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\pagestyle{myheadings}  %%{Authors}{Title}
\markboth{\quad\small \em E.I.~Kats and M.~Monastyrsky \hfill}
{\hfill\small\em Phase transitions in membranes with high genus\quad}

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\begin{document}
%\title{Phase transitions in membranes with high genus}
\title{PHASE TRANSITIONS IN MEMBRANES WITH HIGH GENUS}
%\author{E. I. Kats $^{1 \, ,2}
\author{E.I.~KATS}
\address{Laue-Langevin Institute, F-38042, Grenoble, France \\and\\
Landau Institute for Theoretical Physics, RAS, GSP-1, Moscow, Russia}
\author{M.~MONASTYRSKY}
%\author{M.~Monastyrsky}
%\address{$^1$
% Laue-Langevin Institute, F-38042, Grenoble, France \\
%$^2$ Landau Institute for Theoretical Physics, RAS, GSP-1, MOscow, Russia.}
\address{
Institute of Theoretical and Experimental
Physics, 117259 Moscow, Russia}

%\date{\today}



\maketitle

%\begin{abstract}
\vspace{0.3cm}
\abstracts{
The new geometrical method describing phase transitions and fluctuations in
membranes with nontrivial topology is elaborated.
The method based on the
possibility to represent the real membranes  as minimal surfaces but embedded
in $S^3$. The change of genus of a membrane corresponds to creating
holes in the corresponding minimal surface. In the frame of mean field
theory the model of
phase  transitions like crystallisation of holes is developed.
It can be considered as a model describing
the phase transition from cubic to sponge phase.
}

\vspace{0.4cm}

%\end{abstract}
\tableofcontents

\newpage
\section*{Preface}
\addcontentsline{toc}{section}{\numberline{}Preface}
It is a great honor for us to publish this article as a tribute to
 Misha Marinov, an
outstanding physicist, and a remarkable and brave man. One of us (M.M.) many
 years was a colleague of
Misha in ITEP.He was one of such a rare few  who determined the healthy
moral climate at the Institute.
His advices helped us in many difficult situations.
Misha had very broad interests in many fields of  Physics including
nuclear physics, field theory and mathematical physics. His papers reflected high
professional standards, deep penetration in  the physics of problems
and an excellent command of mathematics.
A clear critical mind and broad knowledge let him
 write remarkable reviews in several fields of theoretical physics.
Let us recall
that he was one of the last students who passed the famous ``Theorminimum Landau''
with Landau himself.
For many years he was interested in the problem of the quantization of systems with
constraints. In this field he published some original papers and nice review
of the Regge-Lund method.\cite{1}
This method, originally appearing in the string theory and
integrable models, has interesting applications in liquid crystals
and superfluid liquids.The potential of this theory is far from being explored.
 In our paper we discuss some slightly different geometrical ideas to determine phase
transitions in complex liquid membranes.
Such membranes exist in many physical media.It is worthwhile to note that
the description of the phase transition in liquid membranes represents
from the physical point of view  a far reaching generalization of the theory of holes introduced
by Yakov Frenkel as early as the 1920's.\cite{2}

Beside the great physical interest their study leads to highly nontrivial mathematical
problems related to very new fields of geometry and topology.

\section{Introduction}

The membranes that we shall consider here are thin and flexible sheets of
amphilic molecules. Such a construction distinguishes this object from
similar systems like liquid interfaces or shells. On the interface of
liquids the exchange of particles between both volume phases occurs.
It
corresponds to finite surface tension $\approx 100 ~{\rm erg}/{\rm cm}^2$. For membranes
which are in equilibrium with the solvent of amphilic molecules the number of
molecules is fixed.
That means zero surface tension. So, the
energy of membranes is characterized mainly by bending energy
but not by a surface
tension.
For liquid membranes which we are discussing the characteristic bending
module is of the order of temperature, $\kappa \simeq 10^{-14}$ erg.
This is the
principal distinction with shells where characteristic bending energy
is of the order $10^{11}$ erg, 25 orders
higher than in the case of liquid membranes. So, for membranes we must use
statistical mechanics approach, whereas for shells we apply elasticity theory.
In solvents lipids or surfactants bilayers typically form closed surfaces or
vesicles.
The characteristic length scale of vesicle is of order $1 - 10 ~\mu m$.
There are a diversity of forms of membranes.\cite{3,5}
Better known membranes with simple
topology like sphere and torus but in experiments one observed membranes with
higher genuses, e.g. with $g=2,3$.\,\cite{6,7} Here $g$ is the number of handles.

