# Coherence times of precisely depth controlled NV centers in diamond

Junfeng Wang  Hefei National Laboratory for Physical Science at Microscale,
and Department of Physics, University of Science and Technology of China,
Hefei, Anhui, 230026, P. R. China  Wenlong Zhang  Hefei National Laboratory
for Physical Science at Microscale, and Department of Physics, University of
Science and Technology of China, Hefei, Anhui, 230026, P. R. China  Jian Zhang
Hefei National Laboratory for Physical Science at Microscale, and Department
of Physics, University of Science and Technology of China, Hefei, Anhui,
230026, P. R. China  Jie You  Key lab of Quantum Information, CAS, University
of Science and Technology of China, Hefei, Anhui, 230026, P. R. China  Yan Li
Key lab of Quantum Information, CAS, University of Science and Technology of
China, Hefei, Anhui, 230026, P. R. China  Guoping Guo  Key lab of Quantum
Information, CAS, University of Science and Technology of China, Hefei, Anhui,
230026, P. R. China  Fupan Feng  Hefei National Laboratory for Physical
Science at Microscale, and Department of Physics, University of Science and
Technology of China, Hefei, Anhui, 230026, P. R. China  Xuerui Song  Hefei
National Laboratory for Physical Science at Microscale, and Department of
Physics, University of Science and Technology of China, Hefei, Anhui, 230026,
P. R. China  Liren Lou  Hefei National Laboratory for Physical Science at
Microscale, and Department of Physics, University of Science and Technology of
China, Hefei, Anhui, 230026, P. R. China  Wei Zhu  Hefei National Laboratory
for Physical Science at Microscale, and Department of Physics, University of
Science and Technology of China, Hefei, Anhui, 230026, P. R. China  Guanzhong
Wang<EMAIL_ADDRESS>Hefei National Laboratory for Physical Science at
Microscale, and Department of Physics, University of Science and Technology of
China, Hefei, Anhui, 230026, P. R. China

###### Abstract

We investigated the depth dependence of coherence times of nitrogen-vacancy
(NV) centers through precisely depth controlling by a moderately oxidative at
580 ∘C in air. By successive nanoscale etching, NV centers could be brought
close to the diamond surface step by step, which enable us to trace the
evolution of the number of NV centers remained in the chip and to study the
depth dependence of coherence times of NV centers with the diamond etching.
Our results showed that the coherence times of NV centers declined rapidly
with the depth reduction in their last about 22 nm before they finally
disappeared, revealing a critical depth for the influence of rapid fluctuating
surface spin bath. By monitoring the coherence time variation with depth, we
could make a shallow NV center with long coherence time for detecting external
spins with high sensitivity.

In recent years, shallow NV center has attracted increasing attention owing to
its applications in nanoscale spin detectionkey-1 ; key-2 ; key-3 ; key-4 ;
key-5 ; key-6 ; key-7  and surface spin noise investigationkey-8 ; key-9 ;
key-10 ; key-11 . Since the minimum detectable magnetic dipole moment scales
as $\delta\mu\propto r^{3}/\sqrt{T_{2}}$ key-9 ; key-12 ; key-13 , where r is
the NV-target spin distance and $T_{2}$ is the coherence times of the NV
center, the nanoscale control of the NV center depth and the investigation of
depth dependence of NV center coherence time are important.

Recently, two methods, plasma etching key-14  and oxidative etching key-13 ;
key-15 , have been developed to control the depth of the NV center. Cui et al.
found that the conventional oxygen plasma etching had low damage to diamond
key-14 . For oxidative etching, Kim et al. demonstrated that the diamond could
be etched at the temperature, ranging from 550 ∘C to 620 ∘C, in pure oxygen
gas key-13 . Loretz et al. reported that the oxidative etching rate for
diamond was about 10 $nm/h$ at 650 ∘C in air and by using this method they
realized a 1.9-nm-deep NV center key-15 . However, the etching rate was not
short enough to precisely control the depth of NV center .

In this work, we performed oxidative etching in air on diamond at a reduced
oxidative temperature 580 ∘C, obtaining about 1.1 nm/h etching rate. By
successive nanoscale etching, NV centers could be brought close to the diamond
surface step by step, which enable us to investigate the depth dependence of
coherence times of NV centers. The coherence time of NV center declined
rapidly with the depth reduction in their last about 22 nm before they finally
disappeared, which was attributed to the influence of the rapid fluctuating
surface spin bath. By monitoring the coherence time variation with depth, we
could make a shallow NV center with long coherence time for detecting external
spins with high sensitivitykey-5 ; key-7 ; key-15 ; key-16 .

