# Existence of entire solutions to a fractional Liouville equation in
$\mathbb{R}^{n}$

Ali Hyder
Universität Basel
<EMAIL_ADDRESS>The author is supported by the Swiss National Science
Foundation.

###### Abstract

We study the existence of solutions to the problem

$(-\Delta)^{\frac{n}{2}}u=Qe^{nu}\quad\text{in }\mathbb{R}^{n},\quad
V:=\int_{\mathbb{R}^{n}}e^{nu}dx<\infty,$

where $Q=(n-1)!$ or $Q=-(n-1)!$. Extending the works of Wei-Ye and Hyder-
Martinazzi to arbitrary odd dimension $n\geq 3$ we show that to a certain
extent the asymptotic behavior of $u$ and the constant $V$ can be prescribed
simultaneously. Furthermore if $Q=-(n-1)!$ then $V$ can be chosen to be any
positive number. This is in contrast to the case $n=3$, $Q=2$, where Jin-
Maalaoui-Martinazzi-Xiong showed that necessarily $V\leq|S^{3}|$, and to the
case $n=4$, $Q=6$, where C-S. Lin showed that $V\leq|S^{4}|$.

##  1 Introduction to the problem

In this paper we consider the equation

$\displaystyle(-\Delta)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad\text{in
$\mathbb{R}^{n}$},$ (1)

where $n\geq 1$ and

$\displaystyle V:=\int_{\mathbb{R}^{n}}e^{nu}dx<\infty.$ (2)

The operator $(-\Delta)^{\frac{n}{2}}$ can be defined as follows. For $s>0$ we
set

$\widehat{(-\Delta)^{s}\varphi}(\xi):=|\xi|^{2s}\hat{\varphi}(\xi),\quad\text{for
}\varphi\in\mathcal{S}(\mathbb{R}^{n}),$

where

$\mathcal{S}(\mathbb{R}^{n}):=\left\\{u\in
C^{\infty}(\mathbb{R}^{n}):\sup_{x\in\mathbb{R}^{n}}|x|^{N}|D^{\alpha}u(x)|<\infty\text{
for all }N\in\mathbb{N}\text{ and }\alpha\in\mathbb{N}^{n}\right\\}$

is the Schwartz space. Define the space

$L_{s}(\mathbb{R}^{n}):=\left\\{v\in
L^{1}_{loc}(\mathbb{R}^{n}):\int_{\mathbb{R}^{n}}\frac{|v(x)|}{1+|x|^{n+2s}}dx<\infty\right\\}.$

Then we have the following definition:

###### Definition 1.1

Let $f\in L^{1}(\mathbb{R}^{n})$. A function $u\in
L_{\frac{n}{2}}(\mathbb{R}^{n})$ is said to be a solution of

$(-\Delta)^{\frac{n}{2}}u=f\quad\text{in }\mathbb{R}^{n},$

if

$\displaystyle\int_{\mathbb{R}^{n}}u(-\Delta)^{\frac{n}{2}}\varphi
dx=\int_{\mathbb{R}^{n}}f\varphi dx,\text{ for every
}\varphi\in\mathcal{S}(\mathbb{R}^{n}),$ (3)

where the integral on the left-hand side of (3) is well-defined thanks to
Proposition A.2 in the appendix.

Geometrically if $u$ is a smooth solution of (1)-(2) then the conformal metric
$g_{u}:=e^{2u}|dx|^{2}$ on $\mathbb{R}^{n}$ ($|dx|^{2}$ is the Euclidean
metric on $\mathbb{R}^{n}$) has constant $Q$-curvature equal to $(n-1)!$.
Moreover the volume and the total $Q$-curvature of the metric $g_{u}$ are $V$
and $(n-1)!V$ respectively.

It is well-known that

$u_{\lambda,x_{0}}(x):=\log\frac{2\lambda}{1+\lambda^{2}|x-x_{0}|^{2}},$ (4)

is a solution of (1)-(2) with $V=|S^{n}|$ for every $\lambda>0$ and
$x_{0}\in\mathbb{R}^{n}$. Any such $u_{\lambda,x_{0}}$ is called spherical
solution, because it can be obtained by pulling back the round metric of
$S^{n}$ onto $\mathbb{R}^{n}$ via stereographic projection. When $n=2$ W.
Chen-C. Li [4] showed that these are the only solutions to (1)-(2). On the
other hand in higher even dimension non-spherical solutions do exist as shown
by A. Chang-W. Chen [3]:

Theorem A ([3]) _Let $n\geq 4$ be any even number. Then for every
$V\in(0,|S^{n}|)$ there exists at least one solution to (1)-(2). _

Moreover J. Wei and D. Ye [14] in dimension $4$ and A. Hyder and L. Martinazzi
[7] for arbitrary even dimension $n\geq 4$ proved the existence of solution to
(1)-(2) with prescribed volume and asymptotic behavior in the following sense:

Theorem B ([14], [7]) _Let $n\geq 4$ be even. For a given $V\in(0,|S^{n}|)$,
and a given polynomial $P$ such that degree$(P)\leq n-2$ and_

$x\cdotp\nabla P(x)\to\infty\quad\text{as }|x|\to\infty,$ (5)

_there exists a solution $u$ to (1)-(2) having the asymptotic behavior_

$\displaystyle u(x)=-P(x)-\alpha\log|x|+C+o(1),$ (6)

_where $\alpha:=\frac{2V}{|S^{n}|}$, and $o(1)\to 0$ as $|x|\to\infty$._

When $n$ is odd things are more complex as the operator
$(-\Delta)^{\frac{n}{2}}$ is nonlocal. In a recent work T. Jin, A. Maalaoui,
L. Martinazzi, J. Xiong [8] have proven:

Theorem C ([8]) _For every $V\in(0,|S^{3}|)$ there exists at least one smooth
solution to (1)-(2)_ with $n=3$.

Extending the results of [3], [14], [7] and [8] to arbitrary odd dimension
$n\geq 3$ we prove the following theorem about the existence of solutions to
(1)-(2) with prescribed asymptotic behavior:

###### Theorem 1.1

Let $n\geq 3$ be an odd integer. For any given $V\in(0,|S^{n}|)$ and any given
polynomial $P$ of degree at most $n-1$ such that

$P(x)\to\infty\quad\text{as }|x|\to\infty,$ (7)

there exists a $u\in C^{\infty}(\mathbb{R}^{n})\cap
L_{\frac{n}{2}}(\mathbb{R}^{n})$ solution to (1)-(2) having the asymptotic
behavior given in (6) with $\alpha=\frac{2V}{|S^{n}|}$.

Notice that, contrary to the result of Theorem C, in Theorem 1.1 we can now
prescribe both the asymptotic behaviour and the volume, similar to Theorem B,
but in fact in more generality, since the condition (5) has been replaced by
the weaker condition (7). Actually with minor modifications one can prove that
the condition (7) also suffices in even dimension. On the other hand we do not
expect this assumption to be optimal, compare to Theorem D below.

We also remark that the condition $0<V<|S^{n}|$ is necessary for the existence
of non-spherical solution to (1)-(2) in dimension 3 and 4 as shown in [8] and
[10] respectively, but in higher dimension solutions could exist for large
$V$, as shown for instance in dimension 6 by L. Martinazzi [13].

Also the condition $n\geq 3$ in Theorem 1.1 is necessary, since for $n=1$ any
solution of (1)-(2) is spherical, i.e. as in (4), see F. Da Lio, L. Martinazzi
and T. Rivière [5].

Now we move from the problem of existence to the problem of studying the most
general asymptotic behavior of solutions to (1)-(2).

