# On Maximum Complexity in Holography

Shahrokh Parvizi<EMAIL_ADDRESS>Department of Physics, School of
Sciences, Tarbiat Modares University, P.O.Box 14155-4838, Tehran, Iran
Mojtaba Shahbazi<EMAIL_ADDRESS>Department of Physics, School
of Sciences, Tarbiat Modares University, P.O.Box 14155-4838, Tehran, Iran

###### Abstract

In a quantum circuit, it is believed that complexity itself reaches a maximum
of order exponential in the number of q-bits or equivalently exponential in
entropy of the black hole. However, the current holographic proposals do not
meet this criterion. The holographic proposals find the complexity of the very
late times to be linear in the entropy, while in the quantum circuit, it is
expected that complexity meets within a finite time its maximum value in an
exponential in the entropy. These points are required to be altered in
holographic proposals of complexity. This paper introduces a new holographic
proposal that meets this criterion and consolidates the Lloyd bound.

##  1 Introduction

There are some paradoxes in the combination of quantum mechanics and the
general theory of relativity, mainly when one focuses on the physics of black
holes by quantum mechanics’ eye [1]. In this manner, the study of black hole
physics may provide some meteoric way to a successful quantum gravity theory.
Recently Susskind’s proposal based on the AdS/CFT correspondence made a big
jump considering the interior of black holes [3]. By this conjecture, the
volume of Einstein-Rosen bridges (ERB) corresponds to the complexity of the
conformal field theory on the boundary of an AdS space, which is coined as
Complexity=Volume (CV) proposal. Complexity is the minimum number of quantum
gates required to turn a reference state into a target state.
Later, Brown et al. [4, 5] suggested another holographic proposal dubbed as
Complexity=Action (CA) proposal. It states that the action of bulk theory in
the Wheeler-DeWitt patch, which is the domain of all space-like trajectories
anchored to either side of the Penrose diagram is dual to the complexity of
the boundary theory.

It can be shown that in any quantum circuit, there is an upper bound of
$e^{K}$ for the complexity, where $K$ is the number of q-bits [3]. Because $K$
number of q-bits is proportional to the entropy $s$, we put $K=s$. In this
way, the maximum complexity is $C_{max}=e^{s}$ [3, 2]. Furthermore, the
complexity growth rate is bounded from above by the mass of the system known
as the Lloyd bound [7].

It has been revealed that some theories do not respect the Lloyd bound by CV
and some by CA proposal in the full-time behavior [8], in Einstein-Maxwell-
Dilaton theories [9, 10], in Lifshitz hyper scaling violating exponent space
[11], in local quantum quench [12] and an anisotropic solution of Einstein-
Dilaton-Axion theory [13]. This point leads to new proposals [9, 14] and
modifications [15, 16, 17].

Nonetheless, none of these cases could maintain the criterion of maximum
complexity because they find the complexity for the late times linear in
entropy. Indeed, maximum complexity arises at times of order exponential in
entropy. Besides, the bulk side in AdS/CFT can not afford the quantum effects
at late times regime [19]. The point is that by Poincaré recurrence theorem,
every unitary evolution by finite entropy will return to its initial state at
late but finite times111Non-unitary evolutions with finite entropy are not
required to respect Poincaré recurrence but it could be managed to have the
recurrences by nonperturbative dynamical effects [28].. On the bulk side,
correlation functions at late times are to be returned to a lower bound to
respect the Poincaré recurrence; however, one can not recover Poincaré
recurrence in the presence of black holes [19]. On its face, the maximum
complexity that is reached at late times (the classical Poincaré recurrence
time) is a quantum effect and can not be recovered on the bulk side.
Nonetheless, Maldacena’s proposal in [19] by some averaging pursues recovering
the recurrences at classical Poincare recurrence time; in addition, in two-
dimensional gravitational theory, it could be shown that there would be the
maximum complexity by considering the full path integral. Then it makes sense
to consider maximum complexity in holography.

On the other hand, due to the ambiguity in the gate choice in the holographic
complexity, there would be a class of holographic proposals that all members
share the same expected requirements, then there is not a unique holographic
proposal of the complexity. In this paper, we provide a new class of
holographic proposal given in (13), that can fulfill the maximum complexity
condition, and we will see that it behaves the Lloyd condition well,
particularly for the large masses.

Our proposal, which we call “Hyperbolic Proposal” could be seen either as an
averaged complexity. In Sachdev-Ye-Kitaev (SYK) models that are examples of
chaotic systems, the hyperbolic proposal bears a resemblance to averaged
complexity.

The organization of the paper is as follows: In the next section, the expected
features of holographic complexity candidates are reviewed. Our new proposal
is introduced in section 3, and we discuss its behavior. In section 4, the
relation between the hyperbolic proposal and averaged complexity is drawn, and
we conclude in the final section.

##  2 Review on Expected Features of Holographic Complexity

It is known that the computational complexity represents four expected
features: linear growth, a particular behavior under perturbation, a maximum
value of complexity, and the Lloyd bound [2, 30]. As a consequence, it is
expected that the holographic picture of complexity embraces these features.

One may consider the holographic complexity as a function like $C(\lambda)$,
where $\lambda$ is a dynamical variable in the black hole and $C$ is a
suitable function. In the case of well-known CV and CA proposals, $\lambda$ is
respectively the volume of the Einstein-Rosen bridge and the action in the
Wheeler-DeWitt patch, and $C$ is simply a linear function with constant
coefficients.

First of all, we expect the holographic complexity $C(\lambda)$ to show a
linear growth in time and, secondly, behaves under perturbation as its dual in
quantum circuits. Although it is expected that complexity reaches its maximum
in a finite time, the current holographic proposals lack this condition.

Let us discuss these behaviors in the following subsections and review the
expected features in CV and CA proposals.

###  2.1 Linear growth

In quantum circuits, it can be seen that restricted to a shorter time-scale
when complexity is much smaller than its maximum, its growth is linear [3]:

$C\propto Kt$

where $K$ is the number of q-bits. Although except at the beginning, it could
be non-linear, it is believed that complexity grows all the way to its maximum
linearly [30].

Let us start with the CV proposal and consider a black hole geometry of the
following form:

$\displaystyle ds^{2}$
$\displaystyle=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{D-2}^{2}$ (1)
$\displaystyle f(r)$ $\displaystyle=1+r^{2}-\frac{\mu}{r^{D-3}}$ (2)

where for $D>3$,

$\displaystyle\mu=\frac{16\pi G_{N}M}{(D-2)\omega_{D-2}}$ (3)

in which $\omega_{D-2}$ is the volume of $D-2$ dimensional sphere. The volume
growth rate of ERB which is defined for a two-sided black hole is given by
[3]:

$\frac{dV}{dt}=\omega_{D-2}r^{D-2}\sqrt{f(r)}$

By induced metric for a volume in AdS-Schwarzschild, the volume is written as
[3]:

$V=\int r^{2}\sqrt{-f+\frac{r^{\prime 2}}{f}}dtd\Omega$ (4)

where $r^{\prime}=\frac{dr}{dt}$. It could be shown that for finite times Eq.
(4) turns into [3]:

$V\propto sT|t_{L}+t_{R}|$ (5)

where $s$ and $T$ are entropy and temperature of the black hole, respectively,
and $t_{L}$ and $t_{R}$ are times for the left and right side of the black
hole. With $C\propto V$, Eq. (5) indicates linear growth of the holographic
complexity.