There exist also non-compact membranes like a stack of layers which possess
cylindrical or plane geometry.
Such super-molecular aggregates can be
self-organized at large scales and exhibit phases with different types of
orientational and translational order.

For example spherical vesicles can be organized in three-dimensional cubic
lattice, or infinite cylinders can be packed in two-dimensional hexagonal
lattice. Let us mention that in this case, crystalline order
exists on large scales (in
typical examples of order $10^{-4}$ cm, whereas on the smaller
scales the system
remains liquid.
There exist also other structures builded by infinite
layers.
The simplest example is the one-dimensional lattice of almost plane
layers.
Such a structure $L_{\alpha}$ called laminar is an analogue of smectic
liquid crystal. There exists also the phase $Q_{\alpha}$ with a space cubic
symmetry and the bloc structure generated by infinite periodic surface of
complicated topology. Schematically such structure can be determined in
thermodynamic limit $g\to\infty$, as a three-dimensional cubic set of
``handles.'' This structure is principally different from a topologically 
trivial
cubic lattice of spherical vesicles. The melting process of periodically
distributed phase $Q_{\alpha}$ leads to the isotropic phase,so called sponge
phase $L_3$. The structural bloc of the phase $Q_{\alpha}$ is non-periodic
minimal surface with finite density of handles $g$, however the space
distribution of $g$ is determined by the nearest order. In this paper 
we consider
a new method of classification of different types of membranes and their
phase transitions.

\section{Willmore surfaces and Minimal surfaces in \boldmath{$S^3$}}

Here we discuss some mathematics which we use to describe phase transitions
in membranes.
  
Let $M^2$ be a surface (closed, with boundary or non-compact) embedded in $R^3$.

\begin{definition} A surface $M^2$ is called Willmore surface if it is an 
extremum of the Helfrich-Willmore (HW) functional: 
\be \label{eq:1}
F=\int_{M^2} H^2\;d A
\ee
where $H$ is the mean curvature, and $dA$ is an area element. 
\end{definition}

By Gauss-Bonnet theorem (HW) functional is equivalent to the: 
$$
F=\int_{M^2} (k^2_1+k^2_2)\;dA
$$ 
where $k_1$ and $k_2$ are the principal curvatures of $M$ and  
$$
H=\frac{k_1+k_2}{2}\,.
$$ 
Therefore the extremum of functional (\ref{eq:1}) determine the most plane
surfaces of fixed topology. The following result due to J.~Weiner\,\cite{11}  
is important for future use. \\[2mm]
{\bf Proposition 1.} \\[2mm]
Let $M^2$ be a minimal surface in $S^3$ and $\gamma$ is a stereographic 
projection $S^3 \to R^3$. Then 
\be \label{eq:2}
\gamma(\hat{M}^2)=M^2 \qquad \mbox{and} \qquad F(M^2)=\sigma(\hat{M}^2)
\ee
where $\sigma(\hat{M}^2)$ is the area of the minimal surface.
This statement
is valid both for closed surfaces and for the one with a boundary. 
Willmore surfaces
are not exhausted by the projection of minimal surfaces.
For example, there
exist an infinite set of Willmore torus which are not equivalent to minimal
torus in $S^3$.\,\cite{12} 
These torus are extremum of (\ref{eq:1}) but not a minima. This
result, apparently, is true for a surfaces of a higher genus. Willmore surfaces
of higher genus generated by minimal surfaces in $S^3$ were constructed 
in Refs.~\protect\citebk{13}, \protect\citebk{14}. It is important to emphasize that Willmore surfaces embedded in 
$R^3$ satisfied the following: 
$$
\int_{M^2} H^2\;d A \ge 4\pi n 
$$ 
where $n$ is the number of intersection points ($n$--different pre-images under
projection). From this and the square estimation of minimal surfaces embedded
in $S^3$ follows that all Willmore surfaces  for $g\ge 2$  have at least 
points of intersections.\footnote{The equivalence between the special Willmore
surfaces and minimal surfaces following from the Proposition 2 leads to a 
deep analogy with the string theory. Minimal surfaces $M^2 \subset S^3$ are
the ``world sheets'' of the string determined in
$S^3$ with Nambu-Goto action.
The surfaces $M^2$ play the role of instantons in the Yang-Mills theory
determined in the space $S^3$.} 