For $n$ even we have this result due to C.S. Lin for $n=4$ and L. Martinazzi
when $n\geq 6$:

Theorem D ([10], [12]) _Any solution $u$ of (1)-(2) with $n$ even has the
asymptotic behavior_

$\displaystyle u(x)=-P(x)-\alpha\log|x|+o(\log|x|)$ (8)

_where $\alpha=\frac{2V}{|S^{n}|}$, $\frac{o\left(\log|x|\right)}{\log|x|}\to
0$ as $|x|\to\infty$ and $P$ is a polynomial bounded from below and of degree
at most $n-2$._

Under certain regularity assumptions T. Jin, A. Maalaoui, L. Martinazzi, J.
Xiong [8] extended the above result to dimension $3$, and among other things
they proved:

Theorem E ([8]) _Let $u\in W^{2,1}_{loc}(\mathbb{R}^{3})$ be such that $\Delta
u\in L_{\frac{1}{2}}(\mathbb{R}^{3})$ and $u$ satisfies (2). If $u$ solves (1)
in the sense that_

$\int_{\mathbb{R}^{3}}(-\Delta)u(-\Delta)^{\frac{1}{2}}\varphi
dx=2\int_{\mathbb{R}^{3}}e^{3u}\varphi dx,\quad\text{for every
}\varphi\in\mathcal{S}(\mathbb{R}^{3}),$

_then $u$ has the asymptotic behavior given by (8) with
$\alpha=\frac{2V}{|S^{3}|}$ and $P$ is a polynomial bounded from below and of
degree $0$ or $2$._

In our upcoming paper [6] extending Theorem E we study the asymptotic behavior
of solutions to (1)-(2) in arbitrary odd dimension under much weaker
regularity assumptions:

Theorem F([6]) _Let $n\geq 3$ be any odd integer. Let $u\in
L_{\frac{n}{2}}(\mathbb{R}^{n})$ be a solution to (1)-(2) in the sense of
Definition 1.1. Then $u$ has the asymptotic behavior given by (8) with
$\alpha=\frac{2V}{|S^{n}|}$ and $P$ a polynomial bounded from below and of
degree at most $n-1$._

Now we shall discuss the case when the $Q$-curvature is negative. We consider
the equation

$\displaystyle(-\Delta)^{\frac{n}{2}}u=-(n-1)!e^{nu}\quad\text{in
$\mathbb{R}^{n}$}.$ (9)

Geometrically a smooth solution of (9) corresponds to a conformally flat
metric $g_{u}=e^{2u}|dx|^{2}$ on $\mathbb{R}^{n}$ which has constant
$Q$-curvature $-(n-1)!$.

In even dimension $n\geq 4$ L. Martinazzi [11] has shown that any solution $u$
to (9)-(2) has the asymptotic behavior given by (8) with
$\alpha=-\frac{2V}{|S^{n}|}$, while for any $V>0$ and any given polynomial $P$
of degree at most $n-2$ satisfying (5), A. Hyder-L. Martinazzi [7] have proven
the existence of solutions to (9)-(2) having the asymptotic behavior given by
(6) with $\alpha=-\frac{2V}{|S^{n}|}$. As in the positive case, we shall
extend this existence result to arbitrary odd dimension $n\geq 3$, again
replacing condition (5) with the weaker condition (7).

###### Theorem 1.2

Let $n\geq 3$ be an odd integer. For any given $V>0$ and any given polynomial
$P$ of degree at most $n-1$ satisfying (7) there exists a $u\in
C^{\infty}(\mathbb{R}^{n})\cap L_{\frac{n}{2}}(\mathbb{R}^{n})$ solution to
(9)-(2) having the asymptotic behavior given in (6) with
$\alpha=-\frac{2V}{|S^{n}|}$.

Finally we remark that in dimension $1$ and $2$ (9)-(2) has no solution
(compare to [11] and [5]).

Acknowledgements I would like to thank Prof. Luca Martinazzi for introducing
me to this problem and his constant help throughout the work.

##  2 Proof of Theorem 1.1 and Theorem 1.2

The proof of Theorem 1.1 and 1.2 rests on the following theorem:

###### Theorem 2.1

Let $w_{0}(x)=\log\frac{2}{1+|x|^{2}}$ and let
$\pi:S^{n}\setminus\\{N\\}\to\mathbb{R}^{n}$ be the stereographic projection
and $N=(0,\dots,0,1)\in S^{n}$ be the North pole. Take any number
$\alpha\in(-\infty,0)\cup(0,2)$ and consider two functions $K,\varphi\in
C^{\infty}(\mathbb{R}^{n})$ such that

$\displaystyle\int_{\mathbb{R}^{n}}\varphi
dx=\gamma_{n}:=\frac{(n-1)!}{2}|S^{n}|,$ (10)

$\alpha K>0$ everywhere in $\mathbb{R}^{n}$ and whenever $\alpha<0$ then
$|K|>\delta e^{-\delta|x|^{p}}$ for some $\delta>0$, $0<p<n$. If both of
$Ke^{-nw_{0}}$ and $\varphi e^{-nw_{0}}$ can be extend as $C^{2n+1}$ function
on $S^{n}$ via the stereographic projection $\pi$ then the problem

$(-\Delta)^{\frac{n}{2}}w=Ke^{n(w+c_{w})}-\alpha\varphi\quad\text{in
$\mathbb{R}^{n}$},\quad
c_{w}:=-\frac{1}{n}\log\left(\frac{1}{\alpha\gamma_{n}}\int_{\mathbb{R}^{n}}Ke^{nw}dx\right),$
(11)

has at least one solution $w\in C^{\infty}(\mathbb{R}^{n})\cap
L_{\frac{n}{2}}(\mathbb{R}^{n})$ (in the sense of Definition 1.1) so that
$\lim_{|x|\to\infty}w(x)\in\mathbb{R}$.

Now the proof of Theorem 1.1 and Theorem 1.2 follows at once by taking

$u:=-P+\alpha u_{0}+w+c_{w},$

where $u_{0}\in C^{\infty}(\mathbb{R}^{n})$ is given by Lemma 2.2 with
$k=2n+3$, $w$ is the solution in Theorem 2.1 with
$\varphi=(-\Delta)^{\frac{n}{2}}u_{0}$ which satisfies (10) thanks to Lemma
2.3, and $K:=\operatorname{sign}(\alpha)(n-1)!e^{-nP+n\alpha u_{0}}$. Notice
that $Ke^{-nw_{0}}$ can be extend smoothly on $S^{n}$ via the stereographic
projection $\pi$ where as $\varphi e^{-nw_{0}}$ can be extend as a $C^{2n+1}$
function.

###### Lemma 2.2

For every positive integer $k$ there exists a $u_{0}\in
C^{\infty}(\mathbb{R}^{n})$ such that

$\displaystyle u_{0}(x)=\log\frac{1}{|x|}\quad\text{for }\,|x|\geq 1\text{ and
}\quad|D^{\alpha}(-\Delta)^{\frac{n}{2}}u_{0}(x)|\leq\frac{C}{|x|^{2n+k+|\alpha|}}\quad\text{for
}x\neq 0,$ (12)

for any multi-index $\alpha\in\mathbb{N}^{n}$.

###### Proof.

Inductively we define

$v_{j}(x)=\int_{0}^{x_{1}}v_{j-1}((t,\bar{x}))dt,\quad\text{for
}x=(x_{1},\bar{x})\in\mathbb{R}\times\mathbb{R}^{n-1},\,j=1,2,\dots,k,$

where

$v_{0}(x)=\log\frac{1}{|x|}.$

Let $\chi\in C^{\infty}(\mathbb{R}^{n})$ be such that

$\chi(x)=\left\\{\begin{array}[]{ll}0&for\,|x|\leq\frac{1}{2}\\\
1&for\,|x|\geq 1.\end{array}\right.$

We claim that $u_{0}=\frac{\partial^{k}}{\partial x_{1}^{k}}(\chi v_{k})$
satisfies (12). It is easy to see that $u_{0}(x)=\log\frac{1}{|x|}$ for
$|x|\geq 1$. By Lemma A.1
$\frac{1}{\gamma_{n}}(-\Delta)^{\frac{n-1}{2}}\frac{\partial^{k}}{\partial
x_{1}^{k}}v_{k}$ is a fundamental solution of $(-\Delta)^{\frac{1}{2}}$ on
$\mathbb{R}^{n}$ and hence for $x\neq 0$,
$(-\Delta)^{\frac{1}{2}}(-\Delta)^{\frac{n-1}{2}}\frac{\partial^{k}}{\partial
x_{1}^{k}}v_{k}(x)=0$. For $|x|>2$ using integration by parts we compute

