\begin{corollary}\label{CorollaryOnSheaves}
    Let $F$ be a torsion-free $H$-slope-semistable sheaf on $S_5$, then the numerical Chern characters of $F$ satisfy \eqref{Ch1Ch2Inequal}.
\end{corollary}

\begin{corollary}\label{MainCorollary}
    Let $F$ be a torsion-free $H$-slope-semistable sheaf on $X$, then the numerical Chern characters of $F$ satisfy \eqref{InequalityOnThreefolds}.
\end{corollary}

\begin{theorem}\label{MainTheorem}
     Let $X\subset \mathbb{P}^4_{\mathbb{C}}$ be a smooth quintic $3$-fold and $H: = c_1(\mathcal{O}_X(1))$, $F$ be a $\nu_{\alpha,0,H}$-tilt-stable object in $\Coh^{0,H}(X)$ for some $\alpha >0$. Suppose $\mu_H(F)\in [-1,1]$. then
    \begin{equation}\label{InequalityOnThreefolds}
        \xi_H(F)\leq
    \begin{cases}
    \frac{-1}{2}|\mu_H(F)| & \text{when } 0\leq |\mu_H(F)| \leq \frac{1}{4}; \\[0ex]
    \frac{1}{2}|\mu_H(F)| - \frac{1}{4} & \text{when } |\mu_H(F)|\in[\frac{1}{4},\frac{5}{13}]\cup[\frac{8}{13},\frac{3}{4}]; \\[0ex]
    \frac{-3}{20}|\mu_H(F)|  & \text{when } \frac{5}{13} \leq |\mu_H(F)|\leq \frac{6}{13}; \\[0ex]
    \frac{1}{2}|\mu_H(F)|-\frac{3}{10} & \text{when } \frac{6}{13}\leq |\mu_H(F)| \leq \frac{7}{13}; \\[0ex]
    \frac{23}{20}|\mu_H(F)| - \frac{13}{20} & \text{when } \frac{7}{13}\leq |\mu_H(F)| \leq \frac{8}{13}; \\[0ex]
    \frac{3}{2}|\mu_H(F)| - 1 & \text{when } \frac{3}{4}\leq |\mu_H(F)| \leq 1.
\end{cases}
    \end{equation}
\end{theorem}

\begin{proposition}\label{ChInequalityOnSurface}
    Let $S_5\subset \mathbb{P}^3$ be a smooth irreducible quintic surface, $H = c_1(\mathcal{O}_{S_5}(1))$, and let $F$ be an object in $\Db(S_5)$ such that $\mu_H(F)\in (0,1)$. Suppose $F$ is $\nu_{\alpha,0,H}$-tilt-stable (or $\nu_{\alpha',1,H}$-tilt-stable) for some $\alpha > 0$ (resp. or $\alpha' > \frac{1}{2}$), then
    \begin{equation}\label{Ch1Ch2Inequal}
        \xi_H(F)\leq
    \begin{cases}
    \frac{17}{26}\mu_H(F) - \frac{2}{13} & \text{when } 0 < \mu_H(F) \leq \frac{7}{46}; \\[0ex]
    -\frac{5}{14}\mu_H(F)  & \text{when } \frac{7}{46} \leq  \mu_H(F)\leq \frac{7}{20}; \\[0ex]
    \frac{1}{2}\mu_H(F)-\frac{3}{10} & \text{when } \frac{7}{20}\leq \mu_H(F) \leq \frac{13}{20}; \\[0ex]
    \frac{19}{14}\mu_H(F) - \frac{6}{7} & \text{when } \frac{13}{20}\leq \mu_H(F)< \frac{39}{46}; \\[0ex]
    \frac{9}{26}\mu_H(F) & \text{when } \frac{39}{46}\leq \mu_H(F)< 1.
\end{cases}
    \end{equation}
\end{proposition}

The classical result of Bogomolov and Gieseker provides a fundamental inequality for stable sheaves on smooth projective varieties.

\maketitle

The Bogomolov--Gieseker (BG) inequality plays an important role in the study of stability conditions and moduli spaces of sheaves. It is a natural question to seek refinements of the BG inequality for special varieties; see for instance \cite[Question 5.1]{BM23}.

Motivated by Gepner type stability conditions on graded matrix factorizations, Toda proposed in \cite{Toda2017GepnerPoint} a stronger inequality on quintic threefolds, and proved it for $\rk(E) = 2$ case.
\begin{theorem}[{\cite[Conjecture $1.1$]{Toda2017GepnerPoint}}]\label{TodaConj0}
 Let $X\subset \mathbb{P}^4_{\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\mathcal{O}_X(1))$. Then for any torsion-free $H$-slope-stable sheaf $E$ on $X$ with $\ch{1}(E)/\rk (E) = -H/2$, we have
 \begin{equation}\label{TodaConj1}
 \frac{H\cdot \Delta(E)}{\rk(E)^2} > 1.5139\cdots,
 \end{equation}
 where the right-hand side of is an irrational real number lying in $\mathbb{Q}(e^{2\pi \sqrt{-1}/5})$. Equivalently,
 \begin{equation}\label{TodaConj2}
 \frac{H\ch{2}(E)}{H^3\rk(E)} < -0.02639\cdots
 \end{equation}
\end{theorem}

Along the way, we also obtain a stronger Bogomolov--Gieseker type inequality for quintic surfaces.
\begin{theorem}[See \Cref{ChInequalityOnSurface} and \Cref{CorollaryOnSheaves}]
 Let $S_5\subset \mathbb{P}^3_{\mathbb{C}}$ be a smooth quintic surface and $H\coloneqq c_1(\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\Coh(X)$ with
 \begin{equation*}
 \mu_H(F)\in (0,1).
 \end{equation*}
 Then the following piecewise inequality holds:
 \begin{equation}\notag
 \xi_H(F)\leq
 \begin{cases}
 \frac{17}{26}\mu_H(F) - \frac{2}{13} & \text{when } 0 < \mu_H(F) \leq \frac{7}{46}; \\[0ex]
 -\frac{5}{14}\mu_H(F) & \text{when } \frac{7}{46} \leq \mu_H(F)\leq \frac{7}{20}; \\[0ex]
 \frac{1}{2}\mu_H(F)-\frac{3}{10} & \text{when } \frac{7}{20}\leq \mu_H(F) \leq \frac{13}{20}; \\[0ex]
 \frac{19}{14}\mu_H(F) - \frac{6}{7} & \text{when } \frac{13}{20}\leq \mu_H(F)< \frac{39}{46}; \\[0ex]
 \frac{9}{26}\mu_H(F) & \text{when } \frac{39}{46}\leq \mu_H(F)< 1.
\end{cases}
 \end{equation}
\end{theorem}

The classical BG inequality extends to tilt-semistable objects.
\begin{theorem}[{\cite[Theorem 7.3.1]{bayer_bridgeland_2013}}, {\cite[Proposition 2.21]{piyaratne_moduli_2019}}]
 Let $X$ be a smooth projective variety, and $H\in \mathrm{NS}(X)_{\mathbb{R}}$ an ample class. If $E$ is $\nu_{\alpha,\beta,H}$-tilt-semistable for some $\alpha > \frac{1}{2}\beta^2$, then $\overline{\Delta}_H(E) \geq 0$.
\end{theorem}

