The geometric classification of $2$-step nilpotent algebras
and applications 111 The work of the first author was completed as a part of
the implementation of the development program of the Scientific and
Educational Mathematical Center Volga Federal District, agreement no.
075-02-2020-1488/1. The work is supported by FAPESP 19/00563-1, RFBR
20-01-00030, CNPq 302980/2019-9. The authors thank Sergey Pchelintsev for
sharing the manuscript [25].

Mikhail Ignatyeva, Ivan Kaygorodovb & Yury Popovc

a Samara National Research University, Samara, Russia

b CMCC, Universidade Federal do ABC, Santo André, Brazil

c Saint Petersburg State University, Russia

E-mail addresses:

Mikhail Ignatyev<EMAIL_ADDRESS>
Ivan Kaygorodov<EMAIL_ADDRESS>
Yury Popov<EMAIL_ADDRESS>
Abstract. We give a geometric classification of complex $n$-dimensional
$2$-step nilpotent (all, commutative and anticommutative) algebras. Namely, it
has been found the number of irreducible components and their dimensions. As a
corollary, we have a geometric classification of complex $5$-dimensional
nilpotent associative algebras. In particular, it has been proven that this
variety has $14$ irreducible components and $9$ rigid algebras.

Keywords: Nilpotent algebra, 2-step nilpotent algebra, associative algebra,
geometric classification, irreducible components, degeneration.

MSC2020: 17A30, 16S80, 16W22, 17A99, 17D30.

## Introduction

Geometric properties of a variety of algebras defined by a family of
polynomial identities have been an object of study since 1970’s. Gabriel
described the irreducible components of the variety of $4$-dimensional unital
associative algebras [9]. Mazzola classified algebraically and geometrically
the variety of $5$-dimesional unital associative algebras [24]. Cibils
considered rigid associative algebras with $2$-step nilpotent radical [6].
Burde and Steinhoff constructed the graphs of degenerations for the varieties
of $3$-dimensional and $4$-dimensional Lie algebras [4]. Grunewald and
O’Halloran calculated the degenerations for the variety of $5$-dimensional
nilpotent Lie algebras [12]. Chouhy proved that in the case of finite-
dimensional associative algebras, the $N$-Koszul property is preserved under
the degeneration relation [5]. Degenerations have also been used to study the
level of complexity of an algebra [11, 21]. Given algebras ${\bf A}$ and ${\bf
B}$ in the same variety, we write ${\bf A}\to{\bf B}$ and say that ${\bf A}$
degenerates to ${\bf B}$, or that ${\bf A}$ is a deformation of ${\bf B}$, if
${\bf B}$ is in the Zariski closure of the orbit of ${\bf A}$ (under the base-
change action of the general linear group). The study of degenerations of
algebras is very rich and closely related to deformation theory, in the sense
of Gerstenhaber [10]. It offers an insightful geometric perspective on the
subject and has been the object of a lot of research. In particular, there are
many results concerning degenerations of algebras of small dimensions in a
variety defined by a set of identities (see, for example, [17, 19, 14, 12, 1,
4, 3] and references therein). One of the main problems of the geometric
classification of a variety of algebras is a description of its irreducible
components. From the geometric point of view, in many cases, the irreducible
components of the variety are determined by the rigid algebras, i.e., algebras
whose orbit closure is an irreducible component. It is worth mentioning that
this is not always the case and Flanigan had shown that the variety of
$3$-dimensional nilpotent associative algebras has an irreducible component
which does not contain any rigid algebras — it is instead defined by the
closure of a union of a one-parameter family of algebras [7].

The variety of $2$-step nilpotent algebras is the intersection of all
varieties of algebras defined by a family of polynomial identities of degree
equal or more than three (for example, varieties of associative,
assosymmetric, Novikov, Leibniz, Zinbiel, and so on). And it plays an
important role in the geometric classification of such algebras, because each
non-$2$-step nilpotent algebra from this variety degenerates to some $2$-step
nilpotent algebras. The same situation appears in commutative and
anticommutative cases. On the other hand, the variety of $2$-step nilpotent
Lie algebras has a proper interest (for example, see [22, 2, 8] and references
therein). A systematic study of $2$-step nilpotent algebras from the geometric
point started from a paper of Shafarevich, where he described number and
dimensions of irreducible components of the variety of $2$-step nilpotent
commutative algebras [25]. Recently, it was proven that the variety of
$n$-dimensional (all, commutative or anticommutative) nilpotent algebras is
irreducible and the dimensions of these varieties were also calculated [18].
The full graphs of degenerations in the varieties of $4$-dimensional $2$-step
nilpotent all algebras [20] and $8$-dimensional $2$-step nilpotent
anticommutative algebras [1] are constructed. In the first part of our paper,
we are following the ideas from [25] for finding the number and dimensions of
irreducible components in the variety of $n$-dimensional $2$-step (all,
commutative and anticommutative) nilpotent algebras (Theorem A). After that,
we are discussing in detail the case of $5$-dimensional $2$-step nilpotent
algebras. Algebraic classification of $5$-dimensional $2$-step nilpotent
algebras is a very ”wild” problem, but using the results from the first part
of our paper, it will be possible to take a geometric classification of these
algebras (Theorem B), which is the principal tool for future geometric
classifications of $5$-dimensional (nilpotent, solvable or all) associative,
assosymmetric, Novikov, symmetric Leibniz, etc. varieties of algebras.

Recently, an algebraic classification of complex $5$-dimensional nilpotent
(non-$2$-step nilpotent) associative algebras was obtained in [15]. Early, a
tentative geometric classification of these algebras was given in [23], but
unfortunately, as we can see from our result, it was completely wrong. In the
present paper, using a geometric classification of complex $5$-dimensional
$2$-step nilpotent algebras (Theorem B) and an algebraic classification of
complex $5$-dimensional nilpotent (non-$2$-step nilpotent) associative
algebras [15], we give the complete geometric classification of complex
$5$-dimensional nilpotent associative algebras (Theorem C).

##  1\. Around some results of Shafarevich

For $k\leq n$ consider the (algebraic) subset $\mathfrak{Nil}_{n,k}^{2}$ of
the variety $\mathfrak{Nil}_{n}^{2}$ of 2-step nilpotent $n$-dimensional
algebras defined by

$\mathfrak{Nil}_{n,k}^{2}=\\{A\in\mathfrak{Nil}_{n}^{2}:\dim A^{2}\leq k,\
\dim\operatorname{Ann}{(A)}\geq k\\}.$

It is easy to see that
$\mathfrak{Nil}_{n}^{2}=\cup_{k=1}^{n}\mathfrak{Nil}_{n,k}^{2}.$ Analogously
for the varieties
$\mathfrak{Nil}_{n}^{2}{}^{\mathrm{c}},\mathfrak{Nil}_{n}^{2}{}^{\mathrm{ac}}$
of commutative and and anticommutative 2-step nilpotent algebras we define the
subsets $\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{c}}$ and
$\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{ac}},$ respectively.

This theorem is completely analogous to [25, Theorem 1], but we provide its
proof to ensure that the paper is self-contained.

######  Theorem A.

The sets $\mathfrak{Nil}_{n,k}^{2}$ are irreducible and

(1)
$\mathfrak{Nil}_{n}^{2}=\bigcup_{k}\mathfrak{Nil}_{n,k}^{2},\quad\text{for}\quad
1\leq k\leq\left\lfloor n+\frac{1-\sqrt{4n+1}}{2}\right\rfloor$

is the decomposition of $\mathfrak{Nil}_{n}^{2}$ into irreducible components.
Analogously, we have decompositions:

$\mathfrak{Nil}_{n}^{2}{}^{\mathrm{c}}$ | $=$ | $\bigcup_{k}\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{c}},$ | for | $1\leq k\leq\left\lfloor n+\frac{3-\sqrt{8n+9}}{2}\right\rfloor,$
---|---|---|---|---
$\mathfrak{Nil}_{n}^{2}{}^{\mathrm{ac}}$ | $=$ | $\bigcup_{k}\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{ac}},$ | for | $1+(n+1)\operatorname{mod}2\leq k\leq\left\lfloor n+\frac{1-\sqrt{8n+1}}{2}\right\rfloor\text{ for }n\geq 3.$

Moreover,

$\dim\mathfrak{Nil}_{n,k}^{2}$ | $=$ | $(n-k)^{2}k+(n-k)k,$
---|---|---
$\dim\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{c}}$ | $=$ | $\frac{(n-k)(n-k+1)}{2}k+(n-k)k,$
$\dim\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{ac}}$ | $=$ | $\frac{(n-k)(n-k-1)}{2}k+(n-k)k.$

###### Proof.

We prove the result for the variety $\mathfrak{Nil}_{n}^{2},$ for the
varieties $\mathfrak{Nil}_{n}^{2}{}^{\mathrm{c}}$ and
$\mathfrak{Nil}_{n}^{2}{}^{\mathrm{ac}}$ the proofs are analogous. Let
$A\in\mathfrak{Nil}_{n}^{2}$ be an algebra and let $k=\dim A^{2}.$ Since the
square of $A$ lies in its annihilator, it must be spanned by other $n-k$ basis
vectors. Therefore, we must have $k\leq(n-k)^{2}$ and we have the
decomposition (1). Note that

Let us show that the sets $\mathfrak{Nil}_{n,k}^{2}$ are irreducible. For $k$
as in (1) consider the set $S_{n,k}\subset\mathfrak{Nil}_{n,k}^{2}$ of
algebras with the following multiplication tables:

(2) $\begin{gathered}e_{i}e_{j}=\sum_{p=1}^{k}c_{ij}^{n-k+p}e_{n-k+p},\quad
e_{i}e_{n-k+p}=0,\quad e_{n-k+p}e_{n-k+s}=0,\\\ i,j=1,\ldots,n-k,\quad
p,s=1,\ldots,k.\end{gathered}$

Because $S_{n,k}$ is irreducible (indeed, it is isomorphic to
$\mathbb{C}^{(n-k)^{2}k}$), it must lie in a unique irreducible component of
$\mathfrak{Nil}_{n,k}^{2}.$ However, it is easy to see that every algebra from
$\mathfrak{Nil}_{n,k}^{2}$ is $\operatorname{GL}_{n}(\mathbb{C})$-conjugated
to an algebra from $\mathfrak{Nil}_{n,k}^{2}$ (that is, one can choose a basis
in that algebra such that the multiplication table in this basis is of the
form (2)). Since the group $\operatorname{GL}_{n}(\mathbb{C})$ is connected,
its action on $\mathfrak{Nil}_{n,k}^{2}$ must preserve irreducible components.
This shows that $\mathfrak{Nil}_{n,k}^{2}$ is itself irreducible.

Consider a subset $U_{n,k}\subset\mathfrak{Nil}_{n,k}^{2}$ given by
$U_{n,k}=\\{A\in\mathfrak{Nil}_{n}^{2}:\dim A^{2}=k,\
\dim\operatorname{Ann}{(A)}=k\\}.$ As we have seen, the sets
$\\{A\in\mathfrak{Nil}_{n}^{2}:\dim A^{2}\leq l\\}$ and
$\\{A\in\mathfrak{Nil}_{n}^{2}:\dim\operatorname{Ann}{(A)}\geq s\\}$ are
algebraic for all $l$ and $s,$ thus the set $U_{n,k}$ is open in
$\mathfrak{Nil}_{n,k}^{2}.$ However, the sets $U_{n,k}$ and $U_{n,k^{\prime}}$
have empty intersection for $k^{\prime}\neq k,$ therefore
$\mathfrak{Nil}_{n,k}^{2}\not\subseteq\cup_{k^{\prime}\neq
k}\mathfrak{Nil}_{n,k^{\prime}}^{2}$ and the decomposition (1) is indeed the
decomposition of $\mathfrak{Nil}_{n}^{2}$ into irreducible components. The
same proof is valid for $\mathfrak{Nil}_{n}^{2}{}^{c}.$ However, the set
$U_{n,1}\cap\mathfrak{Nil}^{2}{}^{\mathrm{ac}}$ is empty for an even $n$:
indeed, let $A^{2}=\langle e\rangle$ for $e\in A,$ where $(A,\mu)$ is an
anticommutative 2-step nilpotent algebra. Then the matrix corresponding to the
map $\operatorname{proj}_{e}\circ\mu:V\otimes V\to\mathbb{F},$ where
$\operatorname{proj}_{e}$ is the projection to the space generated by $e$ and
$V$ is a vector space complement of $\langle e\rangle$ to $A$ is skew-
symmetric and is of odd size, therefore it must be degenerate. This means that
there is a nonzero vector in $V\cap\operatorname{Ann}{(A)}$ and
$\dim\operatorname{Ann}{(A)}\geq 2.$ Particularly, we must have
$\mathfrak{Nil}_{n,1}^{2}{}^{\mathrm{ac}}\subseteq\mathfrak{Nil}_{n,2}^{2}{}^{\mathrm{ac}}$
if $n\geq 3.$ However, for any $n,2\leq k\leq(n-k)(n-k-1)/2$ the sets
$U_{n,k}\cap\mathfrak{Nil}^{2}{}^{\mathrm{ac}}$ are nonempty (and open in
$\mathfrak{Nil}_{n,k}^{2}{}^{\mathrm{ac}}$). Indeed, if $A^{2}=\langle
e_{n-k+1},\ldots,e_{n}\rangle$ where $k$ is as above and $V$ is a vector space
complement of $A^{2}$ to $A$, then considering the matrices $A_{k}$
corresponding to the maps $\operatorname{proj}_{e_{n-k+p}}\circ\mu:V\otimes
V\to\mathbb{F},\ p=1,\ldots,k,$ one can see that the condition $A\in U_{n,k}$
is equivalent to the condition that matrices $A_{k}$ are linearly independent
and their kernels have zero intersection. Choosing an appropriate basis in the
space of the skew-symmetric matrices of size $(n-k),$ it is easy to see that
this condition indeed defines an open subset.