$\displaystyle(-\Delta)^{\frac{n}{2}}u_{0}(x)$
$\displaystyle=(-\Delta)^{\frac{1}{2}}(-\Delta)^{\frac{n-1}{2}}\frac{\partial^{k}}{\partial
x_{1}^{k}}(\chi v_{k}-v_{k})(x)$
$\displaystyle=C_{n}\int_{|y|<1}\frac{(-\Delta)^{\frac{n-1}{2}}\frac{\partial^{k}}{\partial
y_{1}^{k}}(\chi v_{k}-v_{k})(y)}{|x-y|^{n+1}}dy$
$\displaystyle=C_{n}\int_{|y|<1}\left(\chi
v_{k}-v_{k}\right)(y)\frac{\partial^{k}}{\partial
y_{1}^{k}}(-\Delta)^{\frac{n-1}{2}}\left(\frac{1}{|x-y|^{n+1}}\right)dy,$

and

$D^{\alpha}(-\Delta)^{\frac{n}{2}}u_{0}(x)=C_{n}\int_{|y|<1}\left(\chi(y)v_{k}(y)-v_{k}(y)\right)D_{x}^{\alpha}\frac{\partial^{k}}{\partial
y_{1}^{k}}(-\Delta)^{\frac{n-1}{2}}\left(\frac{1}{|x-y|^{n+1}}\right)dy.$

Hence

$|D^{\alpha}(-\Delta)^{\frac{n}{2}}u_{0}(x)|\leq
C\frac{\|v_{k}\|_{L^{1}(B_{1})}}{|x|^{2n+k+|\alpha|}}.$

∎

###### Lemma 2.3

Let $u_{0}\in C^{\infty}(\mathbb{R}^{n})$ be as given by Lemma 2.2 for a given
$k\in\mathbb{N}$. Then

$\int_{\mathbb{R}^{n}}(-\Delta)^{\frac{n}{2}}u_{0}(x)dx=\gamma_{n}.$

###### Proof.

Let $\eta\in C^{\infty}(\mathbb{R}^{n})$ be such that

$\eta(x)=\left\\{\begin{array}[]{ll}1\quad\text{if }|x|\leq 1\\\
0\quad\text{if }|x|\geq 2.\end{array}\right.$

We set $\eta_{k}(x)=\eta(\frac{x}{k})$. Then noticing that
$(-\Delta)^{\frac{n}{2}}u_{0}\in L^{1}(\mathbb{R}^{n})$ one has

$\displaystyle\int_{\mathbb{R}^{n}}(-\Delta)^{\frac{n}{2}}u_{0}(x)dx$
$\displaystyle=\lim_{k\to\infty}\int_{\mathbb{R}^{n}}(-\Delta)^{\frac{n}{2}}u_{0}(x)\eta_{k}(x)dx$
$\displaystyle=\lim_{k\to\infty}\int_{B_{1}}\left(u_{0}(x)-\log\frac{1}{|x|}\right)(-\Delta)^{\frac{n}{2}}\eta_{k}(x)dx+\gamma_{n}$
$\displaystyle=\gamma_{n},$

where in the second equality we used the fact that
$\frac{1}{\gamma_{n}}\log\frac{1}{|x|}$ is a fundamental solution of
$(-\Delta)^{\frac{n}{2}}$ and the third equality follows from the locally
uniform convergence of $(-\Delta)^{\frac{n}{2}}\eta_{k}\to 0$. ∎

It remains to prove Theorem 2.1. In order to do that we recall the definition
of $H^{n}(S^{n})$ .

###### Definition 2.1

Let $n\geq 3$ be an odd integer. Let $\\{Y^{m}_{l}\in C^{\infty}(S^{n}):1\leq
m\leq N_{l},\,l=0,1,2,\dots\\}$ be a orthonormal basis of $L^{2}(S^{n})$ where
$Y^{m}_{l}$ is an eigenfunction of the Laplace-Beltrami operator
$-\Delta_{g_{0}}$ ($g_{0}$ denotes the round metric on $S^{n}$) corresponding
to the eigenvalue $\lambda_{l}=l(l+n-2)$ and $N_{l}$ is the multiplicity of
$\lambda_{l}$. The space $H^{n}(S^{n})$ is defined by

$H^{n}(S^{n})=\left\\{u\in
L^{2}(S^{n}):\|u\|_{\dot{H}^{n}(S^{n})}<\infty\right\\},$

where for any

$u=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}u^{m}_{l}Y^{m}_{l}$

we set

$\|u\|_{\dot{H}^{n}(S^{n})}^{2}:=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}\left(\lambda_{l}+\left(\frac{n-1}{2}\right)^{2}\right)\prod_{k=0}^{\frac{n-3}{2}}(\lambda_{l}+k(n-k-1))^{2}(u^{m}_{l})^{2}.$

Notice that the norm $\|u\|_{\dot{H}^{n}(S^{n})}^{2}$ is equivalent to the
simpler norm
$\|u\|^{2}:=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}\lambda_{l}^{n}(u_{l}^{m})^{2}$,
but has the advantage of taking the form

$\|u\|_{\dot{H}^{n}(S^{n})}=\|P_{g_{0}}^{n}u\|_{L^{2}(S^{n})},$

where for $n$ odd the Paneitz operator $P^{n}_{g_{0}}$ can be defined on
$H^{n}(S^{n})$ by

$P^{n}_{g_{0}}u=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}\left(\lambda_{l}+\left(\frac{n-1}{2}\right)^{2}\right)^{\frac{1}{2}}\prod_{k=0}^{\frac{n-3}{2}}(\lambda_{l}+k(n-k-1))u^{m}_{l}Y^{m}_{l}.$

Since the operator $P^{n}_{g_{0}}$ is positive we can define its square root,
namely

$(P^{n}_{g_{0}})^{\frac{1}{2}}u:=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}\left(\lambda_{l}+\left(\frac{n-1}{2}\right)^{2}\right)^{\frac{1}{4}}\prod_{k=0}^{\frac{n-3}{2}}(\lambda_{l}+k(n-k-1))^{\frac{1}{2}}u^{m}_{l}Y^{m}_{l},\quad
u\in H^{\frac{n}{2}}(S^{n}),$

where the space $H^{\frac{n}{2}}(S^{n})$ is defined by

$H^{\frac{n}{2}}(S^{n}):=\left\\{u\in
L^{2}(S^{n}):\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}\left(\lambda_{l}+\left(\frac{n-1}{2}\right)^{2}\right)^{\frac{1}{2}}\prod_{k=0}^{\frac{n-3}{2}}(\lambda_{l}+k(n-k-1))(u^{m}_{l})^{2}<\infty\right\\},$

endowed with the norm

$\displaystyle\|u\|^{2}_{H^{\frac{n}{2}}(S^{n})}$
$\displaystyle:=\|u\|^{2}_{L^{2}(S^{n})}+\|u\|^{2}_{\dot{H}^{\frac{n}{2}}(S^{n})}$
$\displaystyle:=\|u\|_{L^{2}(S^{n})}^{2}+\|(P^{n}_{g_{0}})^{\frac{1}{2}}u\|_{L^{2}(S^{n})}^{2}$

###### Definition 2.2

Let $f\in H^{-\frac{n}{2}}(S^{n})$ the dual of $H^{\frac{n}{2}}(S^{n})$. A
function $u\in H^{\frac{n}{2}}(S^{n})$ is said to be a weak solution of

$P^{n}_{g_{0}}u=f,$

if

$\displaystyle\int_{S^{n}}(P^{n}_{g_{0}})^{\frac{1}{2}}u(P^{n}_{g_{0}})^{\frac{1}{2}}\varphi
dV_{0}=\langle f,\varphi\rangle,\quad\text{for every }\varphi\in
H^{\frac{n}{2}}(S^{n}).$ (13)

The following estimate of Beckner is crucial in the proof of Theorem 2.1.