Figure 1: Fluorescence images of the remaining NV centers in the diamond chip
after different oxidative etching times. (a) Image of a representative area of
the chip before oxidative etching. (b) and (c) The same area after oxidative
etching 26 h and 53 h, respectively. The circled single NV centers in the (a)
((b) and (c)) were used for tracing. All the scale bars in (a) ((b), (c)) are
5 $\mu m$. (d) The total number of the remaining NV centers in the selected
area on the diamond chip after different oxidative etching times. (e) The SRIM
simulations of the depth profile of the implanted nitrogen atoms at an energy
of 30 keV.  Figure 2: Coherence times measurements of the NV centers after
different oxidative etching times. (a) and (b) Spin echo and Ramsey
measurements, respectively, of NV-10 after three different oxidative etching
times. The blue lines were the fits of the $I_{PL}$. (c) and (d) The spin echo
and Ramsey measurements of NV-10 and NV-13 after different oxidative etching
times, respectively. (e) Coherence times vs. etching time derived from spin
echo measurements for all the 20 single NV centers. (f) Coherence times vs.
etching time derived from Ramsey measurements for 9 representative single NV
centers. Figure 3: Dynamical decoupling measurements after different oxidative
etching times. (a) and (b) The coherence decay curves of NV-10 in CPMG-N with
the $\pi$ pulse number N from 1 to 100 for the sample before etching and with
etching 45 h, respectively. The solid lines were the fits to the data using
the exponentials $e^{-(t/{T_{2}})^{n}}$,where $T_{2}$ is the coherence times.
(c) The coherence times $T_{2}$ of the CPMG of NV-10 as functions of N for the
chip etched for different times. The inset (bottom right) was the zoom of the
CPMG-N (N from 1 to 16) results for the chip after etching 49h. (d) The
coherence times $T_{2}$ of the CPMG of NV-13 as functions of N after different
oxidative etching times. The solid lines in the (c) and (d) were the fits to
the data using the power-law $T_{2,CPMG-N}/T_{2,echo}=N^{\lambda}$ and the
decoupling efficiency $\lambda$ as a function of the etching time for both
cases were shown in the inset (top left) of the corresponding figures,
respectively.

A $2\times 2\times 0.5$ mm3 (100) electronic grade diamond chip from Element
Six ([${}^{13}C$]=1.1%, [N]<5 ppb) was used for the experiments. At first, a
300-nm-thick polymethyl methacrylate (PMMA) layer was deposited on the diamond
chip surface key-17 ; key-18 . Then a series of 60 nm diameter apertures, as
well as some 10-$\mu m$ -wide strips used as position marks key-18 , were
patterned on the PMMA layer using electron beam lithography. The NV centers in
the chip were generated by implanting 60 keV ${}^{14}N_{2}^{+}$ molecules with
a fluence $0.55\times 10^{11}$ ${}^{14}N_{2}^{+}/cm^{2}$ through the apertures
and strips on the PMMA layer.key-17 ; key-18 . After implantation, the sample
was annealed at 1050 ∘C in a vacuum at $2\times 10^{-5}$ Pa for 2 h to form
long spin coherence time NV centers key-19 . After oxidation at 430 ∘C in air
for 2.5 h, the sample was cleaned in a 1:1:1 boiling mixture of sulfuric,
nitric, and perchloric acid at 200 ∘C for 2 hours.

The oxidative etching was performed in air on the diamond chip situated in a
box furnace with its implanted side up. In order to have a slow etching rate,
we used a lower oxidative temperature of 580 ∘C. The diamond chip was put in
the furnace when the furnace temperature had already stabilized at 580 ∘C. The
process for the furnace temperature being stabilized at 580 ∘C only took about
1 minute. Similarly, when the etching process was finished, we directly took
the diamond chip out to the air for cool down. We traced the evolution of the
number of single NV centers remaining in the chip with the etching time and
investigated the relationship between the coherence times of the NV centers
and the etching times. Figure 1(a) showed the fluorescence image of a
representative area of the diamond chip before oxidative etching, with the
traced single NV centers marked with circles in the figure. The larger bright
specks in the image were NV center clusters produced in the implantation due
to imperfection of the PMMA templet layer deposited on the diamond chip key-18
. After oxidative etching for 26 h, the image of the same area showed less
single NV centers and the bright specks of the NV center clusters also became
less bright and smaller [Fig. 1(b)]. As shown in Fig. 1(c), after oxidative
etching for 53 h, there were only few single NV centers remained on the chip
and the bright specks became even less bright and smaller. These results
clearly demonstrated that the NV centers originally situated at different
depths of the diamond chip had been removed successively by oxidative etching
key-13 ; key-15 .