Let us now calculate the dimension of $\mathfrak{Nil}_{n,k}^{2}.$ Considering
the map

$\displaystyle\operatorname{GL}_{n}(\mathbb{C})\times S_{n,k}$
$\displaystyle\to\operatorname{GL}_{n}(\mathbb{C})\cdot
S_{n,k}=\mathfrak{Nil}_{n,k}^{2},$ $\displaystyle(g,A)$ $\displaystyle\mapsto
g\cdot A,$

we get

(3) $\dim\mathfrak{Nil}_{n,k}^{2}=\dim
S_{n,k}+\dim\operatorname{GL}_{n}(\mathbb{C})-\dim F,$

where $F$ is the preimage of a generic point of $\mathfrak{Nil}_{n,k}^{2}$
with respect to this mapping. Consider the preimage of an algebra $A\in
S_{n,k}\cap U_{n,k}.$ If $g\cdot A^{\prime}=A$ for some
$g\in\operatorname{GL}_{n}(\mathbb{C}),\ A^{\prime}\in S_{n,k},$ then in the
basis
$g^{-1}(e_{1}),\ldots,g^{-1}(e_{n-k}),g^{-1}(e_{n-k+1}),\ldots,g^{-1}(e_{n})$
the algebra $A^{\prime}$ has mutliplication table of the form (2). But since
$A\in U_{n,k},$ we must have $A^{2}=\langle
e_{n-k+1},\ldots,e_{n}\rangle=\langle
g^{-1}(e_{n-k+1}),\ldots,g^{-1}(e_{n})\rangle.$ Therefore, $g$ must lie in the
group preserving the space $A^{2}$ whose dimension is $n^{2}-k(n-k)$ (note
that it acts transitively on $A^{2}$). Substituting all dimensions in (3), we
get the dimension of $\mathfrak{Nil}_{n,k}^{2}$ as stated. ∎

##  2\. The geometric classification of complex $5$-dimensional $2$-step
nilpotent algebras

###  2.1. $\mathfrak{Nil}_{n+1,1}^{2}$

Let $A$ be an algebra from $\mathfrak{Nil}_{n+1,1}^{2}.$ If $A$ has the
multiplication table $e_{i}e_{j}={\mathfrak{a}}_{ij}e_{n+1},$ then it can be
identified with an $n\times n$ matrix ${\bf A}=({\mathfrak{a}}_{ij}).$ The
problem of description of isomorphism classes of $\mathfrak{Nil}_{n+1,1}^{2}$
is then reduced to the calculation of equivalence classes of $n\times n$
matrices under the conjugation action, which was considered in [13]. Thanks to
[13, Theorem 2.1], over the complex field, every square matrix is congruent to
a direct sum of matrices of the form (determined uniquely up to permutation of
summands, see [13]):

  1. (1)

$J_{k}(0);$

  2. (2)

$[J_{k}(\lambda)\backslash I_{k}];$

  3. (3)

$\Gamma_{k},$

where $J_{k}(\lambda)$ is the Jordan block of size $k$ corresponding to the
eigenvalue $\lambda,$ $I_{k}$ is the identity matrix of size $k,$
$[A\backslash B]=\begin{pmatrix}0&B\\\ A&0\end{pmatrix}$ is the skew sum of
matrices $A$ and $B$ and

$\Gamma_{n}=\begin{pmatrix}0&&&&&\udots\\\ &&&&1&\udots\\\ &&&-1&1&\\\
&&1&1&&\\\ &-1&-1&&&\\\ 1&1&&&&\end{pmatrix}.$

Recalling that $[J_{1}(\lambda_{i})\backslash
I]=\left(\begin{array}[]{cc}0&1\\\ \lambda_{i}&0\end{array}\right)$ and let us
define a matrix ${\mathfrak{A}}_{n}$ as

  1. (1)

${\mathfrak{A}}_{2t}={\rm diag}\\{[J_{1}(\lambda_{1})\backslash
I],\ldots,[J_{1}(\lambda_{t})\backslash I]\\}$

  2. (2)

${\mathfrak{A}}_{2t+1}={\rm diag}\\{[J_{1}(\lambda_{1})\backslash
I],\ldots,[J_{1}(\lambda_{t})\backslash I],1\\}.$

Then ${\bf H}(\lambda_{1},\ldots,\lambda_{\lceil{n/2}\rceil})$ is the family
of $(n+1)$-dimensional algebras defined by the matrix ${\mathfrak{A}}_{n}.$
After a careful calculation of the dimension of the space of derivations of
all algebras from this family we conclude that $\dim\operatorname{Der}{\bf
H}(\lambda_{1},\ldots,\lambda_{\left\lfloor\frac{n}{2}\right\rfloor})$ is
generically $\left\lfloor\frac{3n}{2}\right\rfloor+1,$ therefore, the
dimension of algebraic variety which defined this family is

$(n+1)^{2}-\dim\operatorname{Der}{\bf
H}(\lambda_{1},\ldots,\lambda_{\left\lfloor\frac{n}{2}\right\rfloor})+\left\lfloor\frac{n}{2}\right\rfloor=n(n+1)$

Hence, Theorem A gives the following obvious corollary

######  Corollary.

The variety $\mathfrak{Nil}_{n,1}^{2}$ is defined by the family of algebras
${\bf H}(\lambda_{1},\ldots,\lambda_{\left\lfloor\frac{n}{2}\right\rfloor}).$
In particular, $\mathfrak{Nil}_{4,1}^{2}$ is defined by

${\mathfrak{V}}_{4+1}\cong{\bf H}(\lambda,\mu)\ :\ e_{1}e_{2}=e_{5}\ \
e_{2}e_{1}=\lambda e_{5}\ \ e_{3}e_{4}=e_{5}\ \ e_{4}e_{3}=\mu e_{5}.$

###  2.2. $\mathfrak{Nil}_{5,2}^{2}$

Let $A$ be an algebra from $\mathfrak{Nil}_{3,2}^{2},$ then if $A$ has the
following multiplication table
$e_{i}e_{j}={\mathfrak{a}}_{ij}e_{4}+{\mathfrak{b}}_{ij}e_{5},$ it can be
identified with two $3\times 3$ matrices ${\bf A}=({\mathfrak{a}}_{ij})$ and
${\bf B}=({\mathfrak{b}}_{ij}).$ Let us denote $A$ as $\langle\langle{\bf
A}:{\bf B}\rangle\rangle.$ The problem of description of isomorphism classes
of $\mathfrak{Nil}_{3,2}^{2}$ is reduced to the description of equivalence
classes of vector spaces generated by two $3\times 3$ matrices under the
action of ${\rm GL}_{3}.$

The variety $\mathfrak{Nil}_{3,2}^{2}$ is defined by an $18$-dimensional
family of algebras constructed from matrices ${\bf
A}^{g}=({\mathfrak{a}}_{ij})$ and ${\bf B}^{g}=({\mathfrak{b}}_{ij}),$ where
${\mathfrak{a}}_{ij}$ and ${\mathfrak{b}}_{ij}$ are parameters. Thanks to
corollary Corollary and [13], the variety $\langle\langle{\bf A}^{g}:{\bf
B}^{g}\rangle\rangle$ is in orbit closure of the $10$-parameters family of
algebras $\langle\langle\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\
0&\lambda&0\\\ \end{array}\right):{\bf B}^{g}\rangle\rangle,$ where $\lambda$
and ${\mathfrak{b}}_{ij}$ are parameters. Obviously, the last variety is in
the orbit closure of the $8$-dimensional family of algebras
$\langle\langle\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\ 0&\lambda&0\\\
\end{array}\right):\left(\begin{array}[]{ccc}0&\mu_{1}&\mu_{2}\\\
\mu_{3}&\mu_{4}&\mu_{5}\\\ \mu_{6}&\mu_{7}&1\\\
\end{array}\right)\rangle\rangle,$ where $\lambda$ and $\mu_{i}$ are
parameters. The last family of algebras we will denote as $\mathfrak{V}.$
Hence, the variety $\mathfrak{Nil}_{3,2}^{2}$ is defined by
${\mathfrak{V}}_{3+2}.$ After a careful calculation of the dimension of the
algebra of derivations of all algebras from ${\mathfrak{V}}_{3+2},$ we
conclude, that between its $8$ parameters, there are $6$ independent
parameters and $2$ dependent of others.

###  2.3. $\mathfrak{Nil}_{5,3}^{2}$

Thanks to [3], the variety of $\mathfrak{Nil}_{5,3}^{2}$ is defined by the
following family of algebras

${\mathfrak{V}}_{2+3}\cong A_{133}(\lambda)\ :\ e_{1}e_{1}=e_{3}+\lambda
e_{5}\ \ e_{1}e_{2}=e_{3}\ \ e_{2}e_{1}=e_{4}\ \ e_{2}e_{2}=e_{5}$

###  2.4. Classification theorem

Summarizing, we have the following statement.

######  Theorem B.

The variety of complex $5$-dimensional $2$-step nilpotent algebras has
dimension $24$ and it has $3$ irreducible components defined by

$\mathcal{C}_{1}=\overline{\\{\mathcal{O}({\mathfrak{V}}_{2+3})\\}},$
$\mathcal{C}_{2}=\overline{\\{\mathcal{O}({\mathfrak{V}}_{3+2})\\}},$
$\mathcal{C}_{3}=\overline{\\{\mathcal{O}({\mathfrak{V}}_{4+1})\\}}.$

##  3\. The geometric classification of complex $5$-dimensional nilpotent
associative algebras

###  3.1. Algebraic classication of complex $5$-dimensional nilpotent
associative algebras

Thanks to [15, 16, 14], we have the classification of all complex
$5$-dimensional nilpotent (non-$2$-step nilpotent) non-commutative associative
algebras:

${\mathcal{A}}^{4}_{05}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{4}$ | $e_{1}e_{3}=e_{4}$ | $e_{2}e_{1}=e_{4}$ | $e_{3}e_{3}=e_{4}$
---|---|---|---|---|---|---
${\mathcal{A}}^{4}_{06}(1)$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{4}$ | $e_{1}e_{3}=e_{4}$ | $e_{2}e_{1}=e_{4}$ |
$\mu_{1,3}^{5}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{3}=e_{4}$ | $e_{1}e_{5}=e_{4}$ |
|  | $e_{2}e_{1}=e_{3}$ | $e_{2}e_{2}=e_{4}$ | $e_{3}e_{1}=e_{4}$ |  |
$\mu_{1,4}^{5}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{3}=e_{4}$ | $e_{1}e_{5}=e_{4}$ |
|  | $e_{2}e_{1}=e_{3}$ | $e_{2}e_{2}=e_{4}$ | $e_{3}e_{1}=e_{4}$ | $e_{5}e_{5}=e_{4}$ |
$\lambda_{2}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{3}$ |  |
|  | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{5}=e_{3}$ | $e_{5}e_{4}=e_{3}$ |  |
$\lambda_{3}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{3}$ | $e_{2}e_{1}=e_{3}$ | $e_{5}e_{5}=e_{3}$
$\lambda_{4}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{3}$ |  |
|  | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{4}=e_{3}$ | $e_{5}e_{5}=e_{3}$ |  |
$\lambda_{5}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{2}e_{1}=e_{3}$ |  |
|  | $e_{4}e_{5}=e_{3}$ | $e_{5}e_{4}=-e_{3}$ | $e_{5}e_{5}=e_{3}$ |  |
$\lambda_{6}^{\alpha\neq 1}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{5}=e_{3}$ | $e_{5}e_{4}=\alpha e_{3}$
$\mu_{1}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=e_{5}$ |
$\mu_{2}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=e_{5}$ | $e_{4}e_{4}=e_{3}$
$\mu_{3}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{2}e_{1}=e_{3}$ |  |
|  | $e_{4}e_{1}=e_{5}$ | $e_{4}e_{2}=e_{3}$ | $e_{5}e_{1}=e_{3}$ |  |
$\mu_{4}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=e_{5}$ |
|  | $e_{4}e_{2}=e_{3}$ | $e_{4}e_{4}=e_{3}$ | $e_{5}e_{1}=e_{3}$ |  |
$\mu_{5}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=e_{3}+e_{5}$
$\mu_{6}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ |  |
|  | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=e_{3}+e_{5}$ | $e_{4}e_{4}=e_{3}$ |  |
$\mu_{7}^{\alpha\neq 1}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=\alpha e_{5}$
$\mu_{8}^{\alpha\neq 1}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ |  |
|  | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=\alpha e_{5}$ | $e_{4}e_{4}=e_{3}$ |  |
$\mu_{9}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=e_{3}$ | $e_{4}e_{4}=e_{5}$
$\mu_{10}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{4}=e_{5}$
$\mu_{11}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{4}=e_{3}+e_{5}$
$\mu_{12}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ |  |
|  | $e_{2}e_{1}=e_{3}$ | $e_{4}e_{1}=e_{2}-e_{5}$ | $e_{5}e_{1}=e_{3}$ |  |
$\mu_{13}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=e_{2}-e_{5}$ | $e_{4}e_{4}=e_{3}$ | $e_{5}e_{1}=e_{3}$ |  |
$\mu_{14}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=e_{2}+e_{5}$ | $e_{4}e_{2}=2e_{3}$ | $e_{4}e_{4}=2e_{5}$ | $e_{5}e_{1}=e_{3}$ |
$\mu_{15}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=e_{2}+e_{5}$ | $e_{4}e_{2}=2e_{3}$ | $e_{4}e_{4}=e_{3}+2e_{5}$ | $e_{5}e_{1}=e_{3}$ |
$\mu_{17}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=e_{3}+e_{5}$ | $e_{4}e_{4}=e_{2}$ | $e_{4}e_{5}=e_{3}$ | $e_{5}e_{4}=e_{3}$ |
$\mu_{18}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=-e_{5}$ | $e_{4}e_{4}=e_{2}$ | $e_{4}e_{5}=-e_{3}$ | $e_{5}e_{4}=e_{3}$ |
$\mu_{19}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=e_{2}$ | $e_{4}e_{2}=e_{3}$ | $e_{4}e_{4}=e_{3}+e_{5}$ | $e_{5}e_{1}=e_{3}$ |
$\mu_{20}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=e_{3}+e_{5}$ | $e_{4}e_{4}=-e_{2}+2e_{5}$ | $e_{4}e_{5}=e_{3}$ | $e_{5}e_{4}=-e_{3}$ |
$\mu_{21}^{i}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=(1-i)e_{2}+ie_{5}$ | $e_{4}e_{2}=2e_{3}$ | $e_{4}e_{4}=-ie_{2}+e_{3}+(1+i)e_{5}$
|  | $e_{4}e_{5}=e_{3}$ | $e_{5}e_{1}=(1-i)e_{3}$ | $e_{5}e_{4}=-ie_{3}$ |  |
$\mu_{21}^{-i}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=(1+i)e_{2}-ie_{5}$ | $e_{4}e_{2}=2e_{3}$ | $e_{4}e_{4}=ie_{2}+e_{3}+(1-i)e_{5}$
|  | $e_{4}e_{5}=e_{3}$ | $e_{5}e_{1}=(1+i)e_{3}$ | $e_{5}e_{4}=ie_{3}$ |  |
$\mu_{22}^{\alpha}$ | $:$ | $e_{1}e_{1}=e_{2}$ | $e_{1}e_{2}=e_{3}$ | $e_{1}e_{4}=e_{5}$ | $e_{2}e_{1}=e_{3}$ |
|  | $e_{4}e_{1}=(1-\alpha)e_{2}+\alpha e_{5}$ | $e_{4}e_{2}=(1-\alpha^{2})e_{3}$ | $e_{4}e_{4}=-\alpha e_{2}+(1+\alpha)e_{5}$
|  | $e_{4}e_{5}=-\alpha^{2}e_{3}$ | $e_{5}e_{1}=(1-\alpha)e_{3}$ | $e_{5}e_{4}=-\alpha e_{3}$ |  |

###  3.2. Geometric classication of complex $5$-dimensional nilpotent
associative algebras

####  3.2.1. Degenerations of algebras

Given an $n$-dimensional vector space ${\bf V}$, the set ${\rm Hom}({\bf
V}\otimes{\bf V},{\bf V})\cong{\bf V}^{*}\otimes{\bf V}^{*}\otimes{\bf V}$ is
a vector space of dimension $n^{3}$. This space inherits the structure of the
affine variety $\mathbb{C}^{n^{3}}.$ Indeed, let us fix a basis
$e_{1},\dots,e_{n}$ of ${\bf V}$. Then any $\mu\in{\rm Hom}({\bf V}\otimes{\bf
V},{\bf V})$ is determined by $n^{3}$ structure constants
$c_{i,j}^{k}\in\mathbb{C}$ such that $\mu(e_{i}\otimes
e_{j})=\sum_{k=1}^{n}c_{i,j}^{k}e_{k}$. A subset of ${\rm Hom}({\bf
V}\otimes{\bf V},{\bf V})$ is Zariski-closed if it can be defined by a set of
polynomial equations in the variables $c_{i,j}^{k}$ ($1\leq i,j,k\leq n$).

