicts (27).

Case (iii) $\deg P_{0}=3$.

In order for $\\{P_{0}=0\\}$ to be area minimizing, we have $l=14$ and $l=26$.

If $l=14$, then $\lambda_{1}=-24$ and thus $(l-3)^{2}+4\lambda_{1}=121-96=25$.
By (25) $k^{-1}\deg Q_{m}=1$. This contradicts (27).

If $l=26$, then $\lambda_{1}=-48$ and $(n-3)^{2}+4\lambda_{1}=337$ is not a
perfect square. This contradicts (26).

Case (iv) $\deg P_{0}=4$.

In order for $\\{P_{0}=0\\}$ to be area minimizing, we have $l\geq 16$. By
(26), there is $a\in\mathbb{N}$ so that

$(l-3)^{2}-12(l-2)=a^{2}.$

This is equivalent to

$(l-9-a)(l-9+a)=48.$

The positive integer solutions to this equation are

$(l,a)=(16,1),(17,4),(22,11).$

Since $l$ must be even by Lemma 5.3, the solution $(l,a)=(17,4)$ is not
possible.

If $(l,a)=(16,1)$, from (25) we have $k^{-1}\deg Q_{m}=3$.

If $(l,a)=(22,11)$, from (25) we have $k^{-1}\deg Q_{m}=1$. Both violate (27).

Case (v) $\deg P_{0}=6$.

By Lemma 5.3, $\\{P_{0}=0\\}$ is not area minimizing in this case. This
finishes the proof. ∎

Now we prove our second “non-existence” theorem.

######  Theorem 5.6.

If there is a nonlinear polynomial solution to the minimal surface equation on
$\mathbb{R}^{8}$, then there exists a cubic, strictly stable, area minimizing
cone in $\mathbb{R}^{8}$ which is not strictly minimizing.

###### Proof.

We take $n=8$ throughout the proof. By Theorem 2.9, we have

$P=P_{0}^{k}Q_{m}+P_{m-1}+\cdots+P_{2}+P_{1}.$

Since $\\{P_{0}=0\\}$ is 7 dimensional, it is a regular cone.

By Lemma 3.1, the function $u=\frac{1}{|\nabla(P_{0}Q_{m}^{1/k})|}$ is a
positive Jacobi field on $\\{P_{0}=0\\}$. By [23, Lemma 6.1.7], we have
$\lambda_{1}\leq-(n-2)$. Thus

(28) $\deg P_{0}\leq\gamma_{+}+1-k^{-1}\deg
Q_{m}\leq\frac{n-1+\sqrt{(n-3)^{2}-4(n-2)}}{2}=4.$

By Theorem 5.4, we know that $\deg P_{0}\geq 3$.

If $\deg P_{0}=4$, then equality holds in (28) and in particular
$\lambda_{1}=-(n-2)=-6$. By the rigidity part of Theorem A in [28] (see also
[18], [30]), we know that $\Sigma$ must be the Clifford torus. Hence
$\\{P_{0}=0\\}$ is a quadratic cone. This contradicts $\deg P_{0}=4$.

Thus $\deg P_{0}=3$. We show that $\\{P_{0}=0\\}$ is strictly stable and not
strictly minimizing. Indeed suppose $\\{P_{0}=0\\}$ were not strictly stable,
then $\lambda_{1}=-\frac{25}{4}$. Since
$u=\frac{1}{|\nabla(P_{0}Q_{m}^{1/k})|}$ is a positive Jacobi field decaying
without $\log$ factor, we have

$\deg P_{0}-1+k^{-1}\deg Q_{m}=\frac{5}{2}.$

Hence $k^{-1}\deg Q_{m}=\frac{1}{2}$. This contradicts the fact that $k$ is
odd and $\deg Q_{m}$ is even by Theorem 2.9. Now suppose $\\{P_{0}=0\\}$ is
strictly minimizing, then we can apply [21, Theorem 5] to see that

$\deg P\leq\gamma_{-}+1\leq\frac{7}{2}.$

This contradicts Theorem 3.6 where cubic polynomial solutions are ruled out. ∎

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# Population-Level Inference of Strong Gravitational Lenses with Neural
Network-Based Selection Correction

Ronan Legin  Connor Stone  Yashar Hezaveh  Laurence Perreault-Levasseur

###### Abstract

A new generation of sky surveys is poised to provide unprecedented volumes of
data containing hundreds of thousands of new strong lensing systems in the
coming years. Convolutional neural networks are currently the only state-of-
the-art method that can handle the onslaught of data to discover and infer the
parameters of individual systems. However, many important measurements that
involve strong lensing require population-level inference of these systems. In
this work, we propose a hierarchical inference framework that uses the
inference of individual lensing systems in combination with the selection
function to estimate population-level parameters. In particular, we show that
it is possible to model the selection function of a CNN-based lens finder with
a neural network classifier, enabling fast inference of population-level
parameters without the need for expensive Monte Carlo simulations.

