# The rigidity of eigenfunctions’ gradient estimates

Guoyi Xu, Xiaolong Xue  Guoyi Xu
Department of Mathematical Sciences
Tsinghua University, Beijing
P. R. China, 100084<EMAIL_ADDRESS>Xiaolong Xue
Department of Mathematical Sciences
Tsinghua University, Beijing
P. R. China<EMAIL_ADDRESS>
###### Abstract.

On compact Riemannian manifolds with non-negative Ricci curvature and smooth
(possibly empty), convex (or mean convex) boundary, if the sharp Li-Yau type
gradient estimate of an Neumann (or Dirichlet) eigenfunction holds at some
non-critical points of the eigenfunction; we show that the manifold is
isometric to the product of one lower dimension manifold and a round circle
(or a line segment).
Mathematics Subject Classification: 35J15, 58J05, 58J50.

G.Xu was partially supported by NSFC 12141103.

section[0em] *[1pc]. subsection[1.5em] *[1pc].

###### Contents

  1. 1 Introduction
  2. 2 The sharp gradient estimates for eigenfunctions
  3. 3 The local splitting induced by eigenfunctions
  4. 4 The global splitting

##  1\. Introduction

For manifolds with non-negative Ricci curvature ($Rc\geq 0$), Cheeger-
Gromoll’s splitting theorem [CG] says: the existence of geodesic line results
in the isometric splitting of the manifold. It can be shown that the Busemann
function with respect to the geodesic line is a harmonic function, whose
gradient has constant norm. Then the isometry is induced by the gradient curve
of this harmonic function.

######  Remark 1.1.

Note the square norm of harmonic functions’ gradient is a subharmonic function
in manifolds with $Rc\geq 0$. If the maximum norm of harmonic functions’
gradient is obtained at some point $p\in M^{n}$, then the strong maximum
principle implies that the norm of harmonic functions’ gradient is constant,
which guarantees that the manifold isometrically splits.

Therefore from the pure analytic view point, the above isometric splitting
result can be viewed as the rigidity part of harmonic functions’ gradient
estimate.

Figure 1. The Figure for Cheeger-Gromoll’s splitting theorem with
$M^{n}=N^{n-1}\times\mathbb{R}$

For the global positive Green’s function, the sharp estimate and the related
rigidity are known to Colding [Colding]. For the global positive Green’s
function $G$ on a complete Riemannian manifold with nonnegative Ricci
curvature, it is shown that $|\nabla(G^{\frac{1}{2-n}})|\leq 1$, and $M^{n}$
is isometric to $\mathbb{R}^{n}$ if and only if the equality holds at some
point in $M^{n}$.

The harmonic functions in Cheeger-Gromoll’s splitting theorem and Colding’s
rigidity are global. The study of local harmonic functions in manifolds with
non-negative Ricci curvature, was initiated by Yau’s Liouville theorem [Yau]
and Cheng-Yau’s gradient estimate for positive harmonic functions [CY], where
the gradient of the logarithm of positive harmonic functions is considered.

In [Xu-growth], the sharp Cheng-Yau type gradient estimate in two dimensional
manifolds with $Rc\geq 0$ is obtained. Furthermore, the corresponding rigidity
is established in [HXY] as follows: the realization of maximum norm of
positive harmonic function’s gradient in geodesic disk implies, the geodesic
disk is isometric to an Euclidean disk. However, the sharp Cheng-Yau type
gradient estimate is unknown for higher dimension cases so far.

Li [Li] and Li-Yau [LY] applied the gradient estimates of eigenfunctions to
study the bound of the eigenvalues on manifolds. The simplest model for
eigenfunctions are sine or cosine functions. The Li-Yau type gradient estimate
of eigenfunctions (the estimate of $|D\sin^{-1}u|$, where $u$ is one
normalized eigenfunction of compact manifold) is firstly obtained in [LY,
Theorem $10$].

Through a delicate refined gradient estimate of eigenfunctions, Zhong-Yang
[ZY] firstly obtain the sharp lower bound of the first eigenvalue for compact
Riemannian manifolds with $Rc\geq 0$. Hang-Wang [HW] answered Sakai’s question
in [Sakai], they showed that if the sharp lower bound of the $1$st eigenvalue
is obtained, then manifold is isometric to a round circle. Zhong-Yang’s type
gradient estimate of eigenfunction is closely related to the solution of
suitable ODE (also see [Kroger] and [AX]). There is also log-concavity
property of the first eigenfunctions established by two-points method in [AC]
(also see [Ni]). For Riemannian manifold satisfying $Rc\geq-(n-1)$, the sharp
gradient estimate of the $1$st eigenfunction and the rigidity of sharp $1$st
eigenvalue is studied in [SW] (also see [Li12]).

Although Li-Yau type gradient estimate of eigenfunctions is not sharp enough
to get the sharp lower bound of the first eigenvalue, the equality in the
estimate can be achieved for model spaces; which splits a round circle factor
isometrically. In this paper, we study the rigidity part with respect to Li-
Yau type gradient estimates of eigenfunctions, which is partially motivated by
Remark 1.1.

The main result of this paper is as follows, which can be viewed as an
isometric splitting theorem for compact manifolds comparing Cheeger-Gromoll’s
splitting Theorem for complete non-compact manifolds.

######  Theorem 1.2.

For compact Riemannian manifold $(M^{n},g)$ with $Rc\geq 0$ and smooth
(possibly empty) boundary . Assume $\Delta u=-\lambda
u,\displaystyle\max_{x\in M^{n}}|u(x)|=1$, where $\lambda>0$. If
$|\nabla\sin^{-1}u|^{2}(p)=\lambda$ holds at some point $p$, where $|u(p)|<1$.
Define $\mathcal{N}(u)=\\#(M^{n}-u^{-1}(0))$ as the number of connected
components of $M^{n}-u^{-1}(0)$ (the number of nodal domains of $u$); then:

  1. (1)

. If $\partial M^{n}=\emptyset$, then $M^{n}$ is isometric to
$\mathbb{S}^{1}(\frac{\mathcal{N}(u)}{2\sqrt{\lambda}})\times N^{n-1}$, where
$\mathbb{S}^{1}(\frac{\mathcal{N}(u)}{2\sqrt{\lambda}})$ is a round circle
with radius $\frac{\mathcal{N}(u)}{2\sqrt{\lambda}}$ and $\mathcal{N}(u)=2k$
for some $k\in\mathbb{Z}^{+}$;

Figure 2. The Figure for Case (1) of Theorem 1.2

  2. (2)

. If $\partial M^{n}\neq\emptyset$ is convex and $\frac{\partial
u}{\partial\vec{n}}\big{|}_{\partial M^{n}}=0$, then $M^{n}$ is isometric to
$[0,\frac{\mathcal{N}(u)-1}{\sqrt{\lambda}}\cdot\pi]\times N^{n-1}$ and
$\mathcal{N}(u)\geq 2$;

Figure 3. The Figure for Case (2) of Theorem 1.2

  3. (3)

. If $\partial M^{n}\neq\emptyset$ is mean convex and $u\big{|}_{\partial
M^{n}}=0$; then $M^{n}$ is isometric to
$[0,\frac{\mathcal{N}(u)}{\sqrt{\lambda}}\cdot\pi]\times N^{n-1}$ and
$\mathcal{N}(u)\geq 1$;

Figure 4. The Figure for Case (3) of Theorem 1.2

where $N^{n-1}$ is a compact Riemannian manifold with $Rc\geq 0$ and $\partial
N^{n-1}=\emptyset$.

######  Remark 1.3.

The gradient curve of Busemann function is used in [CG72] to show the
homeomorphism type of the manifolds. Furthermore the gradient curve of
harmonic functions is used to obtain the isometric splitting in [CG]. The
proof of the above theorem relies on the gradient curve of $\sin^{-1}u$.
However, because of the possible boundary of manifolds and the critical points
of eigenfunctions, we firstly obtain a locally splitting result, then extend
this splitting to the global manifold.

The organization of this paper is as follows. In Section 2, we firstly give an
alternative proof of the sharp gradient estimate of $\sin^{-1}u$, where $u$ is
one normalized eigenfunction. The sharp gradient estimate is known in
equivalent form (see [Li12], [Yang]), however our method is uniform and
consistent in dealing with the corresponding rigidity part later.

In Section 3, we obtain the vanishing Hessian of $\sin^{-1}u$, then using the
gradient curve of this function to show the locally splitting in the connected
component without critical points of eigenfunctions.

Finally we show the locally splitting can be extended to the globally
splitting in Section 4. Note the canonical strong maximum principle (see[GT],
[PW]) requires the uniformly elliptic property of the differential operators.
Because the elliptic operator related to $\sin^{-1}u$ is degenerate at the
critical points of $u$, we can not apply the canonical strong maximum
principle directly.

The key observation is that using the Taylor expansion of the Riemannian
metric and eigenfunction at the critical points, we find a suitable make-up
function, which plays the similar role of the make-up function used in the
proof of classical Hopf’s Strong Maximum Principle (see [GT, Lemma $3.4$]).
Therefore, we obtain the implication of Strong Maximum Principle for this
degenerate elliptic operator, which guarantees the extension of locally
splitting.

Figure 5. The Figure of Theorem 4.1 from local splitting to global splitting

##  2\. The sharp gradient estimates for eigenfunctions

We firstly study the behavior of eigenfunctions’ gradient at the boundary,
where Neumann boundary condition and smooth convex boundary (respectively
Dirichlet boundary condition and smooth, mean convex boundary) are used.

######  Lemma 2.1.

For compact Riemannian manifold $(M^{n},g)$, assume $\Delta u=-\lambda u$ with
$|u|\leq 1$, where $\lambda>0$. If $\partial M^{n}$ is smoothly convex and
$\frac{\partial u}{\partial\vec{n}}\big{|}_{\partial M^{n}}=0$, or if
$\partial M^{n}$ is smooth, mean convex and $u\big{|}_{\partial M^{n}}=0$;
then

$\displaystyle\frac{\operatorname{\partial}|\nabla\sin^{-1}u|^{2}}{\operatorname{\partial}\vec{n}}(x)\leq
0,\quad\quad\quad\quad\forall x\in\partial M^{n}\ \text{satisfying}\
|u(x)|<1;$

where $\vec{n}(x)$ is the outward unit normal vector on
$\operatorname{\partial}M^{n}$.