The general linear group ${\rm GL}({\bf V})$ acts by conjugation on the
variety ${\rm Hom}({\bf V}\otimes{\bf V},{\bf V})$ of all algebra structures
on ${\bf V}$:

$(g*\mu)(x\otimes y)=g\mu(g^{-1}x\otimes g^{-1}y),$

for $x,y\in{\bf V}$, $\mu\in{\rm Hom}({\bf V}\otimes{\bf V},{\bf V})$ and
$g\in{\rm GL}({\bf V})$. Clearly, the ${\rm GL}({\bf V})$-orbits correspond to
the isomorphism classes of algebras structures on ${\bf V}$. Let $T$ be a set
of polynomial identities which is invariant under isomorphism. Then the subset
$\mathbb{L}(T)\subset{\rm Hom}({\bf V}\otimes{\bf V},{\bf V})$ of the algebra
structures on ${\bf V}$ which satisfy the identities in $T$ is ${\rm GL}({\bf
V})$-invariant and Zariski-closed. It follows that $\mathbb{L}(T)$ decomposes
into ${\rm GL}({\bf V})$-orbits. The ${\rm GL}({\bf V})$-orbit of
$\mu\in\mathbb{L}(T)$ is denoted by $O(\mu)$ and its Zariski closure by
$\overline{O(\mu)}$.

Let ${\bf A}$ and ${\bf B}$ be two $n$-dimensional algebras satisfying the
identities from $T$ and $\mu,\lambda\in\mathbb{L}(T)$ represent ${\bf A}$ and
${\bf B}$ respectively. We say that ${\bf A}$ degenerates to ${\bf B}$ and
write ${\bf A}\to{\bf B}$ if $\lambda\in\overline{O(\mu)}$. Note that in this
case we have $\overline{O(\lambda)}\subset\overline{O(\mu)}$. Hence, the
definition of a degeneration does not depend on the choice of $\mu$ and
$\lambda$. It is easy to see that any algebra degenerates to the algebra with
zero multiplication. If ${\bf A}\to{\bf B}$ and ${\bf A}\not\cong{\bf B}$,
then ${\bf A}\to{\bf B}$ is called a proper degeneration. We write ${\bf
A}\not\to{\bf B}$ if $\lambda\not\in\overline{O(\mu)}$ and call this a non-
degeneration. Observe that the dimension of the subvariety $\overline{O(\mu)}$
equals $n^{2}-\dim{\mathfrak{D}er}({\bf A})$. Thus if ${\bf A}\to{\bf B}$ is a
proper degeneration, then we must have $\dim{\mathfrak{D}er}({\bf
A})>\dim{\mathfrak{D}er}({\bf B})$.

Let ${\bf A}$ be represented by $\mu\in\mathbb{L}(T)$. Then ${\bf A}$ is rigid
in $\mathbb{L}(T)$ if $O(\mu)$ is an open subset of $\mathbb{L}(T)$. Recall
that a subset of a variety is called irreducible if it cannot be represented
as a union of two non-trivial closed subsets. A maximal irreducible closed
subset of a variety is called an irreducible component. It is well known that
any affine variety can be represented as a finite union of its irreducible
components in a unique way. The algebra ${\bf A}$ is rigid in $\mathbb{L}(T)$
if and only if $\overline{O(\mu)}$ is an irreducible component of
$\mathbb{L}(T)$.

In the present work we use the methods applied to Lie algebras in [12]. To
prove degenerations, we will construct families of matrices parametrized by
$t$. Namely, let ${\bf A}$ and ${\bf B}$ be two algebras represented by the
structures $\mu$ and $\lambda$ from $\mathbb{L}(T)$, respectively. Let
$e_{1},\dots,e_{n}$ be a basis of ${\bf V}$ and $c_{i,j}^{k}$ ($1\leq
i,j,k\leq n$) be the structure constants of $\lambda$ in this basis. If there
exist $a_{i}^{j}(t)\in\mathbb{C}$ ($1\leq i,j\leq n$, $t\in\mathbb{C}^{*}$)
such that the elements $E_{i}^{t}=\sum_{j=1}^{n}a_{i}^{j}(t)e_{j}$ ($1\leq
i\leq n$) form a basis of ${\bf V}$ for any $t\in\mathbb{C}^{*}$, and the
structure constants $c_{i,j}^{k}(t)$ of $\mu$ in the basis
$E_{1}^{t},\dots,E_{n}^{t}$ satisfy $\lim\limits_{t\to
0}c_{i,j}^{k}(t)=c_{i,j}^{k}$, then ${\bf A}\to{\bf B}$. In this case
$E_{1}^{t},\dots,E_{n}^{t}$ is called a parametric basis for ${\bf A}\to{\bf
B}$ and it will be denote as ${\bf
A}\xrightarrow{(E_{1}^{t},\dots,E_{n}^{t})}{\bf B}$. Let us denote the
subalgebra generated by $\langle e_{i},\ldots,e_{n}\rangle$ as $A_{i}.$

To prove a non-degeneration ${\bf A}\not\to{\bf B}$ we will use the following
Lemma (see [12]).

######  Lemma.

Let $\mathcal{B}$ be a Borel subgroup of ${\rm GL}({\bf V})$ and
$\mathcal{R}\subset\mathbb{L}(T)$ be a $\mathcal{B}$-stable closed subset. If
${\bf A}\to{\bf B}$ and ${\bf A}$ can be represented by $\mu\in\mathcal{R}$
then there is $\lambda\in\mathcal{R}$ that represents ${\bf B}$.

In particular, it follows from Lemma that ${\bf A}\not\to{\bf B}$, whenever
$\dim({\bf A}^{2})<\dim({\bf B}^{2})$.

When the number of orbits under the action of ${\rm GL}({\bf V})$ on
$\mathbb{L}(T)$ is finite, the graph of primary degenerations gives the whole
picture. In particular, the description of rigid algebras and irreducible
components can be easily obtained. Since the variety of $5$-dimensional
nilpotent associative algebras contains infinitely many non-isomorphic
algebras, we have to fulfill some additional work. Let ${\bf A}(*):=\\{{\bf
A}(\alpha)\\}_{\alpha\in I}$ be a family of algebras and ${\bf B}$ be another
algebra. Suppose that, for $\alpha\in I$, ${\bf A}(\alpha)$ is represented by
a structure $\mu(\alpha)\in\mathbb{L}(T)$ and ${\bf B}$ is represented by a
structure $\lambda\in\mathbb{L}(T)$. Then by ${\bf A}(*)\to{\bf B}$ we mean
$\lambda\in\overline{\cup\\{O(\mu(\alpha))\\}_{\alpha\in I}}$, and by ${\bf
A}(*)\not\to{\bf B}$ we mean
$\lambda\not\in\overline{\cup\\{O(\mu(\alpha))\\}_{\alpha\in I}}$.

Let ${\bf A}(*)$, ${\bf B}$, $\mu(\alpha)$ ($\alpha\in I$) and $\lambda$ be as
above. To prove ${\bf A}(*)\to{\bf B}$ it is enough to construct a family of
pairs $(f(t),g(t))$ parametrized by $t\in\mathbb{C}^{*}$, where $f(t)\in I$
and $g(t)=\left(a_{i}^{j}(t)\right)_{i,j}\in{\rm GL}({\bf V})$. Namely, let
$e_{1},\dots,e_{n}$ be a basis of ${\bf V}$ and $c_{i,j}^{k}$ ($1\leq
i,j,k\leq n$) be the structure constants of $\lambda$ in this basis. If we
construct $a_{i}^{j}:\mathbb{C}^{*}\to\mathbb{C}$ ($1\leq i,j\leq n$) and
$f:\mathbb{C}^{*}\to I$ such that $E_{i}^{t}=\sum_{j=1}^{n}a_{i}^{j}(t)e_{j}$
($1\leq i\leq n$) form a basis of ${\bf V}$ for any $t\in\mathbb{C}^{*}$, and
the structure constants $c_{i,j}^{k}(t)$ of $\mu\big{(}f(t)\big{)}$ in the
basis $E_{1}^{t},\dots,E_{n}^{t}$ satisfy $\lim\limits_{t\to
0}c_{i,j}^{k}(t)=c_{i,j}^{k}$, then ${\bf A}(*)\to{\bf B}$. In this case,
$E_{1}^{t},\dots,E_{n}^{t}$ and $f(t)$ are called a parametric basis and a
parametric index for ${\bf A}(*)\to{\bf B}$, respectively. In the construction
of degenerations of this sort, we will write
$\mu\big{(}f(t)\big{)}\to\lambda$, emphasizing that we are proving the
assertion $\mu(*)\to\lambda$ using the parametric index $f(t)$.

Through a series of degenerations summarized in the table below by the
corresponding parametric bases and indices, we obtain the main result of the
second part of the paper.

####  3.2.2. Classification theorem

######  Theorem C.

The variety of complex $4$-dimensional nilpotent associative algebras has
dimension $13$ and it has $4$ irreducible components defined by

$\mathcal{C}_{1}=\overline{\\{\mathcal{O}(\mathfrak{N}_{2}(\alpha))\\}},$
$\mathcal{C}_{2}=\overline{\\{\mathcal{O}(\mathfrak{N}_{3}(\alpha))\\}},$
$\mathcal{C}_{3}=\overline{\\{\mathcal{O}(\mu_{0}^{4})\\}},$
$\mathcal{C}_{4}=\overline{\\{\mathcal{O}({\mathcal{A}}^{4}_{05})\\}}.$

In particular, we have $2$ rigid algebras: $\mu_{0}^{4}$ and
${\mathcal{A}}^{4}_{05}.$

The variety of complex $5$-dimensional nilpotent associative algebras has
dimension $24$ and it has $14$ irreducible components defined by

$\mathcal{C}_{1}=\overline{\\{\mathcal{O}({\mathfrak{V}}_{2+3})\\}},$
$\mathcal{C}_{2}=\overline{\\{\mathcal{O}({\mathfrak{V}}_{3+2})\\}},$
$\mathcal{C}_{3}=\overline{\\{\mathcal{O}({\mathfrak{V}}_{4+1})\\}},$
$\mathcal{C}_{4}=\overline{\\{\mathcal{O}(\mu_{0}^{5})\\}},$
$\mathcal{C}_{5}=\overline{\\{\mathcal{O}(\mu_{1,4}^{5})\\}},$

$\mathcal{C}_{6}=\overline{\\{\mathcal{O}(\lambda_{6}^{\alpha})\\}},$
$\mathcal{C}_{7}=\overline{\\{\mathcal{O}(\mu_{11})\\}},$
$\mathcal{C}_{8}=\overline{\\{\mathcal{O}(\mu_{15})\\}},$
$\mathcal{C}_{9}=\overline{\\{\mathcal{O}(\mu_{17})\\}},$
$\mathcal{C}_{10}=\overline{\\{\mathcal{O}(\mu_{18})\\}},$

$\mathcal{C}_{11}=\overline{\\{\mathcal{O}(\mu_{20})\\}},$
$\mathcal{C}_{12}=\overline{\\{\mathcal{O}(\mu_{21}^{i})\\}},$
$\mathcal{C}_{13}=\overline{\\{\mathcal{O}(\mu_{21}^{-i})\\}}$ and
$\mathcal{C}_{14}=\overline{\\{\mathcal{O}(\mu_{22}^{\alpha})\\}}.$

In particular, we have $9$ rigid algebras:
$\mu^{5}_{0},\mu_{1,4}^{5},\mu_{11},\mu_{15},\mu_{17},\mu_{18},\mu_{20},\mu_{21}^{i}$
and $\mu_{21}^{-i}.$

  * Proof.

It is easy to see that the complex $n$-dimensional null-filiform associative
algebra

$\mu_{0}^{n}=\\{e_{i}e_{j}=e_{i+j},2\leq i+j\leq n\\}$

is rigid in the variety of complex $n$-dimensional nilpotent associative
algebras. Thanks to [17], all complex $5$-dimensional (or $4$-dimensional)
nilpotent commutative associative algebras are degenerated from $\mu_{0}^{5}$
(or $\mu_{0}^{4}$). The variety of complex $4$-dimensional $2$-dimensional
nilpotent algebras is defined by the following families of algebras

${\mathfrak{N}}_{2}(\alpha)$ | $:$ | $e_{1}e_{1}=e_{3}$ | $e_{1}e_{2}=e_{4}$ | $e_{2}e_{1}=-\alpha e_{3}$ | $e_{2}e_{2}=-e_{4}$ |
---|---|---|---|---|---|---
${\mathfrak{N}}_{3}(\alpha)$ | $:$ | $e_{1}e_{1}=e_{4}$ | $e_{1}e_{2}=\alpha e_{4}$ | $e_{2}e_{1}=-\alpha e_{4}$ | $e_{2}e_{2}=e_{4}$ | $e_{3}e_{3}=e_{4}$

The algebra ${\mathcal{A}}^{4}_{05}$ satisfies the following conditions
$\\{A_{1}A_{2}\subseteq
A_{4},c_{22}^{4}c_{33}^{4}=c_{32}^{4}c_{23}^{4},c_{23}^{4}=c_{32}^{4}\\},$ but
$\\{\mathfrak{N}_{2}(\alpha)$ and $\mathfrak{N}_{3}(\alpha)$ do not. Hence
${\mathcal{A}}^{4}_{05}\not\to\\{\mathfrak{N}_{2}(\alpha),\mathfrak{N}_{3}(\alpha)\\}.$
Since
${\mathcal{A}}^{4}_{05}\xrightarrow{(te_{1},t^{2}e_{2},t^{2}e_{3},t^{3}e_{4},e_{5})}{\mathcal{A}}^{4}_{06}(1)$
we have that the variety of complex $4$-dimensional nilpotent associative
algebras is defined by $\mu_{0}^{4},$ ${\mathcal{A}}^{4}_{05},$
$\mathfrak{N}_{2}(\alpha)$ and $\mathfrak{N}_{3}(\alpha).$

For the rest of our proof, we need to present the list of the following
degenerations:

$\lambda_{6}^{{0}}$ | $\xrightarrow{(e_{1}-e_{5},e_{2},e_{2}+e_{4}+e_{5},e_{3},te_{5})}$ | ${\mathcal{A}}^{4}_{05}$ | $\mu_{1,4}^{5}$ | $\xrightarrow{(te_{1},t^{2}e_{2},t^{3}e_{3},t^{4}e_{4},t^{3}e_{5})}$ | $\mu_{1,3}^{5}$
---|---|---|---|---|---
$\mu_{4}$ | $\xrightarrow{(t^{-1}e_{1},t^{-2}e_{2},t^{-3}e_{3},t^{-1}e_{4},t^{-2}e_{5})}$ | $\mu_{3}$ | $\mu_{11}$ | $\xrightarrow{(t^{-1}e_{1},t^{-2}e_{2},t^{-3}e_{3},t^{-1}e_{4},t^{-2}e_{5})}$ | $\mu_{10}$
$\lambda_{6}^{{1+t}}$ | $\xrightarrow{(te_{1}+e_{5},t^{2}e_{2},t^{3}e_{3},-te_{2}+t^{2}e_{4}+\frac{t^{2}}{2+t}e_{5},te_{5})}$ | $\lambda_{2}$ | $\lambda_{4}$ | $\xrightarrow{(te_{1},t^{2}e_{2},t^{3}e_{3},t^{2}e_{4},t^{\frac{3}{2}}e_{5})}$ | $\lambda_{3}$
$\lambda_{1,4}^{5}$ | $\xrightarrow{(te_{1}+e_{2}+\frac{1}{2t}e_{3},-t^{2}e_{2}+2e_{4},t^{2}e_{4},te_{5},-te_{2}-e_{3})}$ | $\lambda_{4}$ | $\lambda_{6}^{-\frac{1}{1+t}}$ | $\xrightarrow{(te_{1},t^{2}e_{2},t^{3}e_{3},te_{5},-t^{2}(t+1)e_{4}-e_{5})}$ | $\lambda_{5}$
$\mu_{6}$ | $\xrightarrow{(te_{1},t^{2}e_{2},t^{3}e_{3},t^{2}e_{4},t^{3}e_{5})}$ | $\mu_{5}$ | $\mu_{13}$ | $\xrightarrow{(t^{-1}e_{1},t^{-2}e_{2},t^{-3}e_{3},t^{-1}e_{4},t^{-2}e_{5})}$ | $\mu_{12}$
$\mu_{2}$ | $\xrightarrow{(e_{1},e_{2},e_{3},te_{4},te_{5})}$ | $\mu_{1}$ | $\mu_{11}$ | $\xrightarrow{(t^{2}e_{1}-t^{2}e_{4},t^{4}e_{2}+t^{4}e_{3},t^{6}e_{3},-t^{3}e_{2}-t^{3}e_{4},t^{5}e_{5})}$ | $\mu_{2}$
$\mu_{8}^{\alpha}$ | $\xrightarrow{(e_{1},e_{2},e_{3},te_{4},te_{5})}$ | $\mu_{7}^{\alpha}$ | $\mu_{15}$ | $\xrightarrow{(t^{-1}e_{1},t^{-2}e_{2},t^{-3}e_{3},t^{-1}e_{4},t^{-2}e_{5})}$ | $\mu_{14}$
$\mu_{22}^{t^{-1}}\to\mu_{4}$ | $E^{t}_{1}=t^{2}(t^{4}-1)e_{1}+t^{3}(t^{2}-1)^{4}(t^{2}+1)e_{5}$ | $E^{t}_{2}=t^{4}(t^{4}-1)^{2}e_{2}+t^{4}(t-1)^{6}(t+1)^{5}(t^{2}+1)^{2}e_{3}$
---|---|---
| $E^{t}_{3}=t^{6}(t^{4}-1)^{3}e_{3}$ | $E^{t}_{4}=t^{4}(t-1)^{4}(t+1)^{3}(t^{2}+1)e_{2}+(t^{4}+t^{6})e_{4}$
|
$E^{t}_{5}=t^{5}(t-1)^{2}(t+1)(t^{2}+1)^{2}e_{2}+t^{5}(t^{2}-1)^{4}(t^{2}+1)^{2}(t^{2}-t-1)e_{3}+t^{5}(t^{2}-1)(t^{2}+1)^{2}e_{5}$
$\mu_{15}\to\mu_{6}$ | $E^{t}_{1}=\frac{t^{2}}{t+1}e_{1}$ | $E^{t}_{2}=\frac{t^{4}}{(t+1)^{2}}e_{2}$
$E^{t}_{3}=\frac{t^{6}}{(t+1)^{3}}e_{3}$ | $E^{t}_{4}=\frac{t^{3}}{t+1}e_{4}+\frac{t^{4}}{(t+1)^{2}}e_{5}$ | $E^{t}_{5}=\frac{t^{5}}{(t+1)^{2}}e_{5}$
$\mu_{22}^{\alpha}\to\mu_{8}^{\alpha}$ | $E^{t}_{1}=t(\alpha+\alpha^{3})e_{1}+(\alpha+\alpha^{3})e_{5}$ | $E^{t}_{2}=t^{2}(\alpha+\alpha^{3})^{2}e_{2}+t(1-\alpha)(\alpha+\alpha^{3})^{2}e_{3}$
$E^{t}_{3}=t^{3}(\alpha+\alpha^{3})^{3}e_{3}$ | $E^{t}_{4}=t(1-\alpha)(\alpha+\alpha^{3})e_{2}+t^{2}(\alpha+\alpha^{3})e_{4}$ | $E^{t}_{5}=t^{2}(1-2\alpha)(\alpha+\alpha^{3})^{2}e_{3}+t^{3}(\alpha+\alpha^{3})^{2}e_{5}$
$\mu_{11}\to\mu_{9}$ | $E_{1}^{t}=\frac{t^{4}+1}{t-t^{2}}e_{1}-\frac{(t^{4}+1)(1+t^{5})}{2(1-t)^{3}t^{4}}e_{2}+\frac{t^{4}+1}{(1-t)^{2}}e_{4}$ | $E_{2}^{t}=\frac{t^{4}+1}{(1-t)^{2}t^{2}}e_{2}-\frac{(t^{4}+1)^{2}}{(1-t)^{4}t^{5}}e_{3}$
$E_{3}^{t}=\frac{(t^{4}+1)^{2}}{(1-t)^{3}t^{3}}e_{3}$ | $E_{4}^{t}=\frac{t^{4}+1}{(1-t)^{2}t^{2}}e_{2}+\frac{t^{4}+1}{(t-1)^{2}t^{2}}e_{4}$ | $E_{5}^{t}=\frac{t^{4}+1}{(1-t)^{2}t}e_{2}+\frac{(t^{4}+1)^{2}}{(t-1)^{4}t^{4}}e_{5}$
$\mu_{18}\to\mu_{13}$ | $E^{t}_{1}=\sqrt{4t-1}e_{1}-2e_{2}+e_{4}$ | $E^{t}_{2}=4te_{2}-4\sqrt{4t-1}e_{3}$
$E^{t}_{3}=4t\sqrt{4t-1}e_{3}$ | $E^{t}_{4}=-2e_{2}+2te_{4}$ | $E^{t}_{5}=2te_{2}-2\sqrt{4t-1}e_{3}+2t\sqrt{4t-1}e_{5}$
$\mu_{22}^{t}\to\mu_{19}$ | $E^{t}_{1}=(t-1)(t+t^{3})e_{1}+(t-1)^{4}t(1+t^{2})e_{5}$ | $E^{t}_{2}=(t-1)^{2}(t+t^{3})^{2}e_{2}-(t-1)^{6}(t+t^{3})^{2}e_{3}$
$E^{t}_{3}=(t-1)^{3}(t+t^{3})^{3}e_{3}$ | $E^{t}_{4}=(t-1)^{4}(t+t^{3})e_{2}-t(1+t^{2})e_{4}$ | $E^{t}_{5}=(t-1)^{4}(-1+2t)(t+t^{3})^{2}e_{3}-(t-1)(t+t^{3})^{2}e_{5}$

After a careful checking dimensions of orbit closures for the rest of
algebras, we have

$\dim\mathcal{O}(\mu_{1,4}^{5})=\dim\mathcal{O}(\lambda_{6}^{\alpha})=\dim\mathcal{O}(\mu_{11})=\dim\mathcal{O}(\mu_{15})=\dim\mathcal{O}(\mu_{17})=$

$\dim\mathcal{O}(\mu_{18})=\dim\mathcal{O}(\mu_{20})=\dim\mathcal{O}(\mu_{22}^{\alpha})=\mathcal{O}({\mathfrak{V}}_{4+1})=20,$

$\dim\mathcal{O}({\mathfrak{V}}_{3+2})=24,\
\dim\mathcal{O}(\mu_{21}^{i})=\dim\mathcal{O}(\mu_{21}^{-i})=19,\
\mathcal{O}({\mathfrak{V}}_{2+3})=18.$

Hence,
$\mu_{1,4}^{5},\lambda_{6}^{\alpha},\mu_{11},\mu_{15},\mu_{17},\mu_{18},\mu_{20},\mu_{22}^{\alpha},{\mathfrak{V}}_{4+1},{\mathfrak{V}}_{3+2}$
give $10$ irreducible components.

The reasons why $\mu_{21}^{i},\mu_{21}^{-i}$ and ${\mathfrak{V}}_{2+3}$ give
more three irreducible components are given below. Let us denote a set of
conditions which satisfies an algebra ${\mathcal{A}}$ as
${\mathcal{R}}_{\mathcal{A}}.$

  1. (1)

Let us choose the following new basis for the algebra $\mu_{1,4}^{5}$:

$E_{1}=e_{1},\ E_{2}=e_{5},\ E_{3}=e_{2},\ E_{4}=e_{3},\ E_{5}=e_{4}.$

It is easy to see that in this new basis the algebra $\mu_{1,4}^{5}$ satisfies
the following conditions

${\mathcal{R}}_{\mu_{1,4}^{5}}=\left\\{\begin{array}[]{l}A_{1}^{2}\subseteq
A_{3},\ A_{1}A_{2}\subseteq A_{4},\ A_{2}^{2}+A_{4}A_{1}+A_{1}A_{4}\subseteq
A_{5}\end{array}\par\right\\},$
---

but algebras $\mu_{21}^{i},\mu_{21}^{-i}$ and ${\mathfrak{V}}_{2+3}$ not.
Which follows that $\mu_{21}^{i},\mu_{21}^{-i}$ and ${\mathfrak{V}}_{2+3}$ are
not in the orbit closure of $\mu_{1,4}^{5}.$

  2. (2)

Let us choose the following new basis for the algebras $\lambda_{6}^{\alpha},$
$\mu_{11},$ $\mu_{15},$ $\mu_{17},$ $\mu_{18},$ $\mu_{20}$ and
$\mu_{22}^{\alpha}:$

$E_{1}=e_{1},\ E_{2}=e_{4},\ E_{3}=e_{5},\ E_{4}=e_{2},\ E_{5}=e_{3}.$

Below we have the special conditions which are satisfying the cited algebras
in the new basis:

${\mathcal{R}}_{\lambda_{6}^{\alpha}}=\left\\{\begin{array}[]{l}A_{1}^{2}\subseteq
A_{4}\end{array}\par\right\\}$
---
${\mathcal{R}}_{\mu_{11}}=\left\\{\begin{array}[]{l}A_{1}^{2}\subseteq A_{3},\
A_{1}A_{3}+A_{3}A_{1}\subseteq A_{5},\ A_{2}A_{3}=0,\\\
c_{14}^{5}=c_{41}^{5},\
c_{11}^{3}c_{22}^{4}=c_{12}^{3}c_{21}^{4}=c_{21}^{3}c_{12}^{4},\
c_{22}^{3}c_{12}^{4}=c_{22}^{4}c_{12}^{3},\
c_{12}^{3}c_{21}^{3}=c_{11}^{3}c_{22}^{3}\end{array}\par\right\\}$
${\mathcal{R}}_{\mu_{15}}=\left\\{\begin{array}[]{l}A_{1}^{2}\subseteq A_{3},\
A_{1}A_{3}\subseteq A_{5},\ A_{3}^{2}+A_{2}A_{4}+A_{4}A_{2}=0,\\\
c_{12}^{3}=c_{21}^{3},\
c_{12}^{4}c_{22}^{3}=c_{12}^{3}c_{22}^{4}\par\end{array}\par\right\\}$
${\mathcal{R}}_{\mu_{17}}=\left\\{\begin{array}[]{l}A_{1}^{2}\subseteq A_{3},\
A_{2}^{2}\subseteq A_{4},\ \ A_{1}A_{3}+A_{3}A_{1}\subseteq A_{5},\\\
c_{12}^{3}=c_{21}^{3},\ c_{12}^{4}=c_{21}^{4},\ c_{23}^{5}=c_{32}^{5},\
c_{14}^{5}=c_{41}^{5},\ c_{13}^{5}=c_{31}^{5}\end{array}\par\right\\}$
${\mathcal{R}}_{\mu_{18}}=\left\\{\begin{array}[]{l}A_{1}^{3}\subseteq A_{3},\
A_{2}^{4}\subseteq A_{4},\ A_{1}A_{3}+A_{3}A_{1}\subseteq A_{5},\\\
c_{21}^{3}=-c_{12}^{3},\ c_{11}^{3}=0,\
c_{23}^{5}=c_{32}^{5}\end{array}\par\right\\}$
${\mathcal{R}}_{\mu_{20}}=\left\\{\begin{array}[]{l}A_{1}^{3}\subseteq A_{3},\
A_{1}A_{3}+A_{3}A_{1}\subseteq A_{5},\ A_{2}A_{4}=0,\\\
c_{21}^{3}=c_{12}^{3},\ c_{21}^{4}=c_{12}^{4},\ c_{14}^{5}=c_{41}^{5},\
c_{23}^{5}=c_{32}^{5}\end{array}\par\right\\}$
${\mathcal{R}}_{\mu_{22}^{\alpha}}=\left\\{\begin{array}[]{l}A_{1}^{3}\subseteq
A_{3},\ A_{1}A_{3}+A_{3}A_{1}\subseteq A_{5},\\\
(c_{12}^{3}c_{21}^{4}-c_{12}^{4}c_{21}^{3})(c_{11}^{4}(c_{12}^{3})^{2}-c_{11}^{3}c_{12}^{3}c_{12}^{4}-c_{11}^{4}(c_{21}^{3})^{2}+c_{11}^{3}c_{21}^{3}c_{21}^{4})=\\\
\lx@intercol\hfil
c_{22}^{3}(c_{11}^{4}c_{12}^{3}-c_{11}^{3}c_{12}^{4}-c_{11}^{4}c_{21}^{3}+c_{11}^{3}c_{21}^{4})^{2}\lx@intercol,\\\
(c_{12}^{3}c_{21}^{4}-c_{12}^{4}c_{21}^{3})(c_{11}^{4}c_{12}^{3}c_{12}^{4}-c_{11}^{3}(c_{12}^{4})^{2}-c_{11}^{4}c_{21}^{3}c_{21}^{4}+c_{11}^{3}(c_{21}^{4})^{2})=\\\
\lx@intercol\hfil
c_{22}^{4}(c_{11}^{4}c_{12}^{3}-c_{11}^{3}c_{12}^{4}-c_{11}^{4}c_{21}^{3}+c_{11}^{3}c_{21}^{4})^{2}\lx@intercol\par\par\end{array}\par\right\\}$

But algebras $\mu_{21}^{i},\mu_{21}^{-i}$ and ${\mathfrak{V}}_{2+3}$ are not
satisfying it. Which follows that $\mu_{21}^{i},\mu_{21}^{-i}$ and
${\mathfrak{V}}_{2+3}$ are not in the orbit closure of these algebras. Hence,
$\mu_{21}^{i}$ and $\mu_{21}^{-i}$ give two new irreducible components.