We traced 265 single NV centers that were initially in the diamond chip. As
shown in Fig. 1(d), the number of the remaining single NV centers in the chip
reduced with the oxidative etching times. We found that when the etching time
was less than 10 h, the number of the remaining NV centers reduced very
little. Then, when the etching times increased from 10 h to about 60 h, the NV
centers number reduced rapidly. After that, the number reduced slowly and
finally almost all the NV centers finally disappeared after 85 h etching. The
results showed that the number of the remaining single NV centers reduced
about 70% after etching 42 h. By comparing the above results with the SRIM
simulations for an implantation nitrogen atom energy of 30 keV [Fig. 1(e)], we
found that the etching depth corresponding to 42 h etching was 46 nm.
Therefore, we estimated that the etching rate was about 1.1 nm/h at 580 C in
air.

The slow etching rate made it possible to control the depth of the NV center
at nanoscale. Thus, we could precisely trace the evolution of the coherence
time of a single NV center with its depth, which reflected the variation in
its local environment, from the bulk ${}^{13}C$ spin bath to the surface spin
bath. A set of randomly selected 20 single NV centers [named NV-1 through 20]
with their axes parallel to the external magnetic field (47 G) was used for
the tracing. We performed spin echo measurements for all the 20 single NV
centers before the chip etching, and found that the $T_{2}$ of 19 NV centers
among them were between 134 and 234 $\mu s$, and the other one was 3.3 $\mu
s$. The mean value of $T_{2}$ of the 19 single NV centers was about 194 $\mu
s$, which was comparable to that of the native NV center inside electronic
grade diamond key-20 , implying that the NVs were initially deep inside the
chip and the initial major spin impurities were ${}^{13}C$.

Then, we measured the coherence times of the 20 NV centers after successively
etching for various times step by step. Fig. 2(a) showed three representative
results of the spin echo measurements for NV-10 (which just disappeared after
etching 50 h), after different oxidative etching times. In the first two
figures, we found that the photoluminescence (PL) profile collapsed and
revived at ${}^{13}C$ Larmor frequency (about 50.3 kHz), which was due to the
incoherent precession of the 13C nuclei spins in our natural ${}^{13}C$
abundance sampleskey-21 . It was found that for the sample before etching, the
$T_{2}$ of NV-10 was 234.6 $\mu s$. And, after an etching of 42 h, the $T_{2}$
declined to 80 $\mu s$. However, when the etching time reached 49 h, its PL
intensity was not stable and was reduced to about half of the initial value,
indicating that the center had been very shallow and its charge state had
become unstable key-22 ; key-23 . (The similar phenomenon was also observed
for other NV centers, when they were shallow enough by long time etching.)
Moreover, its $T_{2}$ was reduced to only about 0.7 $\mu s$. The above results
revealed that the NV center became very shallow and suffered a strong
influence of the rapid fluctuating surface spin bath key-8 ; key-9 ; key-10 ;
key-24 . The $T_{2}$ of NV-10 in the sample etched for various times were
presented in Fig. 2(c) top. This etching time dependence can be converted to
the depth dependence as the depth varied with the etching time. As shown from
the figure, for short etching time (less than about 30 h), the $T_{2}$
decreased slowly with the etching time, then declined rapidly in its last
about 20 h etching before it finally disappeared (i.e. when its depth was less
than about 22 nm). Finally, when the etching time was longer than 50 h, NV-10
was finally disappeared.

Three representative Ramsey measurements for NV-10 were also shown in Fig.
2(b), and its $T_{2}^{*}$ for the sample after various etching times were
summarized in Fig. 2(c) bottom. We could see that the $T_{2}^{*}$ had a
similar evolution to the $T_{2}$. As another example, Figure 2(d) gave the
results of the coherence time measurements for NV-13 (just disappeared after
etching 62 h). It is clear that the coherence time decreased very slowly with
the etching time in the initial $\sim$ 40 h, while declined rapidly in the
last $\sim$ 20 h etching before it finally disappeared (i.e. when its depth
was less than about 22 nm). The evolution were similar to that of NV-10. Fig.
2(e) summarized the dependences of $T_{2}$ with the etching time for all the
20 NV centers. We found that there was a similar dependence for all these NV
centers: the $T_{2}$ of each NV centers decreased rapidly when the etching
time was about 20 h before the centers finally disappeared (i.e. when their
depth became less than about 22 nm). The results of Ramsey measurements of the
9 representative single NV centers after different oxidative etching times
were shown in Fig. 2(f), which exhibited a similar variation as that of their
$T_{2}$ in Fig. 2(e). The results suggested that, when the depth decreased to
about 22 nm, rapidly fluctuating surface spins, compared with the slow bulk
${}^{13}C$ spin bath, began to play an important role in NV decoherence,
leading to a rapid decreasing in coherence time with a further decrease in
centers depth, which was consistent with the results of the delta doping NV
centers key-9 .