######  Theorem 2.4 ([1])

For every $u\in H^{\frac{n}{2}}(S^{n})$ one has

$\displaystyle\log\left(\frac{1}{|S^{n}|}\int_{S^{n}}e^{u-\overline{u}}dV_{0}\right)\leq\frac{1}{2|S^{n}|n!}\int_{S^{n}}|(P^{n}_{g_{0}})^{\frac{1}{2}}u|^{2}dV_{0},\quad\overline{u}:=\frac{1}{|S^{n}|}\int_{S^{n}}udV_{0}.$

_Proof of Theorem 2.1._ Let $\tilde{K}=K\circ\pi$, $\varphi_{1}=\varphi
e^{-nw_{0}}$ and $\tilde{\varphi_{1}}=\varphi_{1}\circ\pi$. Define the
functional $J$ on $H^{\frac{n}{2}}(S^{n})$ by

$J(w):=\int_{S^{n}}\left(\frac{1}{2}|(P^{n}_{g_{0}})^{\frac{1}{2}}w|^{2}+\alpha\tilde{\varphi_{1}}w\right)dV_{0}-\frac{\alpha\gamma_{n}}{n}\log\left(\int_{S^{n}}|\tilde{K}|e^{nw}e^{-nw_{0}\circ\pi}dV_{0}\right).$

Using Theorem 2.4 we bound

$\displaystyle\log\left(\int_{S^{n}}|\tilde{K}|e^{nw}e^{-nw_{0}\circ\pi}dV_{0}\right)$
$\displaystyle=\left(\log\left(\frac{1}{|S^{n}|}\int_{S^{n}}e^{nw-n\overline{w}}|\tilde{K}|e^{-nw_{0}\circ\pi}dV_{0}\right)+n\overline{w}+C\right)$
$\displaystyle\leq\log\left(\frac{1}{|S^{n}|}\int_{S^{n}}e^{nw-n\overline{w}}dV_{0}\right)+\log\left(\|\tilde{K}e^{-nw_{0}\circ\pi}\|_{L^{\infty}}\right)+n\overline{w}+C$
$\displaystyle\leq\frac{n^{2}}{2|S^{n}|n!}\int_{S^{n}}|(P^{n}_{g_{0}})^{\frac{1}{2}}w|^{2}dV_{0}+n\overline{w}+C.$
(14)

Since for any $c\in\mathbb{R}$ $J(w+c)=J(c)$ we can assume $\overline{w}=0$.
Then from (2) we have

$\displaystyle
J(w)\geq\min\bigg{\\{}\frac{1}{2},\bigg{(}\underbrace{\frac{1}{2}-\frac{\alpha\gamma_{n}}{n}\frac{n^{2}}{2|S^{n}|n!}}_{=(2-\alpha)/4}\bigg{)}\bigg{\\}}\|w\|_{\dot{H}^{\frac{n}{2}}}^{2}-\varepsilon\|w\|_{\dot{H}^{\frac{n}{2}}}^{2}-\frac{1}{\varepsilon}\|\tilde{\varphi}_{1}\|_{L^{2}}^{2}-C,$

where $0<\varepsilon<\frac{1}{2}$ is sufficiently small such that
$\frac{2-\alpha}{4}-\varepsilon>0$ and for $\alpha<0$ using $|K|>\delta
e^{-\delta|x|^{p}}$ one has

$\log\left(\int_{S^{n}}|\tilde{K}|e^{nw}e^{-nw_{0}\circ\pi}dV_{0}\right)\geq\frac{1}{|S^{n}|}\int_{S^{n}}\log\left(|\tilde{K}|e^{-nw_{0}\circ\pi}\right)dV_{0}+n\overline{w}+\log|S^{n}|\geq-C.$

Thus a minimizing sequence $\\{w_{k}\\}$ of $J$ with $\overline{w}_{k}=0$ is
bounded in ${\dot{H}^{\frac{n}{2}}({S^{n}})}$. With the help of Poincaré
inequality

$\|w-\overline{w}\|_{L^{2}(S^{n})}\leq\|(P^{n}_{g_{0}})^{\frac{1}{2}}w\|_{L^{2}(S^{n})},\quad\text{for
every }w\in H^{\frac{n}{2}}(S^{n}),$

which easily follows from the definition of
$\|(P^{n}_{g_{0}})^{\frac{1}{2}}w\|_{L^{2}(S^{n})}$, we conclude that the
sequence $\\{w_{k}\\}$ is bounded in ${H^{\frac{n}{2}}({S^{n}})}$. Then up to
a subsequence $w_{k}$ converges weakly to $u$ for some
$u\in{H^{\frac{n}{2}}({S^{n}})}$. From the compactness of the map $v\mapsto
e^{v}$ from ${H^{\frac{n}{2}}({S^{n}})}$ to $L^{P}(S^{n})$ for any
$p\in[1,\infty)$ (for a simple proof see Proposition 7 in [8] which holds in
higher dimension as well) we have (up to a subsequence)

$\lim_{k\to\infty}\log\left(\int_{S^{n}}|\tilde{K}|e^{nw_{k}}e^{-nw_{0}\circ\pi}dV_{0}\right)=\log\left(\int_{S^{n}}|\tilde{K}|e^{nu}e^{-nw_{0}\circ\pi}dV_{0}\right).$

Moreover from the weak convergence of $w_{k}$ to $u$ we have

$\lim_{k\to\infty}\int_{S^{n}}\tilde{\varphi}_{1}w_{k}dV_{0}=\int_{S^{n}}\tilde{\varphi}_{1}udV_{0}\quad\text{and}\quad\|u\|_{H^{\frac{n}{2}}(S^{n})}\leq\liminf_{k\to\infty}\|w_{k}\|_{H^{\frac{n}{2}}(S^{n})},$

and from the compact embedding $H^{\frac{n}{2}}(S^{n})\hookrightarrow
L^{2}(S^{n})$ we get

$\lim_{k\to\infty}\|w_{k}\|_{L^{2}(S^{n})}=\|u\|_{L^{2}(S^{n})}.$

Thus
$\|(P^{n}_{g_{0}})^{\frac{1}{2}}u\|_{L^{2}(S^{n})}\leq\liminf_{k\to\infty}\|(P^{n}_{g_{0}})^{\frac{1}{2}}w_{k}\|_{L^{2}(S^{n})}$
which implies that $u$ is a minimizer of $J$ and hence $u$ is a weak solution
of (in the sense of Definition 2.2)

$P^{n}_{g_{0}}u+\alpha\tilde{\varphi_{1}}=\frac{\alpha\gamma_{n}}{\int_{S^{n}}\tilde{K}e^{nu}e^{-nw_{0}}dV_{0}}\tilde{K}e^{-nw_{0}\circ\pi}e^{nu}=:C_{0}\tilde{K}e^{-nw_{0}\circ\pi}e^{nu}.$

Since $\tilde{\varphi_{1}}\in C^{2n+1}(S^{n})$ and
$\tilde{K}e^{-nw_{0}\circ\pi}\in C^{\infty}(S^{n})$ we have

$P^{n}_{g_{0}}u=C_{0}\tilde{K}e^{-nw_{0}\circ\pi}e^{nu}-\alpha\tilde{\varphi_{1}}\in
L^{2}(S^{n}),$

and by Lemma 2.5 below $u\in H^{n}(S^{n})$ and a repeated use of Lemma 2.6
gives $u\in C^{2n+1}(S^{n})$.

We set $w:=u\circ\pi^{-1}$ and $w_{k}:=u_{k}\circ\pi^{-1}$ where $u_{k}\in
C^{\infty}(S^{n})$ be such that $u_{k}\xrightarrow{C^{2n+1}(S^{n})}u$. It is
easy to see that
$(-\Delta)^{\frac{n}{2}}w_{k}\xrightarrow{C^{0}(\mathbb{R}^{n})}(-\Delta)^{\frac{n}{2}}w$
and $P^{n}_{g_{0}}u_{k}\xrightarrow{C^{0}(S^{n})}P^{n}_{g_{0}}u$ which easily
follows from

$P^{n}_{g_{0}}u_{k}\xrightarrow{H^{n+1}(S^{n})}P^{n}_{g_{0}}u\text{ and
}H^{n+1}(S^{n})\hookrightarrow C^{0}(S^{n}).$

Now using the following identity of T. Branson (see [2])

$(-\Delta)^{\frac{n}{2}}(v\circ\pi^{-1})=e^{nw_{0}}(P^{n}_{g_{0}}v)\circ\pi^{-1}\quad\text{for
every }v\in C^{\infty}(S^{n}),$

we get

$\displaystyle(-\Delta)^{\frac{n}{2}}w=(-\Delta)^{\frac{n}{2}}(u\circ\pi^{-1})$
$\displaystyle=e^{nw_{0}}\left(C_{0}\tilde{K}e^{-nw_{0}\circ\pi}e^{nu}-\alpha\tilde{\varphi_{1}}\right)\circ\pi^{-1}$
$\displaystyle=C_{0}Ke^{nw}-\alpha\varphi=Ke^{n(w+c_{w})}-\alpha\varphi.$

Since $(-\Delta)^{\frac{n}{2}}w\in L_{\frac{1}{2}}(\mathbb{R}^{n})\cap
C^{2n+1}(\mathbb{R}^{n})$ we have

$(-\Delta)^{\frac{n+1}{2}}w=(-\Delta)^{\frac{1}{2}}(-\Delta)^{\frac{n}{2}}w\in
C^{2n}(\mathbb{R}^{n}),$

and by bootstrap argument we conclude that $w\in C^{\infty}(\mathbb{R}^{n})$.