Machine Learning, ICML

##  1 Introduction

Strong gravitational lensing is a phenomenon where light rays of a distant
background source are deflected by the gravity of foreground matter, resulting
in the production of multiple images. These lensing systems can be used for
important applications in cosmology, such as constraining the mass of dark
matter present in foreground galaxies (e.g., Vegetti et al., 2012; Hezaveh et
al., 2016), providing precise measurements of the expansion rate of the
universe (e.g., Wong et al., 2020) and learning more about the morphology of
distant high redshift galaxies (e.g., Coe et al., 2013).

In the next few years, a new generation of wide-area sky surveys from the
Rubin Observatory’s Legacy Survey of Space and Time (LSST) and the Euclid
Space Telescope (EST) will provide an unprecedented volume of data containing
hundreds of thousands of new strong lensing systems (Collett, 2015).

Recent works have focused on developing new and efficient computational
methods to analyze these large sets of data. In particular, machine learning
has been proposed to efficiently perform various strong lensing analysis
tasks. Convolutional neural networks (CNN) have been trained in classification
tasks to detect strong lenses from survey data (Schaefer et al., 2018; Lanusse
et al., 2018). They have been used to discover new strong lensing systems from
the Kilo Degree Survey (Li et al., 2020, KiDS) and the Ultraviolet Near
Infrared Optical Northern Survey (Savary et al., 2021), and have generally
surpassed traditional methods in terms of speed and accuracy according to
benchmark tests (Metcalf et al., 2019). These networks have also been used to
perform fast and automated inference of the lensing parameters from the
observations of individual strong lensing systems (Hezaveh et al., 2017;
Perreault Levasseur et al., 2017; Wagner-Carena et al., 2021; Park et al.,
2021; Legin et al., 2021), speeding up the analysis by more than $\sim 10$
million times compared to traditional methods.

However, many important science applications involving strong gravitational
lenses rely on population-level statistics of lensing systems (e.g.,
Sonnenfeld & Cautun, 2021; Sonnenfeld, 2021, 2022b, 2022a). Generally, the
task of population-level inference is best framed within a hierarchical
framework, where the problem is broken down in multiple stages. In a
hierarchical framework, a clear separation between the different components of
the analysis is provided, (e.g., between the theoretical models underlying the
population distribution and the parameters describing the individual
observations themselves). Generally, the problem includes the inference of
global hyperparameters $\lambda$ describing the population distribution of
model parameters $\theta$ i.e the parameters describing individual strong
lenses. It is important, however, for these hierarchical models to take into
account the selection mechanisms of the observed data sets to avoid biased
inference results (Sonnenfeld & Cautun, 2021; Sonnenfeld, 2022a). The
selection function would depend on the model parameters of individual strong
lensing observations. Traditionally, this makes population-level inference
difficult in standard settings, where complex selections are difficult to
quantify and the corrections require a large number of simulations at every
step of the posterior sampling process.

In this work, we model the selection function of a CNN-based strong lensing
detector using neural networks. We propose an efficient and flexible
likelihood-free inference framework for performing accurate large scale
population-level inference of strong lensing parameters that can readily
incorporate learned selection effects. In addition, our framework circumvents
issues regarding the interpretability of typical black-box inference machines
by allowing clear separation between the different components of our
hierarchical population model. In section 2, we describe our Bayesian
hierarchical inference framework for inferring the posterior distribution of
hyperparameters $\lambda$ describing the population of strong gravitational
lensing model parameters $\theta$. We also detail our strong lensing
simulations and the modeling of the CNN-based selection bias using neural
networks. In section 3, we test our hierarchical method to infer the
population of strong lensing model parameters based on a set of 1000 simulated
strong lensing observations.