\subsection{Useful Lemmas}
Let $X$ be a smooth projective variety of dimension $n$ and $H\in \text{NS}(X)_{\mathbb{R}}$ be a real ample divisor class. For $E\in \Db(X)$, define
\begin{equation*}
 \overline{v}_H(E) \coloneqq (H^n\ch{0}(E),H^{n-1}\ch{1}(E),H^{n-2}\ch{2}(E)),
\end{equation*}
and, when $H^n\ch{0}(E) \neq 0$,
\begin{equation*}
 \text{and } p_H(E) \coloneqq \left(\frac{H^{n-1}\ch{1}(E)}{H^n\ch{0}(E)}, \frac{H^{n-2}\ch{2}(E)}{H^n\ch{0}(E)}\right) = (\mu_H(E),\xi_H(E)).
\end{equation*}
\begin{remark}
 For a smooth hypersurface $Y\in |mH|$ for $m > 0$, and for $0\leq i\leq n - 1$, we have
 \begin{equation*}
 mH^{n-i}\ch{i}(F) = H^{n-i-1}_Y\ch{i}(F|_Y).
 \end{equation*}
 Hence, $\mu_H$ and $\xi_H$ are invariant under restriction to hypersurfaces. In particular, we have
 \begin{equation*}
 p_H(F) = p_{H_Y} (F|_Y).
 \end{equation*}

\begin{lemma}\label{lem:convexfunction}
 Let $(X,H)$ be a polarized smooth variety such that every $H$-slope-semistable torsion-free sheaf $E$ satisfies
 \begin{align*}
 \frac{h^0(E)}{\rk(E)}\leq h(\mu_H(E))
 \end{align*}
 for some function $h$. Then for every torsion-free sheaf $E$, we have
 \begin{align*}
 \frac{h^0(E)}{\rk(E)} \leq \overline{h}_{\mu^-_{H}(E),\mu^+_H(E)}(\mu_H(E)).
 \end{align*}
\end{lemma}
\begin{proof}
 Let
 \begin{equation*}
 0=E_0\hookrightarrow E_1\hookrightarrow\cdots\hookrightarrow E_n = E,
 \end{equation*}
 be the Harder--Narasimhan filtration of $E$.
 For each quotient $E_i/E_{i-1}$, semi-stability and our assumption yield
 \begin{equation*}
 \frac{h^0(E_i/E_{i-1})}{\rk(E_i/E_{i-1})} \leq h(\mu_H(E_i/E_{i-1})).
 \end{equation*}
 Using these inequalities and additivity of rank and degree, we have
 \begin{align*}
 \frac{h^0(E)}{\rk(E)}&\leq \sum\limits_{i=1}^n\frac{1}{\rk(E)}h^0(E_i/E_{i-1}) = \sum\limits_{i=1}^n\frac{\rk(E_i/E_{i-1})}{\rk(E)}\frac{h^0(E_i/E_{i-1})}{\rk(E_i/E_{i-1})} \\
 &\leq \sum\limits_{i=1}^n\frac{\rk(E_i/E_{i-1})}{\rk(E)}h(\mu_H(E_i/E_{i-1})) \\
 &\leq \sum\limits_{i=1}^n\frac{\rk(E_i/E_{i-1})}{\rk(E)}\overline{h}_{\mu^-_{H}(E),\mu^+_H(E)}(\mu_H(E_i/E_{i-1})) \\
 &\leq \overline{h}_{\mu^-_{H}(E),\mu^+_H(E)}(\mu_H(E)),
 \end{align*}
 where the last line follows from convexity of $\overline{h}_{\mu^-_{H}(E),\mu^+_H(E)}$.
\end{proof}

We will use the following variant of \cite[Lemma 3.3]{Naoki23_BG_hypersurfaces}, which allows us to assume $F$ is everywhere stable:
\begin{lemma}\label{LemmaReductionToStableObjects}
 Let $d\geq 1$ be an integer, and $f:(0,1)\rightarrow \mathbb{R}$ be a star-shaped function along the lines $\beta = 0, d$, satisfying
 \begin{equation*}
 f(t) \leq \tfrac{1}{2}t^2
 \end{equation*}
 for every $t\in (0,1)$. Assume that there exists objects $F'\in \Db(X)$ satisfying the following conditions:
 \begin{enumerate}[{\ \ }(a)]
 \item $F'$ is either $\nu_{0,\alpha}$-tilt-semistable for some $\alpha > 0$, or $\nu_{d,\alpha'}$-tilt-semistable for some $\alpha' > d^2/2$.
 \item $\mu_H(F')\in (0,1)$, $\xi_H(F') > f(\mu_H(F'))$.
 \end{enumerate}
 Then we can choose such an object $F$ so that:
 \begin{enumerate}
 \item $F$ is a reflexive coherent sheaf.
 \item $F\in\Coh^{0,H}(X)$ is $\nu_{\alpha,0,H}$-tilt-stable for any $\alpha > 0$.
 \item $F[1] \in \Coh^{d,H}(X)$ is $\nu_{\alpha',d,H}$-tilt-stable for any $\alpha' > d^2/2$.
 \end{enumerate}
\end{lemma}
\begin{proof}
 Take such an $F'$. Since the value of discriminant is a nonnegative integer, we may assume that $F'$ is an object with the minimal $\overline{\Delta}_H$ of all such objects. Suppose $F'$ becomes strictly $\nu_{\alpha,0,H}$-tilt-semistable for some $\alpha > 0$ (or strictly $\nu_{\alpha',1,H}$-tilt-semistable), then we consider the Jordan--H\"older filtration of $F'$. Since $f$ is star-shaped along the line $\beta = 0$ and $\beta = d$, there is at least one Jordan--H\"older factor $F'_i$ with $\mu_H(F'_i)\in(0,1)$ also satisfying $\xi_H(F'_i) > f(\mu_H(F'_i))$. By \Cref{RemarkOnDiscriminats}, we have $\overline{\Delta}_H(F') > 0$. Hence $\overline{\Delta}_H(F'_i) < \overline{\Delta}_H(F')$ by \Cref{LemmaDiscriminant}, and this violates the minimum assumption on $\overline{\Delta}_H(F')$. As a consequence, we may assume $F'$ is $\nu_{\alpha,0,H}$-tilt-stable (or $\nu_{\alpha',1,H}$-tilt-stable) for any $\alpha > 0$ (or $\alpha'>\frac{1}{2}$).

The Bogomolov--Gieseker (BG) inequality plays an important role in the study of stability conditions and moduli spaces of sheaves. It is a natural question to seek refinements of the BG inequality for special varieties; see for instance \cite[Question 5.1]{BM23}.

Motivated by Gepner type stability conditions on graded matrix factorizations, Toda proposed in \cite{Toda2017GepnerPoint} a stronger inequality on quintic threefolds, and proved it for $\operatorname{rk}(E) = 2$ case.
\begin{theorem}[{\cite[Conjecture $1.1$]{Toda2017GepnerPoint}}]\label{TodaConj0}
    Let $X\subset \mathbb{P}^4_{\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\mathcal{O}_X(1))$. Then for any torsion-free $H$-slope-stable sheaf $E$ on $X$ with $\ch{1}(E)/\operatorname{rk} (E) = -H/2$, we have
    \begin{equation}\label{TodaConj1}
        \frac{H\cdot \Delta(E)}{\operatorname{rk}(E)^2} > 1.5139\cdots,
    \end{equation}
    where the right-hand side of is an irrational real number lying in $\mathbb{Q}(e^{2\pi \sqrt{-1}/5})$. Equivalently,
    \begin{equation}\label{TodaConj2}
    \frac{H\ch{2}(E)}{H^3\operatorname{rk}(E)} < -0.02639\cdots
    \end{equation}
\end{theorem}

In this paper, we establish a stronger Bogomolov--Gieseker type inequality on quintic threefolds, which in particular implies Toda's conjecture.