Dynamical decoupling was important in NV center based high sensitivity
magnetic field key-25  and temperature key-18 ; key-26  sensing. In
particular, the shallow NV centers, combined with dynamical decoupling, could
also be used to detect external spins key-3 ; key-5 ; key-6 ; key-7  and
investigate the surface spin noise key-9 ; key-10 ; key-11 . In view of this,
we performed CPMG measurements on three arbitrarily selected NV centers (NV-1,
10, 11) on the chip after different oxidative etching times. The results were
shown in Fig. 3. As shown in Fig. 3(a), before etching, the $T_{2}$ of the
CPMG-N of NV-10 were always increasing with the $\pi$ pulse number N, in
particular, the $T_{2}$ of the CPMG-100 was about 360.4 $\mu s$, about 52
times longer than the value 6.9 $\mu s$ for the spin echo. However, in Fig.
3(b), after etching for 45 h, almost all the $T_{2}$ of the CPMG-N were
declined, in particular, the $T_{2}$ of the CPMG-100 was about 120.4 $\mu s$,
which was only about one third of the value before etching. The $T_{2}$ of the
CPMG-N as a function of N for NV-10 with the chip etched for various times
were summarized in Fig. 3(c). The decoupling efficiencies $\lambda$ key-9
varied with the etching time were shown in the inset (top left) of Fig. 3(c).
When the etching time was less than about 30 h, the $T_{2}$ of the CPMG were
decreased very slowly, and the decoupling efficiencies $\lambda$ was nearly a
constant of about 0.82. This was consistent to the fact that the center, for
sample etched for 30 h, was still inside the chip and the main spin bath was a
bulk ${}^{13}C$ bathkey-20 . Then, they both declined rapidly when its depth
decreasing in its last about 20 h etching before it finally disappeared (i.e.
when its depth was less than about 22 nm), which was similar to the results in
Fig. 2(c). The similar phenomena were observed for NV-11 (just disappeared
after etching 48 h) [Fig. 3(d)] and NV-1 (just disappeared after etching 75 h,
and its data was not shown in this paper). The results were consistent with
the fact that when the depth of the NV centers decreased, they would have a
higher coupling strength with surface spin bath that had a fast fluctuation
rate key-9 ; key-10 .

In summary, we investigated the depth dependence of coherence times of NV
centers in diamond with the depth controlled using oxidative etching. We used
a lower oxidative temperature of 580 ∘C in air to reduce the etching rate to
about 1.1 nm/h. Thus, we can more precisely control the depth of the centers.
On the basis of this, we performed the spin echo and Ramsey and CPMG
measurements on the NV centers, of which the depth was decreased by successive
oxidative etching, and found that the coherence times of NV centers declined
rapidly with the depth reduction in their last about 22 nm etching before they
finally disappeared, revealing a critical depth for the influence of rapid
fluctuating surface spin bath, which was consistent with the results of the
delta doping NV centers. key-9 . The obtained results showed how the surface
spin influenced the coherence behavior of NV centers of different depths,
paving the way for investigating surface spins. Particularly, the slow etching
method make it possible to control the depth of the NV center at nanoscale,
which can be used to study the stability of the very shallow NV centers key-24
; key-25 ; key-27 , to investigate the surface spin noise key-8 ; key-9 ;
key-10 ; key-11 , and for high sensitivity (even for single electron and
nuclear) nanoscale spin detection key-5 ; key-7 ; key-15 ; key-16 . Moreover,
it can also be used to control the depth of the NV center in the diamond tip
equipped to the atomic force microscope, which can be used in high sensitivity
nanoscale magnetic imaging of the spins key-28 .

This work was supported by the National Basic Research Program of China
(2013CB921800, 2011CB921400) and the Natural Science Foundation of China
(Grants No. 11374280 and No. 50772110).

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# Triviality Results and Conjugate Radius Estimation of Ricci Solitons

Absos Ali Shaikh∗1, Prosenjit Mandal2 and V. Amarendra Babu3 1,2Department of
Mathematics,
The University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India.  3Department of Mathematics,
Acharya Nagarjuna University, Nagarjuna Nagar,
Guntur, Andhra Pradesh, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The investigation of Ricci solitons is the focus of this work. We have proved
triviality results for compact gradient Ricci soliton under certain
restriction. Later, a rigidity result is derived for a compact gradient
shrinking Ricci soliton. Also, we have estimated the conjugate radius for non-
compact gradient shrinking Ricci solitons with superharmonic potential.
Moreover, an upper bound for the conjugate radius of Ricci soliton with
concircular potential vector field is determined. Finally, it is proved that a
non-compact gradient Ricci soliton with a pole and non-negative Ricci
curvature is non-shrinking.

††footnotetext:  ∗ Corresponding author.
$\mathbf{2020}$ Mathematics Subject Classification: 53C20; 53C25; 53E20.
Key words and phrases: Ricci soliton; scalar curvature; superharmonic
function; conjugate radius; Riemannian manifold.