##  2 Methods

Given a set of observations $\\{x_{i}\\}$, the task is to infer the posterior
distribution of population hyperparameters $p(\lambda|\\{x_{i}\\})$. Using
Bayes’ theorem, this can be expressed as

$\displaystyle p(\lambda|\\{x_{i}\\})$
$\displaystyle=\frac{p(\lambda)\prod_{i}p(x_{i}|\lambda)}{\int
d\lambda^{\prime}p(\lambda^{\prime})\prod_{i}p(x_{i}|\lambda^{\prime})}$
$\displaystyle=\left[\int
d\lambda^{\prime}\frac{p(\lambda^{\prime})}{p(\lambda)}\prod_{i}\frac{p(x_{i}|\lambda^{\prime})}{p(x_{i}|\lambda)}\right]^{-1},$
(1)

In a typical hierarchical model, the individual observations $x_{i}$ may
explicitly depend on intermediate model parameters $\theta$ and only
implicitly on $\lambda$. In this case, the likelihood $p(x_{i}|\lambda)$ can
be expanded as

$\displaystyle p(x_{i}|\lambda)$ $\displaystyle=\int
p(x_{i}|\theta)p(\theta|\lambda)d\theta,$ (2)

where $\lambda$ is ignored in $p(x_{i}|\theta)$ as it only implicitly affects
the observation $x_{i}$.

In practice, it may be more likely to observe certain events $x_{i}$. This may
lead to the chosen set of events $\\{x_{i}\\}$ to instead represent a biased
selection of all possible observations, which would consequently result in a
biased inference of the posterior distribution of the population
hyperparameters $p(\lambda|\\{x_{i}\\})$. In the presence of selection effects
(Mandel et al., 2019; Moore & Gerosa, 2021), the proper expression for the
likelihood $p(x_{i}|\lambda)$ is

$\displaystyle p(x_{i}|\lambda)$ $\displaystyle=\frac{\int
p(x_{i}|\theta)p(\theta|\lambda)d\theta}{\alpha(\lambda)},$ (3)

where $\alpha(\lambda)$ is evaluated by integrating over the product of the
probability of detection $p_{\mathrm{det}}(\theta)$ of an event with model
parameters $\theta$ and the population model $p(\theta|\lambda)$,

$\displaystyle\alpha(\lambda)=\int
p_{\mathrm{det}}(\theta)p(\theta|\lambda)d\theta.$ (4)

As such, accounting for selection bias incorporates an overall normalization
factor $\alpha(\lambda)$ within the likelihood $p(x|\lambda)$ that can be
interpreted as the fraction of samples from the population model
$p(\theta|\lambda)$ that would be observed given the selection function
$p_{\mathrm{det}}(\theta)$ (Mandel et al., 2019). Note that Equation 3 can be
rewritten as

$\displaystyle p(x_{i}|\lambda)$
$\displaystyle=\mathbb{E}_{p(\theta|x_{i})}\left[\frac{p(\theta|\lambda)}{p(\theta)}\right]\frac{p(x_{i})}{\alpha(\lambda)},$
(5)

by applying Bayes’ theorem on the likelihood of the individual lensing
observations $p(x_{i}|\theta)$ to obtain an expectation value over the lensing
model parameter posterior $p(\theta|x_{i})$.

Plugging this new expression back into Equation 1, we obtain

$\displaystyle p(\lambda|\\{x_{i}\\})$ $\displaystyle=\left[\int
d\lambda^{\prime}\frac{p(\lambda^{\prime})}{p(\lambda)}\prod_{i}\frac{\mathbb{E}_{p(\theta|x_{i})}\left[\frac{p(\theta|\lambda^{\prime})}{p(\theta)\alpha(\lambda^{\prime})}\right]}{\mathbb{E}_{p(\theta|x_{i})}\left[\frac{p(\theta|\lambda)}{p(\theta)\alpha(\lambda)}\right]}\right]^{-1},$
(6)

where the marginal probabilities $p(x_{i})$ from applying Bayes’ theorem on
both $p(x_{i}|\lambda)$ and $p(x_{i}|\lambda^{\prime})$ cancel out. As the
expectation values cannot be evaluated analytically, we can instead
approximate them as a mean over samples from the lensing posterior
$p(\theta|x_{i})$. Similarly, we can estimate $\alpha(\lambda)$ by rewriting
Equation 4 as the expectation value of $p_{\mathrm{det}}(\theta)$ over
$p(\theta|\lambda)$ and approximate it with samples from $p(\theta|\lambda)$.