For a torsion-free coherent sheaf $E$ on an $n$-dimensional polarized smooth projective variety $(X,H)$, we denote
\begin{align*}
    \mu_H(E)\coloneqq\frac{H^{n-1}\ch{1}(E)}{H^n\operatorname{rk}(E)};\quad\xi_H(E)\coloneqq\frac{H^{n-2}\ch{2}(E)}{H^n\operatorname{rk}(E)}.
\end{align*}

\begin{theorem}[See \Cref{MainTheorem} and \Cref{MainCorollary}]
    Let $X\subset \mathbb{P}^4_{\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\Coh(X)$. Assume that
    \begin{equation*}
        \mu_H(F)\in [-1,1].
    \end{equation*}
    Then
    \begin{equation}\notag
         \xi_H(F)\leq
    \begin{cases}
    -\frac{1}{2}|\mu_H(F)| & \text{when } 0\leq |\mu_H(F)| \leq \frac{1}{4}; \\[0ex]
    \frac{1}{2}|\mu_H(F)| - \frac{1}{4} & \text{when } |\mu_H(F)|\in[\frac{1}{4},\frac{5}{13}]\cup[\frac{8}{13},\frac{3}{4}]; \\[0ex]
    -\frac{3}{20}|\mu_H(F)|  & \text{when } \frac{5}{13} \leq |\mu_H(F)|\leq \frac{6}{13}; \\[0ex]
    \frac{1}{2}|\mu_H(F)|-\frac{3}{10} & \text{when } \frac{6}{13}\leq |\mu_H(F)| \leq \frac{7}{13}; \\[0ex]
    \frac{23}{20}|\mu_H(F)| - \frac{13}{20} & \text{when } \frac{7}{13}\leq |\mu_H(F)| \leq \frac{8}{13}; \\[0ex]
    \frac{3}{2}|\mu_H(F)| - 1 & \text{when } \frac{3}{4}\leq |\mu_H(F)| \leq 1.
\end{cases}
    \end{equation}
\end{theorem}

Along the way, we also obtain a stronger Bogomolov--Gieseker type inequality for quintic surfaces.
\begin{theorem}[See \Cref{ChInequalityOnSurface} and \Cref{CorollaryOnSheaves}]
    Let $S_5\subset \mathbb{P}^3_{\mathbb{C}}$ be a smooth quintic surface and $H\coloneqq c_1(\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\Coh(X)$ with
    \begin{equation*}
        \mu_H(F)\in (0,1).
    \end{equation*}
    Then the following piecewise inequality holds:
    \begin{equation}\notag
        \xi_H(F)\leq
    \begin{cases}
    \frac{17}{26}\mu_H(F) - \frac{2}{13} & \text{when } 0 < \mu_H(F) \leq \frac{7}{46}; \\[0ex]
    -\frac{5}{14}\mu_H(F)  & \text{when } \frac{7}{46} \leq  \mu_H(F)\leq \frac{7}{20}; \\[0ex]
    \frac{1}{2}\mu_H(F)-\frac{3}{10} & \text{when } \frac{7}{20}\leq \mu_H(F) \leq \frac{13}{20}; \\[0ex]
    \frac{19}{14}\mu_H(F) - \frac{6}{7} & \text{when } \frac{13}{20}\leq \mu_H(F)< \frac{39}{46}; \\[0ex]
    \frac{9}{26}\mu_H(F) & \text{when } \frac{39}{46}\leq \mu_H(F)< 1.
\end{cases}
    \end{equation}
\end{theorem}

Our proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann–-Roch theorem combined with Clifford-type inequalities for plane quintic curves.

This approach parallels the methods of \cite{li_stability_2019,koseki_stability_2022,liu_stability_2022}, which established analogous inequalities for some complete intersection type Calabi--Yau threefolds. The main improvement here lies in a refined restriction theorem (\Cref{RestrictionTheorem}), which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$. As a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates. The method employed here extends to weighted complete intersection hypersurfaces as well, though the computations in each case tend to be considerably more demanding.

Stronger Bogomolov--Gieseker type inequalities on Calabi--Yau threefolds are known to play a crucial role in constructing Bridgeland stability conditions, as demonstrated in \cite{li_stability_2018,li_stability_2019,koseki_stability_2022,liu_stability_2022} and the recent preprint \cite{feyzbakhsh2025stabilityconditionscalabiyauthreefolds}. In fact, Toda's conjecture (\Cref{TodaConj0}) was originally motivated by the search for a Gepner-type stability condition on the quintic threefold. However, as far as the author is aware, the existing inequalities are not yet sufficient for this purpose due to certain technical obstacles.

The construction of such a Gepner-type stability condition on the quintic remains an intriguing direction for future work.

Our proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann--Roch theorem combined with Clifford-type inequalities for plane quintic curves. The main improvement here lies in a refined restriction theorem (\Cref{RestrictionTheorem}), which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$. As a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates.

Our proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann--Roch theorem combined with Clifford-type inequalities for plane quintic curves. The main improvement here lies in the following refined restriction theorem, which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$.

\begin{lemma}[\textbf{Restriction theorem}]\label{RestrictionTheorem}
    Let $(X,H)$ be a polarized smooth projective variety with dimension $n \geq 2$. Let $E\in \Coh(X)$. Suppose there exists $\alpha > 0$ and $m\in\mathbb{Z}_{> 0 }$ such that both $E$ and  $E(-mH)[1]$ are in $\Coh^{0,H}(X)$ and $\nu_{\alpha,0,H}$-tilt-semistable.

    Then for a smooth irreducible subvariety $Y\in |mH|$, the restricted sheaf $E|_Y$ has $\rk(E) = \rk(E|_{Y})$, $H_Y^{n-2}\ch{1}(E|_Y) = mH^{n-1}\ch{1}(E)$ and when $n \geq 3$, $\ch{2}(E|_Y) = mH\ch{2}(E)$. Moreover, we have
    \begin{align*}
    [\mu_{H_Y}^-(E|_Y),\mu^+_{H_Y}(E|_Y)]\subseteq \llbracket \tfrac{m}{2}+\nu_{\alpha,0,H}(E), \tfrac{m}{2}+\nu_{\alpha,0,H}(E(-mH)[1]) \rrbracket.
    \end{align*} In particular, if $\nu_{\alpha,0,H}(E) = \nu_{\alpha,0,H}(E(-mH)[1])$, then $E|_Y$ is $H_Y$-slope-semistable.
\end{lemma}

As a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates.

[Bogomolov--Gieseker Inequality \cite{bogomolov_holomorphic_1979,gieseker_theorem_1979}]
Let $X$ be a smooth projective complex variety and $H$ an ample divisor on $X$. For any torsion-free $H$-slope-semistable sheaf $E$ on $X$, we have
\begin{equation*}
    \Delta(E) \cdot H^{\dim(X) - 2} \geq 0,
\end{equation*}
where the discriminant is defined by
\begin{equation*}
    \Delta(E) \coloneqq \ch{1}(E)^2 - 2\ch{0}(E)\ch{2}(E).
\end{equation*}.

Question:
Let $X$ be a smooth projective complex variety of dimension $\dim(X)$, and let $H$ be an ample divisor on $X$. For a torsion-free sheaf $E$ on $X$ that is slope-semistable with respect to $H$, define its discriminant by
\[
\Delta(E)\coloneqq \ch_1(E)^2-2\ch_0(E)\ch_2(E).
\]
Which quantitative inequality holds for such a sheaf?

For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-2}\ge 0.
\]

Choices:
B.
For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-1}\ge 0.
\]
C.
For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-2}\in \mathbb{R}.
\]
D.
There exists a constant $c_X>0$, depending only on $X$, such that for every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-2}\ge c_X.
\]
E.
For every torsion-free sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-2}\ge 0.
\]

intersection power matches codimension-two class

nonnegativity conclusion dropped, retaining only well-definedness of the intersection number

uniform lower bound strengthened from 0 to a positive constant depending only on X


The fourth stable homotopy group of spheres is trivial, meaning that $\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\in\{2,3,4,5\}$, as we will briefly explain.

Curtis proved that $\pi_n(S^4)\neq 0$ for all $n\geq 4$ \cite{Cur69}. Curtis also proved that $\pi_n(S^2)\neq 0$ for all $n\not\equiv 1\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \cite{MMO75}. The proof that $\pi_n(S^5)\neq 0$ for all $n\geq 5$ was given by Mori \cite{Mor75} and Mahowald \cite{Mah75,Mah82}.