###### Proof.

Let $\\{e_{1},\cdots,e_{n-1},e_{n}\\}$ be a local orthogonal frame such that
$\\{e_{1},\cdots,e_{n-1}\\}$ are tangent to $\operatorname{\partial}M^{n}$ and
$e_{n}=\vec{n}$ is the outward normal vector.

In the rest argument, we assume $x\in\partial M^{n}$ satisfying $|u(x)|<1$. If
$\partial M^{n}$ is smooth, convex and $\frac{\partial
u}{\partial\vec{n}}\big{|}_{\partial M^{n}}=0$, then at $x$,

$\displaystyle\nabla_{\vec{n}}|\nabla u|^{2}$
$\displaystyle=\sum_{i=1}^{n}2\nabla_{\vec{n}}\nabla_{i}u\cdot\nabla_{i}u=\sum_{i=1}^{n}-2(\nabla_{i}\vec{n})u\cdot\nabla_{i}u,$
$\displaystyle=\sum_{1\leq i,j\leq
n}-2\langle\nabla_{i}\vec{n},j\rangle\nabla_{j}u\cdot\nabla_{i}u=-2\operatorname{\langle}\nabla_{\nabla
u}\vec{n},\nabla u\operatorname{\rangle}\leq 0.$

by the convexity of $\operatorname{\partial}M^{n}$.

If $\partial M^{n}$ is smooth, mean convex and $u\big{|}_{\partial M^{n}}=0$,
then note $\Delta u(x)=-\lambda u(x)=0$, we have

$\begin{split}\nabla_{\vec{n}}|\nabla
u|^{2}&=\sum_{i=1}^{n}2\nabla_{\vec{n}}\nabla_{i}u\cdot\nabla_{i}u=2\nabla_{\vec{n}}\nabla_{\vec{n}}u\cdot\nabla_{\vec{n}}u=2\nabla^{2}u(e_{n},e_{n})\cdot\nabla_{\vec{n}}u\\\
&=2(\Delta
u-\sum_{i=1}^{n-1}\nabla^{2}u(e_{i},e_{i}))\cdot\nabla_{\vec{n}}u=-2\sum_{i=1}^{n-1}\nabla^{2}u(e_{i},e_{i})\cdot\nabla_{\vec{n}}u\\\
&=-2\sum_{i=1}^{n-1}(\nabla_{i}\nabla_{i}u-(\nabla_{e_{i}}e_{i})u)\cdot\nabla_{\vec{n}}u=2\sum_{i=1}^{n-1}\operatorname{\langle}\nabla_{e_{i}}e_{i},\nabla
u\operatorname{\rangle}\cdot\nabla_{\vec{n}}u\\\
&=2\sum_{i=1}^{n-1}\operatorname{\langle}\nabla_{e_{i}}e_{i},e_{n}\operatorname{\rangle}\cdot(\nabla_{\vec{n}}u)^{2}\leq
0,\end{split}$

where the last inequality comes from the mean-convex property of
$\operatorname{\partial}M^{n}$.

Now we have

$\nabla_{\vec{n}}|\nabla\sin^{-1}u|^{2}(x)=\frac{(\nabla_{\vec{n}}|\nabla
u|^{2})(1-u^{2})-|\nabla
u|^{2}(-2u\nabla_{\vec{n}}u)}{(1-u^{2})^{2}}=\frac{\nabla_{\vec{n}}|\nabla
u|^{2}}{1-u^{2}}\leq 0.$

∎

######  Notation 2.2.

For $u\in C^{\infty}(M^{n})$, for any domain $\Omega\subseteq M^{n}$ with
$|\nabla u|(x)\neq 0,|u(x)|<1$ in $\Omega$; we define
$\mathscr{L}_{u}:C^{\infty}(\Omega)\rightarrow C^{\infty}(\Omega)$ as follows:

$\displaystyle\mathscr{L}_{u}(f)(x)\vcentcolon=\Delta f(x)+\nabla
f(x)\cdot\Psi(x)-\frac{2|\nabla u|^{2}}{(1-u^{2})^{2}}f(x),\quad\quad\forall
f\in C^{\infty}(\Omega);$ (2.1) $\displaystyle\Psi(x)=\frac{\nabla
u}{1-u^{2}}\cdot\\{-\frac{1}{2}\frac{(1-u^{2})^{2}}{|\nabla u|^{4}}\cdot\nabla
u\cdot\nabla\Big{(}\frac{|\nabla u|^{2}}{1-u^{2}}(x)\Big{)}-2u\\}.$

The following lemma is the key differential inequality for make-up function
$F(x)=\frac{|\nabla u|^{2}}{1-u^{2}}(x)-\lambda$, which will be crucially used
in the proof Theorem 4.1.

######  Lemma 2.3.

Suppose $M^{n}$ is a compact Riemannian manifold with $Rc\geq 0$, let $u$ be
an eigenfunction with $\Delta u=-\lambda u$ in $M^{n}$, where $\lambda>0$. For
any $x\in M^{n}$ with $|u(x)|<1$ and $|\nabla u(x)|\neq 0$, let
$F(x)=\frac{|\nabla u|^{2}}{1-u^{2}}(x)-\lambda$, we have

$\displaystyle\mathscr{L}_{u}(F)(x)\geq 0,$

Proof: Let $\\{e_{1},\cdots,e_{n}\\}$ be an orthonormal basis at $T_{x}M$ with
$\nabla_{e_{i}}e_{j}(x)=0$, $1\leq i,j\leq n$, where $e_{1}=\frac{\nabla
u}{|\nabla u|}(x)$. Denote $u_{i}(x)=\nabla_{e_{i}}u(x)$ and
$u_{ij}=\nabla_{e_{j}}\nabla_{e_{i}}u(x)$, then $u_{1}(x)=|\nabla u|(x)$ and
$u_{i}(x)=0$, where $2\leq i\leq n$.

Direct computation yields

$\displaystyle\nabla F=2|\nabla u|\cdot\Big{(}\frac{\nabla^{2}u\cdot
e_{1}}{1-u^{2}}+\frac{|\nabla u|^{2}\cdot u\cdot
e_{1}}{(1-u^{2})^{2}}\Big{)}.$

Then we have

$\displaystyle u_{11}(x)=\frac{(1-u^{2})\cdot\nabla F\cdot e_{1}}{2|\nabla
u|}-\frac{|\nabla u|^{2}\cdot u}{(1-u^{2})}.$

And we get

(2.2) $\displaystyle|\nabla^{2}u|^{2}\geq u_{11}^{2}=\frac{|\nabla u|^{4}\cdot
u^{2}}{(1-u^{2})^{2}}+\frac{(1-u^{2})\nabla F\cdot e_{1}}{2|\nabla
u|}\cdot\Big{\\{}\frac{(1-u^{2})\nabla F\cdot\nabla u}{2|\nabla
u|^{2}}-2\frac{|\nabla u|^{2}u}{1-u^{2}}\Big{\\}}.$

From the Bochner’s formula and (2.2), we can get

$\displaystyle\Delta F$ $\displaystyle\geq-2\nabla
F\cdot\nabla[\ln(1-u^{2})]+\frac{2\cdot\Big{\\{}|\nabla u|^{2}(|\nabla
u|^{2}-\lambda)+(1-u^{2})\cdot(|\nabla^{2}u|^{2}-\lambda|\nabla
u|^{2})\Big{\\}}}{(1-u^{2})^{2}}\cdot$ $\displaystyle\geq\frac{2|\nabla
u|^{2}}{(1-u^{2})^{2}}F-\nabla F\cdot\Psi(x);$

where $\Psi$ is defined as in (2.1). Then the conclusion follows. ∎

Now we are ready to show the sharp gradient estimate for (Neumann or
Dirichlet) eigenfunctions in a uniform way.

######  Theorem 2.4.

For compact Riemannian manifold $(M^{n},g)$ with $Rc\geq 0$, assume $\Delta
u=-\lambda u,\displaystyle\max_{x\in M^{n}}|u(x)|=1$, where $\lambda>0$. If
$\partial M^{n}$ is smooth, convex and $\frac{\partial
u}{\partial\vec{n}}\big{|}_{\partial M^{n}}=0$, or if $\partial M^{n}$ is
smooth, mean convex and $u\big{|}_{\partial M^{n}}=0$; then

$\displaystyle|\nabla u|^{2}+\lambda u^{2}\leq\lambda.$

######  Remark 2.5.

The Neumann case is originally proved in [LY], and the Dirichlet case is
proved in [Yang, Theorem $3.2$].

###### Proof.

Step (1). Let
$u_{\operatorname{\epsilon}}=\frac{u}{1+\operatorname{\epsilon}}$, where
$\operatorname{\epsilon}>0$, then $|u_{\operatorname{\epsilon}}|<1$. We define
$F_{\operatorname{\epsilon}}(x)=\frac{|\nabla
u_{\operatorname{\epsilon}}|^{2}}{1-u_{\operatorname{\epsilon}}^{2}}-\lambda$.
Suppose $F_{\operatorname{\epsilon}}(x_{0})=\max\limits_{M^{n}}\frac{|\nabla
u_{\operatorname{\epsilon}}|^{2}}{1-u_{\operatorname{\epsilon}}^{2}}$. There
are two cases :

If $x_{0}\in M^{n}\backslash\operatorname{\partial}M^{n}$, then we get $\nabla
F_{\operatorname{\epsilon}}(x_{0})=0$.