The difficulty in estimating $\alpha(\lambda)$, however, is that the selection
function $p_{\mathrm{det}}(\theta)$, which captures the effects of the
selection bias, is typically complex and not explicitly known (e.g.,
Sonnenfeld, 2022a). This is especially true if the selection method is based
on CNN detections of strong lenses, as the entire detection algorithm is
framed as a black-box.

In this work, we circumvent this issue by modeling the selection function
$p_{\mathrm{det}}(\theta)$ of CNN-based detectors of strong lenses. In
particular, we train a neural network to classify between detected and non-
detected samples of strong lensing model parameters $\theta$. The output
probability of this classification network results in the probability of
detection $p_{\mathrm{det}}(\theta)$ of a lensing system with parameters
$\theta$.

Furthermore, we train a convolutional Bayesian Neural Network (BNN) to predict
the posterior distribution of strong lensing model parameters $p(\theta|x)$
and to rapidly sample from it in order to approximate the expectation values
from Equation 6. Given that upcoming sky surveys are estimated to discover
hundreds of thousands of new strong lenses, traditional methods such as
likelihood-based inference, which require time-consuming analyses from strong
lensing experts, will be intractable for this task. Instead, BNNs have been
shown to be orders of magnitude faster than traditional methods and can
provide well-calibrated uncertainties on strong lensing parameter predictions
(Perreault Levasseur et al., 2017; Wagner-Carena et al., 2021; Park et al.,
2021).

###  2.1 Simulations

We simulate two sets of strong lensing observations: low-resolution images
similar to what is expected to be observed by wide-area surveys such as LSST
and the corresponding high-resolution follow-up observations. As a whole, our
simulations can be described by a 13-dimensional model parameter vector
$\theta$. Examples of the lensing simulations are shown in appendix A.

At test time, the CNN detector is applied to the set of mock LSST lensing
observations. Then, for the images that are classified as a detection, we
apply our BNN model to their follow-up observations in order to infer the
posterior distribution of strong lensing model parameters $p(\theta|x)$ with
higher precision. Given the inferred posterior $p(\theta|x)$, we can evaluate
Equation 6 by approximating the expectation values using samples from it.

###  2.2 Selection function

We train a multilayer perceptron (MLP) neural network to predict the
probability of detection $p_{\mathrm{det}}(\theta)$ for CNN-based detectors of
strong gravitational lenses. This neural network is trained to classify
between a set of $\theta$ that results in the detection of strong lensing
systems and a set that results in a non-detection.

To generate training data, we apply a CNN detector to a set of simulated low-
resolution strong lensing images. This reflects the lens search procedure in
LSST-like survey data. Lenses with parameters $\theta$ which are detected by
the CNN are classified as detectable, and the missed lenses are classified as
non-detectable. An illustration of the training data generation is shown in
Figure 1.

Figure 1: Flow chart illustrating the generation of training data used for
training the neural selection function. The CNN-based lens finder is applied
to simulated low-resolution strong lensing images with model parameters
$\theta$ sampled from a prior distribution. Two sets of model parameters are
made based on whether the images were detected as strong lenses. The neural
selection function is trained to classify between these two sets, where the
output classification probability represents the probability of detection
$p_{\mathrm{det}}(\theta)$.

The neural network that predicts the selection function
$p_{\mathrm{det}}(\theta)$ is composed of an input layer and a hidden layer
each with 256 neurons, in addition to a final layer outputting a single value.
We use the Exponential Linear Unit activation function (Clevert et al., 2015,
ELU) for the first two layers, and the sigmoid activation for the output of
the network. The final layer predicts the probability of $\theta$ being
classified as detectable, which gives us a prediction for the probability of
detection $p_{\mathrm{det}}(\theta)$. The weights of the neural network are
updated using the Adam optimizer with a constant learning rate of $10^{-4}$.
Our training data consists of two equal sets of 10,000 detectable and non-
detectable $\theta$ parameters. The network is trained for 10,000 epochs with
a batch size of 1000.