In the CA proposal, the computational complexity is proportional to the action
of the theory in the WDW patch, $C\propto\mathcal{S}$. Since the action is
computed in a bounded manifold, the Gibbons-Hawking-York term should be added
to the action of the theory:

$\mathcal{S}=\mathcal{S}_{theory}+\frac{1}{8\pi G}\int_{\partial M}\sqrt{h}K$

where $h$ is the induced metric on the boundary of the patch and $K$ is
extrinsic curvature constructed by the induced metric. At late but finite
times, the complexity growth rate for a $D$ dimensional AdS-Schwarzschild is
given by [5]:

$\frac{d\mathcal{S}}{dt}=-\frac{\Omega_{D-2}r_{h}^{D-1}}{8\pi
GL_{AdS}^{2}}=2sT$ (6)

This shows the linear behavior as expected.

The expressions (5) and (6) show that the current proposals of holographic
complexity describe the linear growth of complexity.

###  2.2 Perturbation

In [6], Susskind and Maldacena showed that by holography a pair of entangled
particles is dual to an ERB or in short $ER=EPR$. Then a perturbation in
gravity leads to a perturbation in the pair particles. Another test to check
is on the perturbed thermo-field double (TFD) states which are dual of the
geometries with shock waves [3, 29]. In other words, a perturbation in the
evolution of the black hole should hold in the correspondence between gravity
and the quantum circuit. Small perturbations in TFD state could be conducted
by precursors as:

$|\psi(t)\rangle=W(t)|\psi\rangle$

where $W(t)=U^{\dagger}WU$ with $W$ close to one and $U=e^{-iH}$. The schema
of a quantum circuit of the precursors could be seen in Fig. 1.

Figure 1: A circuit of precursor. The horizontal lines are q-bits and the
circles are quantum gates. Figure 2: A Penrose diagram for a two-sided black
hole with a shock wave at time $t_{w}$. This is supposed to be dual to a
precursor.

The complexity of the perturbed TFD state is read as [3]:

$\displaystyle C\propto K(t_{f}-2n_{p}t_{*})$ (7)

where $t_{f}$ is the total time of the perturbation, $n_{p}$ is the number of
stages of the perturbation and $t_{*}$ is the scrambling time. The dual
picture is the geometries with $n_{p}$ number of shock waves as in Fig. 2 [3].

By CV proposal the volume of ERB in the presence of $n_{p}$ number of shock
waves is given by [3]:

$\displaystyle V\propto(t_{f}-2n_{p}t_{*})+O(1)$ (8)

where $O(1)$ is the various time variables terms of the first order.

In CA proposal, where the action of the theory in the presence of $n_{p}$
number of shock waves should be considered, the action at late times is read
as [5]:

$\displaystyle\mathcal{S}\propto(t_{f}-2n_{p}t_{*})$ (9)

At finite times there would be some corrections but the leading term is
linear. Eqs. (8) and (9) indicate that both CV and CA proposals are in
agreement with the perturbed TFD in (7).

###  2.3 Lloyd bound

The complexity growth rate is bounded from above by a conjecture called the
Lloyd bound [7]:

$\frac{dC}{dt}\leq\frac{2M}{\pi}$ (10)

where $M$ is the mass of the black hole. This inequality originates in the
uncertainty principle on the quantum side. In other words, the time a quantum
gate takes to operate on a state is restricted by the uncertainty in energy
and time. The holographic models should respect the inequality. However, there
are some theories that either CV or CA violates the bound [8]-[13]. Although
there is a new proposal called CV-2.0 [14], which can improve the Lloyd bound
condition, particularly for Einstein-Maxwell Dilaton theories [9], it also
suffers from some violating cases [18]. In section 3, we show how it can be
improved.

###  2.4 Maximum complexity

In a quantum circuit, computational complexity reaches its maximum of
exponential in entropy at a finite time and fluctuates around its maximum.
Furthermore, after quantum Poincaré recurrence time, it jumps off to its
initial value as Fig. 4 depicts [2]. It is expected that a holographic picture
of complexity would exhibit the maximum value. However, it is believed that
the general theory of relativity breaks down at the classical Poincaré
recurrence time222The classical Poincaré time is of order exponential in
entropy while quantum Poincaré recurrence time is of order double exponential
in entropy [2]. where complexity reaches its maximum. Since the maximum value
of complexity occurs at the classical Poincaré recurrence time, none of the
holographic proposals of complexity shows the maximum value i.e. exponential
in entropy.

There are two related reasons that general relativity breaks down at the
classical Poincaré recurrence time. The first reason is due to the correlation
functions of opposite boundaries in extended AdS black holes at the classical
Poincaré recurrence time. It seems that the correlation functions at those
times should reach a minimum of order $e^{-\alpha s}$ where $\alpha$ is a
constant and $s$ the entropy of the black hole; however, on the bulk side, it
lacks the minimum and vanishes. Nonetheless, one can recover the minimum at
the classical Poincaré recurrence time by non-perturbative dynamical
effects333In fact, the path integral on the bulk side is over some geometries
with the same boundary condition. It could be seen that there are geometries
that recover the minimum and have small free energy and large contribution
compared to the geometries that do not recover the minimum. and some averaging
[19, 28]444For a breakdown of the general theory of relativity at the quantum
recurrence time see [34]. The second reason stems from the finite value of
entropy of a black hole. The correlation functions of opposite boundaries are
given by $\langle\mathcal{O}_{L}\mathcal{O}_{R}\rangle\sim e^{-\gamma t}$
where $\gamma$ is a constant [2, 19]. It seems that the correlations are
related to ERBs and as time elapses, the length of ERB grows, which makes the
correlations decay. Since the entropy is finite, as time marches on, the
evolution of quantum states passes through a finite series of orthogonal
states. Consequently, the time it takes to run out of orthogonal states is of
order exponential in entropy and after that, because the states become
superposition of previous states, then the correlations stop decreasing but on
the bulk side $e^{-\gamma t}$ continues to zero [2].

While the general theory of relativity breaks down where complexity reaches
its maximum, any attempt to recover maximum complexity in holography could be
seen as an effective description of the quantum phenomenon in a semi-classical
context. In principle, it can be understood as considering the quantum effects
in a path integral approach which yields to an effective action of classical
fields. The details of quantum effects and their calculation are not known,
but there are criteria and evidences which may guide us to the correct form of
effective classical terms. The same procedure takes place where entanglement
entropy of the Hawking radiation is considered. There is a lot in common
between entanglement entropy and complexity when their developments are
viewed. On one hand, the celebrated Ryu-Takayanagi (RT) surface links
entanglement entropy (EE) of the boundary quantum field theory with the bulk
surface, on the other hand, holographic complexity links the complexity of
operators in the quantum field theory with the volume of the ERB (in CV
proposal). It has been seen that RT surface does not account for the loop
correction [20], in other words, there is a quantum correction that the
holographic EE cannot recover. As a counterpart, the holographic complexity
cannot recover the maximum complexity which is a quantum effect. In addition,
there is a prescription in the low energy limit of quantum gravity that
recovers the loop correction of EE known as quantum extremal surface (QES)
[21, 22]. As a counterpart, in the next section, we provide a prescription
that recovers the maximum complexity in the low energy limit of quantum
gravity, we call it Hyperbolic proposal. And finally, the full path integral
computation in the low dimensional gravity shows that the QES naturally arises
[24, 25]. As a counterpart, the full path integral computation in the low
dimensional gravity shows that the maximum complexity arises [23].