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
    Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

In fact, it appears that every positive integer has at least two Toda primes.

\begin{conj}\label{conj:at least two}
    If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
 Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{conj}\label{conj:at least two}
 If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{figure}[p]
    \includegraphics[width=.75\linewidth]{tn.png}
    \caption{$t(n)$ for $n\leq 100000$}\label{fig:tn}
\end{figure}

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\begin{lem}\label{lem:not 5 or not 3 and 5}
 Let $p\in\{3,5\}$. If $p\nmid n$, then $p\in T(n)$. In particular:
 \begin{itemize}
 \item If $5\nmid n$, then $t(n)\geq 1$.
 \item If $3,5\nmid n$, then $t(n)\geq 2$.
 \end{itemize}
\end{lem}
\begin{proof}
 Let $p\in\{3,5\}$. Then $p-1\mid 4n$, and $\gcd(4n,p)=1$ by assumption. Thus $\gcd(\frac{4n}{p-1},p)=1$, so $p\in T(n)$.
\end{proof}

\begin{lem}\label{lem:t(n)=3}
 Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}
\begin{proof}
 We will induct on $\omega(n)$. Our base cases will consist of $\omega(n)\leq 4$. Note that if $p\nmid n$ for each $p\in\{5,7,13\}$, then $T(n)\supseteq\{5,7,13\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\geq t(a)$ for any prime $p\not\in T(a)$ by \cref{cor:multiply by prime}, so we may assume that every prime factor $p\mid n$ is a Toda prime of some divisor of $n$.
 \begin{itemize}
 \item The case of $\omega(n)=1$ is just the calculation $T(3)=\{5,7,13\}$.
 \item For $\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.
 \item For $\omega(n)=3$, we first compute $t(3\cdot 5\cdot 7)=9$ and $t(3\cdot 5\cdot 13)=t(3\cdot 7\cdot 13)=8$. It remains to compute, for each $p\in\{5,7,13\}$, the Toda primes of $3pq$ for each $q\in T(3p)$. Using the code provided in \cref{sec:code}, we find that $t(3pq)\geq 4$ for all such $p,q$.
 \item For $\omega(n)=4$, we first compute $t(3\cdot 5\cdot 7\cdot 13)=16$. For the remaining computations in this case, we use the code in \cref{sec:code}.
 \begin{itemize}
 \item If $\{p,q\}\subseteq\{5,7,13\}$ and $r\in T(3pq)$, then $t(3pqr)\geq 9$.
 \item If $p\in\{5,7,13\}$ and $\{q,r\}\subseteq T(3p)$, then $t(3pqr)\geq 7$.
 \item If $p\in\{5,7,13\}$, $q\in T(3p)$, and $r\in T(3pq)$, then $t(3pqr)\geq 5$.
 \end{itemize}
 \end{itemize}

\begin{cor}\label{prop:divisible by 3}
 Assume \cref{conj:main}. If $3\mid n$, then $t(n)\geq 3$.
\end{cor}
\begin{proof}
 By \cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \cref{lem:t(n)=3}.
\end{proof}

Firstly, if there exists $q'\in T(3p)-\{5,7,13,q\}$ such that $q'\nmid n$, then $\{5,7,13,q,q'\}-\{p\}\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\{2d+1:d\mid 2n\}$ that are not factors of $n$. Thus if
\begin{equation}\label{eq:heuristic}
\{2d+1\text{ prime}:d\mid 2n\}-(\Omega(n)\cup\{5,7,13,q\})
\end{equation}
is non-empty, then $t(n)\geq 4$. Our heuristic for \cref{conj:main} is that the set $\{2d+1:d\mid 2n\}$ consists of $2^{\omega(n)+1}$ elements, while $\Omega(n)\cup\{5,7,13,q\}$ consists of $\omega(n)+3$ elements.

\begin{prop}\label{prop:t(p)}
Assume $p\geq 7$ is a prime. Let $\vphi$ denote the totient function. If $\vphi(x)=4p$ for some integer $x$, then $T(p)=\{3,5,2p+1\}$ or $\{3,5,4p+1\}$. Otherwise, $T(p)=\{3,5\}$.
\end{prop}
\begin{proof}
 One can directly check that $3,5\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\vphi(x)=p_1^{e_1-1}(p_1-1)\cdots p_m^{e_m-1}(p_m-1)$, where $x=\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\vphi(x)=4p$ if and only if one of the following cases holds:
 \begin{enumerate}[(i)]
 \item $x=2^2\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot 3\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=1$.
 \item $x=2^r\cdot 5^2$, where $r\in\{0,1\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\geq 7$.
 \end{enumerate}
 It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).

\begin{lem}\label{lem:strategy for denoms}
 Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\{2pi+1:i\mid 2a\}$ contains a prime number for each $p\in T(a)$, then \cref{conj:general bernoulli} \eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.
\end{lem}
\begin{proof}
 We know that $p-1\mid 4a$ with $\gcd(\frac{4a}{p-1},p)=1$ for all $p\in T(a)$. Thus $p-1\mid 4am$, and we have $\gcd(\frac{4am}{p-1},p)=1$ if and only if $p\nmid m$. It therefore suffices to show that if $p\mid m$ for some $p\in T(a)$, then $D_{4am}>D_{4a}$.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{lem}\label{lem:t(n)=3}
    Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}

\begin{thm}\label{thm:main}
    Assume \cref{conj:main}. If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.
\end{thm}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\subsection*{Acknowledgements}
We thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.

\label{thm:main}
Assume
\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}
If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.,

Question: For a positive integer n, call an odd prime p a Toda prime of n if p-1 divides 4n and \(\gcd\!\left(p,\frac{4n}{p-1}\right)=1\). Let \(T(n)\) be the set of Toda primes of n, and let \(t(n)=|T(n)|\). Assume the following hypothesis holds universally: whenever m is an odd, square-free multiple of 3, there exists \(p\in\{5,7,13\}\) such that \(p\mid m\) and \(r\nmid m\) for every \(r\in\{5,7,13\}\setminus\{p\}\), and there exists \(q\in T(3p)\setminus\{5,7,13\}\) with \(q\nmid m\), then \(t(m)\ge 4\). Under this assumption, which quantitative estimate holds for Toda primes of a positive integer n?

For every positive integer n, \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(3\mid n\), then \(t(n)\ge 3\).

Choices:
B. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(3\nmid n\), then \(t(n)\ge 2\); and if \(5\mid n\), then \(t(n)\ge 3\).
C. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\).
D. For every positive integer \(n\), \(t(n)\ge 2\); moreover, if \(5\nmid n\), then \(t(n)\ge 3\); and if \(3\mid n\), then \(t(n)\ge 4\).
E. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(n\) is an odd, square-free multiple of \(3\), then \(t(n)\ge 4\).

special-role of the primes 3 and 5 in the baseline bounds

dropped the final clause requiring \(3\mid n\Rightarrow t(n)\ge 3\)

numerical lower bounds 1/2/3 replaced by stronger 2/3/4

scope of the \(t(m)\ge 4\) hypothesis enlarged by dropping the auxiliary existence conditions on \(p\) and \(q\)


[$2\pi^2$-Theorem, Marques-Neves~\cite{MN14}]\label{2pi2-theorem}
Let $\Sigma\subset\varmathbb{S}^3$ be a smooth embedded surface of genus at least $1$.
Then its canonical five-parameter family
\[
\{\Sigma_{(v,t)}\}_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]}
\]
satisfies
\[
\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]}
\mathrm{Area}\bigl(\Sigma_{(v,t)}\bigr)\;\geqslant\; 2\pi^2.
\]
Moreover, equality holds if and only if $\Sigma$ is the Clifford torus.

\begin{theorem}\label{Thm: main}
Let $V=\underline{v}(\Sigma,1)$ be an integral $2$-varifold in $\mathbb{S}^3$
with unit density and mean curvature $H\in L^2(d\mu)$, and assume its support
$\Sigma$ has genus at least $1$.
If the Willmore energy
\[
\mathcal{W}(V)=\int_\Sigma\Bigl(1+\tfrac14 |H|^2\Bigr)\,d\mu
\]
satisfies
\[
\mathcal{W}(V)\le 2\pi^2+\delta^2,
\qquad \delta>0 \text{ sufficiently small},
\]
then, after a suitable conformal transformation of $\mathbb{S}^3$, there exists
a homeomorphism
\[
f:\mathbb{S}^1\times\mathbb{S}^1 \longrightarrow \Sigma \subset\mathbb{S}^3
\]
such that in the standard coordinates $(\theta,\varphi)$ on
$\mathbb{S}^1\times\mathbb{S}^1$ the pull-back metric takes the form
\[
f^{*}g_{\mathbb{S}^3}
= e^{2u}\,(a\, d\theta^2 + 2 b\, d\theta\, d\varphi + c\, d\varphi^2).
\]

