\label{thm0}

		Let $M$ be a closed orientable $n$-manifold with an open book decomposition $\textrm{OB}(P^{n-1},\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\partial P$ and let $\textrm{OB}(V_B,\phi_B)$ be an open book decomposition of $B$. Then, $\textrm{OB}(V_B, \phi_B^k)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$ for all $k \in \mathbb{Z}$.

\begin{corollary} \label{cor2}

		Let $Q$ be a closed simply connected spin $5$-manifold. Then, every open book decomposition of $Q$ spun embeds in $\s^7$ with a fixed page bounded by $Q$.

	\end{corollary}

\begin{corollary}\label{cor3}

		Let $M^3$ be a closed orientable $3$-manifold. Then, every open book decomposition of $M^3$ spun embeds in $\s^5$ with a fixed page bounded by $M$.

	\end{corollary}

\begin{enumerate}
			\item  $\Gamma_k$ is supported on $\mathcal{C}(B)$ and $\Gamma_k$ is isotopic to the identity map, relative to $P \setminus \mathcal{C}(B)$.

			\item $\Gamma_k \circ f = f \circ \phi_B^k$.
		\end{enumerate}

		\noindent Since $\rho|_{\mathcal{C}(B)} = \id_P$, $\ob(V_B, \phi^k_B)$ spun embeds in $\ob(P,\rho)$.

	\end{proof}

	\section{The page trick in a more general setup} \label{pagetrickap}

	Let $M^n$ be a component of the binding of on open book $\ob(\Sigma,\phi)$. Let $\ob(P_1^{n-1}, \rho_1)$ and $\ob(P_2^{n-1}, \rho_2)$ be two open book decompositions of a closed oriented manifold $M^n$. Let $g_i$ denote the embedding of $P_i$ in $M$ as a fixed page of the open books for $i = 1,2$. Let $V$ be a compact bounded submanifold of $M$ such that $g_1(P_1) \cup g_2(P_2)$ is a submanifold of $V$.

	\begin{lemma} \label{genpagetrick}
	$\ob(V, \eta)$ spun embeds in $\ob(\Sigma, \phi)$ for any $\eta \in \langle \rho_1, \rho_2\rangle \subset \Diff_\partial(V)$.
	\end{lemma}

	\begin{exmp}\label{expg1}

	The $5$-sphere admits an open book decomposition with page the disk cotangent bundle of $\s^2$ and monodromy a Dehn-seidel twist : $\ob(DT^*\s^2, \tau_2)$ (see \cite{Saha}). Another open book on $\s^5$ has page is given by $\ob(\s^1 \times \D^3 \#_b \s^3 \times [0,1], \sigma)$, where $\sigma$ is the \emph{push} map (see \cite{hsueh1}, \cite{hsueh2}). Let $g_1$ and $g_2$ be the embeddings of $P_1 = DT^*\s^2$ and $P_2 = \s^1 \times \D^3 \#_b \s^3 \times [0,1]$, respectively, in $\s^5$. Then, $g_1(P_1) \#_b g_2(P_2)$ gives a hypersurface in $\s^5 = \s^5 \# \s^5$. By Lemma \ref{genpagetrick}, for any diffeomorphism $\eta$ of $g_1(P_1) \#_b g_2(P_2)$ generated by $\tau_2$ and $\sigma$, $\ob(g_1(P_1) \#_b g_2(P_2), \eta)$ spun embeds in $\ob(\D^6,\id)$.  \noindent Hsueh \cite{hsueh2} has given a large family of new open book decompositions on $\s^n$. Using his open books, one can generate many more examples as the one above.

	\end{exmp}

	\begin{remark}
		It follows from Theorem \ref{thm0} that $M_k = \ob(\s^1 \times \D^2 \#_b \s^2 \times [0,1], \sigma^k)$ spun embeds in $\ob(\D^5,\id) = S^6$. Since $H_1(M_k) = \Z_k$, by a result of Cochran (see Theorem $6.2$ in \cite{coch}), $M_k$ also embeds in $\s^5$ for all $k \geq 0$. It would be interesting to know whether $M_k$ spun embeds in $\s^5$ for all $k \geq 2$.
	\end{remark}

	\begin{exmp}\label{expg2}

	In example \ref{expg1}, we obtained the page as boundary connected sum of pages of two open books. Suppose we have two pages $P^n_1$ and $P^n_2$ such that there exist relative embeddings $e_1 : (\D^k \times \D^{n-k}, \partial \D^k \times \D^{n-k})  \rightarrow (P_1, \partial P_1)$ and $e_2 : (\D^{n-k}\times \D^k, \partial \D^{n-k} \times \D^k) \rightarrow (P_2, \partial P_2)$. Then one can define the plumbing of $P_1$ and $P_2$ by gluing the images of $e_1$ and $e_2$ via the map $e_1(\vec{x},\vec{y}) \mapsto e_2((-1)^k \vec{y}, \vec{x})$. Ozbagci and Popescu-Pampu has defined a generalized notion of plumbing where two manifolds such as $P_1$ and $P_2$ can be \emph{summed} using a common \emph{patch} (see section $7$ in \cite{OP}).

	\end{exmp}

	\begin{figure}[htbp]

		\centering
		\def\svgwidth{10cm}
		\input{fig02.pdf_tex}
		\caption{The purple and green lines represent $g_1(P_1)$ and $g_2(P_2)$, respectively,  in $M \times [-2,1]$ and $s$ is a coordinate on the second factor of $\Sigma \times [0,1]$. We first isotope $g_1(P_1)$ to induce $\rho_1$ at $s = \frac{1}{3}$, and then isotope $g_2(P_2)$ to induce $\rho_2$ at $s =\frac{2}{3}$. This induces the monodromy $\rho_1 \circ \rho_2$ on $V$.}
		\label{gptfig}

	\end{figure}

	\begin{proof}[Proof of Lemma \ref{genpagetrick}]  We identify a collar neighborhood of $M \subset \Sigma$ with $M \times [-2,2]$ such that $\partial \Sigma = M \times \{2\}$. We take a Seifert type proper embedding of $V$ in $M \times [-2,-1]$. Suppose, we want to induce the map $\rho_1$ on $V$ first. For that we simply isotope the submanifold $g_1(P_1) \times \{-1\}$, relative to its boundary, such that after the isotopy $g_1(P_1) \times \{-1\}$ becomes a Seifert type embedding $f_1$ of $P_1$ in $M \times [-1,0]$. We then apply the page trick in the neighborhood $M \times [-1,1]$ to induce the map $\rho_1$, and isotope $f_1(P_1)$ back to $g_1(P_1) \times \{-1\}$. Similarly, one can induce the map $\rho_2$ on the properly embedded $V$ in $M \times [-2,2]$. See Figure \ref{gptfig}. Let $(\Sigma \times [0,1], (x,\theta))$ be a coordinate on the mapping cylinder of $\ob(\Sigma, \phi)$. Say, $\eta$ has a presentation as a word generated by $\rho_1$ and $\rho_2$ with $l$ many letters. We divide the mapping cylinder in $l$ equal part $\Sigma \times [\frac{i}{l}, \frac{i+1}{l}]$ ($0 \leq i \leq l-1$), and apply an isotopy of $\Sigma$ on each part to induce each of the letters. Thus, $\ob(V,\eta)$ spun embeds in $\ob(\Sigma, \phi)$.

	\end{proof}

	\noindent Although Lemma \ref{genpagetrick} is stated for two open books, its proof shows that a similar statement holds true for any number of open books. The page trick can be used for constructing $5$-dimensional open books with connected binding where all $3$-dimensional open books with connected binding admit spun embedding (see Theorem $4.4$ in \cite{ls1}). Examples of such $5$-dimensional open books were first constructed by Etnyre and Lekili \cite{EL} using mapping torus of a Lefschetz fibration.

	\section{The page trick for Morse open books} \label{morseob}

		A \emph{Morse open book} is similar to an open book $(M^n, B^{n-1},\pi)$, except that the fibration map $\pi : M \setminus B \rightarrow \s^1$ may have singular fibers, with singularities in the interiror of a fiber.

	   \noindent  Let $K^{n-2}$ be a closed oriented submanifold of a closed oriented manifold $L^n$ with trivial normal bundle and let $K$ be nullhomologous in $L$. Then, there exists a hypersurface $V^{n-1} \subset L$ such that $\partial V = K$. Given such a \emph{Seifert surface} $V$, there exists a Morse open book on $L$ with a regular page $V$ (see remark $9.6$ in \cite{OP}). In particular, $L \setminus V$ can be seen as a relative cobordism $W^n$ from $V$ to itself. Let $\psi$ be the monodromy map of the regular page $V$. Let $h : W^n \rightarrow [0,1]$ be a Morse function with $h^{-1}(0) = h^{-1}(1) = V$ and $h$ has critical points only in the interior of $W$. According to Milnor's terminology, $h$ is a Morse function on the \emph{triad} $(W, h^{-1}(0), h^{-1}(1))$. The monodromy $\psi$ is then a diffeomorphism (relative to boundary) between $h^{-1}(0)$ and $h^{-1}(1)$. In this case, the analogue of the mapping torus of an open book is the quotient space $\mathcal{M}(W,h,\psi) = \frac{W}{x \sim \psi(x)}$. Thus, the triple $(W,h,\psi)$ characterizes a Morse open book structure on $L$.

	   \begin{definition}[spun embedding for Morse open books]

	   	Let $L_1^n$ and $L_2^{n+k}$ be two closed oriented manifolds. Let $(W_1,h_1, \psi_1)$ and $(W_2,h_2,\psi_2)$ be Morse open book decompositions of $L_1$ and $L_2$, respectively. We say $(W_1, h_1, \psi_1)$ spun embeds in $(W_2,h_2,\psi_2)$ if there exists an embedding $\iota$ of $L_1$ in $L_2$ such that the followings hold.

	   	\begin{enumerate}
	   		\item $h_2 \circ \iota$ restricts to a Morse function on the triad $(W_1, h_1^{-1}(0), h_1^{-1}(1))$.

	   		\item  The embedding $\iota$ takes a regular page of $h_1$ to a regular page of $h_2$ and $\iota \circ \psi_1 | _{h_1^{-1}(1)}= \psi_2 \circ \iota|_{h_1^{-1}(1)}$.

	   	\end{enumerate}

\begin{proposition} \label{morsepagetrick}

		Let $(W, h, \phi)$ be a Morse open book decomposition of a closed oriented manifold $N$. Then, there exists an embedding $F : W \rightarrow N \times [-1,1] \times [0,1]$ with the following property.

		\begin{enumerate}

			\item The projection map $N \times [-1,1] \times [0,1] \rightarrow [0,1]$ restricts to a Morse function on the triad $(F(W), F(h^{-1}(0)), F(h^{-1}(1)))$, and

			\item $F \circ \phi|_{h^{-1}(1)} = F|_{h^{-1}(0)}$.