Entanglement Entropy | Complexity
---|---
RT Surface | CV/CA/CV-2.0
Loop Correction | Maximum Complexity
QES | Hyperbolic Proposal
Full Path Integral | Full Path Integral

##  3 Hyperbolic Proposal

As we discussed in the last section, the CV and CA proposals and their
extensions do not properly fit all criteria for a holographic complexity. Here
we make a modification which can improve the situation. Holographic proposals
for complexity are a function of a dynamical variable $\lambda$,
$\mathcal{C}(\lambda)$ where the dynamical variable can be either volume $V$
or action $A$ as in the CV and CA proposals or any modification of them. In
this new proposal, we consider $\lambda=C$ where $C$ is either previous
proposal. The main point is adopting $\mathcal{C}$ function to be a hyperbolic
function instead of a linear one.

It should be noted that there is an ambiguity in the notion of holographic
complexity which corresponds to the ambiguity in the choice of the gate set in
the complexity theory and there would be a class of holographic proposals that
share the linearity and the particular behavior under the perturbation
discussed in the previous section, put it differently, the time dependence in
holographic complexity is important [26]. Therefore, there is no uniqueness in
the holographic complexity. In [26], a class of CV proposal is introduced as:

$C\propto\int_{\Sigma_{F_{2}}}d^{d}x\sqrt{g_{ind.}}F_{1}(g_{\mu\nu},\sigma^{\mu})$
(11)

where $F_{1}$ is a scalar function of the induced metric $g_{ind.}$ and an
embedding $\sigma^{\mu}$ of the codimension-one surface $\Sigma_{F_{2}}$,
which in turns comes from extremizing a similar functional replacing $F_{1}$
with another scalar function $F_{2}$. As long as the criteria of the
holographic complexity are satisfied, functions $F_{1}$ and $F_{2}$ are
arbitrary. When $F_{1}=F_{2}=1$, (11) reduces to the known CV proposal.

Now we are claiming such an arbitrariness not inside the functional (11) but
outside of it, i. e.,
$\mathcal{C}(C)=\mathcal{C}(\int_{\Sigma_{F_{2}}}d^{d}x\sqrt{g_{ind.}}F_{1}(g_{\mu\nu},\sigma^{\mu}))$.
Here, we may take $F_{1}=F_{2}=1$ and look for a proper function $\mathcal{C}$
to satisfy all criteria mentioned in the previous section.

This is inspired by studying the complexity in a BTZ black hole which can be
computed analytically. This makes it a fertile ground to see the explicit
behavior of the complexity, which is given by [8]:

$C=s\bigg{(}1+log\Big{(}\frac{\omega}{T}\cosh(\omega
Tt)\Big{)}\bigg{)}\tanh(\omega Tt)$ (12)

where $\omega$ is a constant. At early times, the time dependence of (12) is
$C\propto\tanh(\omega Tt)$ which is linear around $t\simeq 0$ at leading
order. On the other hand, the complexity in the linear region behaves as
$C\sim sTt$. So inspired by the BTZ black holes, we can consider a new
proposal with introducing function $\mathcal{C}$ to be a hyperbolic function
of $\omega C$. It would be a new class of holographic complexity which we call
the class of ’Hyperbolic Proposals’, and introduce it as:

$\mathcal{C}=A\frac{\sinh(\omega_{1}C-\alpha)}{\cosh(\omega_{2}C-\beta)}$ (13)

where $C$ is either CV or CA proposal. If $\omega_{1}=\omega_{2}=\omega$ and
$\alpha=\beta=0$ it yields to the complexity in the BTZ black hole (12). This
function would be quite inspiring when we consider Sachdev-Ye-Kitaev (SYK)
model in section 4.

As an example we are going to consider a member of this class with parameters
$A=1$, $\omega_{1}=\omega_{2}=\text{sech}(s)=:\omega$, $\alpha=0$ and
$\beta=s$, the entropy. Then

$\mathcal{C}=\frac{\sinh(\omega C)}{\cosh(\omega C-s)}$ (14)

With these parameters, at late times when CV and CA find complexity infinite,
$\mathcal{C}$ tends to $e^{s}$, so it respects the maximum complexity
criterion. In addition, where $C\sim s/\omega$ the hyperbolic proposal is
reduced to $\mathcal{C}\propto C$, which is linear as Fig. 3 depicts.

Figure 3: The plot is drawn for the value of $s=4$. The horizontal axis is $C$
the former proposals of holographic complexity. The dashed line is the maximum
entropy $e^{s}$. The red line shows the slope of the curve and indicates its
linearity in the ramp region.

Having shown the linearity, then to fulfill the criteria of holographic
complexity we should consider perturbed TFD state. In view of the fact that
hyperbolic proposal is linear where $C\sim s/\omega$ , as a consequence it
satisfies the perturbed case. However, to show that this proposal can make it
in other regions, we expand Eq. (14) around a generic point $p$ as follows:

$\frac{\sinh(\omega C)}{\cosh(\omega C-s)}=\frac{\sinh(\omega p)}{\cosh(\omega
p)}+\omega\frac{\cosh(s)}{\cosh^{2}(\omega
p-s)}(C-p)-\omega^{2}\frac{\cosh(s)~{}\tanh(\omega p-s)}{\cosh^{2}(\omega
p-s)}(C-p)^{2}+...$ (15)

Since $\omega=\text{sech}(s)$ is less than one, for $p$ in the ramp region, we
have
$\mathcal{C}^{\prime}(p)\gg\mathcal{C}^{\prime\prime}(p)\gg\mathcal{C}^{\prime\prime\prime}(p)\gg...$
so nonlinear terms are exponentially suppressed and can be discarded. As an
example, consider $s=4$, $p$ in the ramp region and $(C-p)\sim\mathcal{O}(1)$,
we find the order of terms in right-hand side of (15), respectively to be
$\mathcal{C}\sim\mathcal{O}(10)+\mathcal{O}(1)+\mathcal{O}(10^{-3})+\mathcal{O}(10^{-4})+\cdots$.
This explains the linearity in the ramp region.

There remains one point on the linearity. It is assumed that the linear growth
of complexity except at the beginning is continued to its maximum value, which
is called the region of complexity ramp, and after reaching the maximum, it
fluctuates around the maximum, which is called the plateau and after the time
of order double exponential in entropy, it jumps down as Fig. 4 shows [30].
While other proposals only represent some parts of the complexity ramp, the
hyperbolic proposal can demonstrate the whole ramp region up to the maximum.
In the final stage, the complexity jump can not be explained by either
proposal.

Figure 4: The plot of complexity in a quantum circuit goes with complexity
ramp and the plateau.