Let $f_0:\mathbb{S}^1\times\mathbb{S}^1\to\mathbb{S}^3$ denote the Clifford torus
embedding
\[
f_0(\theta,\varphi)
= \frac{1}{\sqrt{2}}(\cos\theta,\;\sin\theta,\;\cos\varphi,\;\sin\varphi),
\]
and write its image as $\mathbb{T}^2:=f_0(\mathbb{S}^1\times\mathbb{S}^1)$.
Then
\[
f_0^{*} g_{\mathbb{S}^3}
= \tfrac12 (d\theta^2+d\varphi^2).
\]

The following quantitative rigidity estimates hold:
\begin{enumerate}

\item[\rm(1)]
\[
\| f - f_0 \|_{W^{2,2}(\mathbb{S}^1\times\mathbb{S}^1)} \le C\,\delta.
\]

\item[\rm(2)]
\[
\|u\|_{L^\infty(\mathbb{S}^1\times\mathbb{S}^1)} \le C\,\delta.
\]

\item[\rm(3)]
\[
|a-\tfrac12| + |b| + |c-\tfrac12| \le C\,\delta.
\]

\end{enumerate}
Here $C$ is a universal constant independent of $\Sigma$.
\end{theorem}

For an immersed surface $f:\Sigma\to \varmathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by
\[
\mathcal{W}(\Sigma)
=\int_{\Sigma}\!\left(1+\tfrac{1}{4}|H|^2\right)d\mu_f,
\]
where $H$ is the mean curvature and $d\mu_f$ is the area element of $f^{*}g_{\varmathbb{S}^3}$.
A fundamental property of $\mathcal{W}$ is its conformal invariance.
By the theorem of Marques and Neves resolving the Willmore conjecture~\cite{MN14}, any closed oriented surface $\Sigma\subset\varmathbb{S}^3$ with genus at least one satisfies
\[
\mathcal{W}(\Sigma)\geqslant 2\pi^2,
\]
with equality if and only if $\Sigma$ is, up to a conformal transformation, the Clifford torus
\[
\varmathbb{T}^2=\varmathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times
\varmathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset\varmathbb{S}^3.
\]

The purpose of this paper is to study the quantitative rigidity of this theorem.
Given $\delta<\delta_0\ll1$ and a surface $\Sigma$ with
\[
\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2,
\]
we ask how close $\Sigma$ must be to a conformal image of the Clifford torus, in a sense that simultaneously controls the $W^{2,2}$-parametrization, the $L^\infty$-bound of the conformal factor, and the conformal structure.
This parallels the classical quantitative rigidity for the genus zero case, where De Lellis-M\"uller~\cite{dLMu,dLMu2} proved optimal linear estimates for nearly umbilical surfaces, later extended to higher codimension by Lamm-Sch\"atzle~\cite{LS-2014} and to varifolds by Bi-Zhou~\cite{BZ25}.

A first step toward our result is to understand the topology of surfaces whose Willmore energy is close to the Clifford value.
Using existing existence and energy asymptotic results, one obtains that if a surface $\Sigma$ satisfies $g(\Sigma)\ge1$ and $\mathcal{W}(\Sigma)$ is sufficiently close to $2\pi^2$, then in fact $g(\Sigma)=1$.
This follows directly from the existence theory for Willmore minimizers (Simon for genus 1~\cite{LS93} and Bauer-Kuwert for genus $\ge2$~\cite{BK03}), together with the large-genus limit of Kuwert-Li-Sch\"atzle~\cite{KLS10} and the Willmore conjecture~\cite{MN14}.

Once the genus has been identified, Leon Simon's results~\cite{LS93} imply that any sequence of tori
\(\Sigma_k \subset \varmathbb{S}^3\) with \(\mathcal{W}(\Sigma_k)\to 2\pi^2\) admits, after suitable conformal reparametrization,
a subsequence converging as varifolds to a limit surface whose genus is at least one.
By lower semicontinuity of the Willmore energy, this limit has Willmore energy at most \(2\pi^2\).
In view of the Willmore conjecture, the only possibility is that the limit is itself a torus of Willmore energy \(2\pi^2\).
It is well known to experts (see for instance Kuwert-Li~\cite{KL-2012},  Sch\"atzle~\cite{Sch13} and Rivi\'ere\cite{R14}) that, after parametrizing by a normalized flat torus metric, such a sequence is then qualitatively close to the Clifford torus. Section~2 contains a detailed proof of this qualitative convergence. We in fact formulate and prove a slightly more general statement: the model
surface may be any fixed genus~$\ge1$ minimal surface (not only the Clifford
torus), provided that no energy concentration occurs---in the precise sense
that the mean curvature has small $L^{2}$-norm and the area ratio is close to~$1$.
Our approach proceeds by first establishing convergence of the surfaces
\emph{as subsets of $\varmathbb{S}^{3}$} under these assumptions, and only
afterwards upgrading the convergence to that of the associated conformal
parametrizations.  This is somewhat different in spirit from earlier
treatments, which work directly with compactness of $W^{2,2}$ conformal
immersions into Euclidean space.

The main difficulty is to promote this qualitative convergence to a quantitative, in particular linear, stability estimate.

A natural idea would be to expand the Willmore functional around the Clifford torus: one considers the second variation together with the higher-order remainder, analyzes the spectral properties of the resulting symmetric quadratic form, and then derives quantitative stability from the coercivity of this linearized operator.  However, in order for this approach to be effective, the remainder must be genuinely dominated by the quadratic term, which requires the surface to be $C^1$ or at least Lipschitz close to the Clifford torus.  Establishing such a priori closeness is delicate in our weak $W^{2,2}$-conformal setting.

For this reason, following the strategy of Marques-Neves, we avoid a direct perturbative analysis of the Willmore functional.  Instead, using the Heintze-Karcher-type inequality to relate the Willmore energy to the area of the canonical family $\Sigma_{(v,t)}$, we reformulate the problem in Section~3 as a minimal-surface stability problem.

To describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\cite{MN14}.
For $v\in\mathring{B}^4$, let
\[
F_v(x)=(1-|v|^2)\frac{x-v}{|x-v|^2}-v
\]
be the corresponding conformal transformation.
If $\varmathbb{S}^3\setminus\Sigma=A\cup A^*$ is the decomposition into connected components, we denote $\Sigma_v:=F_v(\Sigma)$ and $\varmathbb{S}^3\setminus\Sigma_v=A_v\cup A_v^*$ with $A_v=F_v(A)$.
Let $d_v$ be the signed distance to $\Sigma_v$, and define
\[
\Sigma_{(v,t)}=\partial\{x\mid d_v(x)<t\}, \qquad (v,t)\in\mathring{B}^4\times[-\pi,\pi].
\]
The Heintze-Karcher-type inequality~\cite{MN14,HK78,R99} gives
\[
\mathrm{Area}(\Sigma_{(v,t)})\leqslant \mathcal{W}(\Sigma),
\]
with equality only at $t=0$ when $\Sigma_v$ is minimal.

Marques and Neves further established the following fundamental result, often referred to as the \emph{$2\pi^2$-Theorem}:

For an immersed surface $f:\Sigma\to \varmathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by
\[
\mathcal{W}(\Sigma)
=\int_{\Sigma}\!\left(1+\tfrac{1}{4}|H|^2\right)d\mu_f,
\]
where $H$ is the mean curvature and $d\mu_f$ is the area element of $f^{*}g_{\varmathbb{S}^3}$.
A fundamental property of $\mathcal{W}$ is its conformal invariance.
By the theorem of Marques and Neves resolving the Willmore conjecture~\cite{MN14}, any closed oriented surface $\Sigma\subset\varmathbb{S}^3$ with genus at least one satisfies
\[
\mathcal{W}(\Sigma)\geqslant 2\pi^2,
\]
with equality if and only if $\Sigma$ is, up to a conformal transformation, the Clifford torus
\[
\varmathbb{T}^2=\varmathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times
\varmathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset\varmathbb{S}^3.
\]

The purpose of this paper is to study the quantitative rigidity of this theorem.
Given $\delta<\delta_0\ll1$ and a surface $\Sigma$ with
\[
\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2,
\]
we ask how close $\Sigma$ must be to a conformal image of the Clifford torus, in a sense that simultaneously controls the $W^{2,2}$-parametrization, the $L^\infty$-bound of the conformal factor, and the conformal structure.
This parallels the classical quantitative rigidity for the genus zero case, where De Lellis-M\"uller~\cite{dLMu,dLMu2} proved optimal linear estimates for nearly umbilical surfaces, later extended to higher codimension by Lamm-Sch\"atzle~\cite{LS-2014} and to varifolds by Bi-Zhou~\cite{BZ25}.