$\square$

###### Lemma 2.5

Let $f\in L^{2}(S^{n})$. Let $u\in H^{\frac{n}{2}}(S^{n})$ be a weak solution
(in the sense of Definition 2.2) of

$P^{n}_{g_{0}}u=f\quad\text{on }S^{n}.$

Then $u\in H^{n}(S^{n})$.

###### Proof.

Let

$u=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}u^{m}_{l}Y^{m}_{l}\quad\text{ and
}f=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}f^{m}_{l}Y^{m}_{l}.$

Taking the test function $\varphi=Y^{m}_{l}$ in (13) we get

$f^{m}_{l}=\int_{S^{n}}(P^{n}_{g_{0}})^{\frac{1}{2}}u(P^{n}_{g_{0}})^{\frac{1}{2}}\varphi
dV_{0}=\left(\lambda_{l}+\left(\frac{n-1}{2}\right)^{2}\right)^{\frac{1}{2}}\prod_{k=0}^{\frac{n-3}{2}}(\lambda_{l}+k(n-k-1))u^{m}_{l}.$

Hence

$\displaystyle\|P^{n}_{g_{0}}u\|_{L^{2}(S^{n})}=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}\left(\lambda_{l}+\left(\frac{n-1}{2}\right)^{2}\right)\prod_{k=0}^{\frac{n-3}{2}}(\lambda_{l}+k(n-k-1))^{2}(u^{m}_{l})^{2}=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}(f^{m}_{l})^{2}<\infty,$

and we conclude the proof. ∎

###### Lemma 2.6

Let $u\in H^{s}(S^{n})$ and $f\in H^{s-n+t}(S^{n})$ for some $s\geq n$ and
$t\geq 0$. If $u$ solves

$\displaystyle P^{n}_{g_{0}}u=f\quad\text{on }S^{n},$

then $u\in H^{s+t}(S^{n})$.

###### Proof.

Let

$u=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}u^{m}_{l}Y^{m}_{l},$

and

$(-\Delta_{g_{0}})^{\frac{s-n}{2}}f=:h=\sum_{i=0}^{\infty}\sum_{j=1}^{N_{i}}h^{j}_{i}Y^{j}_{i},$

where for any $r>0$

$(-\Delta_{g_{0}})^{r}v=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}v^{m}_{l}\lambda_{l}^{r}Y^{m}_{l}\quad\text{for
}v=\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}v^{m}_{l}Y^{m}_{l}\in H^{2r}(S^{n}).$

Then

$\displaystyle(-\Delta_{g_{0}})^{\frac{s-n}{2}}P^{n}_{g_{0}}u=h\quad\text{on
}S^{n}.$ (15)

Multiplying both sides of (15) by $Y^{i}_{j}$ and integrating on $S^{n}$ one
has

$\displaystyle\left(\lambda_{j}+\left(\frac{n-1}{2}\right)^{2}\right)^{\frac{1}{2}}\prod_{k=0}^{\frac{n-3}{2}}(\lambda_{j}+k(n-k-1))\lambda_{j}^{\frac{s-n}{2}}u^{i}_{j}=h^{i}_{j}.$

Since $h\in H^{t}(S^{n})$ we have

$\sum_{l=0}^{\infty}\sum_{m=1}^{N_{l}}\left(\lambda_{l}+\left(\frac{n-1}{2}\right)^{2}\right)\prod_{k=0}^{\frac{n-3}{2}}(\lambda_{l}+k(n-k-1))^{2}\lambda_{l}^{s-n}\lambda_{l}^{t}(u^{m}_{l})^{2}<\infty,$

and hence $u\in H^{s+t}(S^{n})$. ∎

##  Appendix A Appendix

######  Lemma A.1 (Fundamental solution)

For $n\geq 3$ odd integer the function

$\Phi(x):=\frac{(\frac{n-3}{2})!}{2\pi^{\frac{n+1}{2}}}\frac{1}{|x|^{n-1}}=\frac{1}{\gamma_{n}}(-\Delta)^{\frac{n-1}{2}}\log\frac{1}{|x|}$

is a fundamental solution of $(-\Delta)^{\frac{1}{2}}$ in $\mathbb{R}^{n}$ in
the sense that for all $f\in L^{1}(\mathbb{R}^{n})$ we have $\Phi\ast f\in
L_{\frac{1}{2}}(\mathbb{R}^{n})$ and for all
$\varphi\in\mathcal{S}(\mathbb{R}^{n})$

$\displaystyle\int_{\mathbb{R}^{n}}(-\Delta)^{\frac{1}{2}}(\Phi\ast f)\varphi
dx:=\int_{\mathbb{R}^{n}}(\Phi\ast f)(-\Delta)^{\frac{1}{2}}\varphi
dx=\int_{\mathbb{R}^{n}}f\varphi dx.$ (16)

###### Proof.

To show $\Phi*f\in L_{\frac{1}{2}}(\mathbb{R}^{n})$ we bound

$\displaystyle\int_{\mathbb{R}^{n}}\frac{|\Phi*f(x)|}{1+|x|^{n+1}}dx$
$\displaystyle\leq
C\int_{\mathbb{R}^{n}}\frac{1}{1+|x|^{n+1}}\left(\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n-1}}|f(y)|dy\right)dx$
$\displaystyle=C\int_{\mathbb{R}^{n}}|f(y)|\left(\int_{\mathbb{R}^{n}}\frac{1}{1+|x|^{n+1}}\frac{1}{|x-y|^{n-1}}dx\right)dy$
$\displaystyle\leq
C\int_{\mathbb{R}^{n}}|f(y)|\left(\int_{B_{1}}\frac{dx}{|x|^{n-1}}+\int_{\mathbb{R}^{n}}\frac{dx}{1+|x|^{n+1}}\right)dy$
$\displaystyle\leq C\|f\|_{L^{1}(\mathbb{R}^{n})}.$ (17)

If $f\in\ C_{c}^{\infty}(\mathbb{R}^{n})$ then (16) is true by Theorem 5.9 in
[9]. For the general case $f\in L^{1}(\mathbb{R}^{n})$ choose $f_{k}\in
C_{c}^{\infty}(\mathbb{R}^{n})$ such that $f_{k}\to f$ in
$L^{1}(\mathbb{R}^{n})$. Then using (17) with $f\equiv f_{k}-f$ one has

$\int_{\mathbb{R}^{n}}|\Phi\ast(f_{k}-f)||(-\Delta)^{\frac{1}{2}}\varphi|dx\leq
C\int_{\mathbb{R}^{n}}\frac{|\Phi\ast(f_{k}-f)(x)|}{1+|x|^{n+1}}dx\leq
C\|f_{k}-f\|_{L^{1}(\mathbb{R}^{n})}\to 0,$

that is

$\int_{\mathbb{R}^{n}}(\Phi\ast f_{k})(-\Delta)^{\frac{1}{2}}\varphi
dx\to\int_{\mathbb{R}^{n}}(\Phi\ast f)(-\Delta)^{\frac{1}{2}}\varphi dx.$

Now the proof follows from

$\int_{\mathbb{R}^{n}}(\Phi\ast f_{k})(-\Delta)^{\frac{1}{2}}\varphi
dx=\int_{\mathbb{R}^{n}}f_{k}\varphi dx\to\int_{\mathbb{R}^{n}}f\varphi dx.$

∎

Proof of the following Proposition can be found in [6].