If $x_{0}\in\operatorname{\partial}M^{n}$, we can choose an orthonormal basis
$\\{e_{1},\cdots,e_{n}\\}$ in the neighborhood $U$ of $x_{0}$ such that
$e_{n}(x)=\vec{n}(x)$ for all $x\in\operatorname{\partial}M^{n}\cap U$ and
$\nabla_{e_{n}}e_{i}(x)=0$ for all $x\in U$, $1\leq i\leq n$.

From the definition of $x_{0}$, we know

(2.3)
$\nabla_{i}F_{\operatorname{\epsilon}}(x_{0})=0,\,i=1,\cdots,n-1\,\,\text{and}\,\nabla_{\vec{n}}F_{\operatorname{\epsilon}}(x_{0})\geq
0.$

Note
$\nabla_{\vec{n}}F_{\operatorname{\epsilon}}=\nabla_{\vec{n}}|\nabla\sin^{-1}u_{\epsilon}|^{2}$,
from Lemma 2.1, we get that
$\nabla_{\vec{n}}F_{\operatorname{\epsilon}}(x_{0})\leq 0$. Now
$\nabla_{\vec{n}}F_{\operatorname{\epsilon}}(x_{0})=0$ by (2.3). Therefore,
$\nabla F_{\operatorname{\epsilon}}(x_{0})=0$ for
$x_{0}\in\operatorname{\partial}M^{n}$.

Then we obtain

(2.4) $\nabla F_{\operatorname{\epsilon}}(x_{0})=0.$

Step (2). Then from the choice of $x_{0}$, for any $v\in ST_{x_{0}}M^{n}$ and
$\gamma_{x_{0}}(tv)=\exp_{x_{0}}(tv)$, we get

$\displaystyle\nabla^{2}_{v,v}F_{\operatorname{\epsilon}}(x_{0})$
$\displaystyle=2\lim_{t\rightarrow
0}\frac{F_{\operatorname{\epsilon}}(\gamma_{x_{0}}(tv))-F_{\operatorname{\epsilon}}(x_{0})-\nabla_{v}F_{\operatorname{\epsilon}}(x_{0})\cdot
t}{t^{2}}=2\lim_{t\rightarrow
0}\frac{F_{\operatorname{\epsilon}}(\gamma_{x_{0}}(tv))-F_{\operatorname{\epsilon}}(x_{0})}{t^{2}}$
(2.5) $\displaystyle\leq 0.$

We suppose $\nabla u_{\operatorname{\epsilon}}(x_{0})\neq 0$ in the rest
argument, otherwise $|\nabla u_{\operatorname{\epsilon}}|\equiv 0$ on $M^{n}$,
and the conclusion is trivial.

Note $u_{\epsilon}$ is an eigenfunction with $|u_{\epsilon}|<1$. From Lemma
2.3, we get

$\displaystyle\mathscr{L}_{u_{\epsilon}}F_{\epsilon}(x_{0})\geq 0.$

From the above, (2.4) and (2.5), we get
$F_{\operatorname{\epsilon}}(x_{0})\leq 0$, which implies $|\nabla
u_{\operatorname{\epsilon}}|^{2}(x_{0})+\lambda
u_{\operatorname{\epsilon}}^{2}(x_{0})\leq\lambda$. By the definition of
$x_{0}$, we know $|\nabla u_{\operatorname{\epsilon}}|^{2}+\lambda
u_{\operatorname{\epsilon}}^{2}(x)\leq\lambda$ for all $x\in M^{n}$. Let
$\operatorname{\epsilon}\to 0$, we get $|\nabla u|^{2}+\lambda
u^{2}\leq\lambda$. ∎

##  3\. The local splitting induced by eigenfunctions

######  Definition 3.1.

For $u\in C^{\infty}(M^{n})$ and $p\in M^{n}$, if $\nabla u(p)\neq 0$, let
$M_{u,p}$ be the connected component of $M^{n}\backslash\mathscr{C}(u)$
containing $p$, where $\mathscr{C}(u)=\\{x\in M^{n}:|\nabla u|(x)=0\\}$. When
the context is clear, we also use $M_{p}$ for simplicity.

######  Proposition 3.2.

For compact Riemannian manifold $(M^{n},g)$ with $Rc\geq 0$. Assume $\Delta
u=-\lambda u,\displaystyle\max_{x\in M^{n}}|u(x)|=1$, where $\lambda>0$. If
$|\nabla\sin^{-1}u|^{2}(p)=\lambda$ holds at some point $p$ with $|u(p)|<1$ ,
then $\displaystyle N^{n-1}\vcentcolon=u^{-1}(u(p))\cap M_{p}$ is connected
and $\partial N^{n-1}=\emptyset$. Furthermore,

  1. (N)

. If $\partial M^{n}=\emptyset$, then $M_{p}$ is isometric to $\displaystyle
N^{n-1}\times\left(-\frac{\pi}{2\sqrt{\lambda}},\frac{\pi}{2\sqrt{\lambda}}\right)$
and $\partial N^{n-1}=\emptyset$.

  2. (NB)

. If $\partial M^{n}\neq\emptyset$ is smooth, convex and $\frac{\partial
u}{\partial\vec{n}}\big{|}_{\partial M^{n}}=0$, then $M_{p}$ is isometric to
$\displaystyle
N^{n-1}\times\left(-\frac{\pi}{2\sqrt{\lambda}},\frac{\pi}{2\sqrt{\lambda}}\right)$.

  3. (DB)

. If $\partial M^{n}\neq\emptyset$ is smooth, mean convex and
$u\big{|}_{\partial M^{n}}=0$; then $\partial N^{n-1}=\emptyset$. And

    1. (DB-a)

If $M_{p}\cap\partial M^{n}=\emptyset$, then $M_{p}$ is isometric to
$\displaystyle
N^{n-1}\times\left(-\frac{\pi}{2\sqrt{\lambda}},\frac{\pi}{2\sqrt{\lambda}}\right)$.

    2. (DB-b)

If $M_{p}\cap\partial M^{n}\neq\emptyset$, then $M_{p}$ is isometric to
$\displaystyle N^{n-1}\times\left(-\frac{\pi}{2\sqrt{\lambda}},0\right]$.

Proof: Step (1). First note that for any $x\in M_{p}$, we have $|\nabla
u|(x)\neq 0$ and $|u|(x)<1$ by the definition of $M_{p}$ and Theorem 2.4.

Let $V=\\{x\in M_{p}\,\,|\,\,|\nabla\sin^{-1}u|(x)=\sqrt{\lambda}\\}$, which
is nonempty since $p\in V$. It’s evident that $V$ is a closed set in $M_{p}$.

By Lemma 2.3, we have

$\displaystyle\mathscr{L}_{u}(F)(x)\geq 0,\quad\quad\quad\forall x\in M_{p};$

where $F(x)=|\nabla\sin^{-1}u|^{2}(x)-\lambda$.

Given $x_{1}\in V$, note $F$ attains its maximum at $x_{1}$ with $F(x_{1})=0$
by Theorem 2.4. Note $\frac{\partial F}{\partial\vec{n}}\big{|}_{\partial
M^{n}}\leq 0$ by Lemma 2.1, we apply the Strong Maximum Principle (see [GT,
Lemma $3.4$ and Theorem $3.5$]) to get $F\big{|}_{B_{x_{1}}(r)\cap
M_{p}}\equiv 0$. Therefore $(B_{x_{1}}(r)\cap M_{p})\subset V$, and $V$ is
open in $M_{p}$.

Because $M_{p}$ is connected , we conclude that
$|\nabla\sin^{-1}u|(x)=\sqrt{\lambda}$ for any $x\in M_{p}$.

Let $\Theta(x)=\frac{1}{\sqrt{\lambda}}\sin^{-1}u(x)$ for $x\in M_{p}$, then
direct computation combining $|\nabla\Theta|(x)=1,\Delta u=-\lambda u$ yields

$\displaystyle\Delta\Theta=0,\quad\quad\quad\quad\forall x\in M_{p}.$

From the Bochner formula and $Rc\geq 0$, we can infer that

$\begin{split}|\nabla^{2}\Theta|^{2}&=\frac{1}{2}\Delta|\nabla\Theta|^{2}-\langle\nabla\Delta\Theta,\nabla\Theta\rangle-
Rc(\nabla\Theta,\nabla\Theta)\leq 0.\end{split}$

Hence $|\nabla^{2}\Theta|\equiv 0$ and $|\nabla\Theta|\equiv 1$ in $M_{p}$.

Step (2). Define $\mathcal{L}_{t}=\big{\\{}\ x\in M_{p}:\Theta(x)=t\big{\\}}$,
from $|\nabla\Theta|\equiv 1$ and the Implicit Function Theorem, we know that
$\mathcal{L}_{t}$ is $(n-1)$-dim submanifold of $M^{n}$ if
$\mathcal{L}_{t}\neq\emptyset$.

For any $x\in M_{p}$, let $\gamma_{x}$ be the gradient curve of $\nabla\Theta$
with $\gamma_{x}(0)=x$ in $M_{p}$. Define

$\displaystyle T_{x}\vcentcolon=\sup\\{t\geq 0:\gamma_{x}(t)\in
M_{p}\\},\quad\quad\hat{T}_{x}\vcentcolon=\sup\\{t\geq 0:\gamma_{x}(-t)\in
M_{p}\\},\quad\quad\forall x\in M_{p}.$

Now we prove the following claim:

######  Claim 3.3.

$\mathcal{P}(x)=\gamma_{x}\cap\mathcal{L}_{0}:M_{p}\rightarrow\mathcal{L}_{0}$
is well-defined for any $x\in M_{p}$.

Proof: [Proof of Claim 3.3] If $\Theta(x)=0$, then $\mathcal{P}(x)=x$.