##  1\. Introduction and Results

In the mid 1980’s Hamilton [10, 11] introduced the concept of Ricci flow by
the governing equation:

$\frac{\partial g}{\partial t}=-2Ric.$

The study of Ricci solitons is a crucial part in the treatment of Ricci flow.
Ricci solitons, as self similar solutions to Hamilton’s Ricci flow, are
natural generalizations of Einstein metrics. They play an important role in
the singularity analysis of the Ricci flow (see [12] for more details). In the
past decade, a significant number of results have been obtained in classifying
or understanding the geometry of Ricci solitons. For a study of Ricci flow and
Ricci solitons, we refer the reader to [2, 5, 6, 7, 14, 18] and the references
therein.

An $n$-dimensional Riemannian manifold $(M,g)$ with Ricci tensor $Ric$ is
called a Ricci soliton if the Lie derivative $\mathcal{L}_{X}g$ along a smooth
vector field $X$ satisfies

(1) $\frac{1}{2}\mathcal{L}_{X}g+{\rm Ric}=\lambda g,$

where $\lambda$ is a real constant.

The Ricci soliton is referred to as a gradient Ricci soliton if $X$ is the
gradient of a smooth function $f$ on $M$. Thus, (1) implies

(2) $\nabla^{2}f+Ric=\lambda g,$

where $\nabla^{2}f$ represents the Hessian of $f$. Also, for $\lambda<0$
(resp., $\lambda=0$ or $\lambda>0$), a Ricci soliton is called expanding
(resp., steady or shrinking). Moreover, when the vector field $X$ is Killing
or $f$ is a constant, then the Ricci soliton is called trivial. Throughout the
paper we have assumed $M$ as a complete, connected and orientable Riemannian
manifold without boundary.

For any fixed $x\in M$ and $v\in T_{x}M$, there exists a unique geodesic
$\sigma_{v}$ such that $\sigma_{v}(0)=x$ and $\sigma^{\prime}_{v}(0)=v$, then
the exponential map is defined as follows:

$exp_{x}:T_{x}M\rightarrow M,\text{ $exp_{x}(v)$}=\sigma_{v}(1),v\in T_{x}M.$

Now, we recall the definition of the injectivity radius $r_{inj}(M)$ and the
conjugate radius $r_{conj}(M)$ of $M$ which are defined respectively as
follows (see, [20]):

$r_{inj}(M)=\inf_{x\in M}\sup_{r}\\{r\text{ }|\text{ }exp:B(x,r)\rightarrow
exp(B(x,r))\text{ is a diffeomorphism}\\},$

and

$r_{conj}(M)=\inf_{x\in M}\sup_{r}\\{r\text{ }|\text{ }exp:B(x,r)\rightarrow
exp(B(x,r)),\text{ $B(x,r)$ contains no critical points of $exp$}\\},$

where $B(x,r)$ represents the open ball with center at $x$ and radius $r>0$.

It is well-known that $M$ is said to be a manifold with a pole at $p$ if
$exp_{p}:T_{p}M\rightarrow M$ is a diffeomorphism. Also, we have
$r_{inj}(M)\leq r_{conj}(M),$ based on the definition of injectivity radius
and conjugate radius of $M$.

Zhu [20] studied conjugate and injectivity radius on complete, non-compact
Riemannian manifolds and obtained an integral condition for the scalar
curvature along with the determination of an upper bound of the conjugate
radius. In [9], Green proved the maximal conjugate radius theorem, which
states that if $M$ is a compact Riemannian manifold without boundary, and its
scalar curvature $R$ is bounded below by $n(n-1)$, then $r_{conj}(M)\leq\pi$,
and further, $M$ is isometric to the Euclidean sphere $\mathbb{S}^{n}$ if and
only if the equality holds.

If $m$ is an integer with $0\leq m\leq n$ and $\alpha$ is a scalar, then the
$m$-th invariant of $\nabla^{2}f$, is denoted by $S_{m}(f)$ and is defined by
the condition (see, [16] and [17], p. 461)

$det(I+\alpha\nabla^{2}f)=S_{0}(f)+\alpha S_{1}(f)+\cdots+\alpha^{m}S_{m}(f).$

We recall that Hamilton conjectured in [11, 12] that a compact gradient Ricci
soliton with positive curvature operator is an Einstein manifold (a trivial
Ricci soliton), which is settled in [1]. The next important question is to
find the conditions, under which a compact gradient Ricci soliton becomes an
Einstein manifold. Cao [2] proved that every steady or expanding compact Ricci
soliton is trivial. It is known that compact shrinking Ricci solitons are not
necessarily trivial. In this context Chen and Deshmukh [4] studied compact
shrinking Ricci soliton and obtained some necessary and sufficient conditions
to become the solitons trivial. Now, generally a natural question arises,“Is
it possible to impose some other conditions on compact shrinking Ricci soliton
so that it become trivial ?”. We have given an affirmative answer to this
question in the following theorem:

######  Theorem 1.1.

If $(M,g,f)$ is a compact gradient shrinking Ricci soliton with constant
$S_{2}(f)$, then the soliton is trivial, i.e., $M$ is Einstein.