###  3.1 Lloyd bound

To see how (14) fares better than previous proposals, we find the derivative
of Eq. (LABEL:np):

$\dot{\mathcal{C}}=\frac{d\mathcal{C}}{dt}=\frac{\dot{C}}{\cosh^{2}(\omega
C-s)}$ (16)

Owing to the fact that the maximum value of the term $1/\cosh^{2}(\omega C-s)$
is one at $C=s/\omega$ and for $C\gg s/\omega$, $\dot{\mathcal{C}}$ rapidly
goes away, it follows then:

$\dot{\mathcal{C}}=\frac{\dot{C}}{\cosh^{2}(\omega C-s)}\leq\dot{C}$ (17)

For heavy black holes, $\omega=\text{sech}(s)$ tends to zero, and as a
consequence, $\cosh^{2}(\omega C-s)$ tends to infinity, which means that Lloyd
bound by the hyperbolic proposal at the early stages of complexity growth
tends to zero. In other words, if the bound violation in the former proposals
occurs at the early stages of complexity growth, then hyperbolic proposal
respects the bound. Even if the violation in the former proposals occurs at
the middle or late stages of complexity growth, since
$\frac{\dot{C}}{\cosh^{2}(\omega C-s)}<\dot{C}$, hyperbolic proposal behaves
better than the former. For lighter black holes $\omega=\text{sech}(s)$ tends
to one then at the first stages of complexity growth where $C$ is small, the
Lloyd bound in the hyperbolic proposal is similar to the former proposals.
However, as complexity grows, $\frac{1}{\cosh^{2}(\omega C-s)}$ becomes less
than one, so we can declare that always $\dot{\mathcal{C}}\leq\dot{C}$, which
means that hyperbolic proposal in regards to the Lloyd bound, behaves better
than former proposals.

It is worth comparing the Lloyd bound in AdS-BTZ black hole in CA and the
hyperbolic proposal. In [8], Carmi et al. showed that complexity growth rate
violates Lloyd bound for AdS-BTZ black holes in the first stages of complexity
growth, then we expect that hyperbolic proposal shows the violation in light
black holes and respects it in heavy black holes as in Fig. 5.

(a)  (b)

Figure 5: Lloyd bound in BTZ black hole by CA and hyperbolic proposals. In
panel (a) we set s=1.5 representing a large black hole. Panel (b) with s=0.2
represents a small black hole.

##  4 Sachdev-Ye-Kitaev Model

The Sachdev-Ye-Kitaev (SYK) Model is an example of a chaotic quantum system
that contains $N$ Majorana fermions $\psi_{i}$ with a random Hamiltonian [31]:

$\displaystyle
H=\sum_{i_{1}<...<i_{a}}I_{i_{1}...i_{a}}\psi_{i_{1}}...\psi_{i_{a}}$ (18)

where the couplings $I_{i_{1}...i_{a}}$ are chosen at random from a Gaussian
distribution with mean zero and variance $\sigma^{2}$:

$\displaystyle\sigma^{2}=\frac{(a-1)!\mathcal{I}^{2}}{N^{a-1}}$ (19)

in which $\mathcal{I}$ is the variance setting parameter. In [33] for $N=2$
fermions, the disorder-averaged complexity when $I$ is chosen to be Gaussian-
distributed is defined as:

$\overline{C}(t)=\int_{0}^{\infty}dIC_{QM}(t)\frac{e^{-\frac{I^{2}}{2\sigma^{2}}}}{\sqrt{2\pi}\sigma}$
(20)

where $I=\sqrt{\Sigma_{i}I_{i}^{2}}$ for $I_{i}$ as couplings of Hamiltonian
and $C_{QM}$ is the complexity in the quantum side. For $N=2$ it is given by
[33]:

$C_{QM}(t)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos\big{(}(2n-1)It\big{)}}{(2n-1)^{2}}$
(21)

The holographic dual of the SYK model is JT gravity, which is a two-
dimensional quantum gravity with a dilaton [32]. This gravitational theory is
not dual to a quantum theory with a specific Hamiltonian but to an ensemble of
theories with various $I$. So it is expected that the holographic complexity
may represent the average behavior of the complexity on the quantum side. It
is believed that holographic picture breaks down after time scale of
exponential in entropy [2, 30], while in [33] the averaged complexity is
computed from the beginning until times longer than exponential in entropy and
is given by:

$\overline{C}(t)=\frac{2}{\sqrt{\pi}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}\frac{\tanh\big{(}\frac{(2n-1)^{2}t^{2}\sigma^{2}}{4}\big{)}}{1+\tanh\big{(}\frac{(2n-1)^{2}t^{2}\sigma^{2}}{4}\big{)}}$
(22)

which is plotted in Fig. 6.

Figure 6: The plot of averaged complexity with $I$ chosen to be Gaussian-
distributed.

The plot in Fig. 6, shows that the averaged complexity with $I$ chosen to be
Gaussian-distributed has a linear growth and reaches a maximum plateau. This
suggests that the averaged complexity may be the very quantity that can
represent the right behavior of the complexity in this model [33]. So one
expects that on the bulk side, the holographic complexity may correspond to
this averaged quantity. To investigate this, we can find CV holographic
complexity for JT gravity (see [37]), draw $\bar{C}$ in terms of it and make a
comparison between hyperbolic proposal (13) and $\bar{C}\big{(}C(t)\big{)}$,
averaged complexity in terms of holographic complexity. In Fig. 7, the solid
curve is the averaged complexity and the dotted curve is:

$\mathcal{C}=A\frac{\sinh\big{(}\gamma\omega
C-\alpha\big{)}}{\cosh\big{(}\gamma\omega C-\beta\big{)}}$ (23)

where we considered $\omega_{1}=\omega_{2}=\gamma\omega$ in (13) with
$\omega=\text{sech}(s)$ and the entropy is $s=\beta-\alpha$. Numerical results
show that (23) still satisfies the Lloyd bound.

Figure 7: Comparison between hyperbolic proposal and averaged complexity in
terms of holographic complexity $\bar{C}\big{(}C(t)\big{)}$. The plot is drawn
for parameters $\sigma=\gamma=0.009$, $A=0.307$, $\alpha=-0.118$ and
$\beta=0.696$ in (23) with $\omega=\text{sech}(\beta-\alpha)$. In this vein,
the entropy is given by $s=\beta-\alpha$.

Notice that we argued that the maximum complexity should be in the order of
$\mathcal{O}(e^{s})$, we, therefore, expect the numerical factor $A$ to be in
the order of $\mathcal{O}(1)$ or polynomial in $s$. The parameter $\alpha$
reveals that the holographic complexity has a non-vanishing value at the
beginning while the averaged complexity vanishes. This behavior could be
addressed to the CV and CA proposal at the beginning. It could be seen that
either proposal does not represent a linear growth at the beginning while in
quantum theory we expect complexity or averaged complexity would be linear,
then hyperbolic proposal which uses the former proposals suffers the beginning
stages of complexity evolution. Then the remaining nontrivial parameter would
be $\omega$ which numerical analysis shows to be corresponding to $\sigma$ in
the SYK model. The parameter $\sigma$ is proportional to the slop of the
averaged complexity in the complexity ramp [33] and in holography, the slop of
complexity ramp is proportional to $s.T$ where $s$ is entropy and $T$ the
temperature, then $\sigma$ is proportional to the temperature.

So it is very suggestive to consider the hyperbolic proposal (23) as a
holographic dual to the disorder-averaged complexity with $I$ chosen to be
Gaussian-distributed, i.e., $\overline{C}=\mathcal{C}$. In this manner, the
hyperbolic complexity (23) can go beyond the time scale in CA or CV proposals,
and indeed the hyperbolic proposal as a conjecture for averaged complexity can
broaden holography beyond that time scale to the complexity plateau region.

##  5 Conclusion

The holographic complexity makes a new avenue to explore black hole physics
further. It is expected that any holographic complexity candidate embraces
some features: the linear growth, particular behavior under perturbation, the
maximum of order exponential in entropy, and the Lloyd bound. The current
proposals of holographic complexity do not meet some of these conditions,
particularly the maximum value. In other words, they find the maximum value
linear in entropy. The reason behind that could be referred to the Poincaré
recurrence theorem and the breakdown of the bulk side at late times regime in
AdS/CFT. Nevertheless, having the Poincaré recurrences on the bulk side is
reachable which makes the expectation that a holographic proposal for
complexity would show the maximum value. Moreover, the Lloyd bound in some
models is violated. These situations require that the known proposals of
holographic complexity should be modified.