A first step toward our result is to understand the topology of surfaces whose Willmore energy is close to the Clifford value.
Using existing existence and energy asymptotic results, one obtains that if a surface $\Sigma$ satisfies $g(\Sigma)\ge1$ and $\mathcal{W}(\Sigma)$ is sufficiently close to $2\pi^2$, then in fact $g(\Sigma)=1$.
This follows directly from the existence theory for Willmore minimizers (Simon for genus 1~\cite{LS93} and Bauer-Kuwert for genus $\ge2$~\cite{BK03}), together with the large-genus limit of Kuwert-Li-Sch\"atzle~\cite{KLS10} and the Willmore conjecture~\cite{MN14}.

Once the genus has been identified, Leon Simon's results~\cite{LS93} imply that any sequence of tori
\(\Sigma_k \subset \varmathbb{S}^3\) with \(\mathcal{W}(\Sigma_k)\to 2\pi^2\) admits, after suitable conformal reparametrization,
a subsequence converging as varifolds to a limit surface whose genus is at least one.
By lower semicontinuity of the Willmore energy, this limit has Willmore energy at most \(2\pi^2\).
In view of the Willmore conjecture, the only possibility is that the limit is itself a torus of Willmore energy \(2\pi^2\).
It is well known to experts (see for instance Kuwert-Li~\cite{KL-2012}, Sch\"atzle~\cite{Sch13} and Rivi\'ere\cite{R14}) that, after parametrizing by a normalized flat torus metric, such a sequence is then qualitatively close to the Clifford torus. Section~2 contains a detailed proof of this qualitative convergence. We in fact formulate and prove a slightly more general statement: the model
surface may be any fixed genus~$\ge1$ minimal surface (not only the Clifford
torus), provided that no energy concentration occurs---in the precise sense
that the mean curvature has small $L^{2}$-norm and the area ratio is close to~$1$.
Our approach proceeds by first establishing convergence of the surfaces
\emph{as subsets of $\varmathbb{S}^{3}$} under these assumptions, and only
afterwards upgrading the convergence to that of the associated conformal
parametrizations. This is somewhat different in spirit from earlier
treatments, which work directly with compactness of $W^{2,2}$ conformal
immersions into Euclidean space.

To describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\cite{MN14}.
For $v\in\mathring{B}^4$, let
\[
F_v(x)=(1-|v|^2)\frac{x-v}{|x-v|^2}-v
\]
be the corresponding conformal transformation.
If $\varmathbb{S}^3\setminus\Sigma=A\cup A^*$ is the decomposition into connected components, we denote $\Sigma_v:=F_v(\Sigma)$ and $\varmathbb{S}^3\setminus\Sigma_v=A_v\cup A_v^*$ with $A_v=F_v(A)$.
Let $d_v$ be the signed distance to $\Sigma_v$, and define
\[
\Sigma_{(v,t)}=\partial\{x\mid d_v(x)<t\}, \qquad (v,t)\in\mathring{B}^4\times[-\pi,\pi].
\]
The Heintze-Karcher-type inequality~\cite{MN14,HK78,R99} gives
\[
\mathrm{Area}(\Sigma_{(v,t)})\leqslant \mathcal{W}(\Sigma),
\]
with equality only at $t=0$ when $\Sigma_v$ is minimal.

Marques and Neves further established the following fundamental result, often referred to as the \emph{$2\pi^2$-Theorem}:

For an immersed surface $f:\Sigma\to \mathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by
\[
\mathcal{W}(\Sigma)
=\int_{\Sigma}\!\left(1+\tfrac{1}{4}|H|^2\right)d\mu_f,
\]
where $H$ is the mean curvature and $d\mu_f$ is the area element of $f^{*}g_{\mathbb{S}^3}$.
A fundamental property of $\mathcal{W}$ is its conformal invariance.
By the theorem of Marques and Neves resolving the Willmore conjecture~\cite{MN14}, any closed oriented surface $\Sigma\subset\mathbb{S}^3$ with genus at least one satisfies
\[
\mathcal{W}(\Sigma)\ge 2\pi^2,
\]
with equality if and only if $\Sigma$ is, up to a conformal transformation, the Clifford torus
\[
\mathbb{T}^2=\mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times
\mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset\mathbb{S}^3.
\]

To describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\cite{MN14}.
For $v\in\mathring{B}^4$, let
\[
F_v(x)=(1-|v|^2)\frac{x-v}{|x-v|^2}-v
\]
be the corresponding conformal transformation.
If $\mathbb{S}^3\setminus\Sigma=A\cup A^*$ is the decomposition into connected components, we denote $\Sigma_v:=F_v(\Sigma)$ and $\mathbb{S}^3\setminus\Sigma_v=A_v\cup A_v^*$ with $A_v=F_v(A)$.
Let $d_v$ be the signed distance to $\Sigma_v$, and define
\[
\Sigma_{(v,t)}=\partial\{x\mid d_v(x)<t\}, \qquad (v,t)\in\mathring{B}^4\times[-\pi,\pi].
\]
The Heintze-Karcher-type inequality~\cite{MN14,HK78,R99} gives
\[
\mathrm{Area}(\Sigma_{(v,t)})\le \mathcal{W}(\Sigma),
\]
with equality only at $t=0$ when $\Sigma_v$ is minimal.

Marques and Neves further established the following fundamental result, often referred to as the \emph{$2\pi^2$-Theorem}:

With the five-parameter family $\{\Sigma_{(v,t)}\}$ of
Marques-Neves recalled above, we now indicate how it enters our proof.

In this section we study the almost minimizing property for surfaces whose
Willmore energy is sufficiently close to $2\pi^{2}$.
A central object in our analysis is the canonical five-parameter family
$\{\Sigma_{(v,t)}\}$ introduced by Marques-Neves.
Rather than repeating the full construction and discussion from the
Introduction, we recall only the variational problem that plays a decisive role:
\begin{equation}\label{eq:max-canonical-area}
\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]}
\mathrm{Area}\bigl(\Sigma_{(v,t)}\bigr)\,.
\end{equation}

The fundamental result of Marques-Neves, stated as Theorem~\ref{2pi2-theorem} in
the Introduction, asserts that for any smooth embedded surface of genus at least
$1$, the canonical family satisfies
\[
\max_{(v,t)} \mathrm{Area}(\Sigma_{(v,t)}) \ge 2\pi^{2},
\]
with equality if and only if the surface is the
Clifford torus.