\section{Phase transitions with a change of membrane topology}
 
In the previous section we have shown that  Willmore surfaces providing minimum
of Helfrich energy in the fixed topological class $g=const$ are related 
to minimal
surfaces and provide the minima of functional $F$. Therefore instead of
calculating the partition function of the Hamiltonian (\ref{eq:1}) 
we are able to
calculate the partition function determined by Gibbs factor 
$e^{-\int\sigma(\hat{M}^2)d\hat{A}}$ for minimal surfaces in $S^3$. Such
approach is especially useful to determine the structures of complex topology,
like cubic or sponge phases. Both of these structures are the 
phases with finite
density of handles.
In terminology of minimal surfaces in $S^3$ such structures
according (\ref{eq:2}) relate to minimal surfaces with finite density of
holes.
Before going to more detailed discussion of this correspondence, let us
add some words with respect to physical properties of cubic and sponge phase.

Cubic phase of complex liquids are constructed by triple-periodic
surfaces.
Elementary cell of such a structure of the scale of 10 $\mu m$ to 150
$\mu m$ consists of a large number of molecules which diffuse freely 
along the
membrane. In this case cubic structure determined by periodic distribution of
mass density $\rho(r)$ and other characteristics, including $g(r)$  related
with this
one. The sponge phase $L_3$ is determined by the short-range 
order of distribution
$\rho(r)$ or $g(r)$. In the language of a dual system distribution of holes
in $S^3$ we can speak about periodic in cubic case or amorphous in sponge phase
distribution of holes density $n(r)$.

If the energy $E_h$ of the hole creation in minimal surfaces is finite, then 
the mean density of holes can be estimated by 
\be \label{eq:3} 
n\approx \frac{1}{\xi^2}\exp\left(-\frac{E_h}{T}\right)
\ee
where $\xi^2$ is the mean size of an elementary cell creating by the holes on a
minimal surface. This size is related with the mean value of 
physical cell of a
cubic structure by transformation (\ref{eq:2}). The exponential factor
in Formula (\ref{eq:3}) has the
meaning of the probability of
creation of a hole. To estimate $\xi$ we use the following observation. Let us
write full (HW) functional (\ref{eq:1}) with the Gaussian curvature term:
\be \label{eq:4} 
F=\frac{\kappa}{2}\int H^2\;dA + \bar{\kappa}\int K\;dA
\ee
where two bending modules $\kappa$ and $\bar{\kappa}$ (saddle-splay) 
which determine a 
deformation of
the membranes with respect to the mean and the Gaussian curvature
correspondingly. The creation of a sponge and a cubic phases depends of a
spontaneous rise of a complexity of the topology of the surface, that is a
creation of handles or of passages between a layers. It happens when
$\bar{\kappa}>0$ since according Gauss-Bonnet theorem the second summand in 
(\ref{eq:4}) is equal $4\pi\bar{\kappa}(1-N_h)$, where $N_h$ is the number of 
handles.
The process of a spontaneous creation of
the handles leads to an appearing of an increasing number of handles, but of a
smaller size. The process stops at the equilibrium scale $\xi$ which we
search. It happens when when the positive terms of higher order of the mean
curvature will be equal with the negative term of the Gaussian curvature. For
finding $\xi$ we might estimate the summands of order $H^{-4}$ and $K^{-2}$ 
in the expansion (\ref{eq:4}). Minimization of the energy developed in such a 
way, determine the equilibrium scale $\xi $ 
$$
\xi\simeq\sqrt{\frac{\kappa}{\bar{\kappa}}}a
$$
We used the natural estimate $\kappa a^2$ ($a$ the molecular size) for the
fourth order elasticity moduli.