Now we consider the case $\Theta(x)<0$. We firstly show $T_{x}\geq-\Theta(x)$.
By contradiction, assume $T_{x}<-\Theta(x)$. Then by $|\nabla\Theta|\leq 1$,
we have

$\displaystyle-\frac{\pi}{2\sqrt{\lambda}}<\Theta(x)\leq\Theta(\gamma_{x}(t))\leq\Theta(x)+t\leq\Theta(x)+T_{x}<0,\quad\quad\forall
t\in[0,T_{x}).$

Using $|\nabla\Theta|\equiv 1$ in $M_{p}$ and
$u(z)=\sin(\sqrt{\lambda}\Theta(z))$ for any $z\in M_{p}$, we get

(3.1) $\displaystyle|\nabla
u|(\gamma_{x}(t))\geq\sqrt{\lambda}\cdot\sqrt{1-\sin^{2}(\sqrt{\lambda}\Theta(x))}>0,\quad\forall
t\in[0,T_{x}).$

From (3.1), we get

(3.2) $\displaystyle\lim_{t\rightarrow T_{x}}|\nabla
u|(\gamma_{x}(t))\geq\sqrt{\lambda}\cdot\sqrt{1-\sin^{2}(\sqrt{\lambda}\Theta(x))}>0,$

which implies $\gamma_{x}(T_{x})$ is well-defined and $\gamma_{x}(T_{x})\in
M_{p}$.

Furthermore,

(3.3) $\displaystyle
u(\gamma_{x}(t))=\sin(\sqrt{\lambda}\Theta(\gamma_{x}(t)))\leq\sin(\sqrt{\lambda}[\Theta(x)+T_{x}])<0,\quad\quad\forall
t\in[0,T_{x}).$

For case (DB), from (3.3), we know $\gamma_{x}(T_{x})\notin\partial M^{n}$.
Then there is $\delta>0$ such that $\gamma_{x}(T_{x}+\delta)\in M_{p}$.

For case (NB), from (3.2), we know $|\nabla
u|\left(\gamma_{x}(T_{x})\right)>0$.

  1. (a)

. If $\gamma_{x}(T_{x})\notin\partial M^{n}$, there is $\delta>0$ such that
$\gamma_{x}(T_{x}+\delta)\in M_{p}$.

  2. (b)

. If $\gamma_{x}(T_{x})\in\partial M^{n}$, from
$\gamma_{x}^{\prime}(T_{x})=\nabla\Theta\left(\gamma_{x}(T_{x})\right)=\frac{\nabla
u}{|\nabla u|}\left(\gamma_{x}(T_{x})\right)\neq 0$ and
$\frac{\operatorname{\partial}u}{\operatorname{\partial}\vec{n}}\left(\gamma_{x}(T_{x})\right)=0$;
we obtain $\gamma_{x}^{\prime}(T_{x})\in
T_{\gamma_{x}(T_{x})}\operatorname{\partial}M^{n}$.

Combining $\operatorname{\partial}M^{n}$ is smooth and $(\nabla
u\cdot\vec{n})\big{|}_{\partial M^{n}}=0$, we can extend $\gamma_{x}$ (which
is the gradient curve of $u$) in $\partial M^{n}$. Hence there is $\delta>0$
such that $\gamma_{x}(T_{x}+\delta)\in M_{p}$.

By the above argument, there is $\delta>0$ such that
$\gamma_{x}(T_{x}+\delta)\in M_{p}$. It contradicts the definition of $T_{x}$.
Therefore $T_{x}\geq-\Theta(x)$.

Next we prove that $\gamma_{x}(-\Theta(x))\in M_{p}$. Like (3.1), we have

(3.4) $\displaystyle|\nabla
u|(\gamma_{x}(t))\geq\sqrt{\lambda}\cdot\sqrt{1-\sin^{2}(\sqrt{\lambda}\Theta(x))}>0,\quad\quad\forall
t\in[0,-\Theta(x)).$

This implies $\gamma_{x}(-\Theta(x))\in M_{p}$ is well-defined.

Now $\mathcal{P}(x)=\gamma_{x}(-\Theta(x))$ is well defined.

If $\Theta(x)>0$, consider the gradient curve of $-\nabla\Theta$ as above, we
can get that $\mathcal{P}(x)$ is also well-defined. By all the above, we know
that $\mathcal{P}(x)$ is well-defined for $x\in M_{p}$. ∎

Step (3). Now for cases (N), (NB) and (DB-a), we will show that for each
$x\in\mathcal{L}_{0}$, there is

(3.5) $\displaystyle T_{x}=\hat{T}_{x}=\frac{\pi}{2\sqrt{\lambda}}.$

By contradiction, if $T_{x}<\frac{\pi}{2\sqrt{\lambda}}$, then
$\Theta(\gamma_{x}(T_{x}))<\frac{\pi}{2\sqrt{\lambda}}$.

We get $|\nabla
u|(\gamma_{x}(T_{x}))=\sqrt{\lambda}\cdot\sqrt{1-u^{2}(\gamma_{x}(T_{x}))}>0$.
Hence there is $\delta>0$ such that $\gamma_{x}(T_{x}+\delta)\in M_{p}$, which
contradicts the definition of $T_{x}$.

Similar argument applies on $\hat{T}_{x}$. Therefore (3.5) is proved.

Similarly, for case (DB-b) we can show

(3.6) $\displaystyle T_{x}\cdot\hat{T}_{x}=0,\quad\text{and}\quad
T_{x}+\hat{T}_{x}=\frac{\pi}{2\sqrt{\lambda}},\quad\quad\quad\forall
x\in\mathcal{L}_{0}\cap\partial M^{n}.$

For cases (N), (NB) and (DB-a), we define the map

(3.7)
$\displaystyle\psi(x)=(\Theta(x),\mathcal{P}(x)):M_{p}\rightarrow(-\frac{\pi}{2\sqrt{\lambda}},\frac{\pi}{2\sqrt{\lambda}})\times\mathcal{L}_{0}.$

Because $x=\gamma_{\mathcal{P}(x)}(\Theta(x))$ and (3.5), we get that $\psi$
is a bijective map. Hence $\psi$ is a diffeomorphism between $M_{p}$ and
$(-\frac{\pi}{2\sqrt{\lambda}},\frac{\pi}{2\sqrt{\lambda}})\times\mathcal{L}_{0}$.

For case (DB-b), we get that the map $\psi(x)=(\Theta(x),\mathcal{P}(x))$ is a
diffeomorphism between $M_{p}$ and
$(-\frac{\pi}{2\sqrt{\lambda}},0]\times\mathcal{L}_{0}$ (or
$[0,\frac{\pi}{2\sqrt{\lambda}})\times\mathcal{L}_{0}$).

Step (4). Assume $\xi_{1}^{\prime}(0)=v_{1}\in T_{x}M^{n}$, now using
$|\nabla^{2}\Theta|\equiv 0$ and the definition of $\mathcal{P}$, we get

$\displaystyle\langle\psi_{*}v_{1},\psi_{*}v_{1}\rangle$
$\displaystyle=\frac{d}{dt}\psi(\xi_{1})\cdot\frac{d}{dt}\psi(\xi_{1})=\langle\frac{d}{dt}(\Theta(\xi_{1}(t)),\mathcal{P}(\xi_{1}(t))),\frac{d}{dt}(\Theta(\xi_{1}(t)),\mathcal{P}(\xi_{1}(t)))\rangle$
$\displaystyle=\langle\frac{d}{dt}\Theta(\xi_{1}(t)),\frac{d}{dt}\Theta(\xi_{1}(t))\rangle+\langle\frac{d}{dt}\mathcal{P}(\xi_{1}(t)),\frac{d}{dt}\mathcal{P}(\xi_{1}(t))\rangle$
$\displaystyle=\langle\nabla\Theta\cdot v_{1},\nabla\Theta\cdot
v_{1}\rangle+\langle\nabla\mathcal{P}\cdot v_{1},\nabla\mathcal{P}\cdot
v_{1}\rangle=\langle v_{1},v_{1}\rangle.$

Hence $\psi$ is a local isometry, the conclusion $\psi$ is an isometry follows
from all the above.

Therefore $N^{n-1}=u^{-1}(u(p))\cap M_{p}$ is isometric to $\mathcal{L}_{0}$.
Because $M_{p}$ is connected, we get $N^{n-1}$ is connected, by the above
isometry between $M_{p}$ and the product of $N^{n-1}$ with an interval.

Also we obtain $\partial N^{n-1}=\emptyset$ by the above isometry and
$\partial M^{n}$ is smooth. ∎

##  4\. The global splitting

######  Theorem 4.1.

For compact Riemannian manifold $(M^{n},g)$ with $Rc\geq 0$ and smooth
(possibly empty) boundary . Assume $\Delta u=-\lambda
u,\displaystyle\max_{x\in M^{n}}|u(x)|=1$, where $\lambda>0$. If
$|\nabla\sin^{-1}u|^{2}(p)=\lambda$ holds at some point $p$, where $|u(p)|<1$.
Define $\mathcal{N}(u)=\\#(M^{n}-u^{-1}(0))$ as the number of connected
components of $M^{n}-u^{-1}(0)$ (the number of nodal domains of $u$); then:

  1. (1)

. If $\partial M^{n}=\emptyset$, then $M^{n}$ is isometric to
$\mathbb{S}^{1}(\frac{\mathcal{N}(u)}{2\sqrt{\lambda}})\times N^{n-1}$, where
$\mathbb{S}^{1}(\frac{\mathcal{N}(u)}{2\sqrt{\lambda}})$ is a round circle
with radius $\frac{\mathcal{N}(u)}{2\sqrt{\lambda}}$ and $\mathcal{N}(u)=2k$
for some $k\in\mathbb{Z}^{+}$;

  2. (2)

. If $\partial M^{n}\neq\emptyset$ is convex and $\frac{\partial
u}{\partial\vec{n}}\big{|}_{\partial M^{n}}=0$, then $M^{n}$ is isometric to
$[0,\frac{\mathcal{N}(u)-1}{\sqrt{\lambda}}\cdot\pi]\times N^{n-1}$ and
$\mathcal{N}(u)\geq 2$;

  3. (3)

. If $\partial M^{n}\neq\emptyset$ is mean convex and $u\big{|}_{\partial
M^{n}}=0$; then $M^{n}$ is isometric to
$[0,\frac{\mathcal{N}(u)}{\sqrt{\lambda}}\cdot\pi]\times N^{n-1}$ and
$\mathcal{N}(u)\geq 1$;

where $N^{n-1}$ is a compact Riemannian manifold with $Rc\geq 0$ and $\partial
N^{n-1}=\emptyset$.