###### Proposition A.2

For any $s>0$ and $\varphi\in\mathcal{S}(\mathbb{R}^{n})$ we have

$|(-\Delta)^{s}\varphi(x)|\leq\frac{C}{|x|^{n+2s}}.$

## References

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# The propagation and transformation of light beam on curved surface

ZIQIU SHAO  Zhejiang Province Key Laboratory of Quantum Technology and Device,
Department of Physics, Zhejiang University, Hangzhou 310027, China  ZHAOYING
WANG<EMAIL_ADDRESS>Zhejiang Province Key Laboratory of Quantum
Technology and Device, Department of Physics, Zhejiang University, Hangzhou
310027, China

###### Abstract

Starting from the wave function with abstract index notations, with the help
of Wentzel–Kramers–Brillouin (WKB) approximation, a generalized coordinate-
independent expression for the evolution of light beam on curved space is
derived. By defining the propagation axes, the expression reduces to
integrable Green functions. As a first application, one Gaussian beam on
curved surface is analyzed for its propagation and transformation. The
stationary status and Talbot effect of light field on different shapes of
curved surfaces are discussed. We also notice that a periodical fractional
Fourier transform can be perfectly achieved during the free propagation on
curved surface, which must be implemented by additional optical elements in
flat case. We hope our theoretical results can provide some references for the
practical application in curved space.

###### pacs:

Valid PACS appear here

##  I Introduction

The characteristics of light beam propagation upon curved surface is now
drawing increasing attention. It contains a vast range of applications towards
versatile topics. In astrophysics, the Hanbury Brown and Twiss (HBT)
measurement is a method to examine the angular size of stars, and it can be
used to evaluate the curvature of surfaceSchultheiss  _et al._ (2015)Brown and
Twiss (1956). Similarly, Wolf effect is developed on curved surface, which
could also be a method of surface shape evaluatingXu and Wang (2019). Surface
Plasmon Polaritons (SPPs) are naturally surface bounded wave, for non-planer
surface, the dynamics of it obey the generalized wave function along with
space metricLibster-Hershko  _et al._ (2019). The equivalence between the
metric of surface and the index gradient of waveguide was discussed through
conformal mapping as well, one subtle example is the comparison between
Maxwell’s fisheye and the sphereTyc  _et al._ (2015); Xu  _et al._ (2019); Tyc
_et al._ (2011); Xu and Chen (2014). For classical optics, as the wave length
of optical beams is tiny when compared with macroscopic scale, it is
acceptable to treat the light transmission on curved surface under the frame
of geometrical opticsXu  _et al._ (2019); Tyc  _et al._ (2011); Šarbort and
Tyc (2012); Leonhardt (2006); Xu  _et al._ (2020). But with the increasing
research demand of microstructure whose spatial size is of the same order of
magnitude with respect to the wavelength, the field on it must be treated by
the means of wave optics. It is nowadays a widely discussed topic, wave optics
on the curved space were theoretically revealedBatz and Peschel (2010); Conti
(2016); Perczel  _et al._ (2018); Schultheiss  _et al._ (2020); Batz and
Peschel (2008); Tyc (2013); Lai  _et al._ (2018); Wang  _et al._ (2018);
Ferrari and Cuoghi (2008) and experimentally observedSchultheiss  _et al._
(2010); Patsyk  _et al._ (2018); Wang  _et al._ (2017).

But until now, there has not been a proper method of making connection between
geometric optics and wave optics on curved surface. One may purse a
differential geometric method for geometrical optics then turn to a Sturm-
Liouville (S-L) eigenvalue problem for wave optics, separately. In classical
flat space cases, a widely used method of connecting these two concepts is the
WKB approximationJeffreys (1925), which is a mathematical processing of
neglecting over high order derivative term(s) of amplitude in wave equation.
The WKB approximation is a treatment upon flat manifold originally, to expand
its usage to generalized manifolds, some efforts should be made. Two inspiring
examples are the assumptions of surface of revolutionBatz and Peschel (2008)
and the radially symmetric mediaTyc (2013). The basic method is to take the
effect of space curvature into the derivative operator of wave function, in
consideration of proper-length transformation and eigenvalue condition,
respectively. However, these considerations would not be available near the
poles of rotational symmetric system, limited with the discontinuity of
spherical or polar coordinate at poles. What’s more, the currently used method
allows non-geodesic lines to be the propagation axes, which is not tally with
the facts and it is inevitable if adopting WKB approximation after defining
specific coordinate variables in wave equation.

A new thought for this topic is to evaluate the effect of specific surface
curvature after having solved the wave equation. A generalized WKB
approximation is still utilized but before the definition of specific
coordinate system. By defining the propagation axes, we derived integrable
Green functions for light beams. It’s a concise and effective interpretation
for the propagating field on curved surface. Our theory is confirmed by the
flat space limit. The revolution of typical Gaussian beam is demonstrated
thoroughly, a stationary state and the Talbot effect of light field are
revealed. For a brand-new discovery, we found that a fractional Fourier
transform can be perfectly performed on the surface of revolution without any
additional optical elements, which may provide potential applications in
signal processing. Finally, as an outlook, due to the complete description of
optical phase in our method, the evolution of ultrashort pulses on curved
surface can be readily further evaluated.

##  II Wave function on curved surface

A surface can always be treated as a hypersurface imbedded into a background
manifold. Starting from pseudo-Riemannian space, the wave equation written
with abstract indices givesWald (2010),

${{\partial}^{a}}{{\partial}_{a}}{{A}_{b}}=-4\pi{{J}_{b}}$ (1)

where super- and sub-script obey the Einstein summation convention, that when
an index variable appears twice in a single term and is not otherwise defined,
it implies summation of that term over all the values of the index. The index
$a$ goes through over four independent coordinates
$\left({{q}_{1}},{{q}_{2}},{{q}_{3}},{{c}^{-1}}t\right)$.
${{J}_{b}}=\left(c\rho,\mathbf{J}\right)$ is the covariant electric current
vector, where $\rho$ is the electric charge density and $\mathbf{J}$ is the
electric current vector. $A_{b}$ is the covariant vector potential. For two-
dimensional surface, taking the separationda Costa (1981) with respect to the
tangential part and the normal part
$A\left({{q}_{1}},{{q}_{2}},{{q}_{3}},t\right)={{A}^{T}}\left({{q}_{1}},{{q}_{2}},t\right){{A}^{N}}\left({{q}_{3}},t\right)$,
accompanying with the thin layer assumption, the wave function on two-
dimensional sub-manifold writes:

${{\partial}^{a}}{{\partial}_{a}}{{A}^{T}}_{b}+\left({{H}^{2}}-K\right){{A}^{T}}_{b}=-4\pi{{J}^{T}}_{b}$
(2)

where $H$ is the mean curvature while $K$ is the Gaussian curvature. After the
separation, the index $a$ now runs over
$\left({{q}_{1}},{{q}_{2}},{{c}^{-1}}t\right)$. For macroscopic radii of
curvature, the influence of the left second term of Eq.(2) can be neglected.
By using the relationship between the field and vector potential and
neglecting the polarization effects, the wave equation for propagating field
on two-dimensional surface can be simplified as follows,

${{\partial}^{a}}{{\partial}_{a}}{{E}^{T}}\left(t\right)-{{c}^{-2}}\partial_{t}^{2}{{E}^{T}}\left(t\right)=0$
(3)

with the index $a$ now runs over $\left({{q}_{1}},{{q}_{2}}\right)$. Use the
ansatz $E=C{{e}^{ikL}}$ , substitute it back to Eq.(3)

$\begin{split}&{{\partial}^{a}}{{\partial}_{a}}C+ik\left({{\partial}^{a}}C{{\partial}_{a}}L+{{\partial}^{a}}L{{\partial}_{a}}C+C{{\partial}^{a}}{{\partial}_{a}}L\right)\\\
&+{{k}^{2}}\left(1-{{\partial}^{a}}L{{\partial}_{a}}L\right)C=0\end{split}$
(4)

Under the WKB approximation which works on trivial manifolds, the second
derivation term should be neglected. Extract the real part and imagine part of
Eq.(4), we obtain the following expressions.