		\end{enumerate}

		\end{proposition}

\label{genhopftrick}

	The Hopf annulus trick has a straightforward generalization for any open book decomposition. Let $N^n = \ob(V^{n-1},\phi)$. Consider the Seifert type embedding $h$ of $V$ in $N\

\begin{exmp} \label{brancexmp}

			Let $K^{n-2}$ be a closed oriented submanifold of a closed oriented manifold $L^n$ with trivial normal bundle and let $K$ be nullhomologous in $L$. Thus, there exists a Morse open book $(W, h, \phi)$ on $L$, such that the closure of $L \setminus K$ is the relative cobordism $W$ between a regular page $V$ to itself and the monodromy or the return map is $\phi$. We partition the trivial mapping cylinder $(L \times [-1,1]) \times [0,k]$ in $k$ parts : $L \times [-1,1] \times [i,i+1]$ ($0 \leq i \leq k-1$). By Proposition \ref{morsepagetrick}, we can embed a copy of $W$ in $L \times [-1,1] \times [i,i+1]$ via an embedding $F_i$ such that the projection map $L \times [-1,1] \times [i,i+1] \rightarrow [i,i+1]$ restricts to a Morse function on the triad $(F_i(W), F_i(h^{-1}(0)), F_i(h^{-1}(1)))$, and $F_i \circ \phi|_{h^{-1}(1)} = F_i|_{h^{-1}(0)}$. By concatenating $F_0, F_1,\dots, F_{k-1}$, we get a spun embedding of a Morse open book $(W_k, h_k, \phi^k)$ in $\ob(L \times [-1,1],\id) = L \times \s^2$, where $W_k$ is a relative cobordism obtained by concatenating $k$ copies of $W$ and $h_k$ is the corresponding Morse function given by $h \ast h \ast \cdots \ast h$ ($k$ times). By construction, the total space $\widetilde{L}$ of the Morse open book $(W_k, h_k, \phi^k)$ is a $k$-fold cyclic branched cover of $L$ branched along $K$.

		\end{exmp}

An open book decomposition of a manifold $M^m$ is a pair $(V^{m-1},h)$, such that $M^m$ is diffeomorphic to the quotient space $\mathcal{MT}(V^{m-1}, h) \cup_{id} \partial V^{m-1} \times D^2$. Here, $V^{m-1}$ is called the \emph{page} and $\partial V$ is called the \emph{binding} of the open book. The map $h$, called \emph{monodromy}, is a diffeomorphism of $V^{m-1}$ that restricts to identity near the boundary $\partial V$, and $\mathcal{MT}(V^{m-1}, h)$ denotes the mapping torus of $h$. We denote the total space of such an open book by $\textrm{OB}(V,h)$.

\noindent An embedding $f$ of $\textrm{OB}(V_1,h_1)$ in $\textrm{OB}(V_2,h_2)$ is called a \emph{spun embedding} (or an \emph{open book embedding}), if $f$ takes the page $V_1$ to the page $V_2$ and $f$ is compatible with the monodromies : $h_2 \circ f = f \circ h_1$, up to isotopy. We say $M^n$ spun embeds in $N^{n+k}$ if there exists an open book decomposition of $M$ that spun embeds in an open book decomposition of $N$. In recent times, the study of codimension $2$ \emph{spun embedding} (or \emph{open book embedding}) has attracted much attention (eg. \cite{EF}, \cite{EL}, \cite{pps}, \cite{GPS}, \cite{Saha} ,\cite{ls1}). The notion of spun embedding of an open book decomposition is strictly stronger than ordinary smooth embedding. Obstructions to spun embedding an open book in a given open book has been discussed in \cite{ls1}. Etnyre and Lekili \cite{EL} gave the first constructions of codimension $2$ spun embeddings of $3$-manifolds by embedding pages as fibers in a Lefschetz fibration. Later, an alternate construction of spun embedding was given by Pancholi, Pandit and the fourth author \cite{pps}, based on the observation that there is a proper embedding of a annulus $\mathcal{A}$ in $\mathbb{D}^4$ such that, the Dehn twist along the core of $\mathcal{A}$ is induced by an isotopy of $\mathbb{D}^4$, relative to the boundary. This construction, called the \emph{annulus trick}, was later adapted to the setting of non-orientable $3$-manifolds by Ghanawat, Pandit and Selvakumar \cite{GPS}. In this note, we observe a generalization of this construction and give some applications.

We prove that if a closed manifold $B$ is a connected component of the binding of an open book decomposition of a manifold $M$, then every open book decomposition of $B$ spun embeds in $M$. As an application, we prove that every open book decomposition of a simply connected spin $5$-manifold spun embeds in $\s^7$ and every $3$-dimensional open book spun embeds in $\s^5$. We also define a notion of spun embedding for Morse open books.

An open book decomposition of a manifold $M^m$ is a pair $(V^{m-1},h)$, such that $M^m$ is diffeomorphic to the quotient space $\mathcal{MT}(V^{m-1}, h) \cup_{id} \partial V^{m-1} \times D^2$. Here, $V^{m-1}$ is called the \emph{page} and $\partial V$ is called the \emph{binding} of the open book. The map $h$, called \emph{monodromy}, is a diffeomorphism of $V^{m-1}$ that restricts to identity near the boundary $\partial V$, and $\mathcal{MT}(V^{m-1}, h)$ denotes the mapping torus of $h$. We denote the total space of such an open book by $\textrm{OB}(V,h)$.

\noindent An embedding $f$ of $\textrm{OB}(V_1,h_1)$ in $\textrm{OB}(V_2,h_2)$ is called a \emph{spun embedding} (or an \emph{open book embedding}), if $f$ takes the page $V_1$ to the page $V_2$ and $f$ is compatible with the monodromies : $h_2 \circ f = f \circ h_1$, up to isotopy. We say $M^n$ spun embeds in $N^{n+k}$ if there exists an open book decomposition of $M$ that spun embeds in an open book decomposition of $N$. In recent times, the study of codimension $2$ \emph{spun embedding} (or \emph{open book embedding}) has attracted much attention (eg. \cite{EF}, \cite{EL}, \cite{pps}, \cite{GPS}, \cite{Saha} ,\cite{ls1}). The notion of spun embedding of an open book decomposition is strictly stronger than ordinary smooth embedding. Obstructions to spun embedding an open book in a given open book has been discussed in \cite{ls1}. Etnyre and Lekili \cite{EL} gave the first constructions of codimension $2$ spun embeddings of $3$-manifolds by embedding pages as fibers in a Lefschetz fibration. Later, an alternate construction of spun embedding was given by Pancholi, Pandit and the fourth author \cite{pps}, based on the observation that there is a proper embedding of a annulus $\mathcal{A}$ in $\D^4$ such that, the Dehn twist along the core of $\mathcal{A}$ is induced by an isotopy of $\D^4$, relative to the boundary. This construction, called the \emph{annulus trick}, was later adapted to the setting of non-orientable $3$-manifolds by Ghanawat, Pandit and Selvakumar \cite{GPS}. In this note, we observe a generalization of this construction and give some applications.

Let $M$ be a closed orientable $n$-manifold with an open book decomposition $\ob(P^{n-1},\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\partial P$ and let $\ob(V_B,\phi_B)$ be an open book decomposition of $B$. Then, $\ob(V_B, \phi_B^k)$ spun embeds in $\ob(P^{n-1},\rho)$ for all $k \in \Z$.

\noindent Here, $\phi_B^k$ denotes the $k$-times composition of $\phi_B$. An immediate corollary of Theorem \ref{thm0} is the following.

\noindent Let $Q^5$ be a simply connected spin manifold. Kwon and Van Koert (Corollary $3.9$ in \cite{KoVan}) have shown that every simply connected spin $5$-manifold is a connected sum of Brieskorn manifolds. Thus, $M = B_1 \# B_2 \# \cdots \# B_l$ for some $l \in \Z_{>0}$, where the $B_i$s are $5$-dimensional Brieskorn manifolds. Let $\ob(V_i, \phi_i)$ be an open book decomposition of $\s^7$ with binding $\partial V_i = B_i$. Then, $\ob(V_1 \#_b V_2 \#_b \cdots \#_b V_l, \phi_1 \circ \phi_2 \circ \phi_l) = \ob(V_1,\phi_1) \# \ob(V_2,\phi_2) \# \cdots \# \ob(V_l,\phi_l)$ (see section $2.4$ in \cite{DGK}) is an open book decomposition of $\s^7$ with binding $Q^5$. Thus, Theorem \ref{thm0} implies the following.

\section{preliminaries} \label{prelim}
 \subsection{Open books and embeddings} An open book decomposition of a closed $(2n+1)$-manifold $M$ consists of a codimension $2$ closed submanifold $B$ and a fibration map $\pi : M \setminus B \rightarrow \s^1$, such that in a tubular neighborhood of $B \subset M$, the restriction map $\pi : B \times (\D^2 \setminus \{0\}) \rightarrow \s^1$ is given by $(b,r,\theta) \mapsto \theta$. The fibration $\pi$ determines a unique fiber manifold $N^{2n}$ whose boundary is $B$. The closure $\bar{N}$ is called the \emph{page} and $B$ is called the \emph{binding}. The monodromy of the fibration map $\pi$ determines a diffeomorphism $\phi$ of $\bar{N}$ such that $\phi$ is identity near the boundary $\partial \bar{N}$. In particular, $M = \mathcal{MT}(\bar{N}, \phi) \cup _{id, \partial} \partial \bar{N} \times \D^2$. We denote such an open book decomposition of $M$ by $\textrm{OB}(\bar{N},\phi)$. The map $\phi$ is called the \emph{monodromy} of the open book. This description of an open book decomposition in terms of its page and monodromy is known as an \emph{abstract open book}. However, these two notions are equivalent and we will always describe an open book decomposition of a manifold as an abstarct open book. Two open book decompositions with the same page are \emph{equivalent} if their monodromies are isotopic, relative to a collar neighborhood of the binding in a page.

\begin{proof}[Proof of Theorem \ref{thm0}] Recall that $M^n = \ob(P^{n-1}, \rho)$, where $\rho$ restricts to the identity map on a collar neighborhood $\mathcal{C}(B)$ of $B = \partial P$ in $P$. Let us identify this collar neighborhood with $B \times [-1,1]$ such that $B \times \{-1\} = \partial P$. Consider an open book decomposition $\ob(V_B,\phi_B)$ of $B$ and let $f$ be the Seifert type embedding of $V_B$ in $B \times [-1,1]$ such that $f(\partial V_B) \subset B \times \{-1\}$. This gives a proper embedding of $V_B$ in $P$. Given an integer $k$, by Proposition \ref{pagetrick}, there exists a diffeomorphism $\Gamma_k$ of $P$ with the following properties.

\begin{theorem} \label{thm0}

		Let $M$ be a closed orientable $n$-manifold with an open book decomposition $\ob(P^{n-1},\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\partial P$ and let $\ob(V_B,\phi_B)$ be an open book decomposition of $B$. Then, $\ob(V_B, \phi_B^k)$ spun embeds in $\ob(P^{n-1},\rho)$ for all $k \in \Z$.

	\end{theorem}

\noindent Here, $\phi_B^k$ denotes the $k$-times composition of $\phi_B$. An immediate corollary of Theorem \ref{thm0} is the following.

	\begin{corollary} \label{cor1}
		Every open book decomposition of $\mathbb{S}^{n-2}$ spun embeds in $\mathbb{S}^n = \textrm{OB}(\mathbb{D}^{n-1}, \textrm{id})$ for $n \geq 4$.
	\end{corollary}

	\noindent More generally, one can think of any codimension $2$ fibered submanifold of $\mathbb{S}^m$. In particular, every $(2n-1)$-dimensional Brieskorn manifold appears as the binding of an open book decomposition of $\mathbb{S}^{2n+1}$.