In the following theorem, we have given an application of the Bochner formula
for gradient Ricci solitons and obtained a triviality result:

######  Theorem 1.2.

If $(M,g,f)$ is a compact gradient Ricci soliton with constant $S_{2}(f)$,
then there does not exist any non-trivial gradient Ricci soliton.

If a gradient Ricci soliton is isometric to a quotient of
$N\times\mathbb{R}^{k}$, it is said to be rigid (for more details about
rigidity, see [15] and the references therein), where $N$ is an Einstein
manifold. Petersen and Wylie [15], studied gradient Ricci soliton and
determined certain conditions, under which a gradient Ricci soliton becomes
rigid. It is known that for compact Riemannian manifolds every steady or
expanding Ricci soliton is rigid [13]. Moreover, Hamilton [11] and Ivey [13]
proved that all shrinking compact Ricci solitons are rigid for the dimensions
2 and 3 respectively. Eminenti et al. [8] proved that any $n$-dimensional
compact shrinking Ricci soliton with constant scalar curvature is rigid. In
[15] Petersen and Wylie obtained a little more general result and proved the
following:

######  Theorem 1.3 ([15]).

A compact gradient shrinking Ricci soliton is rigid if

$\int_{M}Ric(\nabla f,\nabla f)\leq 0.$

In the following theorem, we have proved a rigidity result for compact
gradient shrinking Ricci soliton with some condition different from Petersen
and Wylie [15]. In particular, we have proved the following theorem:

######  Theorem 1.4.

A compact gradient shrinking Ricci soliton $(M,g,f)$ with constant $S_{2}(f)$
is rigid.

In the following three theorems, we have estimated upper bound for the
conjugate radius of a Ricci soliton:

######  Theorem 1.5.

Let $(M,g,f)$ be a non-compact gradient shrinking Ricci soliton with $Ric\geq
0$. If $f$ is superharmonic, then the conjugate radius of $M$ is given by

$r_{conj}(M)\leq\pi\sqrt{\frac{n-1}{\lambda}}.$

######  Theorem 1.6.

Let $(M,g,X)$ be an $n(\geq 3)$-dimensional Ricci soliton with the potential
vector field $X$ as concircular. Then the conjugate radius of $M$ is given by

$r_{conj}(M)\leq\pi\sqrt{\frac{n-1}{\lambda-\phi}},$

where $(\lambda-\phi)$ is positive.

######  Theorem 1.7.

Let $(M,g)$ be a complete Riemannian manifold satisfying
$Ric+\nabla^{2}f\geq\lambda g$. If the acceleration of $f$ along a geodesic
ray $\sigma(t)$ is less than or equal to some real number k, then the
conjugate radius of $M$ is given by

$r_{conj}(M)\leq\pi\sqrt{\frac{n-1}{\lambda-k}},$

where $(\lambda-k)$ is positive.

######  Theorem 1.8.

Let $(M,g,f)$ be a non-compact gradient Ricci soliton with a pole at $p$ and
$Ric\geq 0$. If the potential function $f$ is at most linear along a geodesic
ray $\sigma(t)$, with $\sigma(0)=p$, then the soliton is non-shrinking.

##  2\. Proof of the results

In this section, we have presented the proof for the results of the paper.
First we have stated the following lemmas, which will be used in the sequel:

######  Lemma 2.1 ([17]).

Let $(M,g)$ be a compact Riemannian manifold. Then the following holds:

$\int_{M}2S_{2}(f)=\int_{M}Ric(\nabla f,\nabla f).$

######  Lemma 2.2 ([17]).

Let $(M,g)$ be a compact Riemannian manifold. If $f$ is a smooth function on
$M$, then $S_{2}(f)$ can’t be constant unless it vanishes.