$\begin{matrix}1-{{\partial}^{a}}L{{\partial}_{a}}L=0\\\
{{\partial}^{a}}C{{\partial}_{a}}L+{{\partial}^{a}}L{{\partial}_{a}}C+C{{\partial}^{a}}{{\partial}_{a}}L=0\end{matrix}$
(5)

Here we should emphasize, until now, in our analysis, we haven’t specially
referred to the metric of surface, and coordinates
$\left({{q}_{1}},{{q}_{2}}\right)$ remains undefined. In Eq.(5), it is always
possible to define the first coordinate ${{q}_{1}}=L$ itself and the other
coordinate perpendicular to it, writes ${{\partial}_{{{q}_{2}}}}L=0$. The
amplitude term $C$ then satisfies

$4{{\partial}_{{{q}_{1}}}}C+C{{g}^{-1}}{{\partial}_{{{q}_{1}}}}g=0$ (6)

where $g$ is the determinant of space metric upon the argument $q_{1}$ and
$q_{2}$. $q_{1}$ or $L$ in Eq.(5) was named the eikonal in flat space, we
follow this note on curved surface. It is a key in this section to connect the
optical phase with the definition of curved coordinate system. For a two-
dimensional surface, this eikonal can be defined as the length of trajectory
curve. Utilizing the principle of Green function, the distribution of
propagation field will be derived as follows. All the ingredients in Eq.(7)
now have a unambiguous physical correspondence.

${{E}_{output}}={{E}_{incident}}\otimes C{{e}^{ikL}}$ (7)

##  III The evaluation of eikonal $L$

Figure 1: Coordinate system definition of surface of revolution. Red arrowed
lines indicate the vector of wave field.

A common and useful type of surface is called surface of revolution whose
space metric writes
$d{{s}^{2}}=d{{h}^{2}}+\rho{{\left(h\right)}^{2}}d{{\theta}^{2}}$. Drawn as
sphere but without loss of generality, the sketch of surface and the
corresponding coordinate system grids are plotted in Fig.1. The propagation
trajectories are denoted as the red arrows, with $A$ to be the incident point
and $B$ the exit point. $\theta$ is the angle of rotation, $h$ is the arc
length from the alternative point towards the maximum rotational circuit, and
$\rho\left(h\right)$ is the radius of surface toward the rotation axis. For a
specific surface with constant Gaussian curvature, the reliance between $\rho$
and $h$ gives

$\rho\left(h\right)={{r}_{0}}\cos\left(h/r\right)$ (8)

For $r={{r}_{0}}$ one gets the sphere, for $r>{{r}_{0}}$ a spindle type, and
for $r<{{r}_{0}}$ a bulge type surface. Once the space metric is given, the
geodesic ray function or the optical trajectory writes

$\begin{split}d\theta&=\frac{constdh}{\rho\left(h\right)\sqrt{{{\rho}^{2}}\left(h\right)-cons{{t}^{2}}}}\\\
&=\frac{{{r}_{0}}\cos\psi
dh}{\rho\left(h\right)\sqrt{{{\rho}^{2}}\left(h\right)-{{r}_{0}}^{2}{{\cos}^{2}}\psi}}\end{split}$
(9)

where the constant parameter
$const={{r}_{0}}\cos\left(h/r\right)\cos\varphi={{r}_{0}}\cos\psi$ determines
the trajectory of light ray. In this parametrization, we are now able to
complete the integration $\theta$.

$\theta=-\frac{r}{{{r}_{0}}}\arcsin\frac{\tan{{h}_{2}}/r}{\tan\psi}+\frac{r}{{{r}_{0}}}\arcsin\frac{\tan{{h}_{1}}/r}{\tan\psi}$
(10)

The eikonal function $L$ is the arc length between point $A$ and $B$, which
could be calculated with the help of metric.

$L=\int_{{{h}_{1}}}^{{{h}_{2}}}{ds}=-r\arcsin\frac{\sin{{h}_{2}}/r}{\sin\psi}+r\arcsin\frac{\sin{{h}_{1}}/r}{\sin\psi}$
(11)

For the paraxial optical system with the propagation axis to be the $h$
direction and starts at $h=0$, the constant now satisfies
${{r}_{0}}\cos\psi={{r}_{0}}\cos\left({{h}_{1}}/r\right)\cos{{\varphi}_{1}}={{r}_{0}}\cos{{\varphi}_{1}}$.
For the paraxial optical system with the propagation axis to be the $\theta$
direction, eikonal $L$ could be expanded with the axial length
${{r}_{0}}\theta$ and a deviation term, which is expressed as follows.

$\begin{split}L&={{r}_{0}}\theta-\frac{r\tan\frac{{{h}_{2}}}{r}}{\sin{{\varphi}_{2}}}\left(\frac{1}{\cos{{\varphi}_{2}}}-1\right)+\frac{r\tan\frac{{{h}_{1}}}{r}}{\sin{{\varphi}_{1}}}\left(\frac{1}{\cos{{\varphi}_{1}}}-1\right)\\\
&+O{{\left(\frac{{{h}_{2}}}{r}\right)}^{3}}+O{{\left(\frac{{{h}_{1}}}{r}\right)}^{3}}\\\
&={{r}_{0}}\theta-\frac{{{h}_{2}}{{\varphi}_{2}}}{2}+\frac{{{h}_{1}}{{\varphi}_{1}}}{2}+O{{\left(\frac{{{h}_{2}}}{r}\right)}^{3}}+O{{\left(\frac{{{h}_{1}}}{r}\right)}^{3}}\end{split}$

With the relationship between incident(output) angles and the trajectory
parameters,

$\displaystyle{{\varphi}_{1}}=\tan\frac{{{h}_{2}}}{r}\csc\frac{{{r}_{0}}}{r}\theta-\tan\frac{{{h}_{1}}}{r}\cot\frac{{{r}_{0}}}{r}\theta$
$\displaystyle{{\varphi}_{2}}=\tan\frac{{{h}_{1}}}{r}\csc\frac{{{r}_{0}}}{r}\theta-\tan\frac{{{h}_{2}}}{r}\cot\frac{{{r}_{0}}}{r}\theta$

the eikonal function finally gets

$L={{r}_{0}}\theta+\frac{1}{2r}\left(-2{{h}_{1}}{{h}_{2}}\csc\frac{{{r}_{0}}\theta}{r}+{{h}_{1}}^{2}\cot\frac{{{r}_{0}}\theta}{r}+{{h}_{2}}^{2}\cot\frac{{{r}_{0}}\theta}{r}\right)$
(12)

The Eikonal function Eq.(12) is the main result of this paper. Regardless of
the reliance of eikonal definition on coordinate system, this calculation is
flexible in conformal mapping on curved space. We are now able to describe
several intuitive properties of the light propagation on two-dimensional
curved surface.

For flat surface limit when $r$ and $r_{0}$ approaches infinity, it’s able to
define ${{r}_{0}}\theta=z$ as the propagation distance. ${{r}_{0}}\theta/r$
remains a small angle, then the eikonal function Eq.(12) can be simplified as

$L=z+\frac{{{\left({{h}_{1}}-{{h}_{2}}\right)}^{2}}}{2z}$ (13)

which is the eikonal function under paraxial approximation in flat surface.

For a more complete description, the variation of amplitude term is also
investigated. From Eq.(6), the amplitude term can also be defined simply from
the determinate of space metric, or equivalently speaking, the Gaussian
(intrinsic) curvature. After a little algebra, for surface with constant
Gaussian curvature $K=1/{{r}^{2}}$ defined in this paper, the amplitude term
gives $C=1/\sqrt{r\sin\left(L/r\right)}$. When choosing the equator to be the
propagation axis, it can be naturally rewritten as
$C=1/\sqrt{r\sin\left({{r}_{0}}\theta/r\right)}$.

##  IV Field propagation

Figure 2: The stationary Gaussian light beam with the size
${{\sigma}_{0}}=\sqrt{2r/k}$ on surface of revolution. The parameters of
surfaces are (a)$r=0.5{{r}_{0}}$, (b)$r=0.75{{r}_{0}}$, (c)$r=1.5{{r}_{0}}$
and (d)$r=2{{r}_{0}}$, respectively.

Figure 3: The propagations of light beam on sphere. The initial incident
positions are noted in red, with incident field as a Gaussian shape whose
initial beam size (a)${{\sigma}_{0}}=0.2\sqrt{2r/k}$ and
(b)${{\sigma}_{0}}=5\sqrt{2r/k}$. The Talbot effect property holds for
alternative incident field.