Proof: Step (1). Define $N^{n-1}=u^{-1}(-1)\cap\overline{M_{p}}$, which is an
$(n-1)$-dim hypersurface in $M^{n}$ by Proposition 3.2 (if
$u^{-1}(-1)\cap\overline{M_{p}}=\emptyset$, we define
$N^{n-1}=u^{-1}(1)\cap\overline{M_{p}}$, similar argument applies); and
$U_{\epsilon}(N^{n-1})=\\{x\in M^{n}:d(x,N^{n-1})<\epsilon\\}$. Then we have a
coordinate chart $\\{x_{1},x^{\prime}\\}$ for $U_{r_{0}}(N^{n-1})$ locally,
where $x^{\prime}=(x_{2},\cdots,x_{n})$ is a coordinate chart for $N^{n-1}$
and

$\displaystyle x_{1}\big{|}_{M_{p}\cap
U_{r_{0}}(N^{n-1})}=d(x,N^{n-1}),\quad\quad\quad
x_{1}\big{|}_{U_{r_{0}}(N^{n-1})-M_{p}}=-d(x,N^{n-1}),$

where $r_{0}>0$ is to be determined later. We define
$U_{r_{0}}^{-}(N^{n-1})\vcentcolon=U_{r_{0}}(N^{n-1})\cap\\{x\in
M^{n}:x_{1}<0\\}$.

Consider the metric $g$ in $U_{r_{0}}(N^{n-1})$, then we can write
$\displaystyle g=dx_{1}^{2}+\sum_{2\leq i,j\leq
n}f_{ij}(x_{1},x^{\prime})dx_{i}dx_{j}$ locally. By Proposition 3.2, there is
$a>0$, such that the coefficients $f_{ij}$ satisfy:

$f_{ij}(x_{1},x^{\prime})=\begin{cases}&f_{ij}(0,x^{\prime}),\,\,0\leq
x_{1}\leq a,\\\ &f_{ij}(x_{1},x^{\prime}),\,\,-a\leq x_{1}\leq 0.\end{cases}$

For any $q\in N^{n-1}$, without loss of generality we can assume
$q=(0,0,\cdots,0)$. Now we get

$\displaystyle f_{ij}(0,x^{\prime})=\delta_{ij}+O(|x^{\prime}|).$

Denote $G=\mathrm{det}(f_{ij})_{2\leq i,j\leq n}$, we get

$(\log\sqrt{G})_{x_{1}}=\frac{1}{2}(\log G)_{x_{1}}=O(x_{1}).$

We use $u_{i},u_{ij}$ to denote $\frac{\partial u}{\partial
x_{i}},\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}$ in the rest
argument, similar notation applies for other functions.

Note $u_{11}(q)=\lambda$ because
$u\big{|}_{M_{p}}(x_{1},x^{\prime})=\sin(\sqrt{\lambda}x_{1}-\frac{\pi}{2})$
from Proposition 3.2. Also we have

$\displaystyle
u_{\alpha}(q)=0,\quad\quad\text{if}\quad\quad\alpha=(\alpha_{1},\cdots,\alpha_{k})\neq(1,\cdots,1),$
$\displaystyle u_{\beta}(q)=0,\quad\quad\text{if}\quad\quad\beta=(1,\cdots,1)\
\text{and there are odds number of $1$},$

Note $N^{n-1}$ is compact, there is a finite coordinate chart covering of
$N^{n-1}$. Using the Taylor’s expansion of $u(x_{1},x^{\prime})$ at $q$ about
$x_{1}$, where $q$ can be freely chosen from $N^{n-1}$; we get

(4.1) $\displaystyle u(x)=-1+\frac{1}{2}\lambda
x_{1}^{2}+O(|x_{1}|^{4}),\quad\quad\quad\forall x\in U_{r_{0}}(N^{n-1}).$

By (4.1), we can choose $r_{0}>0$ such that

$\displaystyle|\nabla
u\big{|}_{\overline{U_{r_{0}}(N^{n-1})}-N^{n-1}}|>0,\quad\quad\text{and}\quad\quad|u\big{|}_{\overline{U_{r_{0}}(N^{n-1})}-N^{n-1}}|<1.$

Step (2). Now we define
$F:\overline{U_{r_{0}}^{-}(N^{n-1})}\rightarrow\mathbb{R}$ as follows:

$F(x)\vcentcolon=\left\\{\begin{array}[]{rl}&\frac{|\nabla
u|^{2}}{1-u^{2}}(x)-\lambda\ ,\quad\quad\quad\quad\quad\quad\quad\quad
x\in\overline{U_{r_{0}}^{-}(N^{n-1})}-N^{n-1},\\\ &0\
,\quad\quad\quad\quad\quad\quad\quad x\in N^{n-1}.\end{array}\right.$

From the Taylor’s expansion of $u$ at any point of $N^{n-1}$, we get that $F$
is continuous on $\overline{U_{r_{0}}^{-}(N^{n-1})}$.

Then from (4.1), we have

(4.2) $\displaystyle\frac{-2|\nabla u|^{2}}{(1-u^{2})^{2}}$
$\displaystyle=-\frac{2}{x_{1}^{2}}+O(1).$

Direct computation yields

$\displaystyle\Psi(x)$ $\displaystyle=\frac{\nabla
u}{1-u^{2}}\cdot\\{-\frac{1}{2}\frac{(1-u^{2})^{2}}{|\nabla u|^{4}}\cdot\nabla
u\cdot\nabla\Big{(}\frac{|\nabla u|^{2}}{1-u^{2}}(x)\Big{)}-2u\\}$ (4.3)
$\displaystyle=\Big{(}\frac{2}{x_{1}}+O(x_{1})\Big{)}\cdot\partial x_{1}.$

Consider $v(x)=x_{1}^{2}-x_{1}$, then $v$ satisfies

(4.4) $\displaystyle v\big{|}_{U_{r_{0}}^{-}(N^{n-1})}\geq 0,\quad\quad
v\big{|}_{x_{1}=0}=0,\quad\quad\frac{\partial}{\partial
x_{1}}v\big{|}_{x_{1}=0}=-1<0.$

Furthermore, using (4.3) and (4.2), we have

$\displaystyle\mathscr{L}_{u}(v)$ $\displaystyle=\Delta v(x)+\nabla
v(x)\cdot\Psi(x)-\frac{2|\nabla u|^{2}}{(1-u^{2})^{2}}v(x)$
$\displaystyle=v_{11}+(\log\sqrt{G})_{x_{1}}v_{1}+\sum_{2\leq i,j\leq
n}f^{ij}v_{ij}+\sum_{2\leq i,j\leq
n}\\{\frac{\operatorname{\partial}f^{ij}}{\operatorname{\partial}x_{i}}+(\log\sqrt{G})_{x_{i}}f^{ij}\\}v_{j}$
$\displaystyle+v_{1}\cdot(\frac{2}{x_{1}}+O(x_{1}))-\frac{2}{x_{1}^{2}}v+O(1)v$
$\displaystyle=2+O(x_{1})+(2x_{1}-1)\cdot(\frac{2}{x_{1}}+O(x_{1}))-\frac{2}{x_{1}^{2}}\\{x_{1}^{2}-x_{1}\\}$
$\displaystyle=4+O(x_{1})>0,$

in the last inequality above we use $x_{1}\leq r_{0}<<1$.

Now there is $r_{0}>0$ and $v\in C^{\infty}(U_{r_{0}}(N^{n-1}))$ satisfying

(4.5) $\displaystyle\mathscr{L}_{u}(v)(x)>0,\quad\quad\quad\forall x\in
U_{r_{0}}^{-}(N^{n-1}).$

Step (3). For any $q\in N^{n-1}$, let $\vec{n}=\partial x_{1}$ be the outer
normal unit vector of $\partial U_{r_{0}}^{-}(N^{n-1})$ at $q$. Using (4.1) we
can compute directly that

(4.6) $\displaystyle\frac{\partial F}{\partial\vec{n}}(q)=0.$

We will show that there is $z\in U_{r_{0}}^{-}(N^{n-1})$ such that
$F(z)=F(q)=0$. By contradiction, otherwise

(4.7) $\displaystyle F(z)<F(q)=0,\quad\quad\quad\forall z\in
U_{r_{0}}^{-}(N^{n-1}).$

From Lemma 2.3, we have

(4.8) $\displaystyle\mathscr{L}_{u}(F)\geq 0.$

Now we consider $F+\epsilon\cdot v$. Because of (4.7), we can choose
$\epsilon>0$ such that

(4.9) $\displaystyle(F+\epsilon\cdot v)\big{|}_{x_{1}=-r_{0}}\leq 0.$

Then we have

$\displaystyle(F+\epsilon\cdot v)\big{|}_{\partial U_{r_{0}}^{-}(N^{n-1})}\leq
0.$

By (4.5), we get $\mathscr{L}_{u}(F+\epsilon\cdot
v)\big{|}_{U_{r_{0}}^{-}(N^{n-1})}>0$, then from the Maximum principle, we get

$\displaystyle\sup_{x\in U_{r_{0}}^{-}(N^{n-1})}(F+\epsilon\cdot
v)(x)\leq\max\\{0,\sup_{x\in\partial U_{r_{0}}^{-}(N^{n-1})}(F+\epsilon\cdot
v)(x)\\}=0.$

Note $(F+\epsilon\cdot v)(q)=0$, then we obtain

$\displaystyle\frac{\partial}{\partial x_{1}}\big{|}_{x=q}(F+\epsilon\cdot
v)(x)\geq 0.$

Combining (4.4), this yields

$\displaystyle\frac{\partial}{\partial
x_{1}}\big{|}_{x=q}F(x)\geq-\epsilon\cdot\frac{\partial}{\partial
x_{1}}\big{|}_{x=q}v(x)=\epsilon>0.$

This contradicts (4.6).