	 \noindent Let $Q^5$ be a simply connected spin manifold. Kwon and Van Koert (Corollary $3.9$ in \cite{KoVan}) have shown that every simply connected spin $5$-manifold is a connected sum of Brieskorn manifolds. Thus, $M = B_1 \# B_2 \# \cdots \# B_l$ for some $l \in \Z_{>0}$, where the $B_i$s are $5$-dimensional Brieskorn manifolds. Let $\textrm{OB}(V_i, \phi_i)$ be an open book decomposition of $\mathbb{S}^7$ with binding $\partial V_i = B_i$. Then, $\textrm{OB}(V_1 \#_b V_2 \#_b \cdots \#_b V_l, \phi_1 \circ \phi_2 \circ \phi_l) = \textrm{OB}(V_1,\phi_1) \# \textrm{OB}(V_2,\phi_2) \# \cdots \# \textrm{OB}(V_l,\phi_l)$ (see section $2.4$ in \cite{DGK}) is an open book decomposition of $\mathbb{S}^7$ with binding $Q^5$.  Thus, Theorem \ref{thm0} implies the following.

	\begin{corollary} \label{cor2}

		Let $Q$ be a closed simply connected spin $5$-manifold. Then, every open book decomposition of $Q$ spun embeds in $\mathbb{S}^7$ with a fixed page bounded by $Q$.

	\end{corollary}

  \noindent Saeki (Theorem $6.1$ in \cite{Saeki}) has shown that every closed oriented $3$-manifold can be realized as the connected binding of an open book decomposition of $\mathbb{S}^5$. Together with Theorem \ref{thm0} this implies the following.

	\begin{corollary}\label{cor3}

		Let $M^3$ be a closed orientable $3$-manifold. Then, every open book decomposition of $M^3$ spun embeds in $\mathbb{S}^5$ with a fixed page bounded by $M$.

	\end{corollary}

	\noindent We note that an analogue of \ref{cor2} and \ref{cor3} for contact manifolds can not hold. It is known by the work of Thurston--Winkelenkemper and Giroux that every contact structure on a $3$-manifold corresponds to an open book decomposition. By Kasuya \cite{Kasuya}, a contact manifold $(M,\zeta)$ admits a contact embedding in a contact $\mathbb{S}^5$ only if its first Chern class $c_1(\zeta)$ vanishes. However, in general, it is not true that every spun embedding of a $3$-manifold in $\mathbb{S}^5$ corresponds to a contact embedding of the corresponding contact $3$-manifold. In particular, many of the spun embeddings given by Corollary \ref{cor2} and Corollary \ref{cor3} are examples of codimension $2$ proper embeddings (embedding between pages of open books) which are not isotopic to a symplectic embedding.

	\vspace{0.25cm}

	\noindent It is not known yet if there exists a fixed open book decomposition of $\mathbb{S}^5$ where all $3$-dimensional open books admit spun embedding. We call such an open book decomposition of $\mathbb{S}^5$ \emph{universal}. A recent work (Theorem $1.4$ in \cite{ls1}) by the second and the fourth author shows that none of Saeki's open books (given by Theorem $6.1$ in \cite{Saeki}) can be universal.

	\vspace{0.25cm}

	Theorem \ref{thm0} is proved using a simple generalization of the \emph{Hopf annulus trick} (see section \ref{genhopftrick}), which we call the \emph{page trick}. In section \ref{pagetrickap}, we discuss some more applications of the page trick to construct spun embeddings. In particular, the recent finding of various new open book decompositions by Hsueh \cite{hsueh2}, and a generalization of the notion of plumbing by Ozbagci and Popescu-Pampu \cite{OP} give us a large class of novel examples where the page trick can be applied.

	Finally, in section \ref{morseob}, we discuss a notion of spun embedding for \emph{Morse open books} and describe a version of the page trick in this context (see Proposition \ref{morsepagetrick} and Example \ref{brancexmp}).

	Unless stated otherwise, we always work in the category of smooth manifolds.

The only proof-related description given is that \emph{``Theorem \ref{thm0} is proved using a simple generalization of the \emph{Hopf annulus trick} (see section \ref{genhopftrick}), which we call the \emph{page trick}.''} No further steps or outline of the argument are provided in the post-theorem introduction.

Question: Let $M$ be a closed orientable $n$-manifold with an abstract open book decomposition $\textrm{OB}(P^{n-1},\rho)$, where $P^{n-1}$ is the page and $\rho:P\to P$ is a diffeomorphism equal to the identity near $\partial P$. Let $B^{n-2}$ be a connected component of the binding $\partial P$. Assume that $B$ itself has an abstract open book decomposition $\textrm{OB}(V_B,\phi_B)$, where $V_B$ is the page and $\phi_B:V_B\to V_B$ is a diffeomorphism equal to the identity near $\partial V_B$. Here, a spun embedding (open book embedding) of $\textrm{OB}(V_1,h_1)$ into $\textrm{OB}(V_2,h_2)$ means an embedding that sends the page $V_1$ into the page $V_2$ and is compatible with the monodromies in the sense that $h_2\circ f$ and $f\circ h_1$ are isotopic. Under these hypotheses, which conclusion holds?

For every integer $k\in\mathbb{Z}$, the open book $\textrm{OB}(V_B,\phi_B^k)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$.

Choices:
B. For every integer $k\in\mathbb{Z}$, the open book $\textrm{OB}(V_B,\rho^k\circ\phi_B)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$.
C. The open book $\textrm{OB}(V_B,\phi_B)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$.
D. There exists an integer $k\in\mathbb{Z}$ such that the open book $\textrm{OB}(V_B,\phi_B^k)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$.
E. For every integer $k\ge 0$, the open book $\textrm{OB}(V_B,\phi_B^k)$ smoothly embeds in $\textrm{OB}(P^{n-1},\rho)$, and hence spins embeds there.


[{\bf \cite{RazShaDeZee4D, RazShe}}]\label{thm:kvarER}
    Let $k\ge 3$ and let $f\in \mathbb{R}[x_1,\ldots, x_k]$. Then one of the following holds:\\
    (i) For every $A_1,\ldots, A_k\in \mathbb{R}$ each of size $n$ one has
    $$
    f(A_1,\ldots,A_k)|=\Omega\left(n^{3/2}\right)$$\\
    (ii) $f$ is of one of the forms:
    \begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}

\begin{thm}\label{thm:app}
    Let $\nu$ be  the moment curve in $\RR^d$ and let $P\subset \nu$ be any finite set of size $n$. Then, for every $\eps>0$,
    $$
|\Delta(P)|=\Omega\left(n^{\frac{5d-4}{2d}-\eps}\right),
$$
    where the implicit constant depends only on $\eps$ and on $d$.
\end{thm}

In many cases in combinatorial geometry, counting questions involving distances, slopes, collinearity, etc., can be reformulated as analogous counting questions involving grid points lying on certain algebraic varieties. A unified study of such problems began with a question of Elekes~\cite{Ele} about expansion of bivariate real polynomials $f(x,y)$.
Specifically, he asked: For a bivariate polynomial $f\in \mathbb{R}[x,y]$ and given finite sets $A,B\subset \mathbb{R}$, how small can be the image set
$$
f(A,B)=\{f(a,b)\mid a\in A, b\in B\}.
$$

Elekes conjectured that the image of $f$ on a $n\times n$ Cartesian product must be of cardinality superlinear in $n$, unless $f$ has a very concrete {\it special form}. This was confirmed in 2000 by Elekes and R\'onyai~\cite{EleRon00} who proved the following dichotomy:
Either $f$ has one of the forms
\begin{align}
f(x,y)&=h(p(x)+q(y))\quad\text{or}\nonumber\\
f(x,y)&=h(p(x)q(y)),\label{specialbivariate}
\end{align} for some univariate real polynomials $p,q,h$, or, otherwise, for every finite $A,B\subset \mathbb{R}$, each of size $n$, we have
\begin{equation}\label{eq:ER}
|f(A,B)|=\omega(n).
\end{equation}

In case $f$ is not one of the forms in \eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \cite{RazShaDeZee} to be $\Omega(n^{4/3})$, and further improved in \cite{SolZha} to be $\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.

An analogue of the Elekes--R\'onyai problem can be formulated for polynomials in more than two variables. The trivariate case was studied by Raz, Sharir, and De Zeeuw~\cite{RazShaDeZee4D}, and the general $k$-variate case was established by Raz and Shem Tov~\cite{RazShe}. They obtain the following result.

\begin{abstract}
Let $f\in \RR[x_1,\ldots, x_k]$, for $k\ge 2$. For any finite sets $A_1,\ldots, A_k\subset \RR$, consider the set
$$
f(A_1,\ldots, A_k):=\{f(a_1,\ldots, a_k)\mid (a_1,\cdots,a_k)\in A_1\times\cdots \times A_k\},
$$
that is, the image of $A_1\times \cdots\times A_k$ under $f$.
Extending a theorem of Elekes and R\'onyai, which deals with the case $k=2$, and the result of Raz, Sharir, and De Zeeuw~\cite{RazShaDeZee4D}, dealing with the case $k=3$, it is proved in Raz and Shem Tov~\cite{RazShe}, that for every choice of finite $A_1,\ldots, A_k\subset \RR$, each of size $n$, one has
\begin{equation}\label{RSbound}
|f(A_1,\ldots,A_k)|=\Omega(n^{3/2}),
\end{equation}
unless $f$ has some degenerate special form.

In case $f$ is not one of the forms in \eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \cite{RazShaDeZee} to be $\Omega(n^{4/3})$, and further improved in \cite{SolZha} to be $\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.

Note that the bound in Theorem~\ref{thm:kvarER}, for non-special polynomials $f$, is independent of $k$, and in particular coincides with the bound for $k=3$.
At first glance this may appear to be merely a consequence of the proof. Indeed, the argument in \cite{RazShe} reduces the $k$-variate case for $k \ge 4$ to the trivariate case by fixing values for $k-3$ of the variables. They then show that if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \eqref{kspecial}.

We prove the following main result of the paper.
\begin{thm}\label{mainthm}
Let $k\ge 3$ and let $f\in\RR[x_1\ldots,x_k]$.
Assume that ${\rm rank}(f)=r\ge 2$.
Then, for every $\varepsilon>0$, the following holds:
Let $A_1,\ldots,A_k\subseteq \RR$ be finite, each of size $n$. Then $$
\lvert f(A_1,\ldots,A_k)\rvert= \Omega\left( n^{\frac{5r-4}{2r}-\varepsilon}\right),
$$
where the constant of proportionality depends on $\deg(f)$, on $r$, and on $\eps$.
\end{thm}

We observe that ${\rm rank}(f)=1$ corresponds to the special forms from Theorem~\ref{thm:kvarER}. Indeed, we have the following theorem.
 \begin{thm}\label{rank1char}
Let $k\ge3$ and let $f\in\RR\left[x_1 \ldots,x_k\right]$. Assume that $f$ depends non-trivially on each of its variables and that $$
{\rm rank}(f) = 1.$$
Then $f$ has one of the forms
\begin{align}
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)+\cdots+p_{k}\left(x_{k}\right)\right)\ \ \text{ or}\nonumber\\
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)\cdot\ldots\cdot p_{k}\left(x_{k}\right)\right),\label{special1}
\end{align}
for some univariate real polynomials $h(x), p_1(x),\ldots,p_k(x)$.
\end{thm}

\begin{prop}\label{r_eq_k}
Let $k\ge 3$, $f\in\RR\left[x_1\ldots,x_k\right]$, and assume that ${\rm{rank}}(f)=k-1$. Then, for every $\varepsilon>0$,
the following holds: Let $A_0,\ldots,A_k\subseteq \RR$ be finite, each of size $n$. Then
$$
\lvert f(A_1,\ldots,A_k)\rvert
=
\Omega\left(n^{\frac{5(k-1)-4}{2(k-1)}-\varepsilon}\right),
$$
where the constant of proportionality depends on $\deg(f)$, on $k$, and on $\varepsilon$.
\end{prop}

So in order to complete the proof of Proposition~\ref{r_eq_k}
we only need to prove \eqref{upbndS}. Let $T=T_{f,x_1}$ be the coefficient map defined in the introduction. By assumption, ${\rm rank}(J_T)=k-1$. Thus, there exist indices $\left(i_1,\ldots, i_{k-1}\right) $ such that for
$$
\hat{T}:(x_2\ldots,x_k)
\mapsto(\alpha_{i_1}(x_2\ldots,x_k),\ldots,\alpha_{i_{k-1}}(x_2\ldots,x_k)),
$$
we have
$$\det J_{\hat{T}} \not\equiv 0.$$
Define
\begin{align*}
S_0 &=\left\{\left(a_{1},\ldots,a_{k},b\right)\in S \mid\det J_{\hat{T}}\left(a_{2},\ldots,a_{k}\right)=0\right\}, \\
S' & = S\setminus S_0.
\end{align*}
Clearly $|S|= |S_0|+|S'|$.
Observe that,
$$
|S_0|=|A_1|\cdot\left|\left(A_2\times\cdots\times A_k\right) \cap \left\{\det J_{\hat{T}}=0\right\}\right|\le n\cdot \deg(\det J_{\hat T})n^{k-2},
$$
where the inequality is due to the Schwartz--Zippel Lemma (see \cite{Schw,Zipp}).
Thus, we get
\begin{equation}
\label{S0bound}
\left| S_0\right|=O(n^{k-1}),
\end{equation}
where the constant of proportionality depends on $\deg(f)$ and on $k$.