We have introduced a new class of holographic complexity conjecture and called
it the hyperbolic proposal which is a hyperbolic function of previous
proposals in which the maximum complexity of order exponential in entropy is
reached. The new proposal is inspired by the behavior of the complexity in BTZ
black holes where the complexity could be computed analytically. As a proposal
for complexity, the linear growth in time and the behavior under the
perturbation have been shown. Although the Lloyd bound in this new proposal is
not guaranteed to be satisfied, the complexity growth rate in the hyperbolic
proposal is always less than or equal to its former counterparts in CA and CV.
It could be seen that in large masses, the complexity growth rate in the
hyperbolic proposal is much less than its counterparts, and in light masses,
it is comparable to them. Then, if the violation occurs in the early or middle
stages of the complexity growth of a large mass regime, the hyperbolic
proposal behaves the Lloyd bound much better than the former proposals.

In SYK models, which are chaotic quantum mechanical systems, the disorder-
averaged complexity resembles the hyperbolic proposal. Subsequently, it could
be considered that the hyperbolic proposal in such systems is a conjecture for
disorder-averaged complexity with $I$ chosen to be Gaussian-distributed, and
not the complexity itself. Then it can exhibit beyond the time scale of order
exponential in entropy, which has been believed that holographic picture could
not go beyond that time scale. Consequently, the hyperbolic proposal as a
conjecture for disorder averaged complexity can broaden our holographic
perspective to the complexity plateau region.

## Acknowledgement

We are grateful to Mohsen Alishahiha for useful discussions and reading the
manuscript and also to Ahmad Moradpoori for helpful comments.

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#  Interpolation of surfaces with asymptotic curves in Euclidean 3-space

Mustafa Altın
Technical Sciences Vocational School,Bingöl University, Turkey
<EMAIL_ADDRESS>

İnan Ünal
Department of Computer Engineering
Munzur University, Tunceli, Turkey
<EMAIL_ADDRESS>

Fatemah Mofarreh
Mathematical Science Department
Faculty of Science Princess Nourah bint Abdulrahman University
Riyadh 11546 Saudi Arabia.
<EMAIL_ADDRESS>

###### Abstract

In this paper, we investigate the interpolation of surfaces which are obtained
from an isoasymptotic curve in 3D-Euclidean space. We prove that there exist a
unique $C^{0}$-Hermite surface interpolation related to an isoasymptotic curve
under some special conditions on the marching scale functions. Finally, we
present some examples and plot their graphs.

##  1 Introduction

Differential geometry is a branch of mathematics which uses advanced calculus
tools in geometry. In recent years, it has become an applicable area of
mathematics in science and technology. Since the manifold theory has used in
general relativity in the 1900s by Einstein, differential geometry of curves,
surfaces and general manifolds have been improving more. Notably, from
medicine to social science and from artificial intelligence to economy, it is
very clear how the differential geometry is applied. In this manner, one can
consider that the applied mathematics has been changed from numerical and
computational methods to differential geometrical tools. For example, to
understand the meaning of multiple features data we use calculus on manifolds
in machine learning [5]. Also, differential geometry permit us to work on non-
euclidean spaces as most real life problems are define in such spaces. So, it
is a fundamental tool for understanding events in the universe.

A relevant kind of curves is geodesics which play the role of straight lines
in Euclidean space on a manifold. Gauss proved that the differential geometry
of a surface is different from the geometry of ambient space. The well known
example of supporting these ideas is that a geodesic on a unit sphere embedded
in a Euclidean space is not a geodesic in a Euclidean space. In this way, the
differential geometry of a surface has many significant properties that we can
use in applied sciences. The minimal distance between two points on a surface
is called a geodesic and this is considered as an important idea in many
applications [4, 12].

A surface could be constructed using a geodesic. In [23] a general surface is
obtained from a polynomial geodesic. Also, considering a 3-dimensional
polynomial curve which is a pregeodesic, the authors constructed ruled cubic
patched in [21]. In addition, in [19] authors investigated a developable
surface that contains a given Bezier geodesic. Wang at al. defined a
parametric surface which is called surface family, using a geodesic curve
[26]. They used the Frenet frame of the curve and presented necessary
conditions in which the curve is an isogeodesic on a parametric surface
considering Frenet aparatus of the curve. The methods in this paper is a
reverse of some engineering problems. Later, Kasap et al. [14] generalized
their methods and presented examples. Li et al. studied the approximation
minimal surface with geodesics by using the Dirichlet function and they
minimized the area of a surface family by using Dirichlet approach. This
method can be used for obtaining the minimal cost of the material while
building surfaces. The family surfaces have been studied for example as in
[13, 14, 1, 2, 27, 28].

The other special curve which is as important as geodesics is the asymptotic
curves. An asymptotic curve is a curve always tangent to an asymptotic
direction of the surface and it has zero normal curvature. On the other hand,
one can be constructed a surface using an asymptotic curve. Saad et al. [22]
approximated the minimal parametric surface with an asymptotic curve by
minimizing the Dirichlet function. In [20] they examined rational developable
surface pencils through an arbitrary parametric curve as its common asymptotic
curve. Moreover, Güler at al. constructed a surface interpolating a given
curve as the asymptotic curve of it [11]. Similar to geodesics, the asymptotic
curves also have many applications in related sciences. In [24] the authors
presented a method to design strained grid structures along asymptotic curves
to benefit from a high degree of simplification in fabrication and
construction. Also, asymptotic curves has many applications in astronomy as in
[7, 10].

Lee et al. [15] introduced a new method to construct a parametric surface in
terms of curves. They defined a surface interpolation associated with a
spatial curve passing through some $m$-points in Euclidean $3$-space.
Motivating all studies mentioned from here, we consider a surface
interpolation using asymptotic curves in Euclidean space. We organized this
paper as follow: We give some fundamental facts which are used throughout the
paper, in Section 2. Section 3 is devoted to the surface interpolations with
isoparametric curve and examples with their graphs.

##  2 Preliminaries

In this section, we give a short review on curves and surfaces in an
3D-Euclidean space. For details we refer the reader to any classical
differential geometry books, for example [25].

Let $\gamma(\omega)$ be a curve which is arc-length $\omega$ in 3D Euclidean
space. Take the Frenet frame of $\gamma(\omega)$ by
$\left\\{\mathcal{V}_{1}(\omega),\mathcal{V}_{2}(\omega),\mathcal{V}_{3}(\omega)\right\\}$.
Then we have the following well known relations between $\kappa(\omega)$ and
$\tau(\omega)$ are the curvature and the torsion of the curve
$\gamma(\omega)$, respectively.

$\left[\begin{array}[c]{c}{\mathcal{V}_{1}}^{{}^{\prime}}(\omega)\\\
{\mathcal{V}_{2}}{{}^{\prime}}(\omega)\\\
{\mathcal{V}_{3}}^{{}^{\prime}}(\omega)\end{array}\right]=\left[\begin{array}[c]{ccc}0&\kappa(\omega)&0\\\
-\kappa(\omega)&0&\tau(\omega)\\\
0&-\tau(\omega)&0\end{array}\right].\left[\begin{array}[c]{c}\mathcal{V}_{1}(\omega)\\\
\mathcal{V}_{2}(\omega)\\\ \mathcal{V}_{3}(\omega)\end{array}\right]$

.