  3. (3)

For the rest of our proof we should to obtain conditions for the following two
non-degenerations $\mu_{21}^{\pm i}\not\to{\mathfrak{V}}_{2+3},$ which will
complete the proof of the Theorem. The necessary conditions are given below.

${\mathcal{R}}_{\mu_{21}^{-i}}=\left\\{\begin{array}[]{l}A_{1}^{3}\subseteq
A_{3},\ A_{1}A_{3}+A_{3}A_{1}\subseteq A_{5},\\\
2c_{22}^{3}c_{11}^{3}=(c_{12}^{3})^{2}+(c_{21}^{3})^{2},\\\
2c_{11}^{4}(c_{22}^{3})^{2}=(c_{12}^{3}+ic_{21}^{3})(2c_{22}^{3}c_{12}^{4}-c_{22}^{4}c_{12}^{3}-ic_{22}^{4}c_{21}^{3}),\\\
2c_{22}^{4}(c_{11}^{3})^{2}=(c_{12}^{3}-ic_{21}^{3})(2c_{11}^{3}c_{12}^{4}-c_{11}^{4}c_{12}^{3}+ic_{11}^{4}c_{21}^{3}),\\\
c_{22}^{3}(c_{21}^{4}-ic_{12}^{4})=c_{22}^{4}(c_{21}^{3}-ic_{12}^{3}),\\\
c_{11}^{3}(c_{21}^{4}-ic_{12}^{4})=c_{11}^{4}(c_{21}^{3}-ic_{12}^{3})\\\
\end{array}\par\right\\}$
---
${\mathcal{R}}_{\mu_{21}^{i}}=\left\\{\begin{array}[]{l}A_{1}^{3}\subseteq
A_{3},\ A_{1}A_{3}+A_{3}A_{1}\subseteq A_{5},\\\
2c_{22}^{3}c_{11}^{3}=(c_{12}^{3})^{2}+(c_{21}^{3})^{2},\\\
2c_{11}^{4}(c_{22}^{3})^{2}=(c_{12}^{3}-ic_{21}^{3})(2c_{22}^{3}c_{12}^{4}-c_{22}^{4}c_{12}^{3}+ic_{21}^{3}c_{22}^{4}),\\\
2c_{22}^{4}(c_{11}^{3})^{2}=(c_{12}^{3}+ic_{21}^{3})(2c_{11}^{3}c_{12}^{4}-c_{11}^{4}c_{12}^{3}-ic_{21}^{3}c_{11}^{4}),\\\
c_{22}^{3}(c_{21}^{4}+ic_{12}^{4})=c_{22}^{4}(c_{21}^{3}+ic_{12}^{3}),\\\
c_{11}^{3}(c_{21}^{4}+ic_{12}^{4})=c_{11}^{4}(c_{21}^{3}+ic_{12}^{3})\\\
\end{array}\par\right\\}$

$\Box$

######  Remark.

Our Theorem C talks that the variety of complex $5$-dimensional nilpotent
associative algebras has $14$ irreducible components. We defined all
dimensions of irreducible components, generic and rigid algebras in this
variety. It is improving some early results in this direction. Namely, the
result from [23] talks that in this variety there are only 13 irreducible
components. Also, the result from [23] does not talk about dimensions of
irreducible components, generic algebras, and rigid algebras of this variety.
On the other hand, there are some inaccuracies in the definition of
irreducible components in [23]. For example, the algebra from the case [(a),
page 36], has nilindex 2 and the basis $\\{x,x^{2},y,xy,x^{2}y\\}.$

## References

  * [1] Alvarez M.A., Degenerations of $8$-dimensional $2$-step nilpotent Lie algebras, Algebras and Representation Theory, 2020, DOI: 10.1007/s10468-020-09987-5
  * [2] Autenried C., Furutani K., Markina I., Vasil’ev A., Pseudo-metric 2-step nilpotent Lie algebras, Advances in Geometry, 18 (2018), 2, 237–263.
  * [3] Calderón A., Fernández Ouaridi A., Kaygorodov I., On the classification of bilinear maps with radical of a fixed codimension, Linear and Multilinear Algebra, 2020, DOI:10.1080/03081087.2020.1849001
  * [4] Burde D., Steinhoff C., Classification of orbit closures of $4$–dimensional complex Lie algebras, Journal of Algebra, 214 (1999), 2, 729–739.
  * [5] Chouhy S., On geometric degenerations and Gerstenhaber formal deformations, Bulletin of the London Mathematical Society, 51 (2019), 5, 787–797.
  * [6] Cibils C., $2$-nilpotent and rigid finite-dimensional algebras, Journal of the London Mathematical Society (2), 36 (1987), 2, 211–218.
  * [7] Flanigan F. J., Algebraic geography: Varieties of structure constants, Pacific Journal of Mathematics, 27 (1968), 71–79.
  * [8] Farinati M., Jancsa A., Lie bialgebra structures on $2$-step nilpotent graph algebras, Journal of Algebra, 505 (2018), 70–91.
  * [9] Gabriel P., Finite representation type is open, Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), pp. 132–155.
  * [10] Gerstenhaber M., On the deformation of rings and algebras, Annals of Mathematics (2), 79 (1964), 59–103.
  * [11] Gorbatsevich V., Anticommutative finite-dimensional algebras of the first three levels of complexity, St. Petersburg Mathematical Journal, 5 (1994), 505–521.
  * [12] Grunewald F., O’Halloran J., Varieties of nilpotent Lie algebras of dimension less than six, Journal of Algebra, 112 (1988), 2, 315–325.
  * [13] Horn R., Sergeichuk V., Canonical matrices of bilinear and sesquilinear forms, Linear Algebra and its Applications, 428 (2008), 193–223.
  * [14] Ismailov N., Kaygorodov I., Mashurov F., The algebraic and geometric classification of nilpotent assosymmetric algebras, Algebras and Representation Theory, 24 (2021), 1, 135–148.
  * [15] Karimjanov I., The classification of 5-dimensional complex nilpotent associative algebras, Communications in Algebra, 49 (2021), 3, 915–931.
  * [16] Karimjanov I., Ladra M., Some classes of nilpotent associative algebras, Mediterranean Journal of Mathematics, 17 (2020), 70.
  * [17] Kaygorodov I., Lopes S., Popov Yu., Degenerations of nilpotent associative commutative algebras, Communications in Algebra, 48 (2020), 4, 1632–1639.
  * [18] Kaygorodov I., Khrypchenko M., Lopes S., The geometric classification of nilpotent algebras, arXiv:2102.10392
  * [19] Kaygorodov I., Khrypchenko M., Popov Yu., The algebraic and geometric classification of nilpotent terminal algebras, Journal of Pure and Applied Algebra, 225 (2021), 6, 106625.
  * [20] Kaygorodov I., Popov Yu., Pozhidaev A., Volkov Yu., Corrigendum to ”Degenerations of Zinbiel and nilpotent Leibniz algebras”, Linear and Multilinear Algebra, 2020, DOI: 10.1080/03081087.2020.1749543
  * [21] Kaygorodov I., Volkov Yu., Complete classification of algebras of level two, Moscow Mathematical Journal, 19 (2019), 3, 485–521.
  * [22] Lauret J., Oscari D., On non-singular $2$-step nilpotent Lie algebras, Mathematical Research Letters, 21 (2014), 3, 553–583.
  * [23] Makhlouf A., The irreducible components of the nilpotent associative algebras, Revista Matemática Complutense, 6 (1993), 1, 27–40.
  * [24] Mazzola G., The algebraic and geometric classification of associative algebras of dimension five, Manuscripta Mathematica, 27 (1979), 1, 81–101.
  * [25] Shafarevich I., Deformations of commutative algebras of class $2,$ Leningrad Mathematical Journal, 2 (1991), 6, 1335–1351.


# The systematic error and the anomaly in the sensitivity and the probability
of post-selection raised from the plural weak value

Jing-Hui Huang1,5  Xue-Ying Duan2,3,4  Guang-Jun Wang2,3,4 and Xiang-Yun Hu1,5
1 Institute of Geophysics and Geomatics, China University of Geosciences,
Wuhan 430074, China
2School of Automation, China University of Geosciences, Wuhan 430074, China
3Hubei Key Laboratory of Advanced Control and Intelligent Automation for
Complex Systems, Wuhan 430074, China
4 Engineering Research Center of Intelligent Technology for Geo-Exploration,
Ministry of Education, China
5 State Key Laboratory of Geological Processes and Mineral Resources, China
University of Geosciences, Wuhan, Hubei, 430074, China<EMAIL_ADDRESS>
###### Abstract

Recently, weak value $\rm A_{w}$ derived in the pre- and post-selected weak
measurement has been shown to be powerful in measuring minute physical
effects. In principle, the decrease in the post-selection probability will
increase the sensitivity. Besides, the sensitivity which characterizes the
pointer position shift is proportional to the real part of $\rm A_{w}$, and
the sensitivity which characterizes the pointer shift in momentum space is
proportional to the imaginary part of $\rm A_{w}$. However, we find that the
relationship between the post-selection probability and the sensitivity is
true when $\rm A_{w}$ is a purely real number or a purely imaginary number.
The plural $\rm A_{w}$ will lead to the abnormal behavior where the decrease
in the post-selection probability reduce the sensitivity at certain pre- and
post-selection. In addition, considering the device imperfections and the
environmental instability in the WVA protocol, this anomaly raised from plural
$\rm A_{w}$ will reduce the sensitivity and generate a systematic error of the
measurement compared with the original scheme. Finally, three feasible methods
are proposed to reduce these negative effects when the weak measurement
inevitably changes into measurement with a plural weak value.

††journal: osajournal††articletype: Research Article

##  1 Introduction

High precision phase measurements with optical interferometers play a
significant role in both classical measurements[1, 2] and quantum
measurements[3, 4, 5, 6]. Numerous studies show that the quantum measurements
with weak value amplification(WVA) can beat the classical schemes with high
sensitivity and good robustness[7, 8, 9, 10]. The concept of weak value in
weak measurement was first derived from the work on the arrow of time in
quantum theory by Aharonov, Albert and Vaidman in 1988[11]. In particular, the
weak value $\rm A_{w}$ derived by a weak measurement on pre-selected and post-
selected can be arbitrarily large, where the coupling strength between the
system and the meter is sufficiently small[12]. Thus, the famous example of
the weak measurement of a spin particle leading to a value of 100[11] has
opened up a gate for the WVA in the estimation of small parameters, such as
ultra-sensitive beam deflection measurement[7], velocity measurements[3],
ultra-small polarization rotation[13], rotation velocity[10], single-photon
nonlinearity[14], precision thermometry[15].

In principle, the information is gained at principle at the price of
disturbing the system through the interaction and the system will be projected
onto certain eigenstates in quantum mechanics. However, in weak measurement,
the disturbance to the system is reduced at the cost of a similar reduction in
the amount of information provided by the measurement[16]. In this case, the
probe shifts, averaged over many measurement repetitions, have been amplified
by the weak value $\rm A_{w}$ of the observable $\hat{A}$:
$A_{w}:={\langle{\Phi_{f}}|\hat{A}|{\Phi_{i}}\rangle}/{\langle{\Phi_{f}}|{\Phi_{i}}\rangle},$
where $|{\Phi_{i}}\rangle$ and $|{\Phi_{f}}\rangle$ are the pre-selected and
post-selected states of the system. It has been suggested that a standard
interferometer greatly outperforms weak measurement in a scenario involving a
purely real weak value, but the measurement of a phase using a purely
imaginary weak value can outperform a standard interferometer[12]. Hence,
anomalous works[17, 18, 19, 20] of amplifying the detector signals based on
weak measurement with a purely imaginary weak value. So far, the weak
measurements involving a plural weak value have been seldom reported.

In realistic optical interferometers, the pre-selected and post-selected
states of the system are normally obtained by polarizers or the combination of
wave-plate and polarizer. Due to device imperfections and environmental
instability, the pre-selected state $|{\Phi_{i}}\rangle$ and post-selected
state $|{\Phi_{f}}\rangle$ of the system may deviate from the desired quantum
states. In our work, we find that this deviation will change the purely real
or imaginary weak value to a plural one. Specifically, we study three
measurements for detecting a small phase shift introduced by a birefringent
element involving (a) weak measurements with a purely real $\rm A_{w}$, (b)
weak measurements with a purely imaginary $\rm A_{w}$, (c) weak measurements
with a plural $\rm A_{w}$. As a result, in the scheme (a), the lower
probability of post-selection
$\mathcal{P}=|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}$ increases the
sensitivity(proportional to the real part $\rm Re$ $\rm[A_{w}]$ of $\rm
A_{w}$); and the lower $\mathcal{P}$ leads to the higher
sensitivity(proportional to the imaginary part $\rm Im$ $\rm[A_{w}]$ of $\rm
A_{w}$) in the scheme (b). Nevertheless, an anomaly in the sensitivity and the
probability of post-selection will be raised from the plural weak value, and
the anomaly will cause systematic errors when $\mathcal{P}$ is small enough.

This paper is organized as follows. In section 2, we propose the theoretical
scheme of (a) a weak measurement with a purely real $\rm A_{w}$, (b) a weak
measurement with a purely imaginary $\rm A_{w}$ and (c) a weak measurement a
plural $\rm A_{w}$ to measure time delay induced by a birefringent element.
Then in section 3, numerically, we show these results of the three schemes and
analyze the anomaly in the sensitivity and the probability of post-selection
in scheme 3. Then, in section 4, we analyze the systematic error caused by the
anomaly and put forward some possible solutions to reduce the error. Finally,
a brief conclusion is presented in section 5.

##  2 Schemes of weak measurements involving weak values with different
complex properties

Figure 1: Schemes of weak measurements involving weak values with different
complex properties.