It is important to point out that minimal surface on $S^3$ has a surface
tension. The energy determined by the functional (\ref{eq:2}) proportional to
the area and the coefficient of proportionality has the sense of the surface
tension. Therefore, the interaction between the holes will be short-ranged
instead of real membranes where 
due to the absence of a surface tension in the HW
functional this interaction is long-ranged. The natural parameter for
the estimation of an energy is its temperature. If $R$ is a radius of the
interaction then for the holes of the distances less than $R$, the energy of
the interaction $U \sim T$. It is possible to neglect the interaction on the
large distances. We can formalize that was said above by considering the
following formula for free
energy of the system of holes on the minimal surface
in $S^3$.
\be \label{eq:5}
F_h=T n \ln \frac{n}{e n_0}-T n \ln (1-n \xi^2)
\ee
here $n$ is the density of holes.
The first term in
(\ref{eq:5}) determine the free energy of the ideal gas of holes and the second
one determines an increase of a free energy related with the interaction 
creating
by the nearest holes. We use (\ref{eq:5}) to describe a transition from a
dilute phase of the holes to the dense liquid phase.
Let us recall that a
creation of holes on a minimal surfaces in $S^3$ corresponds to an 
increase the
genus of a real  membrane. Therefore, the dilute phase of holes can be assumed
as a laminar phase $L_{\alpha}$ of membranes with rare passages between layers. 
The
liquid holes phase simulates a sponge phase $L_3$. 
To determine of the order
distribution of holes on the minimal surface  (cubic phases generated by
physical membranes) we have to add a summand in (\ref{eq:5}) 
depending not only
on the holes mean density but also of their distribution on the minimal
surface. In the mean field theory we infer one particle
distribution function of holes displacement ${\bf u}$ with respect to their 
equilibrium positions: 
$$
\rho({\bf u})=\frac{1}{Z}\exp\left[-\frac{1}{2T}\sum_q \varepsilon(q)\; 
|{\bf u}(q)|^2 \right]
$$  
where $\varepsilon(q)$ is the deformation energy of the holes distributions 
and  
$$
Z= \int D {\bf u}(q) \exp\left[-\frac{1}{2T}\sum_q \varepsilon(q)\; 
|{\bf u}(q)|^2 \right]\,.
$$  
The energy $\varepsilon(q)$ has the following expansion:
\be \label{eq:6}
\varepsilon(q)=\tau_{\rm eff} + \sigma_{\rm eff}\;q^2
\ee 
where the term $\tau_{\rm eff}$ is determined by the condensation energy of 
dense phase and the deformation energy is related with surface tension 
$\sigma_{\rm eff}$. We can
find the mean squared displacement 
$$
\left<|{\bf u}^2|\right>=\frac{T}{\sigma_{\rm eff}\sqrt{\tau_{\rm eff}}}
$$ 
and applying classical Lindeman melting criterion\,\cite{17}  for crystals obtain: 
$$
\frac{T}{\sigma_{\rm eff}\sqrt{\tau_{\rm eff}}}\simeq 0.1\; \xi^2\,.
$$