The following relations hold for gradient Ricci solitons:

######  Lemma 2.3 ([2, 19]).

Let $(M,g,f)$ be a gradient Ricci soliton. Then the following holds:

(3) $R+\Delta f=\lambda n,$ (4) $\langle\nabla f,\nabla R\rangle=2Ric(\nabla
f,\nabla f).$

###### Proof of Theorem 1.1.

As $S_{2}(f)$ is a constant, by Lemma 2.2, we infer that $S_{2}(f)$ vanishes.
Therefore, by Lemma 2.1, we have

(5) $\int_{M}Ric(\nabla f,\nabla f)=0.$

Using (4) in (5), we obtain

$\int_{M}\langle\nabla f,\nabla R\rangle=0.$

Now, by divergence theorem, we get

(6) $\int_{M}R\Delta f=0.$

Again, as $\lambda$ is a constant, we have

(7) $\int_{M}n\lambda\Delta f=0.$

Thus, combining (6) and (7), we obtain

(8) $\int_{M}(n\lambda-R)\Delta f=0.$

The above equation (8) together with equation (3) yields

$\int_{M}(\Delta f)^{2}=0.$

It follows that $\Delta f=0$. Therefore, $f$ is harmonic in a compact
Riemannian manifold, hence $f$ is constant. This completes the proof. ∎

###### Proof of Theorem 1.2.

From (3) and Bochner formula, we get

$\frac{1}{2}\Delta|\nabla f|^{2}=|\nabla^{2}f|^{2}-\langle\nabla f,\nabla
R\rangle+Ric(\nabla f,\nabla f).$

Now, using equation (4), we obtain

(9) $\displaystyle\frac{1}{2}\Delta|\nabla f|^{2}$ $\displaystyle=$
$\displaystyle|\nabla^{2}f|^{2}-2Ric(\nabla f,\nabla f)+Ric(\nabla f,\nabla
f)$ $\displaystyle=$ $\displaystyle|\nabla^{2}f|^{2}-Ric(\nabla f,\nabla f).$

Again, equation (5) and (9) together yield

$\int_{M}|\nabla^{2}f|^{2}=0.$

It follows that $\nabla^{2}f$ vanishes. As $M$ is compact, hence $f$ is
constant. This completes the proof of the theorem. ∎

###### Proof of Theorem 1.4.

As $S_{2}(f)$ is a constant, by Lemma 2.2, we get that $S_{2}(f)$ vanishes.
Therefore, by Lemma 2.1, we have

(10) $\int_{M}Ric(\nabla f,\nabla f)=0.$

The equation (10) together with Theorem 1.3, concludes the result. ∎

Setting $l=r_{conj}(M)$, we consider any geodesic ball $B(x,l)$, $x\in M$, and
any $q\in\partial B(x,l)$, there exists an arc length parameter
$\sigma:[0,l]\rightarrow M$, which is the shortest geodesic connecting $x$ and
$q$. Then, we have stated the following lemmas, which will be used in the
sequel:

######  Lemma 2.4 ([20]).

Let $(M,g)$ be a complete non-compact Riemannian manifold. Suppose that
$B=B(p,r)\subset M$ is a geodesic ball with center at $p$ and radius $r>0$ and
$B_{-l}=B(p,r-l)$. If $Ric\geq 0$, then

$\int_{B(p,r-l)}{R}\leq n(n-1)\left(\frac{\pi}{l}\right)^{2}Vol(B(p,r)).$

######  Lemma 2.5 ([20]).

Let $(M,g)$ be a complete non-compact Riemannian manifold with a pole at $p$
and $Ric\geq 0$. Then for any geodesic ray $\sigma(t)$ with $\sigma(0)=p,$ the
following holds

$\frac{1}{r}\int_{0}^{r}{t^{2}Ric(\sigma^{\prime}(t),\sigma^{\prime}(t))}dt\leq{n-1}.$

###### Proof of Theorem 1.5.

Tracing the gradient Ricci soliton equation (2), we have

(11) $R+\Delta f=\lambda n.$

Since, $Ric\geq 0$, hence from (11) and Lemma 2.4, we obtain

$\int_{B(p,r-l)}(\lambda n-\Delta f)\leq
n(n-1)\left(\frac{\pi}{l}\right)^{2}Vol(B(p,r)).$

As $f$ is superharmonic, $\Delta f\leq 0$ , thus

(12) $(\lambda n)Vol(B(p,r-l))\leq
n(n-1)\left(\frac{\pi}{l}\right)^{2}Vol(B(p,r)).$

By volume comparison theorem, we have (see, [20])

$\lim\limits_{r\rightarrow\infty}\frac{Vol(B(p,r-l))}{Vol(B(p,r))}=1.$

Therefore, from (12), we get

$\lambda\leq\frac{(n-1)\pi^{2}}{l^{2}}.$

This implies

$l\leq\pi\sqrt{\frac{n-1}{\lambda}}.$

∎

###### Proof of Theorem 1.6.

We have (see [20], p. 239)

$\int_{0}^{l}{sin^{2}\left(\frac{\pi}{l}t\right)Ric(\sigma^{\prime}(t),\sigma^{\prime}(t))}dt\leq\frac{(n-1)\pi^{2}}{2l}.$