In this section we prove Theorem~\ref{rank1char}. For the proof we will use the following lemma from Raz and Shem Tov~\cite{RazShe}.
\begin{lem}[{\bf Raz--Shem Tov~\cite[Lemma 2.3]{RazShe}}]\label{RazShem}
Let $f\in\RR[x_1,\ldots,x_k]$. Assume
that
\begin{equation}\label{rank1diff}
\frac{\frac{\partial f}{\partial x_1}(x_1,\ldots,x_k)
}
{
r_1(x_1)
}
=\cdots =
\frac{\frac{\partial f}{\partial x_k}(x_1,\ldots,x_k)
}
{
r_k(x_k)
},
\end{equation}
for some univariate real polynomials $r_1,\ldots, r_k$.
Then, $f$ is one of the forms
\begin{align*}
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)+\cdots+p_{k}\left(x_{k}\right)\right)\ \ \text{ or}\\
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)\cdot\cdots\cdot p_{k}\left(x_{k}\right)\right),
\end{align*}
for some univariate real polynomials $h(x), p_1(x),\ldots,p_k(x)$.

\begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}

\begin{thm}\label{rank1char}
Let $k\ge3$ and let $f\in\RR\left[x_1 \ldots,x_k\right]$. Assume that $f$ depends non-trivially on each of its variables and that  $$
{\rm rank}(f) = 1.$$
Then $f$ has one of the forms
\begin{align}
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)+\cdots+p_{k}\left(x_{k}\right)\right)\ \ \text{ or}\nonumber\\
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)\cdot\ldots\cdot p_{k}\left(x_{k}\right)\right),\label{special1}
\end{align}
for some univariate real polynomials $h(x), p_1(x),\ldots,p_k(x)$.
\end{thm}

\begin{align}
f(x,y)&=h(p(x)+q(y))\quad\text{or}\nonumber\\
f(x,y)&=h(p(x)q(y)),\label{specialbivariate}
\end{align}

\begin{thm}[{\bf \cite{RazShaDeZee4D, RazShe}}]\label{thm:kvarER}
    Let $k\ge 3$ and let $f\in \RR[x_1,\ldots, x_k]$. Then one of the following holds:\\
    (i) For every $A_1,\ldots, A_k\in \RR$ each of size $n$ one has
    $$
    f(A_1,\ldots,A_k)|=\Omega\left(n^{3/2}\right)$$\\
    (ii) $f$ is of one of the forms:
    \begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}
\end{thm}

Note that the bound in Theorem~\ref{thm:kvarER}, for non-special polynomials $f$, is independent of $k$, and in particular coincides with the bound for $k=3$.
At first glance this may appear to be merely a consequence of the proof. Indeed, the argument in \cite{RazShe} reduces the $k$-variate case for $k \ge 4$ to the trivariate case by fixing values for $k-3$ of the variables. They then show that if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \eqref{kspecial}.

It is natural to expect that increasing the number of variables should force the image of $f$ to grow faster. However, certain polynomials in many variables can in effect behave like polynomials in fewer variables. For example, consider the $(k+2)$-variate polynomial
$$
f(x,y,z_1\ldots,z_k)=xy+z_1+z_2+\cdots +z_k.$$
Let $A,B,C_1,\ldots,C_k\subset \mathbb{R}$, where $A,B$ are arbitrary finite sets of size $n$ and $C_1=\cdots=C_k=[n]$. Let
$$
C:=C_1+\cdots +C_k=\{k,k+1,\ldots,kn\}.
$$
Then $|C|=\Theta(n)$, and letting $g(x,y,z):=xy+z$, we have
$$
f(A,B,C_1,\ldots,C_k)=g(A,B,C).$$
In this case, with the current techniques, it is unclear how to obtain a bound on the expansion of $f$ that improves upon the trivariate result for $g$.

\paragraph{Our results.}
In this paper, we recognize $k$-variate polynomials that are, in a precise sense, truly $k$-variate, and we improve the corresponding expansion bounds for them. More precisely, for a $k$-variate polynomial $f$, we introduce the notion of the {\it rank} of $f$. If $f$ has rank $r$, then, in a rigorous sense, it is essentially $(r+1)$-variate, and the bound on the size of its image can be improved with an exponent that grows with $r$.

We now define the rank of a polynomial and then state our main result.

Let $f\in \mathbb{R}\left[x_1,\ldots,x_k\right]$ and let $d_{x_1}$ stand for the degree of $f$ with respect to the variable $x_1$.
Write $$
f\left(x_1,\ldots,x_k\right)=\sum\limits_{i=0}^{d_{x_1}}\alpha_{i}\left(x_2,\ldots,x_k\right)x_1^i.
$$
We consider the {\it coefficient map} $T=T_{f,x_1}:\mathbb{R}^{k-1}\to \mathbb{R}^{d_{x_1}+1}$ given by $$
\left(x_2,\ldots,x_k\right)\mapsto \left(\alpha_0\left(x_2,\ldots,x_k\right),\ldots,\alpha_{d_{x_1}}\left(x_2,\ldots,x_k\right)\right).
$$
We define the rank of $f$ with respect to the variable $x_1$ to be
$$
{\rm rank}_{x_1}(f):={\rm rank}(J_T),$$
where $J_T$ stands for the Jacobian matrix of $T$.
Note that
$$
0\le {\rm rank}_{x_1}(f)\le k-1.
$$
Similarly, define ${\rm rank}_{x_i}(f)$, for every $i=2,\ldots, k$, where $x_i$ plays the role of $x_1$.

Finally, define the {\it rank} of the polynomial $f$ to be
$$
{\rm rank}(f):=\max_{1\le i\le k}{\rm rank}_{x_i}(f).
$$

\begin{exmp} \label{example}
Let $$
f\left(x_1,x_2,\ldots,x_k\right) =  x_1x_k + x_2x_k^2\dots + x_{k-1}x_k^{k-1}.$$
Then $\text{rank}(f)=\text{rank}_{x_k}(f)=k-1$.
\end{exmp}

\begin{exmp}
Let $$
f(x_1,x_2,\ldots,x_k) =  p_1(x_1)x_k + p_2(x_1,x_2)x_k^2+\dots + p_{k-1}(x_1,\ldots,x_{k-1})x_k^{k-1},$$
where $p_i$ is an $i$-variate polynomial that depends non-trivially on $x_i$.
Then $\text{rank}(f)=\text{rank}_{x_k}(f)=k-1$. Indeed, in this case the matrix $J_{T_{f,x_k}}$ is upper-triangular.
\end{exmp}

We prove the following main result of the paper.
\begin{thm}\label{mainthm}
Let $k\ge 3$ and let $f\in\mathbb{R}[x_1\ldots,x_k]$.
Assume that ${\rm rank}(f)=r\ge 2$.
Then, for every $\varepsilon>0$, the following holds:
Let $A_1,\ldots,A_k\subseteq \mathbb{R}$ be finite, each of size $n$. Then $$
\lvert f(A_1,\ldots,A_k)\rvert= \Omega\left( n^{\frac{5r-4}{2r}-\varepsilon}\right),
$$
where the constant of proportionality depends on $\deg(f)$, on $r$, and on $\varepsilon$.
\end{thm}

 We observe  that ${\rm rank}(f)=1$ corresponds to the special forms from Theorem~\ref{thm:kvarER}. Indeed, we have the following theorem.
 \begin{thm}\label{rank1char}
Let $k\ge3$ and let $f\in\mathbb{R}\left[x_1 \ldots,x_k\right]$. Assume that $f$ depends non-trivially on each of its variables and that  $$
{\rm rank}(f) = 1.$$
Then $f$ has one of the forms
\begin{align}
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)+\cdots+p_{k}\left(x_{k}\right)\right)\ \ \text{ or}\nonumber\\
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)\cdot\ldots\cdot p_{k}\left(x_{k}\right)\right),\label{special1}
\end{align}
for some univariate real polynomials $h(x), p_1(x),\ldots,p_k(x)$.
\end{thm}

 Finally,
 we present an application of our results to the following Erd\H{o}s-type combinatorial geometric problem.
 Let $\nu$ denote the moment curve in $\mathbb{R}^d$, parameterized by
 $$\nu(t)=(t,t^2,\ldots,t^d),\quad t\in \mathbb{R}.$$
 Let $P\subset \nu$ be a finite set of $n$ points. For any distinct $p_1,\ldots, p_{d+1}\in \nu$, let $\sigma=\sigma(p_1,\ldots,p_{d+1})$ denote the $d$-simplex which is the convex hull of $p_1,\ldots,p_{d+1}$ in $\mathbb{R}^d$, and let  ${\rm vol}(\sigma)$ denote its $d$-dimensional  volume.
 Define
 $$
 \Delta(P)= \left\{{\rm vol}(\sigma(p_1,\ldots,p_{d+1}))\mid p_1,\ldots,p_{d+1}\in P\right\}.$$

 We have the following theorem.
\begin{thm}\label{thm:app}
    Let $\nu$ be  the moment curve in $\mathbb{R}^d$ and let $P\subset \nu$ be any finite set of size $n$. Then, for every $\varepsilon>0$,
    $$
|\Delta(P)|=\Omega\left(n^{\frac{5d-4}{2d}-\varepsilon}\right),
$$
    where the implicit constant depends only on $\varepsilon$ and on $d$.
\end{thm}

 Theorem~\ref{thm:app} is obtained by identifying a $(d+1)$-variate polynomial $f$ whose expansion over a certain $n\times \cdots \times n$ grid in $\mathbb{R}^{d+1}$ corresponds to the number of distinct volumes of $d$-simplices spanned by $P$. We then show that $f$ has rank $d$ and apply our main Theorem~\ref{mainthm}.

\paragraph{Organization of the paper.} The paper is organized as follows. In Section~\ref{sec:pre} we recall an incidence bound that will serve as a key tool in our arguments. In Section~\ref{sec:special}, we establish a special case of our main result, Theorem~\ref{mainthm}, and in Section~\ref{sec:proofmain}, we complete its proof. The proof of Theorem~\ref{rank1char} is provided in Section~\ref{sec:char}. Finally, Section~\ref{sec:app} contains the proof of Theorem~\ref{thm:app}.