Therefore, we find $z\in U_{r_{0}}^{-}(N^{n-1})$ such that $F(z)=F(q)=0$,
which implies $\displaystyle
F(z)=\max_{x\in\overline{U_{r_{0}}^{-}(N^{n-1})}}F(x)=0$. Note $|\nabla
u(z)|>0$, $|u(z)|<1$ and $|\nabla\sin^{-1}u(z)|^{2}=\lambda$; from Proposition
3.2, we get the extended splitting domain on $M^{n}$.

The main conclusion follows by the above and the induction on
$\mathcal{N}(u)$.

∎

## Acknowledgments

We thank Xiaodong Wang for his comments on the early version of this paper.

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National Cheng Kung University]Department of Physics, National Cheng Kung
University, Tainan, 70101, TaiwanNational Cheng Kung University]Department of
Physics, National Cheng Kung University, Tainan, 70101, Taiwan National Cheng
Kung University]Hierarchical Green Energy Materials, Hi- esearch Center,
National Cheng Kung University, Taiwan National Cheng Kung
University]Department of Physics, National Cheng Kung University, Tainan,
70101, Taiwan National Cheng Kung University]Department of Physics, National
Cheng Kung University, Tainan, 70101, Taiwan National Cheng Kung
University]Department of Physics, National Cheng Kung University, Tainan,
70101, Taiwan National Cheng Kung University]Department of Physics, National
Cheng Kung University, Tainan, 70101, Taiwan National Cheng Kung
University]Hierarchical Green Energy Materials, Hi- esearch Center, National
Cheng Kung University, Taiwan

# Orbital-hybridization-created optical excitations in Li2GeO3

Vo Khuong Dien<EMAIL_ADDRESS>[  Hai Duong Pham  [  Ngoc Thanh Thuy
Tran  [  Nguyen Thi Han  [  Thi My Duyen Huynh  [  Thi Dieu Hien Nguyen
<EMAIL_ADDRESS>[  Ming Fa-Lin<EMAIL_ADDRESS>[  [

###### Abstract

Li2GeO3, a ternary electrolyte compound of Li+-based battery, presents the
unusual essential properties. The main features are thoroughly explored from
the first-principles calculations. The concise pictures, the critical orbital
hybridizations in Li-O and Ge-O bonds, are clearly examined through the
optimal $Moir\acute{e}$ superlattice, the atom-dominated electronic energy
spectrum, the spatial charge densities, the atom- and orbital-decomposed van
Hove singularities, and the strong optical responses. The unusual optical
transitions cover the red-shift optical gap, 16 frequency-dependent absorption
structures and the most prominent plasmon mode in terms of the dielectric
functions, energy loss functions, reflectance spectra, and absorption
coefficients. Optical excitations, depending on the directions of electric
polarization, are strongly affected by excitonic effects. The close
combinations of electronic and optical properties can identify a significant
orbital hybridization for each available excitation channel. The developed
theoretical framework will be very useful in fully understanding the diverse
phenomena of cathode/electrolyte/anode materials in Lithium ion-based
batteries.

##  1 INTRODUCTION

Lithium-ion batteries (LIBs) dominate in commercial purposes due to their
high-performance energy resources, e.g. high power, energy densities, and high
reliability, as well as other certain fundamental merits including affordable
price, long life cycle, and friendly environment 1, 2, 3. The recent
experimental progress shows many applications of LIBs, such as electric
vehicles (EV) and hybrid EV and mobile devices, which require the optimum in
producing efficient energy. A LIB principally consists of a cathode (positive
electrode), an anode (negative electrode), and an ionically conductive
Li+-containing electrolyte, in which the third component is closely related to
the unusual transport of the positive lithium ions (Li+) between two
electrodes 1, 2, 3. Depending on the combined alternative materials of three
components, LIBs can provide various performances, e.g, a specific energy
density of 100 to 250 W.h/Kg 4, 5, the volumetric energy density from 250 to
680 W.h/L 6, a specific power density in the range of 300 to 1500 W/kg 7, 8,
and a faster charging time (80 % of charge of states in 15 mins) 9. However,
recently used electrolytes in LIBs are not compatible with later developed
high-voltage positive electrodes, which are one of the most effective ways of
increasing the energy density.
Nowadays, solid electrolytes are rapidly emerging as promising alternatives
given their wider electrochemical window of stability. As potential
electrolyte candidates, Li-Ge-O compounds exhibit a large ionic conductivity
(1.5$\times$10${}^{-5}\Omega$.cm-1 for Li2GeO3), in which electronic
conductivity is negligible with a high ionic transference number 10.
Furthermore, Li2GeO3 shows a wide cycling stability with a reserved charge
capacity of 725 mAhg-1 after 300 cycles at 50 mAg-111. As for battery safety,
Li2GeO3 is suitable for the selection of solid electrolytes according to the
reasonable decreasing interface resistance. On the other hand, previous
reports evidence that the Li2GeO3 sample also exhibit good optical properties
such as the high transparency in the visible region, the strong Stock shift of
the self-trap exciton 12, the polar orthorhombic symmetry 13, 14 suggests that
this material is useful for piezoelectric, pyroelectric, and electrochromic
energy storage devices.
Up to now, a lot of high-resolution measurements on essential physical
properties of anodes/cathodes/electrolytes in Li+-based batteries, e.g,
optimized geometric structures, band structures, optical properties and
transport have been conducted. X-Ray diffraction 11 and low-energy electron
diffraction (LEED 15) are available for investigating the 3D lattice
symmetries, especially Li-X-O related systems. Angle-resolved photoemission
spectroscopy (ARPES 16) is a powerful experimental technique directly
measuring the single particle spectral function, depending on wave vectors and
frequency. Optical properties, such as energy loss functions, reflectance,
absorptance, refractivity,and excitation 17, 18, 19 could be determined
through spectroscopy measurements. For example, analysis of the specular
reflection from the polished surface of a solid can give the complex index of
refraction using the Kramers-Kronig method.
As for Li-Ge-O compounds, a few delicate experimental measurements, such as
X-Ray diffraction 11 and photoluminescence measurement 12 were carried out. Up
to date, concise physical and chemical pictures that closely relate to the
performance of an energy storage system have not been proposed for the
essential properties of the Li-Ge-O compound. Furthermore, the theoretical
electrolytes related to Li-Ge-O compounds with band structures, optical
properties and other predictions about chemical/physical combinations are
still absent. In this work, the close connection of the geometric structure,
electronic and optical excitations of Li2GeO3 solid states electrolyte was
identified for the first time.
The theoretical framework, being based on the significant orbital
hybridizations in chemical bonds 20, 21, is developed by examining the
essential properties in electrolyte materials of Li+-related batteries 22.
This strategy is based on the first-principles calculations on an optimal
lattice symmetry with position-dependent chemical bondings, the atom-dominated
band structure at different energy ranges, the spatial charge densities due to
various orbitals, and the atom- and orbital-projected van Hove singularities
related to orbital overlaps. The energy-decomposed single-/multi-orbital
hybridizations will be utilized to account for the optical threshold
frequency, a lot of prominent absorption structures, a very strong plasmon
response in terms of the dielectric functions, energy loss functions,
reflectance spectra, and absorption coefficients under the distinct electric
polarizations. This is the first work that successfully combines the
geometric, electronic and optical properties with the energy-dependent orbital
hybridizations. Most predicted results require high-resolution experimental
examinations.

##  2 COMPUTATIONAL DETAILS

We used the density functional theory (DFT) method via the Vienna Ab-initio
Simulation Package (VASP) 23 to perform the optimization of the structure and
calculation the electronic and optical properties. The Perdew-Burke-Ernzerhof
(PBE) generalized gradient approximation was used for the exchange-correlation
functional 24. The interaction between the ions and valence electrons was
described by the projector augmented wave (PAW) method 25. The cutoff energy
for the expansion of the plane wave basis was set to 500 eV. The Brillouin
zone was integrated with a special k-point mesh of 15$\times$15$\times$15 in
the Monkhorst-Pack sampling technique 26 for geometric optimization. The
convergence condition of the ground-state is set to be 10-8 eV between two
consecutive simulation steps, and all atoms were allowed to fully relax during
geometric optimization until the Hellmann-Feynman force acting on each atom
was smaller than 0.01 eV.
Subsequent to the DFT results, the quasi-particle Green’s function and the
screened Coulomb interactions (G0W0) approach 27 using 250 eV energy cutoff
for the response functions and the Brillouin zone was integrated with a
special k-points mesh of 8$\times$8$\times$8 in the Monkhorst-Pack sampling
technique to obtain the corrected density of states and electronic band
structure and quasi-particle excitations of Li2GeO3. Regarding the optical
response beyond the independent particle approach, the electron-hole
interaction was taken into account by solving the standard Bethe-Salpeter
equation (BSE) within the Tamm-Dancoff approximation 28. The k-point sampling,
energy cutoff, and number of bands setting the same as in the GW calculation.
In this calculation, the 20 highest occupied valence bands (VBs) and 8 lowest
unoccupied conduction bands (CBs) are included as a basis for the excitonic
states with a photon energy region from 0 eV to 28 eV. In addition, the
broadening parameter $\gamma$, which arises from various de-excitation
mechanisms, was set at 0.1 eV to make the spectrum more accurate for the given
k-point meshes. All parameters in this study have been checked for convergence
of the calculations.

##  3 RESULTS AND DISCUSSIONS

Figure 1: The optimal geometric structure of Li2GeO3 ternary compound, (b)
oxygen atoms around each Li/Ge one, and (c) the first Brillouin zone.