Let $\Sigma\subset\mathbb{S}^3$ be a smooth closed torus
and assume that $\Sigma$ is $(\gamma,r_0)$-regular and $\mathcal{W}(\Sigma)\leq 8\pi$
Consider the canonical family
\[
\{\Sigma_{(v,t)}\}_{(v,t)\in\mathring{B}^4\times[-\pi,\pi]},
\]
and let $(v_0,t_0)$ be any maximizer of the area , i.e.
\[
\mathrm{Area}(\Sigma_{(v_0,t_0)})
 = \max_{(v,t)\in\mathring{B}^4\times[-\pi,\pi]}
 \mathrm{Area}(\Sigma_{(v,t)}).
\]
Then
\[
|v_0|\le 1-\eta_1.
\]
\end{proposition}

Now let $(v_{0},t_{0})$ be a maximizer of the canonical area functional as in
Proposition~\ref{away from boundary of conformal group}.
By the $2\pi^{2}$-Theorem~\ref{2pi2-theorem} and \eqref{Area formula},
\begin{align*}
2\pi^{2}
\;\le\; \mathrm{Area}(\Sigma_{(v_{0},t_{0})})
\;\le\; \int_{\Sigma_{v_{0}}^{*}(t_{0})} J_{\varphi_{t_{0}}}\, d\mathcal{H}^{2}.
\end{align*}
Using again \cite[Lemma~3.5]{MN14} (see also \cite{R99}), we have
\[
J_{\varphi_{t_{0}}}(p)
=\Bigl(1+\frac{|H|^{2}}{4}\Bigr)
 -\Bigl(\frac{H}{2}\cos t_{0}-\sin t_{0}\Bigr)^{2}
 -\sin^{2}t_{0}\, |\mathring{A}|^{2},
\]
where $H$ is the mean curvature and $ \mathring{A}=A-\frac{H}{2}g$. Hence
\begin{align*}
2\pi^{2}
&\le \int_{\Sigma_{v_{0}}^{*}(t_{0})}
\Bigl(1+\frac{|H|^{2}}{4}\Bigr)
 -\Bigl(\frac{H}{2}\cos t_{0}-\sin t_{0}\Bigr)^{2}
 -\sin^{2}t_{0}\, |\mathring{A}|^{2}\, d\mathcal{H}^{2} \\
&\le \int_{\Sigma_{v_{0}}^{*}(t_{0})} \Bigl(1+\frac{|H|^{2}}{4}\Bigr)\, d\mathcal{H}^{2} \\
&\le \mathcal{W}(\Sigma_{v_{0}}) \\
&=\mathcal{W}(\Sigma)\;\le\; 2\pi^{2}+\delta^{2}.
\end{align*}
This yields the three estimates
\begin{align}
\int_{E_{v_{0}}(t_{0})}\Bigl(1+\frac{|H|^{2}}{4}\Bigr)\, d\mathcal{H}^{2}
&\le \delta^{2}, \label{negative small} \\
\int_{\Sigma_{v_{0}}^{*}(t_{0})}
\Bigl(\frac{H}{2}\cos t_{0}-\sin t_{0}\Bigr)^{2} d\mathcal{H}^{2}
&\le \delta^{2}, \label{mean curvature small} \\
\sin^{2}t_{0} \int_{\Sigma_{v_{0}}^{*}(t_{0})} |\mathring{A}|^{2}\, d\mathcal{H}^{2}
&\le \delta^{2}. \label{small time}
\end{align}

\begin{proposition}\label{away from boundary of conformal group}
There exist constants $\eta_1>0$, $\delta_0>0$, and a function
$\gamma=\gamma(\delta_0)>0$ with $\gamma(\delta_0)\to 0$ as $\delta_0\to 0$,
and a universal radius $r_0>0$, such that the following holds.

Let $\Sigma\subset\mathbb{S}^3$ be a smooth closed torus
and assume that $\Sigma$ is $(\gamma,r_0)$-regular and $\mathcal{W}(\Sigma)\leq 8\pi$
Consider the canonical family
\[
\{\Sigma_{(v,t)}\}_{(v,t)\in\mathring{B}^4\times[-\pi,\pi]},
\]
and let $(v_0,t_0)$ be any maximizer of the area , i.e.
\[
\mathrm{Area}(\Sigma_{(v_0,t_0)})
 = \max_{(v,t)\in\mathring{B}^4\times[-\pi,\pi]}
   \mathrm{Area}(\Sigma_{(v,t)}).
\]
Then
\[
|v_0|\le 1-\eta_1.
\]
\end{proposition}

With the five-parameter family $\{\Sigma_{(v,t)}\}$ of
Marques-Neves recalled above, we now indicate how it enters our proof.

The key point is to show that if a genus--one surface $\Sigma$ satisfies
$\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2$, then a \emph{conformal}
transformation of $\varmathbb{S}^3$ yields a surface $\Sigma'$ with
\[
\mathrm{Area}(\Sigma')\;\geqslant\;2\pi^2 - C\delta^2.
\]

The $2\pi^2$ theorem guarantees that the full $(v,t)$-family contains a slice
whose area is at least $2\pi^2$.  Since the variables $v$ encode conformal
transformations and $t$ is the signed distance parameter, our aim is to
realize such a high-area slice \emph{within the conformal}
four-parameter subfamily.
For this, it is necessary to control the distance parameter $t$ in terms of
the energy gap~$\delta$, ensuring that the maximizing slice must lie close to
the $t=0$ slice.

A closer examination of the Heintze-Karcher inequality used by
Marques-Neves shows that, \emph{once the five-parameter family is built
from the quantitatively normalized surface obtained in Section~2}, the
problem reduces to proving that the maximizing parameter $(v_0,t_0)$ stays a
definite distance away from the boundary of the conformal group.
Equivalently, there exists a universal constant $\eta_1>0$ such that
\[
|v_0|\;\leqslant\;1-\eta_1 .
\]
This reduction is carried out in Section~3, where the technical
proof of Proposition~\ref{away from boundary of conformal group} is deferred to the Appendix.

To clarify why such a uniform bound is true, we now outline the geometric
intuition behind it.  When the conformal parameter $v$ approaches the boundary of
$\mathring{B}^4$, the transformed surface $\Sigma_{(v,t)}$ becomes
quantitatively close to a geodesic sphere, which forces its
area to stay strictly below $2\pi^2$.
Thus such $v$ cannot maximize the canonical family.

More precisely, when $t=0$, the qualitative stability implies that $\Sigma_v= \Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, and the region enclosed by $\Sigma_v$ is similarly close to that enclosed by~$S$.
A direct computation then shows that
\[
\mathrm{Area}(\Sigma_v)
\]
is close to $\mathrm{Area}(S)$, which is at most $4\pi$. Since $\Sigma_{(v,t)}$ is obtained from $\Sigma_v$ by shifting along its normal directions, the area difference satisfies an estimate of the form
\[
\bigl|\mathrm{Area}(\Sigma_{(v,t)})-\mathrm{Area}(\Sigma_v)\bigr|
    \;\lesssim\; |t|\,\mathcal{W}(\Sigma_v).
\]
Hence, for sufficiently small $|t|$, the area remains below~$5\pi$.

For larger $|t|$, we distinguish two cases.
If $\Sigma_v$ stays a definite distance from the poles, we may apply the
Euclidean model: a region trapped between two nearly parallel half–spaces of
separation $\varepsilon\ll1$ has the property that, for any fixed
$t_0\sim1$, the boundaries of its $t$-parallel sets ($t\geqslant t_0$) become Lipschitz graphs over one
bounding plane with Lipschitz constant $\to0$ as $\varepsilon/t_0\to0$.
Transferring this to the spherical setting yields that $\Sigma_{(v,t)}$ is a
small Lipschitz graph over a geodesic sphere and hence has area $<5\pi$.

If instead $\Sigma_v$ lies very close to a pole, its diameter is already
tiny, and a simple comparison–geometry argument shows that
$\Sigma_{(v,t)}$ remains of area $<5\pi$.