This determines the melting temperature of crystal of holes on a minimal
surface or, according to our principle of equivalence of minimal surfaces and
real membranes,
the temperature of phase transition between cubic and sponge
phase. 
$$
T\simeq 0.1\; \xi^2\; \sigma_{\rm eff}\sqrt{\tau_{\rm eff}}
$$ 
Let us mentioned that such a phase transition (ordering of holes on minimal
surfaces or handles on Willmore surfaces) does not need a substantial
alteration of the mean density. Therefore, it might be considered as a weak
first order transition, so called weak crystallization phase transition.\cite{kats}

Let us recall that the characteristics of a minimal surface in $S^3$, effective
surface tension $\sigma _{eff}$ and effective energy of holes condensation
are related
with physical characteristics of a real membrane in a very sophisticated
manner.
Some restrictions follow by the very possibility of creation of stable
holes in $S^3$; it is easy to see, e.g., that for $\sigma _{eff} =const$, 
i.e., $\sigma _{eff}$ 
independent of the shape and distribution of holes the creation of holes is
impossible. 

\section{Conformal modes of membranes and vesicles} 

Let us discuss the properties of conformal symmetry for the HW functional. 
Namely, conformal modes of the HW functional determine anomalous 
strong conformal fluctuation of membranes.\\[2mm] 
A. The main property of HW functional is its invariance with respect to
conformal transformations of an ambient space.
The following statement due to
G.~Tomsen was proved in the 20th years of the XX century.\\[2mm]
{\bf Proposition 2}\\[2mm]
Let $M^2$ be a Willmore surface and $M^2 R^3$ is a smooth embedding. The HW
functional is invariant under the conformal group $G(R^3)$. The
group $G(R^3)$, according to the classical Liouville theorem is generated by
dilatation,rotations, translations and inversion. The group $G(R^3)$ is
isomorphic to $O(4,1)$. It follows from the Proposition 2 that the number of
geometrically nontrivial parameters of $G(R^3)$ that is not changing the shape of
a surface is equal 4. To estimate the whole number of independent conformal
modes of the HW functional we need to add some restrictions imposed by physical
conditions. The liquid inside the vesicle is incompressible. So we get 
$$
V=const\;.
$$ 
Amphilic molecules
are densely packed on a surface. Therefore 
$$
A=const\;.
$$
Using these properties it is not difficult to determine conformal modes and the
related conformal diffusion.\\[2mm]
B. {\em The general case}\\[2mm]
Let us remind that the number of conformally nonequivalent compact surfaces is
determined by the genus of a surface 
\be \label{eq:3.4}
\left\{
\begin{array}{ll}
dim_R\; M^2_g=6g-6 & \mbox{for $g\ge 2$} \\[1mm]
dim_R\; M^2_g=2 & \mbox{for $g=1$} \\[1mm]
dim_R\; M^2_g=0 & \mbox{for $g=0$}
\end{array}\right.
\ee      
Comparing (\ref{eq:3.4}) with Liuoville theorem we obtain the number of
conformal mode as $6g-10$ for $g\ge 2$.\\[2mm]
C.  {\em Minimal Willmore surfaces}\\[2mm]
The answer for Willmore surfaces obtained by
projection of  minimal surfaces is substantially different. First of all the
conditions (6)  and (7) now are not independent. The estimation of moduli
parameters of Willmore surfaces also needs more sophisticated technique. In the
compact case we use the arguments of Ref.~\protect\citebk{22}.

Let us denote the Moduli space of Willmore surfaces $M^2_g$ as ${\cal M}$.