Figure 4: Two typical examples of stationary solutions on the surface of
revolution, (a)$r=0.5{{r}_{0}}$ and (b)$r=1.5{{r}_{0}}$. It shows that the
translational symmetric circuit number are (a)$m=0.5$ and (b)$m=1.5$,
respectively, indicating that the field will retrieve back to the incident
form after $m$ circles of traveling.

In laboratory, a most common and practical propagation field is the Gaussian
beam, writes
$E\left({{h}_{1}}\right)={{E}_{0}}\exp\left(-{{h}_{1}}^{2}/{{\sigma}_{0}}^{2}\right)$,
where $\sigma_{0}$ is the initial beam size. On the basis of Eq.(7) and
Eq.(12), an analytic expression of propagation field with Gaussian shape on
curved surface is deduced

$\begin{split}&E\left({{h}_{2}}\right)={{E}_{0}}\sqrt{\frac{{{\sigma}_{0}}}{{{\sigma}_{\theta}}}}\exp\left(-\frac{{{h}_{2}}^{2}}{{{\sigma}_{\theta}}^{2}}\right){{\operatorname{e}}^{i{{\phi}_{1}}}}{{\operatorname{e}}^{i{{\phi}_{2}}}}{{\operatorname{e}}^{ik{{r}_{0}}\theta}}\\\
&{{\sigma}_{\theta}}=\sqrt{\frac{4{{r}^{2}}}{{{k}^{2}}\sigma_{0}^{\text{2}}}{{\sin}^{2}}\frac{{{r}_{0}}}{r}\theta+\sigma_{0}^{\text{2}}{{\cos}^{2}}\frac{{{r}_{0}}}{r}\theta}\\\
&{{\phi}_{1}}=\frac{1}{2}\arctan\left(\frac{k\sigma_{0}^{2}\tan{{r}_{0}}\theta/r}{2r}\right)\\\
&{{\phi}_{2}}=-\frac{\left(4{{r}^{2}}-{{k}^{2}}{{\sigma}_{0}}^{4}\right)\tan{{r}_{0}}\theta/r}{4{{r}^{2}}{{\tan}^{2}}{{r}_{0}}\theta/r+{{k}^{2}}{{\sigma}_{0}}^{4}}{{h}_{2}}^{2}\end{split}$
(14)

$\sigma_{\theta}$ is the beam size at a propagation angle $\theta$.
${{\phi}_{1}}$ is the Gouy phase resulted from the transverse
compressionAndresen  _et al._ (2013). ${{\phi}_{2}}$ is also a propagation
phase, which describes a bending of wavefront. The revolution of not only the
illumination but also the optical phase on curved surface is revealed.

A special case of the initial beam size is ${{\sigma}_{0}}=\sqrt{2r/k}$. For
this type of incidence, no matter how the surface shape with constant-
Gaussian-curvature is, the beam size will keep a constant
${{\sigma}_{\theta}}={{\sigma}_{0}}$ during the propagation and the amplitude
of beams remains unaltered, as shown in Fig.2. The stationary solution only
depends on the relationship between the wave number and the
Gaussian(intrinsic) curvature and have nothing to do with the extrinsic
curvature, namely how this surface is embedded in background manifold. The
figures are plotted with a certain amount of transparency to show the
revolution properties of field. The wave number in these figures are set to be
$k=1000/{{r}_{0}}$ when the variation of beam size is observable with respect
to the radius of curvature. On the other aspect of understanding,
$k{{\sigma}_{0}}^{2}/2$ is the Rayleigh(diffraction) length of Gaussian beam,
while $r$ is the principle radius of surface, the propagation stabilization
requires the equivalence between them. At the situation of stable
transmission, the phase on axis varies as
${{\phi}_{axial}}=\left(1/2r+k\right){{r}_{0}}\theta$, which principally
explores a Gouy phase shift beyond an ordinary propagation phase. Different
from a logarithm reliance on flat space, the Gouy phase now behaves a linear
superposition over propagation distance.

Another special case is the beam propagation on sphere (Fig.3). The sphere is
a kind of surface with perfect isotropy, and this characteristic is inherited
by the light field on it. No matter how the incident field is, it will return
to the initial status after one circle of traveling. This corresponds to the
fact that the spherical harmonic function is a complete set on sphere, any
type of field could be expanded as a linear superposition of it. The Talbot
effect is clearly revealed in Fig.3 that the field diffuses and focuses
spontaneously, and this property holds for alternative incident field. From
this Talbot effect, we find that the eigenvalue condition is contained
naturally in the geometric of curved surface. As proved in geometrical
opticsXu  _et al._ (2020) and now in wave optics as a major progress, the
surface of revolution which satisfies $r=m{{r}_{0}}$ will always show its
translational symmetry after going $m$ circles. Two typical stationary
solutions with limited circles are plotted in Fig.4 for $r=0.5{{r}_{0}}$ and
Fig.4 for $r=1.5{{r}_{0}}$. The self-interference effect is neglected in our
simulation.

A more subtle feature of the curved surface is its possibility of performing a
fractional Fourier transformation. The emergent field
$E\left({{h}_{2}}\right)$ with an incidence $E\left({{h}_{1}}\right)$ after a
propagation angle $\theta$ gives

$\begin{split}&E\left({{h}_{2}}\right)=\frac{\exp\left(ik{{r}_{0}}\theta\right)}{\sqrt{r\sin\left({{r}_{0}}\theta/r\right)}}\exp\left(\frac{ik{{h}_{2}}^{2}}{2r}\cot\frac{{{r}_{0}}\theta}{r}\right)\times\\\
&\int{\exp\left(\frac{ik{{h}_{1}}^{2}}{2r}\cot\frac{{{r}_{0}}\theta}{r}-\frac{ik{{h}_{1}}{{h}_{2}}}{r\sin\left({{r}_{0}}\theta/r\right)}\right)E\left({{h}_{1}}\right)d{{h}_{1}}}\end{split}$
(15)

On the other hand, the definition of fractional Fourier transformationNamias
(1980)

$\begin{split}&{{\text{F}}_{\alpha}}f\left(x\right)=\frac{\exp\left(\frac{i\pi-2i\alpha}{4}\right)}{\sqrt{2\pi\sin\alpha}}\exp\left(-\frac{i{{x}^{2}}}{2}\cot\alpha\right)\times\\\
&\int{\exp\left(-\frac{i{{x}^{\prime}}^{2}}{2}\cot\alpha+\frac{ix{{x}^{\prime}}}{\sin\alpha}\right)f\left({{x}^{\prime}}\right)d{{x}^{\prime}}}\end{split}$
(16)

By means of the substitution ${{x}^{\prime}}={{h}_{1}}\sqrt{k/r}$,
$x={{h}_{2}}\sqrt{k/r}$ and $\alpha={{r}_{0}}\theta/r$, in spite of a constant
coefficient and an axial propagation phase, they are essentially the same
thing. When $\alpha=\pi/2$ and $-\pi/2$, Eq.(16) regains the usual Fourier
transforms, and it corresponds to a propagation distance $L=-r\pi/2$ and
$r\pi/2$ on curved surface. As a closely connection between the fractional
Fourier transform and a parabolic index profile microlensDragoman  _et al._
(1999), Wigner rotationLohmann (1993); Mendlovic and Ozaktas (1993), etc., it
can now find a new correspondence as a curved surface. As an example, the
diffraction pattern of light field with a plane incidence is plotted in Fig.5,
the broken red line indicates an imaginary diagram. The revolution properties
of transformation and diffraction pattern on curved surface are clearly
revealed.

Figure 5: The diffraction pattern of curved surface with a plane incidence.

##  V conclusion

We focused on the light beam propagation on two-dimensional curved surface. In
this paper, through the formalization of eikonal and consequently the point
spread function, a connection between the source contribution and the field
distribution on curved surface was made. This connection is a generalized
theory for an alternative ratio between the principle radius of surface and
the wave length of light beam. Taking the Gaussian initial field into account,
the evolution of not only the amplitude but also the phase is discussed. We
find that the periodic boundary condition is contained naturally inside the
geometric of surface of revolution. Another intriguing feature of fractional
Fourier transformation is explored. This may provide more possibilities in
laser optics and measurements in condensed matter and astronomical physics.

## Funding

This work was supported by the National Key Research and Development Program
of China (Grant No. 2017YFC0601602).

## Disclosures

The authors declare no conflicts of interest.

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