The post-theorem text explains that the proof in \cite{RazShe} for Theorem~\ref{thm:kvarER} “reduces the $k$-variate case for $k \ge 4$ to the trivariate case by fixing values for $k-3$ of the variables.” It then argues that “if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \eqref{kspecial}.”

The post-theorem text explains that the proof in Raz and Sheffer “reduces the $k$-variate case for $k \ge 4$ to the trivariate case by fixing values for $k-3$ of the variables.” It then argues that “if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of
\begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}
.”

[{
\bf \cite{RazShaDeZee4D, RazShe}}]\label{thm:kvarER}
    Let $k\ge 3$ and let $f\in \mathbb{R}[x_1,\ldots, x_k]$. Then one of the following holds:\\
    (i) For every $A_1,\ldots, A_k\in \mathbb{R}$ each of size $n$ one has
    $$
    f(A_1,\ldots,A_k)|=\Omega\left(n^{3/2}\right)$$\\
    (ii) $f$ is of one of the forms:
    \begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}

Question:
Let $k\ge 3$ and let $f\in \mathbb{R}[x_1,\ldots,x_k]$ be a real polynomial in $k$ variables. For finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, define
$$
f(A_1,\ldots,A_k):=\{f(a_1,\ldots,a_k):(a_1,\ldots,a_k)\in A_1\times\cdots\times A_k\}.
$$
Which conclusion holds for such an $f$?

Either for every choice of finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, one has
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{3/2}\bigr),
$$
or there exist univariate real polynomials $h,p_1,\ldots,p_k$ such that $f$ has one of the two forms
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)+\cdots+p_k(x_k)\bigr)
$$
or
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)\cdots p_k(x_k)\bigr).
$$

Choices:
B.
Either for every choice of finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, one has
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{3/2}\bigr),
$$
or there exist univariate real polynomials $h,p_1,\ldots,p_k$ such that after fixing some $k-3$ variables to real constants, the resulting trivariate polynomial has one of the two forms
$$
g(x_i,x_j,x_\ell)=h\bigl(p_i(x_i)+p_j(x_j)+p_\ell(x_\ell)\bigr)
$$
or
$$
g(x_i,x_j,x_\ell)=h\bigl(p_i(x_i)p_j(x_j)p_\ell(x_\ell)\bigr).
$$
C.
If $f$ is not of one of the two forms
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)+\cdots+p_k(x_k)\bigr)
$$
or
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)\cdots p_k(x_k)\bigr),
$$
then for every choice of finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, one has
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{3/2}\bigr).
$$
D.
Either for every choice of finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, one has
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{2}\bigr),
$$
or there exist univariate real polynomials $h,p_1,\ldots,p_k$ such that $f$ has one of the two forms
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)+\cdots+p_k(x_k)\bigr)
$$
or
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)\cdots p_k(x_k)\bigr).
$$
E.
Either there exist finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, for which
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{3/2}\bigr),
$$
or there exist univariate real polynomials $h,p_1,\ldots,p_k$ such that $f$ has one of the two forms
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)+\cdots+p_k(x_k)\bigr)
$$
or
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)\cdots p_k(x_k)\bigr).
$$

global-special-from-all-fixings replaced by existence of one special trivariate fixing

dropped the converse direction in the dichotomy

lower-bound exponent $3/2$ strengthened to $2$

universal quantifier over all sets replaced by existential quantifier


Given positive integers $k$ and $\ell$, we say a graph $G$ is
\emph{Ramsey} for $(K_k,K_\ell)$ if every colouring of the edges of
$G$ with red and blue contains a red copy of $K_k$ or a blue copy of
$K_\ell$ and we denote this property by $G \rightarrow (K_k,K_\ell)$.
In a seminal work~\cite{Ramsey}, Ramsey proved that for all positive
integers $k$ and $\ell$, there exists a positive integer $n$ such that
$K_n \rightarrow (K_k, K_{\ell})$. In the special case $k=\ell$, we
simply write $G \rightarrow K_k$ and we define the \emph{Ramsey
 number} $R(k)$ as the minimum $n$ such that $K_n \rightarrow K_k$.

Estimating $R(k)$ has proven to be notoriously difficult and a central
problem in Ramsey theory is to determine the Ramsey number $R(k)$.
Classical results due to Erd\H{o}s~\cite{Erd} and
Erd\H{o}s–-Szekeres~\cite{Erd.Szek} established the bounds
$2^{k/2} \leq R(k) \leq 2^{2k}$. Despite several refinements, these
exponents remained essentially unchanged for decades. Only in recent
years, significant breakthroughs have been achieved (see,
e.g.,~\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking
advance, Campos, Griffiths, Morris, and Sahasrabudhe~\cite{CGMS}
proved that there exists an $\varepsilon > 0$ such that
$R(k)\leqslant (4-\varepsilon)^k$ for sufficiently large $k$. This
result provides an exponential improvement over the classical
Erd\H{o}s–-Szekeres upper bound. More generally, one can think of the
minimum $n$ such that $K_n\rightarrow (K_k,K_\ell)$, for which there
was another major breakthrough recently by Mattheus and
Verstraete~\cite{MV}, who showed that
$n=\Omega\big(t^3/(\log^4 t)\big)$ vertices are enough to force red
copies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the
edges of $K_n$.

Although much effort has been put into estimating Ramsey numbers, a
parallel and rich direction of research investigates the structure of
graphs that are Ramsey for given pairs of graphs. In this context, we
study Ramsey phenomena of the form $G \rightarrow (K_{s},K_{t})$.

A classical result of Ne\v{s}et\v{r}il and R\"odl~\cite{NR} shows
that for every $k\geq 2$ there are graphs with no copies of $K_k$
that are Ramsey for $K_{k-1}$.

\begin{abstract}
 Given positive integers $k$ and $\ell$ we write
 $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of
 $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we
 denote by $R(k)$ the minimum $n$ such that
 $K_n\rightarrow (K_k,K_k)$. By using probabilistic methods and
 hypergraph containers we prove that for every integer $k \geq 3$,
 there exists a graph $G$ such that $G \nrightarrow (K_k,K_k)$ and
 $G \rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as
 a variation of a classical theorem of Ne\v{s}et\v{r}il and R\"odl
 [The Ramsey property for graphs with forbidden complete subgraphs,
 {\em Journal of Combinatorial Theory, Series B}, \textbf{20} (1976),
 243–249], who proved that for every integer $k\geq 2$ there exists a
 graph $G$ with no copies of $K_k$ such that
 $G\rightarrow(K_{k-1}, K_{k-1})$.
\end{abstract}

\section{Introduction}
Given positive integers $k$ and $\ell$, we say a graph $G$ is
\emph{Ramsey} for $(K_k,K_\ell)$ if every colouring of the edges of
$G$ with red and blue contains a red copy of $K_k$ or a blue copy of
$K_\ell$ and we denote this property by $G \rightarrow (K_k,K_\ell)$.
In a seminal work~\cite{Ramsey}, Ramsey proved that for all positive
integers $k$ and $\ell$, there exists a positive integer $n$ such that
$K_n \rightarrow (K_k, K_{\ell})$. In the special case $k=\ell$, we
simply write $G \rightarrow K_k$ and we define the \emph{Ramsey
 number} $R(k)$ as the minimum $n$ such that $K_n \rightarrow K_k$.

Although much effort has been put into estimating Ramsey numbers, a
parallel and rich direction of research investigates the structure of
graphs that are Ramsey for given pairs of graphs. In this context, we
study Ramsey phenomena of the form $G \rightarrow (K_{s},K_{t})$.

We remark that Theorem~\ref{thm:main} also relates to the theory of
Ramsey equivalence. Szab\'o, Zumstein, and Z\"urcher \cite{Szabo}
introduced the notion of \emph{Ramsey-equivalent graphs}: two graphs
$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we
have $G \rightarrow H_1$ if and only if $G \rightarrow H_2$
(see~\cite{Bloom, Fox}) for results on Ramsey equivalence). More
generally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are
{Ramsey-equivalent} if for every graph $G$ we have
$G \rightarrow (F_1, H_1)$ if and only if $G \rightarrow (F_2, H_2)$.
In other words, the two pairs share exactly the same family of Ramsey
graphs. In this direction, our result implies that the pairs $(K_k,K_k)$
and $(K_{s},K_{k-1})$ for any $s\leq R(k)-1$ are not
Ramsey-equivalent.

\begin{theorem}
 \label{thm:part1}
 For all integers $s\geq 2$ and $k\geq 3$, there exists \(C>0\) such that the
 following holds with high probability for $\cH = \hnp$ when
 $\pp \geq C n^{2-s-1/m_2(K_{k-1})}$. For every subhypergraph
 $\ho \subseteq \cH$ with at least $(1 - o(1))e(\cH)$ hyperedges, we
 have
 \[
 G[\ho] \rightarrow (K_{s}, K_{k-1}).
 \]
\end{theorem}

From Lemma~\ref{lem:quasi}, we have
$\PP{S_i \sqsubset \cH} \leq \pt^{|S_i|}$
for $\pt = \pp \binom{n-2}{s-2}$. Let $m = Dn^{2- 1/m_2(K_{k-1})}$ and
note that from the choice of $C$ we have
$m \leq (D/C)\pp n^{s} \leq \pt n^ 2$. Since $|S_i| \leq m$ for every
$i \in [t]$ and there are at most ${n^2\choose \ell}$ sources $S_i$
with exactly $\ell$ edges, we have
 $$
 \sum_{i=1}^{t} \mathbb{P}[S_i \sqsubset \cH]
 \leq \sum_{i=1}^{t} \pt^{|S_i|}
 \leq \sum_{\ell=1}^{m} \binom{n^2}{\ell} \pt^\ell
 \leq \sum_{\ell=1}^{m} \left(\frac{e\pt n^2}{\ell} \right)^{\ell}.
 $$
 Since $(e\pt n^2/\ell)^{\ell}$ is increasing for
 $\ell \leq \pt n^2$, we may replace $m$ with its upper bound
 $(D/C)\pp n^{s}$ in the above estimation. This together with
 $\pt n^2 \leq \pp n^s$ gives
\begin{equation}
 \sum_{i=1}^{t} \mathbb{P}[S_i \sqsubset \cH]
 \leq m \left(\frac{e\pt n^2}{m} \right)^{m}
 \leq n^2 {\left(\frac{eC}{D}\right)}^{(D/C)\pp n^{s}}
 \leq \exp\left(\delta\pp n^{s}/16\right)\label{eq:bound-si},
\end{equation}
where the last inequality follows from the fact that $C$ is
sufficiently large. Finally, using \eqref{eq:bound-xi} and
\eqref{eq:bound-si}, the bound on \eqref{eq:bound-main} becomes
 \begin{align*}
 \PP{\exists i \in [t]: X_i \leq \delta e(\cH) \text{ and } S_i \sqsubset \cH}
 & \leq \exp\left\{-\frac{\delta \pp n^{s}}{16} \right\} = o(1).
 \end{align*}
 Therefore, with high probability, every $\ho\subseteq \cH$ with
 $e(\ho) \geq (1-\delta)e(\cH)$ is such that
 $G[\ho]\rightarrow (K_{s},K_{k-1})$, which finishes the proof.
\end{proof}

In this short section we combine Theorems~\ref{thm:part1}
and~\ref{thm:part2} to prove our main result.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
 Let $k \geq 3$ be an integer and $s = R(k)-1$. Consider $\pp$ such that $n^{2-s-1/m_2(K_{k-1})} \ll \pp \ll n^{2-s - 1/m_2(K_k)}$ and let $\cH = \hnp$.
 By Theorem~\ref{thm:part1}, with high probability, every subhypergraph $\ho \subseteq \cH$ with \(e(\ho) = (1-o(1))e(\cH)\) satisfies
 \begin{align}
 G[\ho]\rightarrow(K_{s},K_{k-1}).
 \label{eq:H0a}
 \end{align}
 On the other hand, by Theorem~\ref{thm:part2}, with high probability
 there exists a subhypergraph $\ho \subseteq \cH$ with
 \(e(\ho) = (1-o(1))e(\cH)\) such that
 \begin{align}
 G[\ho]\nrightarrow K_k.
 \label{eq:H0b}
 \end{align}
 Since both events can occur with high probability, there exists a
 hypergraph $\ho$ such that both~\eqref{eq:H0a}
 and~\eqref{eq:H0b} hold. Therefore, $G[\ho]$ is the desired graph.