Using Ricci soliton equation (1), we get

(13)
$\int_{0}^{l}{sin^{2}\left(\frac{\pi}{l}t\right)\\{\lambda-\frac{1}{2}(\mathcal{L}_{X}g)(\sigma^{\prime}(t),\sigma^{\prime}(t))\\}}dt\leq\frac{(n-1)\pi^{2}}{2l}.$

For all $Y\in\chi(M)$, $\nabla_{Y}X=\phi Y$, as $X$ is concircular, where
$\phi$ is a smooth function on $M$. Therefore

(14)
$\frac{1}{2}(\mathcal{L}_{X}g)(\sigma^{\prime}(t),\sigma^{\prime}(t))=\phi
g(\sigma^{\prime}(t),\sigma^{\prime}(t)).$

By (13) and (14), we get

(15)
$\int_{0}^{l}{sin^{2}\left(\frac{\pi}{l}t\right)(\lambda-\phi)}dt\leq\frac{(n-1)\pi^{2}}{2l}.$

Again, combining Ricci soliton equation (1) and equation (14), we have

$Ric(\sigma^{\prime}(t),\sigma^{\prime}(t))=(\lambda-\phi)g(\sigma^{\prime}(t),\sigma^{\prime}(t)).$

Since $n\geq 3$, the last relation implies that $M$ is Einstein (see [3], p.
1543). Hence, $(\lambda-\phi)$ is a constant. Therefore, from equation (15),
we get

$\frac{l}{2}(\lambda-\phi)\leq\frac{(n-1)\pi^{2}}{2l}.$

This implies

$l\leq\pi\sqrt{\frac{n-1}{\lambda-\phi}}.$

∎

###### Proof of Theorem 1.7.

We have (see [20], p. 239)

(16)
$\int_{0}^{l}{sin^{2}\left(\frac{\pi}{l}t\right)Ric(\sigma^{\prime}(t),\sigma^{\prime}(t))}dt\leq\frac{(n-1)\pi^{2}}{2l}.$

Using $Ric+\nabla^{2}f\geq\lambda g$ in (16), we get

$\int_{0}^{l}{sin^{2}\left(\frac{\pi}{l}t\right)\\{\lambda-\nabla^{2}f(\sigma^{\prime}(t),\sigma^{\prime}(t))\\}}dt\leq\frac{(n-1)\pi^{2}}{2l}.$

As $f^{\prime\prime}(\sigma(t))\leq k$ , we have

(17)
$\int_{0}^{l}{sin^{2}\left(\frac{\pi}{l}t\right)(\lambda-k)}dt\leq\frac{(n-1)\pi^{2}}{2l}.$

Hence, equation (17) entails

$\frac{l}{2}(\lambda-k)\leq\frac{(n-1)\pi^{2}}{2l}.$

This implies

$l\leq\pi\sqrt{\frac{n-1}{\lambda-k}}.$

∎

###### Proof of Theorem 1.8.

For any $r>0$, the Lemma 2.5 together with gradient Ricci soliton equation
(2), yields

$\frac{1}{r}\int_{0}^{r}t^{2}\left\\{\lambda-\nabla^{2}f(\sigma^{\prime},\sigma^{\prime})\right\\}dt\leq{n-1}.$

Thus

(18) $\frac{1}{r}\left\\{\frac{\lambda
r^{3}}{3}-\int_{0}^{r}t^{2}f^{\prime\prime}(\sigma(t))dt\right\\}\leq{n-1}.$

Now, a simple calculation gives

(19)
$\int_{0}^{r}t^{2}f^{\prime\prime}(\sigma(t))dt=r^{2}f^{\prime}(\sigma(r))-2rf(\sigma(r))+2\int_{0}^{r}f(\sigma(t))dt.$

In view of (18) and (19), we obtain

$\frac{1}{r}\left\\{\frac{\lambda
r^{3}}{3}-r^{2}f^{\prime}(\sigma(r))+2rf(\sigma(r))-2\int_{0}^{r}f(\sigma(t))\right\\}dt\leq{n-1}.$

This implies

$\frac{\lambda}{3}-\frac{1}{r}f^{\prime}(\sigma(r))+\frac{2}{r^{2}}f(\sigma(r))-\frac{2}{r^{3}}\int_{0}^{r}f(\sigma(t))dt\leq\frac{n-1}{r^{2}}.$

As $f$ is at most linear along $\sigma$, taking limit as $r\rightarrow\infty$,
we obtain

$\lambda\leq 0.$

It follows that the Ricci soliton is non-shrinking. ∎

## Acknowledgment

The second author gratefully acknowledges to the CSIR(File
No.:09/025(0282)/2019-EMR-I), Govt. of India for the award of Senior Research
Fellow.

##  3\. Data Availability Statement

Our manuscript has no associated data.

##  4\. Declarations: Conflict of Interest

The authors declare that they have no conflict of interest.

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