The ternary compound, Li2GeO3 with 24 atoms within a conventional unit cell,
is chosen for a model study in illustrating the geometric, electronic and
optical properties. The optimal lattice, as clearly shown in Fig. 1, belongs
to an orthorhombic structure. Each Li/Ge atom is surrounded by four O atoms in
a tetrahedral form. There exist 32 Li-/16 Ge-O bonds, in which the former and
the latter display large fluctuations about their lengths ($\sim$ 1.93-2.12 Å
and $\sim$ 1.72-1.84 Å, respectively). The calculated lattice constants along
the $x$-, $y$\- and $z$-directions, (9.61, 5.46, 4.87 Å), are very close to
the X-ray diffraction measurements, (9.60, 5.50, 4.85 Å) in Ref. [11] and
(9.63, 5.48, 4.84 Å) in Ref. [29]. Obviously, the greatly non-uniform
chemical/physical environment indicates the importance of orbital
hybridizations in chemical bondings. Furthermore, it might be very useful for
supporting the ion transport in electrolyte materials. This behavior will be
responsible for the unusual effects of $Moir\acute{e}$ superlattice and the
highly anisotropic optical transitions.

Figure 2: (a) Band structure along the high-symmetry points in the wave-vector
space, with (b) Li- (enlarged five times), (c) Ge- and (d) O-atom dominances
(green, blue and red circles).

Li2GeO3 exhibits a rich and unique band structure. The main features in Fig.
2(a) show a lot of valence and conduction subbands in a wide energy range (-26
eV $<$ E${}^{c,v}<$ 12 eV), the high asymmetry of hole and electron energy
spectra about the Fermi level (E${}_{F}=$ 0), an indirect band gap of
E${}_{g}^{i}=$ 6.9 eV related to the highest occupied state and the lowest
unoccupied one, respectively, at the Z and $\Gamma$ points, the
parabolic/oscillatory/linear/partially flat energy dispersions with various
band-edge states, and the atom-dominated electronic states in the different
energy regions. Very interestingly, such results are associated with many
significant orbital hybridizations in Li-O and Ge-O bonds, being clearly
identified from the close combinations of the atom- and/or orbital-dependent
band structure (Figs. 2(b), 2(c) and 2(d)), charge density (Fig. 3) and van
Hove singularities (Fig. 4).
The electronic structures can be classified into five categories based on the
various atom and orbital contributions, as clearly indicated in Figs. 2(b),
2(c) and 2(d) by the green, blue and red circles for Li, Ge and O,
respectively. The (I), (II), (III), (IV) and (V) regions, respectively,
correspond to E${}^{c}>$ 7.6 eV, 3.2 eV $<$ E${}^{c}<$ 7.6 eV, -6.0 eV $<$
E${}^{v}<$ -3.8 eV, -10.6 eV $<$ E${}^{v}<$ -6.0 eV, and -24.5 eV $<$
E${}^{v}<$ -21.8 eV. It should be noted that valence subbands disappear
between the (IV) and (V) regions. The Li atoms only make very weak, yet still
significant contributions within a whole range of band structures (small green
circles in Fig. 2(b)), i.e., the unusual essential properties disappear in the
absence of Li-O bonds. The energy-spectrum subgroups are qualitatively
characterized by (I) (Ge, O) co-dominance, (II) (Ge, O) co-dominance, (III) O
dominance, (IV) (Ge, O) co-dominance and (V) O dominance, being supported by
the atom- and orbital-projected density of states (discussed later in Fig. 4).
The specific orbital hybridizations in chemical bonds will be identified to be
associated with the critical energy bands in revealing the prominent
absorption structures.
The spatial charge distributions before/after chemical bondings can provide
some very useful information about the first-step orbital hybridizations in
Li-O and Ge-O bonds. An isolated Li atom has an isotropic charge density (Fig.
3(a)), in which the inner and outer regions (the red and white parts) arise
from 1s and 2s orbitals, respectively. The similar, but wider distribution,
which corresponds to the O case (Fig. 3(b)), is associated with (1s, 2s) and
(2px, 2py, 2pz) orbitals. Furthermore, the highest charge density appears
around Ge (Fig. 3(c)), with the separate ranges of (4s, those below it) and
(4px, 4py, 4pz) orbitals. As for the Li-O bonds, the outer/inner regions show
the obvious/minor deformations along three electric-polarization directions
(Figs. 3(d), 3(e) and 3(f)), especially for the neighboring ones. These
clearly indicate the multi-orbital hybridizations of 2s-(2px, 2py, 2pz)/the
single-orbital hybridization of 2s-2s. In addition to the white regions (Figs.
3(g), 3(h) and 3(i)), the red ones near the Ge and O atoms present observable
changes, suggesting the significant bondings of (4s, 4px, 4py, 4pz)-(2s, 2px,
2py, 2pz). While the above-mentioned analyses are combined with the atom- and
orbital-projected van Hove singularities, the orbital-hybridization-dominated
band-edge states will be identified. They are responsible for a lot of strong
absorption structures (Figs. 5 and 6).

Figure 3: Comparison of an isolated (a)/(b)/(c) Li/O/Ge atom, the spatial
charge distributions related to the significant orbital hybridizations in
(d)/(e)/(f) Li-O and (g)/(h)/(i) Ge-O bonds along the $x$-/$y$-/$z$-directions
for the shortest chemical bonds.  Figure 4: Density of states for different
components: (a) the total magnitude with Li-, Ge- and O-atom contributions,
(b) Li- and 2s-decomposed results, (c) Ge- and (4s, 4px, 4py, 4pz)-projected
calculations, and (d) O- and (2s, 2px, 2py, 2pz)-related ones.

The atom- and orbital-projected density of states (Fig. 4), being supported by
the atom-dominated band structure (Fig. 2) and charge density distribution
(Fig. 3), can clearly identify the energy-dependent orbital hybridizations.
They come from the band-edge states of various energy dispersions, such as,
the extreme, saddle and dispersionless ones. Both critical points in the
energy-wave-vector spaces and dimensionality fully determine their special
structures 20, 21. As a result of too many subbands and broadening effects,
Fig. 4(a) only displays the prominent asymmetric peaks and shoulders (the
black curve). The van Hove singularities of different atoms/orbitals can merge
together (Figs. 4(a)-4(d)), clearly indicating the specific orbital
hybridizations. According to their strong co-relations, the five subgroups of
band structure are further examined to be dominant through (I) 2s-(2s, 2px,
2py, 2pz) & (4s, 4px, 4py, 4pz)-(2s, 2px, 2py, 2pz), (II) 2s-(2px, 2py, 2pz) &
4s-(2px, 2py, 2pz), (III) 2s-(2px, 2py, 2pz) & (4s, 4px, 4py, 4pz)-(2px, 2py,
2pz), (IV) 2s-(2px, 2py, 2pz) & 4s-(2px, 2py, 2pz) and (V) 2s-2s & 4s-2s. Such
identifications of significant orbital bondings would become more delicate
under strong optical responses, mainly owing to the greatly reduced energy
width of the excitation frequency.

Figure 5: (a) The imaginary- and real-parts of the dielectric functions with
the excitonic effects under three electric-polarization directions [the black,
red and blue curves]. (b) Comparison of the imaginary parts of the dielectric
function with and without excitonic effects under xx-polarization direction.

After the perturbation of an electromagnetic (EM) wave, electrons are
vertically excited from the occupied to unoccupied states. Under the single-
particle picture (the linear Kubo formula in Ref. [30]), the optical
absorption spectrum is mainly determined by the joint density of states
related to the valence and conduction subbands and the square of the electric
dipole moment, respectively, corresponding to the number and intensity of
available optical transitions. The excited valence holes and conduction
electrons simultaneously come into existence during the optical excitations;
furthermore, they have a tendency to combine through the attractive Coulomb
potentials under the suitable condition, e.g., the large band gap for the
suppression of temperature broadening. The coupled quasiparticles, the stable
excitons, might strongly affect the main features of the optical absorption
spectra since they make important contributions to many-body effects. This
mean that they take part in the various-order excitation processes using the
Coulomb scatterings, leading to the dramatic transformations of photons. The
excitonic effects, which are closely related to the critical orbital
hybridizations, are the focus of this study.
The imaginary and real parts of dielectric functions are very useful in
understanding the available optical transitions. They can fully examine the
orbital-related prominent responses by linking the atom-dominated band
structure (Fig. 2), charge density (Fig. 3), and atom - $\&$ orbital-
decomposed van Hove singularities (Fig. 4). The former
($\epsilon_{2}(\omega)$), as clearly shown in Fig. 5(a), presents the
excitation characteristics of available channels in the absence of a screening
effect of all valence charges. An optical gap (E${}_{g}^{o}$), which
corresponds to the threshold absorption frequency, is about 5.95/5.92/5.87 eV
under the many-body effects for the $x$-/$y$-/$z$-directions of electric
polarizations (the black/red/blue curves). However, it is $\sim$ 7.00 eV
purely through the single-particle optical excitations for
E$\|$$\hat{\textbf{x}}$ (the blue curve in Fig. 5(b)). Compared with band gap
of E${}_{g}^{i}$ = 6.9 eV, the great red shift clearly indicates the very
strong Coulomb couplings between the excited holes and electrons. The
composite quasiparticles should be quite stable, so that they are expected to
survive at room temperature. The theoretical predictions are relatively easily
examined by the various optical spectroscopies 12, 17, 18, 19.
Very interestingly, there exist 16 pronounced peak/shoulder absorptions (the
distinct colored triangles in Figs. 5(b),S1 and S2). In which, the appearance
of these certain optical-spectral structures is associated with the
transitions from band-edge states/Van Hove singularities of electronic band
structure/orbital-projected density of states. Furthermore, the formation of
the stable excitonic states around the respective extrema of the band-edge
state also creates extra prominence peaks below the band gap. However, it can
not strongly modify intrinsic chemical bondings (the identity of the joined of
density of states). Based on the close connection of the concise physical and
chemical pictures in atom-dominated band structure (Fig. 2), atom-, orbital-
decomposed density of states (Fig. 4), and strong optical responses due to
band-edge states. The frequency-dependent vertical transitions and orbital
hybridizations are identified (detail in Table 1). The corresponding
excitation of channels are marked with arrow heads in Fig. 2(a) and the
triangle in Fig. 4. Generally, these channels are major/minor contributions
from the electron transition between the O-2p occupied states and the
Ge-(4s,4p)/Li-2s unoccupied states. For example,the threshold structure is
marked by a red triangle in Fig. 5(b), which is due to the excitation from the
valence flat band to the conduction minimum extremes at the $\Gamma$ point. It
is mainly/slightly dominated by the O-2pz valence states and the Ge-4s/Li-2s
conduction ones, indicated by the red arrow head in Fig. 2(a) and the red
triangle in Fig. 4, respectively. Using this strategy, the specified
mechanisms never previously revealed in studies for the various absorption
structures could be achieved. This viewpoint has been successfully generalized
to anode and cathode materials of Li+-based batteries.