To compare the three schemes of weak measurements involving a purely real $\rm
A_{w}$, a purely imaginary $\rm A_{w}$ and a plural $\rm A_{w}$, we propose
three schemes of a two-level system($|{\Phi}\rangle$ is a component of the
horizontal polarized state $|{H}\rangle$ and the vertical polarized state
$|{V}\rangle$) in Fig. 1 to detect the same time delay $\tau$ induced by a
birefringent element with different pre- and post-selected states of the
system, where the time delay $\tau$ can be measured serves as the coupling
strength of the interaction between the system $|{\Phi}\rangle$ and the probe
$|{\Psi}\rangle$ in standard weak measurements. Thus, the interaction
Hamiltonian $\hat{H}=\tau\hat{A}\otimes\hat{p}$ leads to a unitary operator

$\displaystyle\hat{U}=e^{-i\int\hat{H}dt}=e^{-i\tau\hat{A}\otimes\hat{p}}\,,$
(1)

where $\hat{A}=|{H}\rangle\langle{H}|-|{V}\rangle\langle{V}|$ is the
observable operator, and $\hat{p}$ is the probe momentum operator conjugate to
the position operator $\hat{q}$. In principle, the time delay $\tau$ can be
obtained by the shifts of the probe function $|{\Psi}\rangle$, and the probe
function after the post-selection will take the form:

$\displaystyle|{\Psi_{f}}\rangle$ $\displaystyle=$
$\displaystyle\langle{\Phi_{f}}|\hat{U}|{\Phi_{i}}\rangle|{\Psi_{i}}\rangle$
(2) $\displaystyle=$
$\displaystyle\langle{\Phi_{f}}|e^{-i\tau\hat{A}\otimes\hat{p}}|{\Phi_{i}}\rangle|{\Psi_{i}}\rangle$
$\displaystyle=$
$\displaystyle\langle{\Phi_{f}}|\left[1-i\tau\hat{A}\otimes\hat{p}\right]|{\Phi_{i}}\rangle|{\Psi_{i}}\rangle+O(\tau^{2})$
$\displaystyle=$
$\displaystyle\langle{\Phi_{f}}|{\Phi_{i}}\rangle\left[1-i\tau
A_{w}\hat{p}\right]|{\Psi_{i}}\rangle+O(\tau^{2})$ $\displaystyle=$
$\displaystyle\langle{\Phi_{f}}|{\Phi_{i}}\rangle e^{-i\tau
A_{w}\hat{p}}|{\Psi_{i}}\rangle+O(\tau^{2})\,,$

where $O(\tau^{2})$ is normally neglected when the coupling strength is weak
for $\tau|A_{w}|\ll 1$. Choosing the Gaussian form with the variance $W^{-2}$
for the initial probe function $|{\Psi_{i}}\rangle$ in the position or
momentum spaces, the weak value can be obtained from the shifts of the
expectation value of the probe momentum p and the probe position q as [21]

$\Delta\langle\hat{q}\rangle=\frac{\int dqq\left|\left\langle
q\mid\Psi_{f}\right\rangle\right|^{2}}{\int dq\left|\left\langle
q\mid\Psi_{f}\right\rangle\right|^{2}}=\tau{\rm Re}\,\rm[A_{w}]\,,$ (3)
$\Delta\langle\hat{p}\rangle=\frac{\int dpp\left|\left\langle
p\mid\Psi_{f}\right\rangle\right|^{2}}{\int dp\left|\left\langle
p\mid\Psi_{f}\right\rangle\right|^{2}}=2\tau W^{2}{\rm Im}\,\rm[A_{w}]\,.$ (4)

Equation (3) shows that the shifts of the expectation value of the probe
position are proportional to the real part $\rm Re\,[A_{w}]$ of the weak
value. And equation (4) shows that the shifts of the expectation value of the
probe momentum are proportional to the imaginary part $\rm Im\,[A_{w}]$ of the
weak value. As a trade-off, the probability
$\mathcal{P}=|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}$ of obtaining a post-
selected state that is almost orthogonal to the pre-selected state is very
small. Note that the complex properties of $\rm\,[A_{w}]$ are defined by the
pre- and post-selected states of the system. In the following, we will show
the theoretical scheme of (a) a weak measurement with a purely real $\rm
A_{w}$, (b) a weak measurement with a purely imaginary $\rm A_{w}$ and (c) a
weak measurement a plural $\rm A_{w}$ to measure time delay induced by a
birefringent element.

###  2.1 Scheme of weak measurements involving a purely real weak value

In order to realize the weak measurement involving a purely real $\rm A_{w}$,
the experimental model is shown in Fig. 1(a). Optical modulator generates a
Gaussian-shaped pulse in the time domain, which travels through the first
polarizer (P1) that pre-selected the system in the polarized state
$|{\Phi_{i}}\rangle={\rm sin}(\frac{\pi}{4}+\alpha)|{H}\rangle+{\rm
cos}(\frac{\pi}{4}+\alpha)|{V}\rangle$, after which it enters a birefringent
element to perturb the system weakly. Afterwards, the second polarizer (P2)
post-select the system at $|{\Phi_{f}}\rangle={\rm
sin}(-\frac{\pi}{4})|{H}\rangle+{\rm cos}(-\frac{\pi}{4})|{V}\rangle$. The
corresponding time shifts in the peak out intensity $\delta t=\tau{\rm
Re}\,\rm[A_{w}]$ are detected by an avalanche photodiode (APD), which is
proportional to ${\rm Re}\,\rm[A_{w}]$. Note that the $\rm[A_{w}]$ can be
calculated by

$A_{w,1}=\frac{\langle{\Phi_{f}}|\hat{A}|{\Phi_{i}}\rangle}{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}=\frac{-{\rm
sin}(\frac{\pi}{4}+\alpha)-{\rm cos}(\frac{\pi}{4}+\alpha)}{-{\rm
sin}(\frac{\pi}{4}+\alpha)+{\rm cos}(\frac{\pi}{4}+\alpha)}={\rm
cot}\alpha\,,$ (5)

and Eq. (5) indicates ${\rm Re}\,\rm[A_{w}]$ = ${\rm Ab}\,\rm[A_{w}]$, where
${\rm Ab}\,\rm[A_{w}]$ represents the modulus of $\rm A_{w}$. Meanwhile, the
probability of post-selection is given by

$|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}_{1}={\rm sin}^{2}\alpha\,.$ (6)

In scheme (a), the sensitivity $\delta t/\tau$ is proportional to $\rm
Re\,\rm[A_{w}]$, and the relationship between $\rm A_{w}$ and the probability
of post-selection is shown in Fig. 2(a).

###  2.2 Scheme of weak measurements involving a purely imaginary weak value

Fig. 1(b) depicts the scheme involving a purely imaginary $\rm A_{w}$ in
momentum space. Normally, a white laser and a following Gaussian filter
together constitute the probe state. Then, the light beam passes the first
polarizer (P1) which pre-selected the system at the state
$|{\Phi_{i}}\rangle=\frac{1}{\sqrt{2}}(|{H}\rangle+|{V}\rangle)$. perturb the
system weakly. A birefringent element introduces a time delay $\tau$, which
corresponds to the coupling strength. Following these works[22, 18, 20] to
realize the post-selection, a quarter-wave plate (QWP) and the second
polarizer (P2) post-select the system at the state,

$|{\Phi_{f}}\rangle\simeq\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&-i\\\
-i&1\end{array}\right)\left(\begin{array}[]{c}\cos(\beta-\pi/4)\\\
\sin(\beta-\pi/4)\end{array}\right)=\frac{1}{\sqrt{2}}[\exp(-i\beta)|H\rangle-\exp(i\beta)|V\rangle]\,,$
(7)

where $\beta$ is the angle between the optical axes of P2 and QWP. Hence, $\rm
A_{w}$ and the probability of post-selection can be obtained immediately by
that:

$A_{w,2}=\frac{\langle{\Phi_{f}}|\hat{A}|{\Phi_{i}}\rangle}{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}=-i{\rm
cot}\beta\,,$ (8) $|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}_{2}={\rm
sin}^{2}\beta\,.$ (9)

Note that scheme (2) involves a purely imaginary weak value with a probe in
the frequency(momentum) domain. Therefore, the sensitivity $\delta p/\tau$ is
proportional to $\rm Im\,\rm[A_{w}]={\rm cot}\beta$, and we display the
relationship between $\rm A_{w}$ and the probability of post-selection in Fig.
2(b).

###  2.3 Scheme of weak measurements involving a plural weak value

In this subsection, the scheme of a weak measurement a plural $\rm A_{w}$ is
shown in Fig. 1(c). The main difference comparing to scheme (2) is the pre-
selected state of the system controlled by P1:

$|{\Phi_{i}}\rangle={\rm sin}(\alpha+\frac{\pi}{4})|{H}\rangle+{\rm
cos}(\alpha+\frac{\pi}{4})|{V}\rangle\,.$ (10)

and the pre-selected sate
$|{\Phi_{f}}\rangle=\frac{1}{\sqrt{2}}[\exp(-i\beta)|H\rangle-\exp(i\beta)|V\rangle]$
of the system keeps the same with the form in scheme (2). Due to the deviation
on the pre-selected state of the system, $\rm A_{w}$ and
$|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}$ are calculated as follows:

$A_{w,3}=\frac{\langle{\Phi_{f}}|\hat{A}|{\Phi_{i}}\rangle}{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}=\frac{({\rm
sin}\alpha+{\rm cos}\alpha){\rm exp}(-i\beta)+(-{\rm sin}\alpha+{\rm
cos}\alpha){\rm exp}(i\beta)}{({\rm sin}\alpha+{\rm cos}\alpha){\rm
exp}(-i\beta)-(-{\rm sin}\alpha+{\rm cos}\alpha){\rm exp}(i\beta)}\,,$ (11)
$|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}_{3}=\frac{1}{4}[({\rm
sin}\alpha+{\rm cos}\alpha)^{2}+(-{\rm sin}\alpha+{\rm cos}\alpha)^{2}-2({\rm
cos}^{2}\alpha-{\rm sin}^{2}\alpha){\rm cos}2\beta]\,.$ (12)

The theoretical formula (11) indicates that the deviation on
$|{\Phi_{f}}\rangle$ will lead to a plural weak value in the scheme (3). we
consider a deviation on the pre-selected state of the system through
introducing a small deflection $\alpha$ of the first polarizer (P1), and a
deviation on the post-selected state of the system through introducing a small
deflection $\beta$ of the second polarizer (P2). The scheme with the light
followed by optical modulator is compared to the scheme (1) involving a
temporal probe. Note that when $\beta\to 0$, the scheme (3) will be the same
as the scheme (1) and lead to the same $A_{w}$ and
$|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}$:

$\left\\{\begin{array}[]{l}A_{w,3}=A_{w,1}\\\
|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}_{3}=|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}_{1}\end{array}\quad\text{
when }\beta\rightarrow 0\,.\right.$ (13)

In addition, another scheme with the light followed by a Gaussian filter is
compared to the scheme (2) involving a frequency-domain probe. Accordingly,
when $\beta\to 0$, the scheme (3) will be the same as the scheme (2) and lead
to the same $A_{w}$ and $|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}$:

$\left\\{\begin{array}[]{l}A_{w,3}=A_{w,2}\\\
|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}_{3}=|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}_{2}\end{array}\quad\text{
when }\alpha\rightarrow 0\,.\right.$ (14)

On the other hand, the scheme (a) will translate the scheme (c) mathematically
due to the deviation. Although this translation can not occur in real optical
systems as shown in Fig. 1, in this paper we just compare it with the weak
measurement with a purely imaginary $\rm A_{w}$. Nevertheless, the translation
from the scheme (c) to the scheme (d) will occur in both mathematically and in
real optical systems. In the next section, we will show how the deviation of
the states of the system affects the weak measurement.

Figure 2: The results of our simulation experiment with different models.

##  3 Results

In order to discuss the impact of the deviation on the weak measurement in
position or momentum space. We numerically simulate the results of scheme(a),
scheme(b) and scheme(c). Thereinto, the scheme(c) with the deflection
($\beta$=0.0002 rad and 0.002 rad) of P2 serves as the deviation of scheme(a)
involving a weak measurement with a purely real $\rm A_{w}$, and the scheme(c)
with the deflection ($\alpha$=0.0002 rad and 0.002 rad) of P1 serves as the
deviation of the scheme(b) involving a weak measurement with a purely
imaginary $\rm A_{w}$. The relevant results are shown in Fig. 2.

Comparing the results(left panel of Fig. 2) of the scheme(c) with the
deflection ($\beta$=0.0002, 0.002 rad) and the scheme(a) with a purely real
$\rm A_{w}$, we find that the deflection on $\beta$ will lead to the weak
measurement involving a plural weak value. At this time the function of $\rm
Re$ $\rm[A_{w}]$ in the scheme (c) is not a monotone function of $\alpha$, and
the anomaly in the sensitivity and the probability of post-selection is raised
when $\alpha$ is small on the order of the deflection. For instance, in the
scheme (c) with $\beta$= 0.0002 rad, the sensitivity (proportional to $\rm Re$
$\rm[A_{w}]$) decreases as the probability of post-selection
$\mathcal{P}=|{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}|^{2}$ decreases when
$\alpha$ < 0.0002 rad, while $\rm Re$ $\rm[A_{w}]$ increases as $\mathcal{P}$
decreases when $\alpha$ < 0.0002 rad. Although the anomaly in scheme (c) with
$\beta$= 0.0002 rad will not appear due to the value of $\alpha$ is going to
be much larger than 0.0002 rad in a real experiment, the anomaly in scheme (c)
with $\beta$= 0.002 rad or bigger $\beta$ will occur when $\alpha$ is set on
the order of the deflection($\beta$= 0.002 rad).

Meanwhile, similar results of the comparison of the scheme(c) with the
deflection ($\alpha$=0.0002, 0.002 rad) and the scheme(b) with a purely
imaginary $\rm A_{w}$ can obtain from Fig. 2. In particular, the contrast of
$\rm Im$ $\rm[A_{w}]$ and $\rm Ab$ $\rm[A_{w}]$with different $\alpha$ are
shown in Fig. 3, where $\rm Im$ $\rm[A_{w}]$ with $\alpha$= 0.000 rad is
corresponding to the result of the scheme(b) with a purely imaginary $\rm
A_{w}$. Considering $\beta$ set to be 0.002 rad in order to obtain the value
of $\rm Im$ $\rm[A_{w}]$, there is no difference between $\rm Im$ $\rm[A_{w}]$
with $\alpha$= 0.000 rad and $\rm Im$ $\rm[A_{w}]$ with $\alpha$= 0.0002 rad.
Nevertheless, the significant difference between $\rm Im$ $\rm[A_{w}]$ with
$\alpha$= 0.000 rad and $\rm Im$ $\rm[A_{w}]$ with $\alpha$= 0.002 rad
appears. Note that the deflection on P1 in the scheme (b) will happen in real
experiments due to the device imperfections and environmental instability, and
this deflection on P1 will change the scheme (b) into the scheme (c) involving
a plural weak value. And this anomaly raised from the plural $\rm A_{w}$ will
cause sensitivity decreases as compared with original design indicators.

Figure 3: The comparison of the scheme(c) with the deflection
($\alpha$=0.0002, 0.002 rad) and the scheme(b) with a purely imaginary $\rm
A_{w}$, where $\alpha$=0.000 rad is corresponding to the scheme (b).