\noindent      
1. Each surface $M^2_g$ possesses a conformal metric and quadratic holomorphic
differential $\omega=Q(z)dz^2$. For any $\omega$ there exists conformal
metrics which determines a surface minimally embedded into $S^3$. The dimension
of the moduli space of such surfaces is equal to $12g-12$. It follows from the
estimation of the space of differentials $\omega$ which is $6g-6$ and the dimension of moduli space of
Riemann surfaces  which is also $6g-6$. The number of different embedding of
$M^2_g$ is
determined by the number of different spinor structure on $M^2_g$. More
precisely, if $\phi_1$ and $\phi_2$ are a different solutions of the
Gauss--Weingarten equations for $M^2_g$ with the same $\omega$ and equal
determinants $\det \phi_1 = \det \phi_2$ then they determine the same spinor
structure on $M^2_g$. Let us choose on the space $M^2_g$ a basis of homological
cycles $a_i, b_i$ ($i=1,...,g$) corresponding to the canonical cuts on $M^2_g$.
The spinor structure is determined by the monodromy transformation of solutions
$\phi_i$ 
$$ 
\begin{array}{l}
\phi_i\to (-1)^{\alpha_{i+1}}\phi_i \\
\phi_i\to (-1)^{\beta_{i+1}}\phi_i
\end{array}
$$ 
where $\alpha_i,\beta_i \in \{0,1\}$.

The numbers $(\alpha_i,\beta_i)$, $\alpha=(\alpha_i,..., \alpha_g)$, 
$\beta=(\beta_i,..., \beta_g)$ are called the $\theta$--characteristics of 
$M^2_g$. The $\theta$--characteristics depend on the choice of basis in $M^2_g$.
The invariant is the parity 
$$
[\alpha_i,\beta_i]=\sum_{i=1}^g \alpha_i\, \beta_i
$$ 
Due to the  parity it is possible to classify embedding of surfaces up to a
regular homotopy. U.~Pincal has shown that two surfaces are equivalent via a
smooth homotopy if the parities of their $\theta$ characteristics
coincide.\cite{21}

Using this result it is easy to estimate the number of conditions $L$ which 
determine one valued embedding of a surface $M^2_g$. The monodromy 
representation is the mapping of the fundamental group $\pi_1(M^2_g) \to
SU(2)$. The number  $L$ is determined by the triviality of monodromy along an
arbitrary cycle c. The group $\pi_1(M^2_g)$ is determined by $2g$ generators
with one relation, and taking into an account the motion along each cycle which is
generated by conformal group $SL(2,C)$ we obtain $6(2g-1)$ real conditions. Let
us add the general global transformation of $S^3$ so we obtain $12g-12$
conditions. In this way we have shown that the number of equations and the
number of parameters which determine compact minimal surfaces
coincide.
Apparently, the number of  such surfaces is finite.\footnote{~It follows
from our consideration that toric minimal surfaces does not have conformal
diffusion modes.} Well known examples of such surfaces were founded by 
Lawson\,\cite{13} and Karcher,
Pinkal and Sterling.\cite{14}

\section{Spectral properties}

Most of characteristics of membranes  can be determined, for example, by the
distribution of marked atoms or the temperature on the surface. This problem
reduces to the to the determination of the spectra of the Beltrami-Laplace
operator $\Delta$ on a surface. A determination of the shape and other
geometrical characteristics of surface of a constant Gauss curvature  by the
spectra of $\Delta$ is the classical mathematical problem. For example, from
the spectra it is possible to reconstruct,in a compact case, the volume,genus
and length of closed geodesics. Moreover, in this case it is possible to prove
that the number of surfaces with the same spectra is finite. In our case
Willmore surfaces are determined by the equation 
$$
\Delta H + H(H^2-K)=0
$$ 
and not need to be a surfaces with constant Gauss curvature. However, as we
mentioned above, apparently the number of different compact minimal surfaces in
$S^3$ is finite and so the problem of restoring Willmore surfaces from the
spectra of its Laplacian is accessible. 

\section*{Acknowledgments}
\addcontentsline{toc}{section}{\numberline{}Acknowledgments}

This research was partially supported by the grants RFBR 00-02-17785
and 99-01-00123.         

\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}
         
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\end{thebibliography}

         
\end{document}         