##  3 Results

We compute the posterior $p(\lambda|\\{x_{i}\\})$ using Equation 6 from a set
of 1000 CNN-detected strong lensing observations. As detailed in section 2.2,
we first train a neural network to predict the CNN detector selection function
$p_{\mathrm{det}}$. Then, we use the neural network’s prediction of
$p_{\mathrm{det}}$ in Equation 4 to compute the normalization factor
$\alpha(\lambda)$ over a grid of $\lambda$ values. Separately, we train a BNN
to infer the lensing posterior $p(\theta|x_{i})$ for the individual strong
lensing observations. In the final step, we use our prediction of
$\alpha(\lambda)$ and $p(\theta|x_{i})$ to compute the posterior of population
hyperparameters $p(\lambda|\\{x_{i}\\})$ from Equation 6. To approximate the
expectation values from Equation 6, we use 100,000 samples from the posterior
distribution $p(\theta|x_{i})$ for each observation $x_{i}$. We compute
$\alpha(\lambda)$ by approximating the integral from Equation 4 using 10,000
samples from the population distribution $p(\theta|\lambda)$. The computation
of $\alpha(\lambda)$ is repeated 1000 times using different samples from
$p(\theta|\lambda)$ and averaged over in order to reduce noise in its
prediction. This helps mitigate instabilities that may occur when dividing by
the normalization factor $\alpha(\lambda)$ in Equation 6. A flow chart
illustrating the problem setup for inference is shown in Figure 5.

For our problem, the hyperparameters $\lambda$ that we wish to infer are the
mean $\mu$ and standard deviation $\sigma$ of a log-normal distribution
describing the population distribution of strong lensing Einstein radii
$R_{E}$. As an initial test, we apply a mask on the strong lensing light for
lensing systems with Einstein radius smaller than $0.8$ arcseconds, making it
impossible for the CNN finder to detect the presence of strong lensing. This
leads to a more important selection bias towards systems with larger Einstein
radii, and for the initial test, allows to more intuitively visualize the
impact of selection bias on the final result. In Figure 2, we show results of
the predicted posterior $p(\lambda|\\{x_{i}\\})$ for two different scenarios:
one in which the hierarchical model includes the correction due to selection
bias and the other where it is ignored (i.e. whether $\alpha(\lambda)$ is
included in Equation 6).

Figure 2: Predicted posterior $p(\lambda|\\{x_{i}\\})$ given a set of 1000
strong lensing observations detected using the CNN strong lensing detector
with the selection bias correction included within the hierarchical model (red
contours) and without (blue contours). The $68\%$, $95\%$ and $99.7\%$
probability regions are shown for both posterior distributions. This result
demonstrates that ignoring the effects of the selection bias caused by the CNN
detectors leads to a biased inference of the population hyperparameters
$\lambda$.

We also perform a test to verify if our modeled selection function
$p_{\mathrm{det}}$ accurately portrays the performance of the CNN strong
lensing detector. To do so, we start by generating a new set of 10,000 strong
lensing observations based on a wide range of model parameters $\theta$,
including the Einstein radius $R_{E}$. Then, in the first case, we evaluate
the detection probability $p_{\mathrm{det}}(\theta)$ for these $\theta$
samples, and keep the samples considered as detected if they satisfy
$\rho<p_{\mathrm{det}}(\theta)$, where $\rho$ is a number randomly sampled
between zero and one for each $\theta$. We denote this set of detected
$\theta$ samples as $\theta_{p_{\mathrm{det}}}$. In the second case, we apply
the CNN detector directly on the strong lensing images. For the observations
that passed as a detection, we keep the samples of $\theta$ that were used to
generate them. We denote this set of samples as $\theta_{\text{CNN}}$. In
Figure 3, we plot the distribution of the Einstein radius parameter $R_{E}$
from both sets of $\theta_{p_{\mathrm{det}}}$ and $\theta_{\text{CNN}}$. This
test allows us to verify if our modeled selection function
$p_{\mathrm{det}}(\theta)$ accurately reflects the overall distribution of
strong lenses that are discovered by the CNN detector.

Figure 3: The predicted number of strong lenses detected from a set of 10,000
simulated strong lenses by the CNN detector (black line) and by our modeled
selection function $p_{\mathrm{det}}$ (dotted red line) as a function of the
Einstein radius $R_{E}$. The neural selection function $p_{\mathrm{det}}$ is
capable of accurately portraying the selection mechanism of CNN-based lens
detectors.

##  4 Discussion

The framework presented in this paper allows for the hierarchical inference of
population-level parameters of strong lensing systems, including the effects
of selection functions. In the first stage of inference, the posteriors of the
parameters of individual lensing systems are obtained. In the second stage,
the distribution of the parameters of the general population of lenses is
inferred using the posteriors of individual lens parameters. In this second
stage of inference, it is also essential to include the effect of the
selection function.