Previous equations are called the Frenet apparatus of a curve and they are
important to understand the geometry of the curve. Also, we can classify
curves via the Frenet–Serret frames. In [26] Wang et al. defined pencil
surface which could be obtained using the Frenet–Serret frames of the curve.
This surface is called by a surface family or a pencil surface and it is
defined as follows.

######  Definition 1.

Let $\gamma(\omega)$ be a curve which is arc-length $\omega$ in
$\mathbb{E}^{3}$ and
$\left\\{\mathcal{V}_{1}(\omega),\mathcal{V}_{2}(\omega),\mathcal{V}_{3}(\omega)\right\\}$
be the Frenet frame of $\gamma$. Then the map

$\Psi(\omega,\eta)=\gamma(\omega)+u(\omega,\eta)\cdot\mathcal{V}_{1}(\omega)+v(\omega,\eta)\cdot\mathcal{V}_{2}(\omega)+z(\omega,\eta)\cdot\mathcal{V}_{3}(\omega),$
(1)

is defined a surface in $\mathbb{E}^{3}$, where $\Omega\geq\omega\geq
0,\Lambda\geq\eta\geq 0$ for a real-valued constants $\Omega,\Lambda$, and
$u(\omega,\eta),v(\omega,\eta)$ and $z(\omega,\eta)$ are $C^{1}-$functions.
The surface $\Psi(\omega,\eta)$ is called as surface family or pencil surface
[26].

By the following definition we classify some special curves on a parametric
surface $\Psi(\omega,\eta)$.

######  Definition 2.

Lets take a curve $\gamma(s)$ on a parametric surface $X(s,\eta)$. Then we
have following characterizations [25]:

  1. $\bullet$

$\gamma(s)$ is said to be an isoparametric curve on $X(s,\eta)$ if there
exists a parameter $\eta_{0}\in[0,\Lambda]$ such that
$X(s,\eta_{0})=\gamma(s).$

  2. $\bullet$

$\gamma(s)$ is an asymptotic curve on a parametric surface $X(s,\eta)$ if
$\frac{\partial\mathcal{N}(s,\eta_{0})}{\partial s}.\mathcal{V}_{1}(s)=0,$
where $\mathcal{N}(s,\eta)$ is the normal vector of surface $X(s,\eta)$,
$\mathcal{V}_{1}(s)$ is the tangent vector of curve $\gamma(s)$.

  3. $\bullet$

$\gamma(s)$ is called isoasymptotic of the surface $X(s,\eta)$ if it is both a
asymptotic curve and an isoparametric curve on the surface $X(s,\eta)$.

With following theorem we have the necessary and sufficient conditions for
$\gamma$ to be isoasymptotic curve on the surface $\Psi(\omega,\eta)$ (1).

######  Theorem 3.

[3] Let $\Psi(\omega,\eta)$ be a parametric surface with the marching-scale
functions;

$\displaystyle u(\omega,\eta)$ $\displaystyle=k(\omega)U(\eta),$
$\displaystyle v(\omega,\eta)$ $\displaystyle=m(\omega)V(\eta),$
$\displaystyle z(\omega,\eta)$ $\displaystyle=n(\omega)Z(\eta),$

where $k(\omega),U(\eta),m(\omega),V(\eta),n(\omega)$ and $Z(\eta)$ are
$C^{1}-$functions. Then $\gamma(\omega)$ is an isoasymptotic curve on a
parametric surface $\Psi(\omega,\eta)$ (1) if and only if we have

$\left\\{\begin{array}[c]{c}U(\eta_{0})=V(\eta_{0})=Z(\eta_{0})=0,\\\
n(\omega)=0\text{ or }\frac{\partial
Z(\eta_{0})}{\partial\eta}=0.\end{array}\right.$ (2)

If we take $k(\omega)=m(\omega)=n(\omega)=1$ ,particularly and consider
$U(\eta)$, $V(\eta)$ and $Z(\eta)$ as polynomials of the forms in (2), then we
have

$\displaystyle U(\eta)$
$\displaystyle=\sum\limits_{t=1}^{n}a_{t}(\eta-\eta_{0})^{t},$ (3)
$\displaystyle V(\eta)$
$\displaystyle=\sum\limits_{t=1}^{n}b_{t}(\eta-\eta_{0})^{t},$ $\displaystyle
Z(\eta)$ $\displaystyle=\sum\limits_{t=1}^{n}c_{t}(\eta-\eta_{0})^{t},\text{ \
}c_{1}=0,$

respectively, where $a_{t},b_{t},c_{t}$ are constant. Then the polynomials
$U(\eta)$, $V(\eta)$ and $Z(\eta)$ in (3) satisfy the isoasymptotic condition
(2). Thus, we can determine marching-scale functions of a surface family
$\Psi(\omega,\eta)$ by the polynomial expressions.

##  3 Surface Interpolations with Isoasymptotic Curve

In this section, we construct a surface with isogeodesic curve passing through
finite control points lying on $\mathbb{E}^{3}.$

Now, we give a definition for surface interpolations with isoasymptotic curve
passing through some control points on $\mathbb{E}^{3}.$

######  Definition 4.

Let $A_{1},A_{2},...,A_{m}$ be different points on $\mathbb{E}^{3}$ and
$\Psi(\omega,\eta):B\subset\mathbb{R}^{2}\rightarrow\mathbb{E}^{3}$ be a
parametric pencil surface given by (1). For some different points
$(\omega_{t},\eta_{t})\in B$ $(t=1,...,m),$ we can construct the surface
$\Psi(\omega,\eta)$ such that $\Psi(\omega_{t},\eta_{t})=A_{t}$. It is called
a surface interpolation associated with the given isoasymptotic curve
$\gamma(\omega)$ passing through $m$-control points $A_{t}~{}(t=1,...,m)$,
simply, $C^{0}-$Hermite surface interpolation with an isoasymptotic curve. In
particular, $\left\\{A_{1},A_{2},...,A_{m}\right\\}$ is called $C^{0}$-Hermite
data.

Polynomials $U(\eta)$, $V(\eta)$ and $Z(\eta)$ with degree $n$ in (3) have
$n,n$ and $n-1$ degrees of freedom in terms of coeffcients $a_{t},b_{t}$ and
$c_{t}$ respectively. In this case, there are two extra degrees of freedom. To
determine a unique parametric surface with isoasymptotic curve, we may assume
$a_{n}=b_{n}=0$.