It is possible to adapt our proof to obtain the following
generalization of Theorem~\ref{thm:main} by considering a linear
$k$-conformal subhypergraph of $\cH_{s}(n,p)$, by choosing
$n^{2-s-1/m_2(K_{\ell-1})} \ll p \ll n^{2-s-1/m_2(K_{\ell})}$.
\begin{theorem}\label{thm:generalization}
 For any integers $k \ge \ell \ge 3$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_{\ell})$ and
 $G \rightarrow (K_{s},K_{\ell-1})$ for $s\leq R(k,\ell)-1$.
\end{theorem}
We propose the following conjecture as a variation of the previous
theorem for three colours.
\begin{conjecture}\label{conj:first}
 For any integers $k \ge \ell \ge 2$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_{\ell})$ and
 $G \rightarrow (K_{k-1},K_{k-1},K_{\ell})$.
\end{conjecture}
Note that the case $\ell = 2$ of the above conjecture is precisely the
result of Ne\v{s}et\v{r}il and R\"odl (Theorem~\ref{thm_NR}). We
conclude proposing the following conjecture that relates to
Conjecture~\ref{conj:first} in the same way that
Theorem~\ref{thm:generalization} relates to Theorem~\ref{thm_NR}.
\begin{conjecture}
 For any integers $k \ge \ell \ge 2$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_k,K_{\ell})$, but
 $G \rightarrow (K_{k+1},K_{k-1},K_{\ell})$.
\end{conjecture}

\begin{theorem}\label{thm:main}
	For every integer $k \ge 3$, there exists a graph $G$ such that $G
	\nrightarrow K_{k}$ and $G \rightarrow (K_{s},K_{k-1})$, for
  $s=R(k)-1$.
\end{theorem}

\begin{theorem}[Ne{\v{s}}et{\v{r}}il \& R{\"o}dl, 1976]\label{thm_NR}
	For every $k\geq 2$ there is a graph $G$ such that
	$K_k \nsubseteq G$ and $G\rightarrow K_{k-1}$.
\end{theorem}

Our main result, Theorem~\ref{thm:main} below, can be seen as a
variation of Theorem~\ref{thm_NR}. We prove that for any $k\geq 3$
there exists a graph $G$ that is not Ramsey for $K_k$ but it is Ramsey
for the pair $(K_{s},K_{k-1})$, for $s=R(k)-1$, i.e., we replace the
condition $K_k \nsubseteq G$ in Theorem~\ref{thm_NR} with the weaker
condition $G\not\rightarrow K_k$, which allows $G$ to contain copies of
$K_k$, but still there is a colouring of $E(G)$ avoiding monochromatic
copies of $K_k$; and we strengthen the conclusion
$G\rightarrow K_{k-1}$ by showing that
$G \rightarrow (K_{s},K_{k-1})$, for $s=R(k)-1$ (note that $s$ cannot
be any larger).
\begin{theorem}\label{thm:main}
	For every integer $k \ge 3$, there exists a graph $G$ such that $G
	\not\rightarrow K_{k}$ and $G \rightarrow (K_{s},K_{k-1})$, for
  $s=R(k)-1$.
\end{theorem}

We remark that Theorem~\ref{thm:main} also relates to the theory of
Ramsey equivalence. Szab\'o, Zumstein, and Z\"urcher \cite{Szabo}
introduced the notion of \emph{Ramsey-equivalent graphs}: two graphs
$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we
have $G \rightarrow H_1$ if and only if $G \rightarrow H_2$
(see~\cite{Bloom, Fox}) for results on Ramsey equivalence). More
generally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are
{Ramsey-equivalent} if for every graph $G$ we have
$G \rightarrow (F_1, H_1)$ if and only if $G \rightarrow (F_2, H_2)$.
In other words, the two pairs share exactly the same family of Ramsey
graphs. In this direction, our result implies that the pairs $(K_k,K_k)$
and $(K_{s},K_{k-1})$ for any $s\leq R(k)-1$ are not
Ramsey-equivalent.

The proof of Theorem~\ref{thm:main} combines probabilistic methods
with the hypergraph container
framework~\cite{balogh2015independent,SaxtonThomason2015} and is
inspired by ideas from~\cite{bollobas2001ramsey}. The rest of the
paper is organized as follows. In Section~\ref{sec:mono}, we show that
with high probability\footnote{Meaning with probability going to $1$
  as $n$ tends to infinity.} the graph $G$ obtained in a natural way from
every ``dense'' subhypergraph of a suitable $n$-vertex random
$s$-uniform hypergraph satisfies $G\rightarrow(K_{s},K_{k-1})$ for
$s = R(k)-1$. In Section~\ref{sec:hyper}, we show that with high
probability a suitable random hypergraph $\mathcal{H}$ contains a dense
subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such
that $G\not\rightarrow K_k$. These results are then combined in
Section~\ref{sec:proof}. Finally, in Section~\ref{sec:conc}, we
outline some directions for future research.

The post-theorem introduction gives the following proof outline for Theorem~\ref{thm:main}: it “combines probabilistic methods with the hypergraph container framework~\cite{balogh2015independent,SaxtonThomason2015} and is inspired by ideas from~\cite{bollobas2001ramsey}.” The paper is then organized into steps: (1) In Section~\ref{sec:mono}, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) In Section~\ref{sec:hyper}, “with high probability a suitable random hypergraph $\mathcal{H}$ contains a dense subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such that $G\not\rightarrow K_k$.” (3) “These results are then combined in Section~\ref{sec:proof}” to produce a graph $G$ satisfying both $G\not\rightarrow K_k$ and $G\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).

The post-theorem introduction gives the following proof outline for the main theorem: it “combines probabilistic methods with the hypergraph container framework Balogh--Morris--Samotij, *Independent sets in hypergraphs* (2015) and Saxton--Thomason, *Hypergraph containers* (2015) and is inspired by ideas from Bollobás--Erdős, *Ramsey graphs* (2001).” The paper is then organized into steps: (1) Next, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) Then, “with high probability a suitable random hypergraph $\mathcal{H}$ contains a dense subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such that $G\not\rightarrow K_k$.” (3) “These results are then combined later” to produce a graph $G$ satisfying both $G\not\rightarrow K_k$ and $G\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).

Question: Let $k\ge 2$ be an integer. For a graph $G$, write $G\rightarrow K_{k-1}$ to mean that every 2-coloring of the edges of $G$ with red and blue contains a monochromatic copy of $K_{k-1}$. Which of the following conclusions about graphs $G$ holds for every such $k$?

There exists a graph $G$ such that $K_k\nsubseteq G$ (that is, $G$ contains no copy of $K_k$) and $G\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.

Choices:
B. There exists a graph $G$ such that $K_{k-1}\nsubseteq G$ and $G\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.
C. There exists a graph $G$ such that $G\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.
D. There exists a graph $G$ such that $K_k\nsubseteq G$ and for every red-blue coloring of the edges of $G$, there is a red copy of $K_{k-1}$ and a blue copy of $K_{k-1}$.
E. For every graph $G$ such that $K_k\nsubseteq G$, one has $G\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.


\begin{equation}\label{mH1}
	\mH(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}

\begin{theorem}\label{TH-00}
Let $u$ be a suitable weak solutions. Suppose that
the following two conditions hold:
\begin{equation}
\liminf_{r\to 0}M(r):=\liminf_{r\to 0}\int_{B_r}\!\! \left( \frac{|u|^2}{r^3} + \frac{|\nabla u|^2}{r} \right) <\infty, \quad \liminf_{r\to 0}\frac1{r^2}\int_{B_r}\!\!(|u|^2+2P)u\cdot\frac x{|x|}>0.
\end{equation}
Then  for any $\{r_k\}_{k=1}^\infty, r_k\downarrow 0$ there is a subsequence $r_{k_m}$ such that
the scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to
a homogenous vector field of degree negative one, and hence
$x=0$ is a regular point.
\end{theorem}

\begin{proposition}\label{prop:cubicEst}
	For fixed constants $\delta_{1}, \delta_{2}>0$, there exists $\ep>0$ such that if $(u,p)$ is a Leray-Hopf solution to the Navier-Stokes equations satisfying
	\begin{equation}\label{h-A}
		\frac{1}{R^3} \int_{B_R} \big|u- \mP_R[u] \big|^2 + \frac{1}{R} \int_{ B_R} \big|\nabla u - \nabla \mP_R[u] \big|^2 \le \ep M[u](R)
	\end{equation}
	for some $R>0$, then
	\begin{equation}
		\bigg|\frac{1}{R^2}\int_{B_{R/2}} \big( |u|^2 + 2 p \big) \, u \cdot \frac{x}{|x|} \bigg|  \le \delta_{1} + \delta_{2} \big(M[u](R)\big)^{\frac{3}{2}}.
	\end{equation}
\end{proposition}

In this paper we study the local behavior of the weak solutions of the stationary incompressible
Navier-Stokes equations in five space dimensions
\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \mathbb{R}^5
\end{equation}
where $\Omega\subset \mathbb{R}^5$ is a domain .

The existence of weak solutions under various assumptions on the
boundary data and $\Omega$ has been established in \cite{Galdi}, \cite{FR-Pisa}, \cite{Struwe-per}.
Moreover, in  \cite{FR-Pisa}, \cite{FR-arma}, \cite{Struwe-per} the authors constructed
smooth solutions of \eqref{eq:problem-3d}.

The problem \eqref{eq:problem} has a number of similarities with the
dynamic Navier-Stokes system in three space dimensions
\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \mathbb{R}^3.
\end{equation}
For instance, in both cases $u\in L^{\frac{10}3}_{loc}, P\in L^{\frac 53}_{loc}, \nabla P\in L^{\frac54}_{loc}$, see  \cite{Seregin-book}.
Due to this a number of mathematicians studied the stationary
Navier-Stokes equations in higher dimensions in order to develop
stronger analytical methods which may be applicable to the
dynamic case \eqref{eq:problem-3d}, see \cite{Galdi}.

In this context, of particular interest is the problem of estimating the
dimension of the singular points of suitable weak solutions $u$, i.e. the points where
$u$ is not bounded. Scheffer \cite{Scheffer}
proved such results  for \eqref{eq:problem-3d} and later
Caffarelli, Kohn and Nirenberg \cite{CKN} improved upon it
showing that  the Hausdorff dimension of the singular set in space-time is atmost one.
We note that the latter result can be established by a different method
by looking at the
small perturbations of the Stokes system \cite{Lin}.
For \eqref{eq:problem} the partial regularity is proved in \cite{Struwe}.