It is of course possible to forgo a hierarchical framework and infer
population-level parameters in a one-stage inference setup. Simulation-based
(or likelihood-free) inference methods that directly model target quantities
such as $p(x|\lambda)$ have proven to be promising methods for inferring
population-level parameters in cosmology (Brehmer et al., 2019; Coogan et al.,
2020; Gerardi et al., 2021).

However, Bayesian hierarchical modeling can be a flexible and powerful
alternative, providing several advantages. A distinct advantage of
hierarchical modeling is the fact that the inference of the population-level
parameters is done independently from the inference of individual systems. In
this framework, the posterior of the lens parameters given the data
$p(\theta|x_{i})$ (the noise term) can be calculated once with expert
(lensing) knowledge, either using machine learning based methods (e.g.,
Hezaveh et al., 2017; Wagner-Carena et al., 2021; Legin et al., 2021) or
likelihood-based methods (e.g., Gu et al., 2022). The inference of population-
level parameters could then be carried out and explored without redoing any
analysis of the observed data. The term $p(\theta|\lambda)$, often called the
“theory term”, describes the theoretical prediction of lens parameters under a
proposed hyperparameter. This allows one to explore any theoretical model in a
modular way using the same set of posteriors, $p(\theta|x_{i})$, calculated
earlier.

In this work, we modeled the selection function $p_{\mathrm{det}}(\theta)$
with neural networks, facilitating accurate calculation of the posterior of
the hyperparameters without the need for repeated Monte Carlo simulations.
Once learned, the same neural selection function could be used to recalculate
the denominator of Equation 3 for any new theoretical model under
consideration.

The approach proposed here uses simulation-based inference and machine
learning models to learn surrogate models for the various modules that form a
hierarchical inference model. This allows taking advantage of the power and
the versatility of neural networks while retaining the benefits and the
interpretability of Bayesian hierarchical frameworks, where uncertainties are
correctly propagated from one level to the next.

## Acknowledgements

This research was supported by a generous grant from the Schmidt Futures
Foundation. The work was also enabled in part by computational resources
provided by Calcul Quebec, Compute Canada and the Digital Research Alliance of
Canada. Y.H. and L.P. acknowledge support from the National Sciences and
Engineering Council of Canada grant RGPIN-2020-05102, the Fonds de recherche
du Québec grant 2022-NC-301305, and the Canada Research Chairs Program. We
acknowledge support from the Natural Sciences and Engineering Research Council
of Canada (NSERC), [funding reference number 411299547]. We thank Adam Coogan
and Alexandre Adam for useful discussions in regards to simulation-based
inference methods.

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##  Appendix A Appendix

Figure 4: Examples of generated strong gravitational lensing simulations. The
set of simulations used consists of low-resolution wide-area sky surveys
(left) and the corresponding follow-up high-resolution image (right). The
probability of detection $p_{\mathrm{det}}$ predicted by our neural network
model is shown in the top corner of each low-resolution image. The strong
lensing simulations are composed of a Singular Isothermal Ellipsoid (Kormann
et al., 1994, SIE) with added external shear representing the main lens
deflector and a background source following a Sérsic brightness profile
(Sérsic, 1963). We also add stellar light following a Sérsic profile to the
main lens deflector galaxy in the low-resolution images. This is not done for
the high-resolution follow-up images as we assume that the stellar light from
the main deflector galaxy is removed in a data preprocessing stage. As a
whole, our simulations can be fully described by a 13-dimensional parameter
vector $\theta$. Figure 5: Flow chart illustrating the general setup for the
inference of the population hyperparameter posterior $p(\lambda|\\{x_{i}\\})$
for an example with four generated strong lensing observations
$\\{x_{1},x_{2},x_{3},x_{4}\\}$. The grey colored boxes are the neural
networks used to perform the hierarchical inference task. The lens finder
(CNN-based detector) is applied to the observations and classifies whether
they contain strong lensing. For the images detected as strong lenses, the
posterior of lensing model parameters $p(\theta|x_{i})$ is obtained by
applying a Bayesian Neural Network (BNN) on the followup high-resolution
images. The population distribution $p(\theta|\lambda)$, neural selection
function $p_{\mathrm{det}}(\theta)$ and predicted lensing posteriors (in this
example, $p(\theta|x_{3})$ and $p(\theta|x_{4})$) are used to compute the
posterior of population-level parameters $p(\lambda|x_{3},x_{4})$.