Now, we consider an isoasymptotic curve on surface parametrization

$\Psi(\omega,\eta)=\gamma(\omega)+U(\eta)\cdot\mathcal{V}_{1}(\omega)+V(\eta)\cdot\mathcal{V}_{2}(\omega)+Z(\eta)\cdot\mathcal{V}_{3}(\omega),$
(4) $\Omega\geq\omega\geq 0,\text{ \ \ \ \ \ \ }\Lambda\geq\eta\geq 0,\text{\
\ }$

with the marching-scale functions are given in (3) for $a_{n}=b_{n}=0.$

######  Theorem 5.

Let $A_{1},A_{2},...,A_{m}$ be different points on a parametric surface
$\Psi(\omega,\eta)$ given in (4). For
$\Psi(\omega_{t},\eta_{t})=A_{t},t=1,...,m$ there exists a unique
$C^{0}$-Hermite surface interpolation with an isoasymptotic curve such that
the marching-scale functions are given by

$\displaystyle U(\eta)=\sum\limits_{t=1}^{n-1}a_{t}(\eta-\eta_{0})^{t},\ \
V(\eta)=\sum\limits_{t=1}^{n-1}b_{t}(\eta-\eta_{0})^{t}\ \ \text{and
}Z(\eta)=\sum\limits_{t=2}^{n}c_{t}(\eta-\eta_{0})^{t},$

and

$\displaystyle
det\left(\begin{array}[c]{ccccc}d_{12}&\eta_{1}-\eta_{0}&(\eta_{1}-\eta_{0})^{2}&\cdots&(\eta_{1}-\eta_{0})^{n-1}\\\
d_{22}&\eta_{2}-\eta_{0}&(\eta_{2}-\eta_{0})^{2}&\cdots&(\eta_{2}-\eta_{0})^{n-1}\\\
\vdots\vdots&\vdots&\ddots&\vdots\\\
d_{n2}&\eta_{n-1}-\eta_{0}&(\eta_{n-1}-\eta_{0})^{2}&\cdots&(\eta_{n-1}-\eta_{0})^{n-1}\end{array}\right)$

where $a_{t},b_{t}$ and $c_{t}$ are constant and $d_{t2}=V(\eta_{t2}),\
t=1,2,\ldots,n$.

###### Proof.

Let us define $(n-1)$-points of the surface $\Psi(\omega_{t},\eta_{t})$ by

$\Psi(\omega_{t},\eta_{t})=A_{t}\text{ for
}\omega\geq\eta_{n-1}...\geq\eta_{1}\geq\eta_{0}\geq 0.\text{ }$

So, we have
$\Psi(\omega_{t},\eta_{t})=A_{t}=\gamma(\omega_{t})+\mathcal{V}_{1}(\omega_{t}).U(\eta_{t})+\mathcal{V}_{2}(\omega_{t}).V(\eta_{t})+\mathcal{V}_{3}(\omega_{t}).Z(\eta_{t})$.
By taking inner product with $\mathcal{V}_{1}(\omega)$,
$\mathcal{V}_{2}(\omega)$, and $\mathcal{V}_{3}(\omega)$, respectively we
obtain the coefficients as following

$\displaystyle U(\eta_{t})$ $\displaystyle=\left\langle
A_{t}-\gamma(\omega_{t}),\mathcal{V}_{1}(\omega_{t})\right\rangle,$
$\displaystyle V(\eta_{t})$ $\displaystyle=\left\langle
A_{t}-\gamma(\omega_{t}),\mathcal{V}_{2}(\omega_{t})\right\rangle,$
$\displaystyle Z(\eta_{t})$ $\displaystyle=\left\langle
A_{t}-\gamma(\omega_{t}),\mathcal{V}_{3}(\omega_{t})\right\rangle.$

Using

$U(\eta_{t})=d_{t1},\text{ }V(\eta_{t})=d_{t2},\text{ }Z(\eta_{t})=d_{t3}$

where $d_{t1}$, $d_{t2}$ and $d_{t3}$ are constant, from (3) for
$a_{n}=b_{n}=0,$ we have following matrices;

$\left(\begin{array}[c]{cccc}\eta_{1}-\eta_{0}&(\eta_{1}-\eta_{0})^{2}&\cdots&(\eta_{1}-\eta_{0})^{n-1}\\\
\eta_{2}-\eta_{0}&(\eta_{2}-\eta_{0})^{2}&\cdots&(\eta_{2}-\eta_{0})^{n-1}\\\
\vdots&\vdots&\ddots&\vdots\\\
\eta_{n-1}-\eta_{0}&(\eta_{n-1}-\eta_{0})^{2}&\cdots&(\eta_{n-1}-\eta_{0})^{n-1}\end{array}\right)\left(\begin{array}[c]{c}a_{1}\\\
a_{2}\\\ \vdots\\\
a_{n-1}\end{array}\right)=\left(\begin{array}[c]{c}d_{11}\\\ d_{21}\\\
\vdots\\\ d_{(n-1)1}\end{array}\right),$
$\left(\begin{array}[c]{cccc}\eta_{1}-\eta_{0}&(\eta_{1}-\eta_{0})^{2}&\cdots&(\eta_{1}-\eta_{0})^{n-1}\\\
\eta_{2}-\eta_{0}&(\eta_{2}-\eta_{0})^{2}&\cdots&(\eta_{2}-\eta_{0})^{n-1}\\\
\vdots&\vdots&\ddots&\vdots\\\
\eta_{n-1}-\eta_{0}&(\eta_{n-1}-\eta_{0})^{2}&\cdots&(\eta_{n-1}-\eta_{0})^{n-1}\end{array}\right)\left(\begin{array}[c]{c}b_{1}\\\
b_{2}\\\ \vdots\\\
b_{n-1}\end{array}\right)=\left(\begin{array}[c]{c}d_{12}\\\ d_{22}\\\
\vdots\\\ d_{(n-1)2}\end{array}\right),$
$\left(\begin{array}[c]{cccc}(\eta_{1}-\eta_{0})^{2}&(\eta_{1}-\eta_{0})^{3}&\cdots&(\eta_{1}-\eta_{0})^{n}\\\
(\eta_{2}-\eta_{0})^{2}&(\eta_{2}-\eta_{0})^{3}&\cdots&(\eta_{2}-\eta_{0})^{n}\\\
\vdots&\vdots&\ddots&\vdots\\\
(\eta_{n-1}-\eta_{0})^{2}&(\eta_{n-1}-\eta_{0})^{3}&\cdots&(\eta_{n-1}-\eta_{0})^{n}\end{array}\right)\left(\begin{array}[c]{c}c_{2}\\\
c_{3}\\\ \vdots\\\ c_{n}\end{array}\right)=\left(\begin{array}[c]{c}d_{13}\\\
d_{23}\\\ \vdots\\\ d_{(n-1)3}\end{array}\right),$

for $1\leq\eta\leq n-1$. Lets take

$\displaystyle M_{1}$
$\displaystyle=\left(\begin{array}[c]{cccc}\eta_{1}-\eta_{0}&(\eta_{1}-\eta_{0})^{2}&\cdots&(\eta_{1}-\eta_{0})^{n-1}\\\
\eta_{2}-\eta_{0}&(\eta_{2}-\eta_{0})^{2}&\cdots&(\eta_{2}-\eta_{0})^{n-1}\\\
\vdots&\vdots&\ddots&\vdots\\\
\eta_{n-1}-\eta_{0}&(\eta_{n-1}-\eta_{0})^{2}&\cdots&(\eta_{n-1}-\eta_{0})^{n-1}\end{array}\right)$

and

$\displaystyle M_{2}$
$\displaystyle=\left(\begin{array}[c]{cccc}(\eta_{1}-\eta_{0})^{2}&(\eta_{1}-\eta_{0})^{3}&\cdots&(\eta_{1}-\eta_{0})^{n}\\\
(\eta_{2}-\eta_{0})^{2}&(\eta_{2}-\eta_{0})^{3}&\cdots&(\eta_{2}-\eta_{0})^{n}\\\
\vdots&\vdots&\ddots&\vdots\\\
(\eta_{n-1}-\eta_{0})^{2}&(\eta_{n-1}-\eta_{0})^{3}&\cdots&(\eta_{n-1}-\eta_{0})^{n}\end{array}\right).$