At the possible singular point $(x_0, t_0)$ the scale invariance  $u(x, t)\mapsto ru(x_0+rx, t_0+r^2), r>0$ suggests that at the scale $r$, $u$ behaves like $1/r$ near $(x_0, t_0)$.
A natural question that follows from this observation is whether
one can classify the scale invariant solutions.
This has been the main approach towards understanding the structure of possible singularities.
\v{S}ver\'{a}k's classification for the self-similar solutions \cite{Sverak} for Navier-Stokes equations \eqref{eq:problem} shows that
a solution of the form $h(x)=\frac{\zeta(\frac x{|x|})}{|x|}$, with some smooth vectorfield $\zeta$, must be identically zero.

Another questions following from this result is whether the solutions sufficiently close to the self-similar one
are in fact zero. It is easy to see that the self-similar vectorfields $h=\frac{\zeta(\frac x{|x|})}{|x|}$ form a Hilbert subspace $\mathcal{H}(R)$ of
the Sobolev space $\mathcal W^{1, 2}(B_R)$ in $\mathbb{R}^5$ with an appropriately scaled invariant norms,
\begin{equation}\label{mH1}
	\mathcal{H}(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}
Thus the Hilbert projection theorem yields that $\mathcal W^{1,2}(B_R) = \mathcal{H}(R) \oplus \mathcal{H}(R)^{\perp}$ and we can define the the corresponding projection operator as $\mathcal{P}_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$. Using this, we can measure the error $u-\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.

Our work is motivated by the following question.

{\textit{  If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically  become self-similar?}}

Our  main result in this direction can be stated as follows:

In this paper we study the local behavior of the weak solutions of the stationary incompressible
Navier-Stokes equations in five space dimensions
\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \mathbb{R}^5
\end{equation}
where $\Omega\subset \mathbb{R}^5$ is a domain .

The problem \eqref{eq:problem} has a number of similarities with the
dynamic Navier-Stokes system in three space dimensions
\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \mathbb{R}^3.
\end{equation}
For instance, in both cases $u\in L^{\frac{10}3}_{loc}, P\in L^{\frac 53}_{loc}, \nabla P\in L^{\frac54}_{loc}$, see \cite{Seregin-book}.
Due to this a number of mathematicians studied the stationary
Navier-Stokes equations in higher dimensions in order to develop
stronger analytical methods which may be applicable to the
dynamic case \eqref{eq:problem-3d}, see \cite{Galdi}.

In this context, of particular interest is the problem of estimating the
dimension of the singular points of suitable weak solutions $u$, i.e. the points where
$u$ is not bounded. Scheffer \cite{Scheffer}
proved such results for \eqref{eq:problem-3d} and later
Caffarelli, Kohn and Nirenberg \cite{CKN} improved upon it
showing that the Hausdorff dimension of the singular set in space-time is atmost one.
We note that the latter result can be established by a different method
by looking at the
small perturbations of the Stokes system \cite{Lin}.
For \eqref{eq:problem} the partial regularity is proved in \cite{Struwe}.

Another questions following from this result is whether the solutions sufficiently close to the self-similar one
are in fact zero. It is easy to see that the self-similar vectorfields $h=\frac{\zeta(\frac x{|x|})}{|x|}$ form a Hilbert subspace $\mathcal{H}(R)$ of
the Sobolev space $\mathcal W^{1, 2}(B_R)$ in $\mathbb{R}^5$ with an appropriately scaled invariant norms,
\begin{equation}\label{mH1}
 \mathcal{H}(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}
Thus the Hilbert projection theorem yields that $\mathcal W^{1,2}(B_R) = \mathcal{H}(R) \oplus \mathcal{H}(R)^{\perp}$ and we can define the the corresponding projection operator as $\mathcal{P}_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$. Using this, we can measure the error $u-\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.

{\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically become self-similar?}}

Our main result in this direction can be stated as follows:

\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \R^5
\end{equation}

\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \R^3.
\end{equation}

{\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically become self-similar?}}

If $\liminf_{r\to 0}M(r)<\infty$
then the singularity may occur only if the function
\begin{equation}
\text{\Large$\wp$}(r) =\frac{1}{r^2} \int_{B_r} \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|}) \, dx .
\end{equation}
takes nonpositive values as $r\to 0$. Moreover,
if $u$ is of the form $\frac{\zeta(\frac x{|x|})}{|x|}$ then one can check that
$\text{\Large$\wp$}(r) =0$. This observation motivates the formulation of
a condition in our next result that allows to control $\text{\Large$\wp$}(r)$.
\begin{theorem}\label{TH}
Let $u$ be a suitable weak solution of \eqref{eq:problem}, $B_1\subset \Omega$, and
\[
m:=\liminf_{R\to 0}M(R)<\infty \qquad \text{where} \quad M(R):=\int_{B_R} \Big( \frac{|u|^2}{R^3} + \frac{|\nabla u|^2}{R} \Big).
\] Let $\mP_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mH(R)$ be the projection operator for the space \eqref{mH1}. There exists $\ep(m)>0$
such that
if
\begin{equation}\label{eq:13-alt}
\frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le \ep(m) M(R)
\end{equation}
holds
for all $R\in (0, R_0)$ then $u$ is regular at $x=0$.
\end{theorem}

\begin{proposition}\label{prop:dAdr}
 Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \eqref{st-NS-N}. For $r>0$, define
 \begin{equation}
 \left\{
 \begin{aligned}
 &D(r)\vcentcolon= \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2 + \frac{1}{4r}|\nabla u |^2 + \frac{3}{4r^3}\left| \nabla (|x|u) \right|^2 + \frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,\\
 &A(r) \vcentcolon= \frac{1}{r^3}\int_{B_r}(x\cdot \nabla)\frac{|u|^2}{2}\, dx + \frac{9}{4r^3}\int_{ B_r}|u|^2 \, dx - \frac{1}{r^{2}} \int_{B_r} \big\{ \frac{|u|^2}{2} + P \big\} u\cdot \frac{x}{|x|} \, dx .
 \end{aligned}
 \right.
 \end{equation}
 Then the following differential equation holds for $r>0$,
 \begin{equation*}
 \frac{d A}{d r} \ge \frac{1}{r} D(r) + \frac{2}{r^3} \int_{B_r} \Big\{\frac{|u|^2}{2} + P \Big\} (u\cdot \frac{x}{|x|}) \, dx .
 \end{equation*}
\end{proposition}
\begin{proof}
 Let us consider the function
 \begin{equation}
 \phi(x)=
 \left\{
 \begin{array}{lll}
 1 \quad & \mbox{if} \ |x|<r,\\
 \frac{r-|x|}\epsilon \quad &\mbox{if} \ r\le |x|\le r+\epsilon,\\
 0 \quad & \mbox{if}\ |x|>r+\epsilon.\\
 \end{array}
 \right.
 \end{equation}
 We then mollify this function and take $\psi=\phi*\rho_\delta$,
 where $\rho$ is the mollification kernel.
 Note that $\psi\in \mathcal C_c^\infty(\R^N)$.
 We use $\psi$ as a test function in the local
 energy inequality to obtain
 \begin{equation}
 \int |\nabla u|^2 \psi
 \le
 \int \left(-u \nabla u +(|u|^2+2P)u\right)\cdot \nabla \psi.
 \end{equation}

\begin{lemma}\label{lemma:reg}
Suppose $m:=\liminf_{R\to 0}M(R)<\infty$.
 If there exists a sufficiently small $\ep>0$, depending on $m$, such that $(u,P)$ is a suitable weak solution to the Navier-Stokes equations satisfying
 \begin{equation}\label{H-Assump}
 \ep \frac{1}{R^{3}}\int_{B_R} |p| + \frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le \ep M[u](R),
 \end{equation}
 for all $R\in(0,1]$ then $u$ is regular at $x=0$.
\end{lemma}
\begin{proof}
If $m\le 8C_E$ then we can apply Proposition \ref{prop:fM}, and
hence the result follows.
 Now suppose $m\in(8C_E, \infty)$. In light of Lemma \ref{lemma:MBound}, there exists $\ep_1>0$ such that if $(u,P)$ satisfies \eqref{H-Assump} with $\ep\in (0,\ep_1)$ then $M(R)$ is uniformly bounded in $R\in(0,1]$ and we set
 \begin{equation}\label{Mast}
 M_{\ast} \vcentcolon= \sup\limits_{0<R\le 1} M(R) \le 2^5 C_0(m),
 \end{equation}
 where $C_0(m)$ is given by \eqref{eq:Cmm}.
 At this point we can apply Proposition \ref{prop:fM}
 with $C_0=2^5C_0(m)$ by choosing $\ep_0$ sufficiently small, however we can avoid this by using the monotonicity formula.

\begin{lemma}\label{lem:Lin}
If $u$ is a suitable weak solution and
\[
\int_{B_1} |u|^3+|p|^{\frac32}<\ep^*
\]
for some sufficiently small $\ep^*$,
then
\begin{equation}\label{eq:Lin1}
\frac1{\theta^5}\int_{B_\theta}\frac{|u-[u]_\theta|^3}{\theta^{\alpha_0}}+\frac{|p-[p]_\theta|^\frac32}{\theta^{\alpha_0}}\le \frac12 \int_{B_1} |u|^3+|p|^{\frac32},
\end{equation}
for some positive $\theta$ and $\alpha_0\in (0, \frac12)$.
\end{lemma}
\begin{proof}
If \eqref{eq:Lin1} fails, then there would be a sequence of solutions
$(u_i, p_i)$ such that $\int_{B_1}|u_i|^3+|p_i|^{\frac32}:=\epsilon_i\to 0$ but
\eqref{eq:Lin1} is not valid. Introduce
\[
\bar u_i=\frac{u_i}{\epsilon_i}, \quad \bar p_i=\frac{p_i}{\epsilon_i},
\]
then
\begin{equation}\label{eq:Lin4}
\epsilon_i \bar u_i\cdot \nabla \bar u_i+\nabla \bar p_i=\Delta \bar u_i,
\quad
\frac1{\theta^5}\int_{B_\theta}\frac{|\bar u-[\bar u]_\theta|^3}{\theta^{\alpha_0}}
+
\frac{|\bar p-[\bar p]_\theta|^\frac32}{\theta^{\alpha_0}}> \frac12,
\quad \int_{B_1} |\bar u_i|^3+|\bar p_i|^{\frac32}\le 2.
\end{equation}
From the local energy inequality $u\in \mathcal W^{1, 2}_{loc}(B_1).$
Moreover, the following equation is satisfied in distributional sense
\begin{equation}
\Delta \bar p_i=-\epsilon_i \frac{\partial^2 (\bar u^k\bar u^l)}{\partial x_l\partial x_k},
\quad \mbox {in}\ B_1.
\end{equation}
From the Poisson representation theorem we can write $\bar p_i=h_i+g_i$,
where $h_i$ is harmonic in $B_1$, and
\begin{equation}
\left\{
\begin{array}{lll}
\Delta g_i=-\epsilon_i \frac{\partial^2 (\bar u^k\bar u^l)}{\partial x_l\partial x_k}
\quad \mbox {in}\ B_{\frac23},\\
g_i=0 \quad \mbox {on}\ \partial B_{\frac23}.
\end{array}
\right.
\end{equation}
From the Calder\'on-Zygmund estimates
$g_i$ is uniformly bounded in ${L^{5/3}(B_{2/3})}$.
Consequently, $h_i\in L^{3/2}(B_{2/3})$ uniformly, hence
from the local estimates for the harmonic functions
\begin{align}
\int_{B_\theta}|\bar p_i-[\bar p_i]_\theta|^{\frac32}\le
\int_{B_\theta}| h_i-[h_i]_\theta|^{\frac32}
\int_{B_\theta}|g_i-[g_i]_\theta|^{\frac32}\\
\le C_0 \theta^5\theta^{3/2}+C_0\epsilon_i\int_{B_{2/3}}|\bar u_i|^3.
\end{align}
For a suitable subsequence $\bar u_i\to \bar u$ in $\mathcal W^{1, 2}(B_{2/3})$
and $\bar p_i\to \bar p$ strongly in ${L^{3/2}(B_{2/3})}$.
Consequently, for sufficiently large $i$, we have
\begin{equation}\label{eq:Lin2}
\int_{B_\theta}|\bar p_i-[\bar p_i]_\theta|^{\frac32} \le C_0 \theta^5\theta^{3/2}
\end{equation}
Since the limit $\bar u$ solves the Stokes system, then it follows that
$\bar u$ is H\"older continuous with, say, exponent $2\alpha_0$, and therefore
$\int_{B_\theta}|\bar u-[\bar u]_\theta|^{\frac32}\le \frac14 \theta^5\theta^{\alpha_0}$.
From the strong convergence $\bar u_i\to \bar u$ in $L^3(B_{2/3})$, we infer that
\begin{equation}\label{eq:Lin3}
\int_{B_\theta}|\bar u-[\bar u]_\theta|^{\frac32}\le \frac13 \theta^5\theta^{\alpha_0}.
\end{equation}
Combining \eqref{eq:Lin2} and \eqref{eq:Lin2} we get a contradiction with the
second inequality in \eqref{eq:Lin4}.
\end{proof}