Table 1: Prominent absorption structures: frequencies, colored indicators and identified orbital hybridizations Excitation frequency (eV) | Colors of arrows |  | Specific orbital hybridizations
---
in Ge-O bonds
| With excitonic
---
effect
| Without excitonic
---
effect
|
6 | – | Black | –
6.3 | – | Dark-red | –
7 | 7 | Red | Ge(4s) - O(2pz)
8 | 8 | Orange | Ge(4s) - O(2px,2py,2pz)
8.8 | 8.9 | Purple | Ge(4s) - O(2px,2py,2pz)
9.9 | 10 | Green | Ge(4s) - O(2px,2py,2pz)
10.6 | 10.6 | Maroon | Ge(4s) - O(2px,2py, 2pz)
11.3 | 11.4 | Blue | Ge(4s,4px,4py,4pz) - O(4px,4py, 4pz)
12.5 | 12.5 | Brown | Ge(4s) - O(2px,2py)
13.6 | 13.5 | Yellow | Ge(4s,4px,4py,4pz) - O(2px,2py,2pz)
14.9 | 14.9 | Pink | Ge(4s) - O(2px,2py,2pz)
16 | 16 | Cyan | Ge(4s,4px,4py,4pz) - O(2px,2py,2pz)
17.2 | 17.2 | Blue-violet | Ge(4s,4px,4py,4pz) - O(2px,2py,2pz)
20.3 | 20.3 | Light-green | Ge(4s,4px,4py,4pz) - O(2px,2py,2pz)
24.5 | 24.5 | Dark-gray | Ge(4s) - O(2s)
27.1 | 27.1 | Gray | Ge(4s) - O(2s)

As for $\epsilon_{1}(\omega)$ (Fig. 5(a)), their prominent absorption
structures can be understood from those of $\epsilon_{2}(\omega)$ by the
Kramers-Kronig relation under the event of optical excitations. Apparently,
the van Hove singularities in $\epsilon_{1}(\omega)$ and
$\epsilon_{2}(\omega)$ might be similar or different, being strongly co-
related by the principal-value integration on the complex plane. Below the
threshold frequency, $\epsilon_{1}(\omega)$ weakly depends on $\omega$, in
which its value roughly lies within the range of 2.1-2.2 for $x$-,
$y$-polarizations and the $z$ one. This will determine the low-frequency
reflectance spectrum (Fig. 6(b)) and the vanishing absorption coefficient
(Fig. 6(c)). Very interesting, $\epsilon_{1}(\omega)$ is very sensitive to the
changes of frequency during the creation of the excited holes and electrons.
It can vanish at weak Landau damping, in which its zero point and the small
$\epsilon_{2}(\omega)$ might appear simultaneously, e.g., the vanishing
$\epsilon_{2}(\omega)$ at 18.5/20/13.6 eV under
E$\|\hat{\textbf{x}}$/E$\|\hat{\textbf{y}}$/E$\|\hat{\textbf{z}}$. In
addition, the zero points for the $z$-polarization (the blue curve) at 6 eV
become meaningless because of the combination with very strong electron-hole
excitations.
The energy loss function (ELF), being defined as
Im$\left[\frac{-1}{\epsilon(\omega)}\right]$31, is the screened response
function due to the significant valence charges of the Li, Ge and O atoms. The
charge screenings can determine the coherent carrier oscillations at the long
wave-length limit during the optical transitions. In general, the collective
excitations (a plasmon mode) are revealed as a sufficiently strong peak with
ELF higher than 1, as clearly indicated in Fig. 6(a). The strongest peak comes
into existence at $\omega_{p}$=18.5/20/13.6 eV for the $x$-/$y$-/$z$-direction
electric polarizations, being attributed to the significant Li-2s, O-(2px,
2py, 2pz) $\&$ Ge-(4s, 4px, 4py, 4pz) orbitals. The O-2s contribution for the
most prominent plasma wave is ignored, since its dominating valence states
only appear below -22 eV (the black curve in Fig. 4(d)). In addition, a few
minor plasmon peaks are accompanied by serious Landau dampings. It should be
noted that two manners could be utilized to identify the collective
excitations, but the current peak in ELF is much better than the zero points
in $\epsilon_{1}(\omega)$. The absence of the latter and/or the combination
with a very large $\epsilon_{2}(\omega)$ is the main reason.
The reflectance spectrum is characterized as
R($\omega$)=$\left|\frac{\sqrt{\epsilon(\omega)}-1}{\sqrt{\epsilon(\omega)}+1}\right|^{2}$
32, when an electromagnetic wave is normally incident on Li2GeO3. The total
electric field will be reflected by the surface, absorbed by the valence
electrons and transmitted through a finite-width sample. Both the reflectance
R($\omega)$ and the absorption coefficient $\alpha(\omega)$ are inverse to the
propagation decay length in [33], as shown in Figs. 6(b) and 6(c),
respectively, directly reflecting the main features of the single-particle
(Figs. 5(a) and 5(b)) and the collective excitations (Fig. 6(a)). For
$\omega<$ E${}_{g}^{o}$, a small reflectance weakly depends on the frequency
and is roughly given
by$\left|\frac{\sqrt{\epsilon(0)}-1}{\sqrt{\epsilon(0)}+1}\right|^{2}$. The
absorption coefficient $\alpha(\omega)=0$ because of the vanishing electron-
hole excitations, leading to the non-decay EM-wave propagation and thus a very
efficient transmission. However, within the active region of valence-electron
excitations (E${}_{g}^{o}<\omega<$ 21 eV), the sensitive and significant
frequency dependences come into existence. Reflectance is enhanced and
displays a large fluctuation under the various inter-band excitation channels,
in which a drastic change of the plasmon edge appears at $\omega_{p}$, e.g.,
$\sim$ 40 % variation under the E$\|\hat{\textbf{y}}$ case (the red curve).
Moreover, since absorption coefficient $\alpha(\omega)$ is very large,
especially that related to the plasmon mode, and its inverse is about 50-300
Å, the EM waves are easily absorbed by Li2GeO3 through the rich electronic
excitations.

Figure 6: The various optical properties: (a) energy loss functions, (b)
reflectance spectra, and (c) absorption coefficients.

In addition to X-ray diffractions 12, 29, only very few experimental
examinations on electronic and optical properties exist. In general, the wave-
vector dependences of occupied valence states can be directly tested by angle-
resolved photoemission spectroscopy (ARPES) 16, 34. A lot of unusual energy
subbands (Fig. 2) due to a $Moir\acute{e}$ superlattice would create high
barriers in the ARPES measurements. Scanning tunneling spectroscopy (STS) is
available for the clear identification of van Hove singularities near the
Fermi level 35. However, STS might be suitable only under a thin-film sample
because of very weak quantum currents. Very interestingly, the optical
spectroscopy methods of reflectance 17, absorption 18 and transmission 19 are
reliable in verifying the frequency-dependent optical properties. For example,
the measured reflectance spectrum, being supported by the Kramers-Kronig
relations between $\epsilon_{1}(\omega)$ and $\epsilon_{2}(\omega)$, can
determine both of them. It provides significant information about the initial
excitonic peaks, the greatly reduced threshold excitation frequency, many
prominent absorption structures, and the strongest plasmon mode at
$\omega_{p}>$ 16 eV. Up to now, one photo-luminescence measurement has
verified the optical gap situated at $\sim$ 6 eV 12, being close to the
current prediction of $\sim$ 5.95 eV. Apparently, the diverse optical
properties in cathode/electrolyte/anode materials of Li+-based batteries are
worthy of systematic investigations, both experimentally and theoretically.

##  4 CONCLUSION

The significant orbital hybridizations in chemical bonds, being based on
first-principles calculations, are thoroughly identified for the Li2GeO3
compound in terms of the geometric, electronic and optical properties. They
will play important roles in fully understanding the diversified essential
properties of cathode/electrolyte/anode materials in ion-related materials.
This solid-state electrolyte presents unusual features, i.e. a $Moir\acute{e}$
superlattice with a highly non-uniform chemical environment, the Li-, Ge- or
O-dominated energy bands, orbital-induced spatial charge densities, atom- and
orbital-decomposed van Hove singularities. As a result, the band-edge states,
which might create the prominent optical responses, are well characterized by
the specific orbital interactions through the developed theoretical framework.
The featured optical transitions cover a red-shift optical gap (E${}_{g}^{o}$
= 5.95 eV) much lower than an indirect one (E${}_{g}^{i}$ = 6.9 eV), a very
long transmission length/low reflectance for $\omega<$ E${}_{g}^{o}$, 16
pronounced single-particle absorption structures (short decay lengths), and
the strongest plasmon peak/a quickly decreasing edge in the energy loss
function/reflectance spectrum at $\omega_{p}$, 17-18 eV, and sensitive changes
due to the electric-field directions. The many-body excitonic effects have
strongly modified the single-particle interband excitations. Under current
investigations, the developed theoretical framework can be further generalized
to include other electrolyte/cathode/anode materials of Li+-based batteries.

##  5 Acknowledgments

This work is supported by the Hi-GEM Research Center and the Taiwan Ministry
of Science and Technology under grant number MOST 108-2212-M-006-022-MY3, MOST
109-2811-M-006-505 and MOST 108-3017-F-006-003.

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