Then the determinants of $M_{1}$ and $M_{2}$ is obtained as

$\displaystyle\det(M_{1})$
$\displaystyle=(-1)^{\frac{(n-1)(n-2)}{2}}\prod\limits_{t=1}^{n-1}(\eta_{t}-\eta_{0})\prod\limits_{1\leq
t<j\leq n-1}(\eta_{t}-\eta_{j}),$

and

$\displaystyle\det(M_{2})=(-1)^{\frac{(n-1)(n-2)}{2}}\prod\limits_{t=1}^{n-1}(\eta_{t}-\eta_{0})^{2}\prod\limits_{1\leq
t<j\leq n-1}(\eta_{t}-\eta_{j}).$

Since $\eta_{t}$ and $\eta_{j}$ are non-zero and different from each others,
for $1\leq t<j\leq n-1$, we get $\det(M_{1})\neq 0$ and $\det(M_{1})\neq 0$,
that is, $a_{1},a_{2}$, …,$a_{n-1},$ $b_{1},b_{2}$, …,$b_{n-1}$ and
$c_{2},c_{3}$, …,$c_{n}$ have unique solutions. This gives us that there
exists a uniquely $C^{0}$-Hermite surface interpolation with an isoasymptotic
curve. ∎

######  Example 6.

Consider a curve parametrized by

$\gamma(\omega)=(\frac{\sin\omega}{2},\frac{\cos\omega}{2},\frac{\sqrt{3}\omega}{2}),\text{
\ \ \ \ \ }0\leq\omega\leq 2\pi.$ (5)

The curve (5) is shown in Figure 1. By a direct computation, we have

$\displaystyle\mathcal{V}_{1}(\omega)$
$\displaystyle=(\frac{\cos\omega}{2},-\frac{\sin\omega}{2},\frac{\sqrt{3}}{2}),$
$\displaystyle\mathcal{V}_{2}(\omega)$
$\displaystyle=(-\sin\omega,-\cos\omega,0),$
$\displaystyle\mathcal{V}_{3}(\omega)$
$\displaystyle=(\frac{\sqrt{3}\cos\omega}{2},-\frac{\sqrt{3}\sin\omega}{2},\frac{-1}{2})$

For $A_{1}=(1,7,\frac{\sqrt{3}\pi}{4})$, the point $A_{1}$ lies on the surface
pencil with an isoasymptotic curve given by (4). If we take

$U(\eta)=a_{1}\eta,\text{ }V(\eta)=b_{1}\eta,\text{ }W(\eta)=c_{2}\eta^{2}$

then there is only one surface with an isoasymptotic curve passing the point
$A_{1}.$ We take $\omega_{1}=\frac{\pi}{2},$ $\eta_{1}=2,$ i.e.,
$\Psi_{1}(\frac{\pi}{2},2)=A_{1}(1,7,\frac{\sqrt{3}\pi}{4})$ We obtain the
equations:

$\displaystyle\frac{1}{2}-2b_{1}$ $\displaystyle=1,$ $\displaystyle-
a_{1}-2\sqrt{3}c_{2}$ $\displaystyle=7,$ $\displaystyle\sqrt{3}a_{1}-c_{2}$
$\displaystyle=0,$

which imply

$a_{1}=-1,\text{ }b_{1}=-\frac{1}{2},\text{ }c_{2}=-\sqrt{3}.$

Thus, we can construct the surface with an isoasymptotic curve passing the one
point $A_{1}=(1,7,\frac{\sqrt{3}\pi}{4})$ given by

$\displaystyle\Psi_{1}(\omega,\eta)$
$\displaystyle=\frac{1}{2}(\sin(\omega)-3\eta^{2}\cos(\omega)+\eta\sin(\omega)-\eta\cos(\omega),$
(6)
$\displaystyle(\eta^{2}+\eta)\sin(\omega)+(\eta+1)\cos(\omega),\sqrt{3}(\omega+\eta^{2}-\eta))$

The surface (6) is shown in Figure 2 and the surface (6) with curve (5) is
shown in Figure 3.

Figure 1: Graph of $\gamma$ Figure 2: Graph of surface $\Psi_{1}$ is
cunstructued by $\gamma$ Figure 3: Graph of surface $\Psi_{1}$ with curve
$\gamma$

######  Example 7.

Let the surface with an isoasymptotic curve above Example pass through the
additional point $A_{2}=(\frac{1}{2},0,\frac{\sqrt{3}\pi}{4})$. For the
convenience of calculations, taking $\omega_{2}=\frac{\pi}{2},$ $\eta_{2}=1,$
we have $A_{2}(\frac{1}{2},0,\frac{\sqrt{3}\pi}{4})$ and obtain the system of
linear equations as follows:

$\displaystyle b_{1}+2b_{2}$ $\displaystyle=-\frac{1}{4},$ $\displaystyle
a_{1}+2a_{2}+2\sqrt{3}(c_{2}+2c_{3})$ $\displaystyle=-7,$
$\displaystyle\sqrt{3}(a_{1}+2a_{2})$ $\displaystyle=2(c_{2}+2c_{3}),$
$\displaystyle b_{1}$ $\displaystyle=-b_{2},$ $\displaystyle-(a_{1}+a_{2})$
$\displaystyle=\sqrt{3}(c_{2}+c_{3}),$ $\displaystyle\sqrt{3}(a_{1}+a_{2})$
$\displaystyle=c_{2}+c_{3}.$

So, we obtain

$\displaystyle b_{1}$ $\displaystyle=\frac{1}{4},\text{
}b_{2}=-\frac{1}{4},\text{
}c_{2}=\frac{7\sqrt{3}(3-\sqrt{3})}{12},~{}c_{2}=\frac{7\sqrt{3}(\sqrt{3}-3)}{12},$
$\displaystyle a_{1}$
$\displaystyle=\frac{14(3-\sqrt{3})}{12},~{}a_{2}=\frac{14(\sqrt{3}-3)}{12}.$

Thus the surface with an isoasymptotic curve passing the two points
$A_{1}=(1,7,\frac{\sqrt{3}\pi}{4})$ and
$A_{2}(\frac{1}{2},0,\frac{\sqrt{3}\pi}{4})$ is uniquely given by

$\displaystyle\Psi_{2}(\omega,\eta)$
$\displaystyle=(\frac{\sin(\omega)}{2}+\frac{7\left(3-\sqrt{3}\right)}{12}\left(\eta-\eta^{2}\right)\cos(\omega)$
$\displaystyle-\frac{\sin(\omega)}{4}\left(\eta-\eta^{2}\right)+\frac{7\left(3-\sqrt{3}\right)}{8}\left(\eta^{2}-\eta^{3}\right)\cos(\omega),$
$\displaystyle\frac{\cos(\omega)}{2}-\frac{7\left(3-\sqrt{3}\right)}{12}\left(\eta-\eta^{2}\right)\sin(\omega)$
(7)
$\displaystyle-\left(\eta-\eta^{2}\right)\frac{\cos(\omega)}{4}-\frac{7\left(3-\sqrt{3}\right)}{8}\left(\eta^{2}-\eta^{3}\right)\sin(\omega)+,$
$\displaystyle\frac{\sqrt{3}s}{2}+\frac{7\left(\sqrt{3}-1\right)}{4}\left(\eta-\eta^{2}\right)-\frac{7\left(\sqrt{3}-1\right)}{8}\left(\eta^{2}-\eta^{3}\right)).$

The surface (7) is shown in Figure 4 and the surface (7) with curve (5) is
shown in Figure 5.

Figure 4: Graph of surface $\Psi_{2}$ Figure 5: Graph of surface $\Psi_{2}$
with curve $\gamma$

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