In addition, an interesting conclusion is obtained from the results with a
plural weak value. In Fig. 2, we find that when the weak measurement is
realized involving a plural weak value, the probability of post-selection
$\mathcal{P}$ is inversely proportional to the modulus of the weak value $\rm
Ab$ $\rm[A_{w}]$, rather than $\rm Re$ $\rm[A_{w}]$ or $\rm Im$ $\rm[A_{w}]$.
And this anomaly goes against previous experience which reducing the
probability of post-selection $\mathcal{P}$ can improve the sensitivity of the
weak measurement with a pure real $\rm[A_{w}]$ or a pure imaginal
$\rm[A_{w}]$. We hope that this anomaly can attract the attention of the WVA
community, from both experimental and theoretical interests.

##  4 Systematic error analysis

Figure 4: Systematic error err($\beta$) on the measurement of $\beta$= 0.002
rad with the anomaly at $\alpha$= 0.002 rad.

In this section, we analyze the systematic error caused by the anomaly and put
forward some possible solutions to reduce the error. As an example, we analyze
the systematic error in the transformation from the measurement with scheme
(b) into the measurement with scheme (c). In particular, the anomaly appears
when we set a deflection $\alpha$= 0.002 rad on P1 in scheme (c), and the
deflection $\beta$ on P2 is the physical quantity to be measured, which can be
calculated from the relationship (11) between the $\beta$ and $\rm[A_{w}]$.
Thereinto, the $\rm Im$ $\rm[A_{w}]$ is proportional to the shifts (4) of the
mean momentum of photons. In the last, the systematic error err($\beta$) of
measuring the $\beta$ is shown in Fig. 4.

Figure 5: The weak value (the left panel) and systematic error err($\beta$)
(the right panel) dependence of $\alpha$ on the measurement of $\beta$= 0.002
rad .

The numerical simulation of systematic error of $\beta$ is shown in Fig. 4
when measuring the deflection $\beta$ on P2 by considering a deflection
$\alpha$= 0.002 rad on P1. Thereinto, the measurement with $\alpha$= 0.000 rad
is corresponding to the measurement of scheme (b) as the original design, and
the measurement with $\alpha$= 0.002 rad is corresponding to the measurement
of scheme (c) with systematic errors caused by the deflection on $\alpha$. By
assuming that the true value of $\beta$ is set at 0.002 rad and the value of
$\rm Im$ $\rm[A_{w}]$ is measured at $a_{1}$. However, because of ignoring the
deflection on $\alpha$ in scheme (b), the value of $\beta$ evaluates to 0.0039
rad with the systematic error $err(\beta)_{1}$= 0.0019 rad. On the other hand,
the sensitivity which is proportional to the value of $\rm Im$ $\rm[A_{w}]$
significantly decreases in scheme (c) as $a_{1}$ is much smaller than $a_{4}$.

To sum up, the systematic error caused by the deflection on the optical
element in scheme (B) will change the scheme to the scheme (C), as a result of
the sensitivity reduction and inaccuracy in the expected measurement with
scheme (B). In the next part we will put forward some possible solutions to
minish the systematic error:

1). The first plan that came up is to control the parameters of the optical
elements precisely even in an unstable environment. Fig. 5 shows that the
systematic error err($\beta_{1}$) falls off and the difference between $\rm
Im$ $\rm[A_{w}]$($\alpha$= 0.002 rad) and $\rm Im$ $\rm[A_{w}]$($\alpha$=
0.000 rad) is getting smaller as $\alpha$ decrease. In addition, with the aid
of the closed-loop control technique(CLCT) in the automatic control system,
CLCT[23, 24] is widely used to accurately control the parameters of the
optical elements. Therefore, we can use CLCT to consistently control the
polarization of P1 to eliminate the deflection.

2). Note that the difference $a_{4}$ \- $a_{2}$ between $\rm Ab$
$\rm[A_{w}]$($\alpha$= 0.002 rad) and $\rm Im$ $\rm[A_{w}]$($\alpha$= 0.000
rad) is smaller than the difference $a_{4}$ \- $a_{1}$ between $\rm Im$
$\rm[A_{w}]$($\alpha$= 0.002 rad) and $\rm Im$ $\rm[A_{w}]$($\alpha$= 0.000
rad), while the systematic error err($\beta_{2}$) estimated by $\rm Ab$
$\rm[A_{w}]$ is smaller than the systematic error err($\beta_{1}$) estimated
by $\rm Im$ $\rm[A_{w}]$. Similarly, following the method to estimate
err($\beta_{1}$) and err($\beta_{2}$) in the left panel of Fig. 4, the
err($\beta_{1}$) and err($\beta_{2}$) dependence of $\alpha$ are shown in the
right panel of Fig. 5. Therefore, using $\rm Ab$ $\rm[A_{w}]$ to calculate the
quantity $\beta$ can get a smaller systematic error err($\beta$) compared to
the traditional method with $\rm Im$ $\rm[A_{w}]$. In addition, it is feasible
and convenient to measure both the shifts of the position and the momentum of
the photon in practical measurements, and this method can effectively reduce
the systematic error at certain deflection on the optical element.

3). From Fig. 4 and Fig. 5(right panel), we can conclude that the systematic
error err($\beta_{3}$) estimated by the probability $\mathcal{P}$ of obtaining
a post-selected state is much smaller than err($\beta_{1}$) obtained by $\rm
Im$ $\rm[A_{w}]$. Note that Li etc.[18] have proposed a scheme called the weak
measurement with two pointers to measure time delays with the light frequency
and the post-selected probability $\mathcal{P}$ in dynamic range with high
precision. Thus, the similar scheme with measurement of the probability
$\mathcal{P}$ consulting their work[18] can availably reduce the systematic
error.

In addition, one thing to note in weak measurements is that estimating the
$\rm Im$ $\rm[A_{w}]$ or $\rm Re$ $\rm[A_{w}]$ in the scheme with a plural
weak value is improper, due to the abnormal non-monotone behaviour of $\rm Im$
$\rm[A_{w}]$ or $\rm Re$ $\rm[A_{w}]$. Note that the sensitivity in WVA is
proportional to $\rm Im$ $\rm[A_{w}]$ or $\rm Re$ $\rm[A_{w}]$, the abnormal
non-monotone behaviour will evaluate two physical quantities. As shown in Fig.
4(left panel), if the $\rm Im$ $\rm[A_{w}]$= $a_{5}$ is calculated from the
shift of the momentum of the photons, the value of $\beta$ to be measured will
get two values $\beta_{1}$ and $\beta_{2}$. In conclusion, the weak
measurement should be realized with a pure real $\rm[A_{w}]$ or pure imaginary
$\rm[A_{w}]$, and the above three methods are effective and necessary to
reduce systematic error and enhance the sensitivity when the weak measurement
changed to measurement with the plural $\rm[A_{w}]$.

##  5 Conclusion

We have found the systematic error and the anomaly in the sensitivity and the
probability of post-selection in weak measurement raised from a plural weak
value. When we realize a scheme of weak measurement with a pure real
$\rm[A_{w}]$ or a pure imaginal $\rm[A_{w}]$, the anomaly caused by the
deflection on polarizer will change the scheme into the measurement with a
plural weak value, and the anomaly will significantly reduce the sensitivity
and generate a systematic error of the measurement compared with the original
scheme.

In the end, we provide three feasible methods to reduce the systematic error
and decrease the sensitivity: minimizing the parameter error of optical
elements as much as possible by CLCT, using the modulus of $\rm[A_{w}]$ or the
probability of obtaining a post-selected state to estimate the physical
quantity to be measured rather than using the imaginal part of $\rm[A_{w}]$.
In addition, the joint measurement with the modulus of $\rm[A_{w}]$ and the
probability of obtaining a post-selected state will be our next step to reduce
these negative effects when the weak measurement inevitably changes into
measurement with a plural weak value.

##  6 Backmatter

Acknowledgments This study is financially supported by MOST Special Fund from
the State Key Laboratory of Geological Processes and Mineral Resources, China
University of Geosciences No. MSFGPMR01-4 and the Fundamental Research Funds
for National Universities, China University of Geosciences(Wuhan) (Grant No.
G1323519204).

Disclosures

The authors declare no conflicts of interest.

Data Availability Statement Data underlying the results presented in this
paper are not publicly available at this time but may be obtained from the
authors upon reasonable request.

## References

  * [1] Y. Li and L. Tong, “Mach-zehnder interferometers assembled with optical microfibers or nanofibers,” Opt. Lett. 33, 303–305 (2008).
  * [2] Y. Park, J. A. na, and R. Slavík, “Ultrafast all-optical first- and higher-order differentiators based on interferometers,” Opt. Lett. 32, 710–712 (2007).
  * [3] G. I. Viza, J. Martínez-Rincón, G. A. Howland, H. Frostig, I. Shomroni, B. Dayan, and J. C. Howell, “Weak-values technique for velocity measurements,” Opt. Lett. 38, 2949–2952 (2013).
  * [4] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
  * [5] X. Zuo, Z. Yan, Y. Feng, J. Ma, X. Jia, C. Xie, and K. Peng, “Quantum interferometer combining squeezing and parametric amplification,” Phys. Rev. Lett. 124, 173602 (2020).
  * [6] K. L. Dooley, E. Schreiber, H. Vahlbruch, C. Affeldt, J. R. Leong, H. Wittel, and H. Grote, “Phase control of squeezed vacuum states of light in gravitational wave detectors,” Opt. Express 23, 8235–8245 (2015).
  * [7] P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
  * [8] J. Ren, L. Qin, W. Feng, and X.-Q. Li, “Weak-value-amplification analysis beyond the aharonov-albert-vaidman limit,” Phys. Rev. A 102, 042601 (2020).
  * [9] F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, X. Y. Jin, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Efficient qubit measurement with a nonreciprocal microwave amplifier,” Phys. Rev. Lett. 126, 020502 (2021).
  * [10] Huang.Jing-Hui, D. Xue-Ying, and H. Xiang-Yun, “Amplification of rotation velocity using weak measurements in sagnac’s interferometer,” The European Physical Journal D 75, 114 (2021).
  * [11] Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
  * [12] N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: Weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).
  * [13] B. de Lima Bernardo, S. Azevedo, and A. Rosas, “Ultrasmall polarization rotation measurements via weak value amplification,” Physics Letters A 378, 2029–2033 (2014).
  * [14] A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
  * [15] A. K. Pati, C. Mukhopadhyay, S. Chakraborty, and S. Ghosh, “Quantum precision thermometry with weak measurements,” Phys. Rev. A 102, 012204 (2020).
  * [16] M. Hallaji, A. Feizpour, G. Dmochowski, J. Sinclair, and A. Steinberg, “Weak-value amplification of the nonlinear effect of a single photon,” Nature Physics 13, 540–544 (2017).
  * [17] Y. Yang, Y. Xu, T. Guan, L. Shi, J. Li, D. Li, Y. He, X. Wang, Z. Li, and Y. Ji, “Spectrum intensity ratio detection for frequency domain weak measurement system,” IEEE Photonics Journal 12, 1–12 (2020).
  * [18] Z. Li, L. Xie, Q. Ti, P. Duan, Z. Zhang, and C. Ren, “Increasing the dynamic range of weak measurement with two pointers,” Phys. Rev. A 102, 023701 (2020).
  * [19] P. Yin, W.-H. Zhang, L. Xu, Z.-G. Liu, W.-F. Zhuang, L. Chen, M. Gong, Y. Ma, X.-X. Peng, G.-C. Li, J.-S. Xu, Z.-Q. Zhou, L. Zhang, G. Chen, C.-F. Li, and G.-C. Guo, “Improving the precision of optical metrology by detecting fewer photons with biased weak measurement,” Light: Science & Applications 10, 103 (2021).
  * [20] J. Zhu, Z. Li, Y. Liu, Y. Ye, Q. Ti, Z. Zhang, and F. Gao, “Weak measurement with the peak-contrast-ratio pointer,” Phys. Rev. A 103, 032212 (2021).
  * [21] Y. Susa, Y. Shikano, and A. Hosoya, “Optimal probe wave function of weak-value amplification,” Phys. Rev. A 85, 052110 (2012).
  * [22] X. Qiu, L. Xie, X. Liu, L. Luo, Z. Li, Z. Zhang, and J. Du, “Precision phase estimation based on weak-value amplification,” Applied Physics Letters 110, 071105 (2017).
  * [23] J. Militky, M. Kadulova, and P. Hlubina, “Highly sensitive displacement measurement based on spectral interferometry and vernier effect,” Optics Communications 366, 335–339 (2016).
  * [24] J. Yan, L. Miao, M. Chen, T. Huang, S. Che, and X. Shu, “Research on the feedback control characteristics and parameter optimization of closed-loop fiber optic gyroscope,” Optik 229, 166298 (2021).


# Instantaneous convexity breaking for the quasi-static droplet model

Albert Chau  Department of Mathematics, The University of British Columbia,
1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2. Email:
<EMAIL_ADDRESS> and  Ben Weinkove  Department of Mathematics, Northwestern
University, 2033 Sheridan Road, Evanston, IL 60208, USA. Email:
<EMAIL_ADDRESS>
###### Abstract.

We consider a well-known quasi-static model for the shape of a liquid droplet.
The solution can be described in terms of time-evolving domains in
$\mathbb{R}^{n}$. We give an example to show that convexity of the domain can
be instantaneously broken.

Research supported in part by NSERC grant $\\#$327637-06 and NSF grant
DMS-2005311.

##  1\. Introduction

We consider the following system of equations for a function $u(x,t)$ and
domains $\Omega_{t}\subset\mathbb{R}^{n}$, for $t\geq 0$. This system is used
to model the quasi-static shape evolution of a liquid droplet of height
$u(x,t)$ occupying the region $\Omega_{t}$:

(1.1) $\begin{split}-\Delta u={}&\lambda_{t},\quad\textrm{on }\Omega_{t}\\\
u={}&0,\quad\textrm{on }\partial\Omega_{t}\\\ V={}&F(|Du|),\quad\textrm{on
}\partial\Omega_{t}\\\ \int_{\Omega_{t}}u\,dx={}&1.\end{split}$

In the above, $V$ is the velocity of the free boundary $\partial\Omega_{t}$ in
the direction of the outward unit normal and
$F:(0,\infty)\rightarrow\mathbb{R}$ is an analytic function with
$F^{\prime}(r)>0$ for $r>0$. The constant $\lambda_{t}>0$ is determined by the
integral condition on $u$.

The initial data is given by a domain $\Omega_{0}$ which we assume is bounded
with smooth boundary $\partial\Omega_{0}$. Note that the domains $\Omega_{t}$
(assuming they are bounded with sufficiently regular boundary
$\partial\Omega_{t}$) determine uniquely the solution $x\mapsto u(x,t)$. Thus
we may denote a solution of (1.1) by a family of evolving domains
$\Omega_{t}$. In Section 2 we will explain what is meant by a _classical
solution_ to this problem.

The system of equations (1.1) has long been accepted as a model for droplet
evolution in the physical literature [1, 5, 7, 10, 11]. There have been
results on weak formulations of this equation by Glasner-Kim [6] and
Grunewald-Kim [8]. Feldman-Kim [3] gave some conditions for global existence
and convergence to an equilibrium. Escher-Guidotti [2] proved a short time
existence result for classical solutions, which we describe in Section 2
below.