The proof of Theorem \ref{TH-00} uses the monotonicity
formula introduced in Proposition \ref{prop:dAdr}, and
a scaling argument. See Lemma \ref{lem:positiveinf} for the proof.
Note that there are no smallness assumptions in the
statement of Theorem \ref{TH-00}.

If $\liminf_{r\to 0}M(r)<\infty$
then the singularity may occur only if the function
\begin{equation}
\text{\Large$\wp$}(r)  =\frac{1}{r^2} \int_{B_r}  \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
\end{equation}
takes nonpositive values as $r\to 0$. Moreover,
if $u$ is of the form $\frac{\zeta(\frac x{|x|})}{|x|}$ then one can check that
$\text{\Large$\wp$}(r) =0$. This observation motivates the formulation of
a condition in our next result that allows to control $\text{\Large$\wp$}(r)$.
\begin{theorem}\label{TH}
Let $u$ be a suitable weak solution of \eqref{eq:problem}, $B_1\subset \Omega$, and
\[
m:=\liminf_{R\to 0}M(R)<\infty \qquad \text{where} \quad M(R):=\int_{B_R} \Big( \frac{|u|^2}{R^3} + \frac{|\nabla u|^2}{R} \Big).
\] Let $\mP_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$ be the projection operator for the space \eqref{mH1}. There exists $\varepsilon(m)>0$
such that
if
\begin{equation}\label{eq:13-alt}
\frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le  \varepsilon(m) M(R)
\end{equation}
holds
for all $R\in (0, R_0)$ then $u$ is regular at $x=0$.
\end{theorem}

It is known that if $u\in \mathcal{H}$ then $u=0$ \cite{Sverak}. In this context, Theorem \ref{TH}
states that if $\|u-\mP_R[u]\|_{\mathcal{W}^{1,2}(R)}$ is small compared to $\|u\|_{\mathcal{W}^{1,2}(R)}$ then
$\text{\Large$\wp$}(r)$ is smaller than $M^{\frac32}(r)$, which
after application of Proposition \ref{prop:dAdr} implies that
$u=0$.

As opposed to the main result in \cite{CKN}, we do not assume that $u$ is small in some scale invariant seminorm, reminiscent to
the ``local" Reynolds number $\frac1{r}\int_{B_r}|\nabla u|^2$. This  leads us to the classification of the
self-similar solution of the incompressible Euler equations in $\mathbb{R}^5$.
In fact, we prove that for such solutions
the Bernoulli pressure is zero. This is the first  key point in our proof of the main technical result, Proposition \ref{prop:cubicEst}.

The second key point is the construction of a monotonicity formula for the suitable weak solutions, which follows from
the weak energy inequality.

We compare Theorem \ref{TH} with the well-known
regularity criteria for suitable weak solutions of
\eqref{eq:problem-3d}, which in its most general form, can be stated as follows:
let $Q(R)=B_R\times(-R^2, 0)$ and define the local Reynolds numbers
\[
E(R)=\frac1R\int_{Q(R)}|\nabla u|^2, \quad C(R)=\frac1{R^2}\int_{Q(R)}|u|^3.
\]

Then the following statement holds: for every $M>0$ there is
$\varepsilon(M)>0$ such that
$\limsup_{R\to 0} C(R)<M, \liminf_{R\to 0}E(R)<\epsilon(M)$ imply that the origin is a regular point.

This result can be found in
Seregin's paper \cite{Seregin}, Theorem 1.4. Note that
 $\limsup_{R\to 0} C(R)<M$ implies that there is a constant $C_0(M)$
 depending on $M$ such that $\limsup_{R\to 0} E(R)<C_0(M)$ \cite{Lewis}.

As opposed to this result, we do not impose the finiteness of upper limit
of the Reynolds number. Instead, we assume that the lower limit is finite, i.e.
\[
\liminf_{R\to 0} M(R)<\infty.
\]

This is the replacement of the condition $\limsup_{R\to 0} C(R)<M$ in \cite{Seregin}.

As for the other condition, $\liminf_{R\to 0}E(R)<\epsilon(M)$, it is replaced by closeness assumption: more precisely, we assume that there is a vectorfield $h$
homogeneous of degree negative one such that
$u-h$ has a suitable small norm compared to $M(R)$.
See Section \ref{sec:energy}
for precise definitions.
Hence our conditions are weaker.

The paper is organized as follows:
In Section \ref{sec:mon} we introduce one of our main technical tools, the
monotonicity formula and prove Theorem \ref{TH-00}. In the next section
we classify the self-similar solutions of the incompressible Euler equations in
$\mathbb R^5$, and prove that for such solutions the Bernoulli
pressure  is zero.
In Section \ref{sec:energy} we prove some estimates for the pressure.
Section \ref{sec:technical} contains one of our main estimates of the
cubic term that appears in the local energy inequality.
In order to control the growth of $M(R)$ we prove an iteration result in Section \ref{sec:iteration}, and apply it to obtain a local bound in Section
\ref{sec:sup-est}.
The proof of Theorem \ref{TH} is given in Section.
\ref{sec:proofofmain}.
We also added an appendix at the end of the paper that contains
some estimates and computations used in the proof of Theorem \ref{TH}.

The post-theorem introduction says: “The proof of Theorem \ref{TH-00} uses the monotonicity formula introduced in Proposition \ref{prop:dAdr}, and a scaling argument. See Lemma \ref{lem:positiveinf} for the proof.”

The post-theorem introduction says: “To prove the main theorem, the proof uses the monotonicity formula introduced in the following proposition.

\begin{proposition}\label{prop:dAdr}
	Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \eqref{st-NS-N}. For $r>0$, define
	\begin{equation}
		\left\{
		\begin{aligned}
			&D(r)\vcentcolon=   \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2 + \frac{1}{4r}|\nabla u |^2 + \frac{3}{4r^3}\left| \nabla (|x|u) \right|^2 + \frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,\\
			&A(r) \vcentcolon= \frac{1}{r^3}\int_{B_r}(x\cdot \nabla)\frac{|u|^2}{2}\, dx + \frac{9}{4r^3}\int_{ B_r}|u|^2 \, dx  - \frac{1}{r^{2}} \int_{B_r} \big\{ \frac{|u|^2}{2} + P \big\} u\cdot \frac{x}{|x|}  \, dx .
		\end{aligned}
		\right.
	\end{equation}
	Then the following differential equation holds for $r>0$,
	\begin{equation*}
		\frac{d A}{d r} \ge \frac{1}{r} D(r) + \frac{2}{r^3} \int_{B_r}  \Big\{\frac{|u|^2}{2} + P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
	\end{equation*}
\end{proposition}

It also uses a scaling argument. For the proof, see the following lemma.

\begin{lemma}\label{lem:positiveinf}
Let
\begin{equation}
Q(r):= \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2
+
\frac{1}{4r}|\nabla u |^2
+
\frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,
\end{equation}
and
\begin{equation}
\text{\Large$\wp$}(r)  =\frac{1}{r^2} \int_{B_r}  \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
\end{equation}
If  $\liminf_{r\to 0} M(r)<\infty$ and
\[
\liminf_{r\to0^+}\left[ Q(r)
+\text{\Large$\wp$}(r)
\right]> 0
\]
then $x=0$ is a regular point.
\end{lemma}”,

\label{TH-00}
Let $u$ be a suitable weak solutions. Suppose that
the following two conditions hold:
\begin{equation}
\liminf_{r\to 0}M(r):=\liminf_{r\to 0}\int_{B_r}\!\! \left( \frac{|u|^2}{r^3} + \frac{|\nabla u|^2}{r} \right) <\infty, \quad \liminf_{r\to 0}\frac1{r^2}\int_{B_r}\!\!(|u|^2+2P)u\cdot\frac x{|x|}>0.
\end{equation}
To prove the main theorem, we show that for any $\{r_k\}_{k=1}^\infty, r_k\downarrow 0$ there is a subsequence $r_{k_m}$ such that
the scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to
a homogenous vector field of degree negative one, and hence
$x=0$ is a regular point.,

Question:
Let $(u,P)$ be a suitable weak solution near $x=0$ of the stationary incompressible NavierStokes system in five dimensions,
\[
u^j\partial_j u^i + \partial_i P = \Delta u^i, \qquad \operatorname{div}u=0,
\]
in a domain $\Omega\subset\mathbb{R}^5$. For $r>0$, write
\[
M(r):=\int_{B_r}\left(\frac{|u|^2}{r^3}+\frac{|\nabla u|^2}{r}\right),
\]
where $B_r$ is the ball of radius $r$ centered at $0$. Assume that
\[
\liminf_{r\to 0} M(r)<\infty
\]
and also
\[
\liminf_{r\to 0}\frac1{r^2}\int_{B_r} (|u|^2+2P)\,u\cdot \frac{x}{|x|}>0.
\]
For a scale $r>0$, define the rescaled velocity field by
\[
u_r(x)=r\,u(rx).
\]
Which conclusion holds under these hypotheses?

For every sequence $\{r_k\}_{k=1}^\infty$ with $r_k\downarrow 0$, there exists a subsequence $r_{k_m}$ such that the scaled fields $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$ converge to a vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point.

Choices:
B. There exists a sequence $r_k\downarrow 0$ such that the scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converge to a vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point.
C. There exists a sequence $r_k\downarrow 0$ such that, after passing to a subsequence, the scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converge to a vector field that is homogeneous of degree $-1$.
D. For every sequence $\{r_k\}_{k=1}^\infty$ with $r_k\downarrow 0$, the full sequence of scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converges to a unique vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point.
E. For every sequence $\{r_k\}_{k=1}^\infty$ with $r_k\downarrow 0$, there exists a subsequence $r_{k_m}$ such that the scaled fields $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$ converge to a vector field that is homogeneous of degree $-1$; moreover, this implies that $x=0$ is a singular point.

dropped regularity conclusion and universal quantifier over sequences

subsequence compactness replaced by full-sequence unique limit

sign of the monotonicity-based regularity consequence