In this note we address the following natural question:

######  Question 1.1.

Is the convexity of $\Omega_{t}$ preserved by the system (1.1)?

This question is implicit in the work of Glasner-Kim [6]. It was raised
explicitly by Feldman-Kim [3, p.822], “Let us point out that, in particular,
it is unknown whether the convexity of the drop is preserved in the system
[(1.1)].”

In this note, we answer Question 1.1 by showing that convexity is _not_
generally preserved. We make an assumption on $F$, namely that

(1.2) $\lim_{r\rightarrow
0^{+}}\frac{F^{\prime\prime}(r)}{F^{\prime}(r)}\geq\gamma,\quad\textrm{for
some }\gamma>0.$

This includes the important cases $F(r)=r^{3}-1$ and $F(r)=r^{2}-1$ considered
in [6] and [3, 8] respectively.

We construct an example where $\Omega_{t}$ is convex for $t=0$, but not convex
for $t\in(0,\delta)$ for some $\delta>0$.

######  Theorem 1.1.

Assume $F$ satisfies assumption (1.2). There exists $\delta>0$ and a bounded
convex domain $\Omega_{0}\subset\mathbb{R}^{2}$ with smooth boundary such that
the solution $\Omega_{t}$ to (1.1) with this initial data is not convex for
any $t\in(0,\delta]$.

Escher-Guidotti [2] showed that as long as $\Omega_{0}$ is a bounded domain
with sufficiently smooth boundary, there always exists a unique classical
solution for a short time, and this is what is meant by “the solution
$\Omega_{t}$” in the statement of Theorem 1.1. In Section 2, we describe more
precisely the results of [2].

In Section 3 we give the proof of Theorem 1.1. The starting point is an
explicit solution of the equation $-\Delta u=\lambda_{0}$ on an equilateral
triangle [9]. We smooth out the corners to obtain our convex domain
$\Omega_{0}$, and show that it immediately breaks convexity.

##  2\. Short time existence

In this section, we recall the short time existence result of Escher-Guidotti
[2].

We first give a definition of a solution of (1.1), following [2]. Note that
the domains $\Omega_{t}$ determine uniquely the functions $u$, so we will
describe the solution of (1.1) in terms of varying domains - given as graphs
over the original boundary.

Fix $\alpha\in(0,1)$. Assume $\Omega_{0}$ is a bounded domain in
$\mathbb{R}^{n}$ whose boundary $\Gamma_{0}:=\partial\Omega_{0}$ is a smooth
hypersurface. Let $\nu(x)$ denote the unit outward normal to $\Gamma_{0}$ at
$x$. Then there exists a maximal constant $\sigma(\Omega_{0})>0$ such that for
any given function $\rho\in C^{2+\alpha}(\Gamma_{0})$ with
$\|\rho\|_{C^{1}(\Gamma_{0})}\leq\sigma$, the set

$\Gamma_{\rho}=\\{x+\rho(x)\nu(x)\ |\ x\in\Gamma_{0}\\},$

is a $C^{2+{\alpha}}$ hypersurface in $\mathbb{R}^{n}$ which is the boundary
of a bounded domain $\Omega=\Omega(\rho)$.

We can now describe a solution of (1.1) in terms of a time-varying family
$\rho(x,t)$. Namely, given

$\rho\in C([0,T],C^{2+\alpha}(\Gamma_{0}))\cap
C^{1}([0,T],C^{1+\alpha}(\Gamma_{0})),$

with
$\sup_{t\in[0,T]}\|\rho(\cdot,t)\|_{C^{1}(\Gamma_{0})}<\sigma(\Omega_{0})$,
write $\Omega_{t}$, for $t\in[0,T]$ for the corresponding family of domains,
with boundaries $\Gamma_{t}:=\Gamma_{\rho(t)}$. The velocity $V$ of the
boundary in the direction of the outward normal, at a point
$y=x+\rho(x,t)\nu(x)\in\Gamma_{t}$ is given by

$V=\frac{\partial\rho}{\partial t}(x,t)\nu(x)\cdot n(y,t),$

where $n(y,t)$ is the outward unit normal to $\Gamma_{t}$ at the point $y$.

Since the domains $\Omega_{t}$ have $C^{2+\alpha}$ boundaries, there exists
for each $t$ a unique solution $u(\cdot,t)\in
C^{2+\alpha}(\overline{\Omega}_{t})$ and $\lambda_{t}\in\mathbb{R}$ of

$-\Delta u=\lambda_{t},\quad\textrm{on }\Omega_{t},\quad
u|_{\Gamma_{t}}=0,\quad\int_{\Omega_{t}}u\,dx=1,$

(see for example [4, Theorem 6.14]).

Then we say that such a $\rho$ is a _classical solution_ of (1.1) with initial
domain $\Omega_{0}$ if the velocity $V(y)$ at each $y\in\Gamma_{t}$, for
$t\in[0,T]$ satisfies

$V=F(|Du|).$

The main theorem of Escher-Guidotti [2] implies in particular the following:

######  Theorem 2.1.

There exists a $T>0$ and a unique classical solution

$\rho\in C([0,T],C^{2+\alpha}(\Gamma_{0}))\cap
C^{1}([0,T],C^{1+\alpha}(\Gamma_{0}))$

of the quasi-static droplet model (1.1) with initial domain $\Omega_{0}$ whose
boundary $\Gamma_{0}$ is smooth.

In fact they prove more: they also allow their initial domain to have boundary
in $C^{2+\alpha}$. Note that this result does not require the assumption
(1.2).

##  3\. Proof of Theorem 1.1

In this section we give a proof of Theorem 1.1. We work in $\mathbb{R}^{2}$,
using $x$ and $y$ as coordinates. The heart of the proof is the following
lemma, which makes use of the assumption (1.2).

######  Lemma 3.1.

There exists a bounded convex domain $\Omega_{0}$ with smooth boundary
$\Gamma_{0}$, and real numbers $0<x_{0}<x_{1}$ with the following properties:

  1. (i)

$\Omega_{0}$ is contained in $\\{y\geq 0\\}$.

  2. (ii)

$(x,0)\in\partial\Omega_{0}$ for $x_{0}\leq x\leq x_{1}$.

  3. (iii)

Let $u(x,y)$ solve

$-\Delta u=\lambda_{0},\quad\textrm{on }\Omega_{0},\quad
u|_{\Gamma_{0}}=0,\quad\int_{\Omega_{0}}u\,dxdy=1,$

for a constant $\lambda_{0}$. Then $V(x):=F(|Du(x,0)|)$ satisfies

$\frac{V(x_{0})+V(x_{1})}{2}>V\left(\frac{x_{0}+x_{1}}{2}\right).$

###### Proof.

We begin with the following explicit solution of the “torsion problem,”
$-\Delta v=\textrm{const}$, on the equilateral triangle [9]. Let $D$ be the
equilateral triangle of side length $2a$ given by

$y>0,\ \sqrt{3}|x|>y-a\sqrt{3}.$

The function

$v=cy((y-a\sqrt{3})^{2}-3x^{2}),\quad\textrm{for }c:=\frac{5}{3a^{5}},$

satisfies

$-\Delta v=4ac\sqrt{3},$

vanishes on the boundary of $D$ and satisfies

$\int_{D}v\,dxdy=1.$

On the bottom edge of the triangle

$E_{1}=\\{(x,0)\in\mathbb{R}^{2}\ |\ -a\leq x\leq a\\},$

we have

$v_{y}(x,0)=3c(a^{2}-x^{2}).$

Hence

$V(x):=F(3c(a^{2}-x^{2})),$

and

(3.1)
$V^{\prime\prime}(x)=36c^{2}x^{2}F^{\prime\prime}(3c(a^{2}-x^{2}))-6cF^{\prime}(3c(a^{2}-x^{2})).$

Recalling that $c=5/(3a^{5})$, then we may choose $a>0$ sufficiently small so
that

(3.2) $36c^{2}x^{2}\geq 2\frac{6c}{\gamma},\quad\textrm{for }|x|\geq a/2,$

where $\gamma>0$ is given by our assumption (1.2). From now on we fix this $a$
(and hence $c$).

It follows from (3.1), (3.2) and (1.2) that $V^{\prime\prime}(x)>0$ for $|x|$
sufficiently close to $a$. In particular there exists $0<x_{0}<x_{1}<a$ with

(3.3) $\frac{V(x_{0})+V(x_{1})}{2}>V\left(\frac{x_{0}+x_{1}}{2}\right).$

The above example above readily implies the existence of a smooth domain
$\Omega_{0}$ satisfying the conditions in the Lemma. Indeed, we only have to
“smooth the corners” of the triangle domain $D$.

Denote the vertices of $D$ by $p_{1},p_{2},p_{3}$. Let
$\\{D_{k}\\}_{k=1}^{\infty}$ be a sequence of bounded convex domains with
smooth boundaries such that for each $k\geq 1$:

  1. (1)

$D_{k}\subset D_{k+1}\subset D$ (the sequence is nested and increasing).

  2. (2)

$D\setminus D_{k}\subset\bigcup_{i=1}^{3}B_{k^{-1}}(p_{i})$, where $B_{r}(p)$
denotes the ball of radius $r$ centered at $p$.

Such a sequence $\\{D_{k}\\}$ can be constructed by “rounding out the corners”
of the triangle $D$ in a ball of radius $k^{-1}$ centered at each corner.

For each $k\geq 1$ let $u_{k}$ on $D_{k}$ be the solutions of

$-\Delta u_{k}=4ac\sqrt{3},\quad\textrm{on }D_{k},\quad u|_{\partial
D_{k}}=0,$

where we recall that $a$ and $c$ are fixed constants.

It follows from property (1) above and the maximum principle that for each
$k\geq 1$

(3.4) $0<u_{k}\leq u_{k+1}\leq v,\quad\textrm{on }D_{k},$

from which we conclude a pointwise limit on the triangle $D$

(3.5) $0\leq u_{\infty}(x):=\lim_{k\to\infty}u_{k}(x)\leq
v(x),\quad\textrm{for }x\in D,$

and define $u_{\infty}(x)$ to be zero on $\partial D$.

By standard elliptic estimates (see for example [4, Theorem 6.19] and the
remark after it), the convergence above will hold in $C^{\ell}(K)$ for any
compact set $K\subset\subset(\overline{D}\setminus\\{p_{1},p_{2},p_{3}\\})$
and any $\ell\geq 0$. Hence $u_{\infty}\in
C^{\infty}(\overline{D}\setminus\\{p_{1},p_{2},p_{3}\\})$ and $-\Delta
u_{\infty}=4ac\sqrt{3}$ on $D$. Moreover, by (3.4) and the continuity of $v$
it is easily verified that $u_{\infty}$ is also continuous at the corners
$p_{1},p_{2},p_{3}$ and thus on all of $\overline{D}$. By the maximum
principle, $u_{\infty}=v$. Note also that

$\int_{D_{k}}u_{k}\,dxdy\rightarrow 1,\quad\textrm{as }k\rightarrow\infty.$

Then for sufficiently large $k$ the domain $\Omega_{0}:=D_{k}$ will satisfy
conditions (i), (ii), (iii), with

$u:=\frac{u_{k}}{\int_{D_{k}}u_{k}\,dxdy},\quad\lambda_{0}:=\frac{4ac\sqrt{3}}{\int_{D_{k}}u_{k}\,dxdy}.$

Here we are using (3.3) and the fact that $x\mapsto F(|Du_{k}(x,0)|)$ will
converge uniformly to $x\mapsto F(|Dv(x,0)|)$ on $[x_{0},x_{1}]$ as
$k\rightarrow\infty$. This completes the proof of the lemma. ∎

###### Proof of Theorem 1.1.

Let $\Omega_{0}$ and $u$ be given as in Lemma 3.1. By Theorem 2.1, there
exists a unique classical solution of (1.1) for a short time interval $[0,T]$
with $T>0$.

The boundaries $\Gamma_{t}$ of $\Omega_{t}$ can be written as graphs over
$\Gamma_{0}:=\partial\Omega_{0}$. In particular, using $x$ as a coordinate,
part of $\Gamma_{t}$ is given by a graph $y=g(x,t)$ for $x_{0}\leq x\leq
x_{1}$, with $g(x,0)=0$ for $x_{0}\leq x\leq x_{1}$, with the unit normal to
$\Omega_{0}$ being in the negative $y$ direction.

We may assume that

$g\in C([0,T],C^{2+\alpha}([x_{0},x_{1}]))\cap
C^{1}([0,T],C^{1+\alpha}([x_{0},x_{1}]).$

Moreover, $(\partial g/\partial t)(x,0)$ represents the _negative_ of the
velocity in the normal direction at time $t=0$. Hence by (iii) of Lemma 3.1,

$\frac{1}{2}\left(\frac{\partial g}{\partial t}(x_{0},0)+\frac{\partial
g}{\partial t}(x_{1},0)\right)<\frac{\partial g}{\partial
t}\left(\frac{x_{0}+x_{1}}{2},0\right).$

Then for $t\in(0,\delta]$ for $\delta>0$ sufficiently small, we have

$\frac{1}{2}\left(g(x_{0},t)+g(x_{1},t)\right)<g\left(\frac{x_{0}+x_{1}}{2},t\right).$

In particular, $x\mapsto g(x,t)$ is not convex for
$(x,t)\in[x_{0},x_{1}]\times(0,\delta]$. Hence $\Omega_{t}$ is not a convex
domain for $t\in(0,\delta]$. ∎

## References

  * [1] Cox, R.G. The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous Flow., J. Fluid Mech. 168 (1986), 169–194
  * [2] Escher, J., Guidotti, P. Local well-posedness for a quasi-stationary droplet model, Calc. Var. Partial Differential Equations (2015) 54:1147–1160
  * [3] Feldman, W.M., Kim, I.C. Dynamic stability of equilibrium capillary drops, Arch. Ration. Mech. Anal. 211 (2014), no. 3, 819–878
  * [4] Gilbarg, D., Trudinger, N.S., Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
  * [5] Glasner, K.B. _A boundary integral formulation of quasi-steady fluid wetting_ , J. Comput. Phys. 207 (2005), no. 2, 529–541
  * [6] Glasner, K.B., Kim, I.C. Viscosity solutions for a model of contact line motion, Interfaces Free Bound. 11 (2009), no. 1, 37–60
  * [7] Greenspan, H.P. On the motion of a small viscous droplet that wets a surface, J. Fluid Mech. 84 (1978), 125–143
  * [8] Grunewald, N., Kim, I.C. A variational approach to a quasi-static droplet model, Calc. Var. Partial Differential Equations 41 (2011), no. 1-2, 1–19
  * [9] Keady, G., McNabb, A. The elastic torsion problem: solutions in convex domains, New Zealand J. Math. 22 (1993), no. 2, 43–64
  * [10] Tanner, L. The spreading of silicone oil drops on horizontal surfaces., J. Phys. D 12 (1979), 1473–1484
  * [11] Voinov, O.V. Hydrodynamics of wetting, Fluid Dyn. 11 (1976), 714–721