Thus in all cases $\mathrm{Area}(\Sigma_{(v,t)})<5\pi$. Since $5\pi < 2\pi^2$, no parameter $(v,t)$ with $|v|$ sufficiently close to $1$ can maximize the canonical area.
This yields the desired estimate
\[
|v_0| \leqslant 1 - \eta_1.
\]

With this uniform control of the conformal parameter, the remaining task is to
derive a quantitative stability estimate for minimal surfaces.
Assembling the arguments developed so far leads to our main quantitative
rigidity theorem:

\begin{theorem}\label{Thm: main}
Let $V=\underline{v}(\Sigma,1)$ be an integral $2$-varifold in $\varmathbb{S}^3$
with unit density and mean curvature $H\in L^2(d\mu)$, and assume its support
$\Sigma$ has genus at least $1$.
If the Willmore energy
\[
\mathcal{W}(V)=\int_\Sigma\Bigl(1+\tfrac14 |H|^2\Bigr)\,d\mu
\]
satisfies
\[
\mathcal{W}(V)\leqslant 2\pi^2+\delta^2,
\qquad \delta>0 \text{ sufficiently small},
\]
then, after a suitable conformal transformation of $\varmathbb{S}^3$, there exists
a homeomorphism
\[
f:\varmathbb{S}^1\times\varmathbb{S}^1 \longrightarrow \Sigma \subset\varmathbb{S}^3
\]
such that in the standard coordinates $(\theta,\varphi)$ on
$\varmathbb{S}^1\times\varmathbb{S}^1$ the pull-back metric takes the form
\[
f^{*}g_{\varmathbb{S}^3}
= e^{2u}\,(a\, d\theta^2 + 2 b\, d\theta\, d\varphi + c\, d\varphi^2).
\]

Let $f_0:\varmathbb{S}^1\times\varmathbb{S}^1\to\varmathbb{S}^3$ denote the Clifford torus
embedding
\[
f_0(\theta,\varphi)
= \frac{1}{\sqrt{2}}(\cos\theta,\;\sin\theta,\;\cos\varphi,\;\sin\varphi),
\]
and write its image as $\varmathbb{T}^2:=f_0(\varmathbb{S}^1\times\varmathbb{S}^1)$.
Then
\[
f_0^{*} g_{\varmathbb{S}^3}
= \tfrac12 (d\theta^2+d\varphi^2).
\]

The following quantitative rigidity estimates hold:
\begin{enumerate}

\item[\rm(1)]
\[
\| f - f_0 \|_{W^{2,2}(\varmathbb{S}^1\times\varmathbb{S}^1)} \leqslant C\,\delta.
\]

\item[\rm(2)]
\[
\|u\|_{L^\infty(\varmathbb{S}^1\times\varmathbb{S}^1)} \leqslant C\,\delta.
\]

\item[\rm(3)]
\[
|a-\tfrac12| + |b| + |c-\tfrac12| \leqslant C\,\delta.
\]

\end{enumerate}
Here $C$ is a universal constant independent of $\Sigma$.
\end{theorem}

Recent work of Rupp-Scharrer~\cite{RS25}, making use of the regularity theorem
developed in~\cite{BZ-2022b}, establishes density of smooth surfaces among
integral $2$-varifolds with square-integrable second fundamental form and
Willmore energy below $8\pi$.
Since unit density together with $H\in L^2$ already yields square-integrable
second fundamental form by~\cite{BZ-2022b}, their result is closely related to
the class of varifolds considered here.

In the appendix we prove a density statement suited to our setting:
compactly supported integral $2$-varifolds with unit density and
square-integrable mean curvature can be approximated in $W^{2,2}$ by smooth
embedded surfaces.
Our construction provides global bilipschitz control between the limiting
parametrization $F$ and its smoothings $F_\eta$, which ensures
embeddedness of the approximating surfaces without requiring any additional
smallness assumptions on the Willmore energy.

\medskip
\noindent\textbf{Outline of the paper.}
In Section~2, we establish the topological stability and qualitative geometric stability for surfaces with Willmore energy sufficiently close to $2\pi^2$.
In particular, we show that such a surface must have genus one and is, after suitable conformal normalization, qualitatively close to the Clifford torus.

In Section~3, we analyze the Marques-Neves canonical family and prove the key estimate that under the assumption $\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2$, one can find a conformal transformation of $\varmathbb{S}^3$ such that the transformed surface $\Sigma'$ satisfies
\[
\mathrm{Area}(\Sigma')\geqslant 2\pi^2 - C\delta^2,
\]
for a universal constant $C>0$.
This reduces the Willmore stability problem to a quantitative stability statement for minimal surfaces.

Section~4 contains the linearization and stability analysis.
Using the estimate from Section~3, we derive the quantitative bounds on the conformal factor, the conformal structure, and the $W^{2,2}$-distance to the Clifford torus, thereby proving Theorem~\ref{Thm: main}.

Finally, in the Appendix we provide the proof of Proposition~\ref{away from boundary of conformal group} and we also establish the density and parametrization results for compactly
supported integral $2$-varifolds with unit density and $L^2$-mean curvature,
showing in particular that such varifolds admit global conformal
parametrizations and can be approximated by smooth embedded surfaces.

Using the Marques--Neves five-parameter family $\{\Sigma_{(v,t)}\}$, the key point is to show that if a genus--one surface $\Sigma$ satisfies $\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2$, then after a \emph{conformal} transformation one gets a surface $\Sigma'$ with $\mathrm{Area}(\Sigma')\geqslant 2\pi^2-C\delta^2$.

The $2\pi^2$-Theorem~\ref{2pi2-theorem} ensures the full $(v,t)$-family contains a slice of area at least $2\pi^2$, and since $v$ encodes conformal transformations and $t$ is signed distance, the aim is to realize such a high-area slice \emph{within the conformal} four-parameter subfamily; for this one must control $t$ in terms of the energy gap $\delta$, so the maximizing slice lies close to $t=0$.

A closer examination of the Heintze--Karcher inequality used by Marques--Neves shows that, once the family is built from the quantitatively normalized surface, the problem reduces to proving the maximizing parameter $(v_0,t_0)$ stays a definite distance away from the boundary of the conformal group, i.e. there exists $\eta_1>0$ such that $|v_0|\leqslant 1-\eta_1$ (reduced in Section~3; technical proof deferred to the Appendix).

Geometric intuition for this uniform bound: as $|v|\to1$, $\Sigma_{(v,t)}$ becomes quantitatively close to a geodesic sphere, forcing area strictly below $2\pi^2$, so such $v$ cannot maximize. More precisely, for $t=0$, $\Sigma_v=\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, so $\mathrm{Area}(\Sigma_v)$ is close to $\mathrm{Area}(S)\le 4\pi$, and for small $|t|$ one uses
\[
\bigl|\mathrm{Area}(\Sigma_{(v,t)})-\mathrm{Area}(\Sigma_v)\bigr|\lesssim |t|\,\mathcal{W}(\Sigma_v)
\]
so the area stays $<5\pi$. For larger $|t|$, either (i) if $\Sigma_v$ stays away from the poles, an Euclidean-model argument shows $\Sigma_{(v,t)}$ is a small Lipschitz graph over a geodesic sphere hence has area $<5\pi$, or (ii) if $\Sigma_v$ lies close to a pole, comparison geometry gives $\mathrm{Area}(\Sigma_{(v,t)})<5\pi$. Thus for $|v|$ sufficiently close to $1$, $\mathrm{Area}(\Sigma_{(v,t)})<5\pi<2\pi^2$, proving $|v_0|\le 1-\eta_1$.

With this uniform control of $v$, the remaining task is to derive a quantitative stability estimate for minimal surfaces; the paper then uses the Section~3 area lower bound to reduce Willmore stability to minimal-surface stability, and in Section~4 performs linearization/stability analysis to obtain the quantitative bounds (conformal factor, conformal structure, and $W^{2,2}$-distance to the Clifford torus), thereby proving Theorem~\ref{Thm: main}.

Using the Marques--Neves five-parameter family $\{\Sigma_{(v,t)}\}$, the key point is to show that if a genus--one surface $\Sigma$ satisfies $\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2$, then after a \emph{conformal} transformation one gets a surface $\Sigma'$ with $\mathrm{Area}(\Sigma')\geqslant 2\pi^2-C\delta^2$.

To prove the main theorem, we use that the full $(v,t)$-family contains a slice of area at least $2\pi^2$, and since $v$ encodes conformal transformations and $t$ is signed distance, the aim is to realize such a high-area slice \emph{within the conformal} four-parameter subfamily; for this one must control $t$ in terms of the energy gap $\delta$, so the maximizing slice lies close to $t=0$.

A closer examination of the Heintze--Karcher inequality used by Marques--Neves shows that, once the family is built from the quantitatively normalized surface, the problem reduces to proving the maximizing parameter $(v_0,t_0)$ stays a definite distance away from the boundary of the conformal group, i.e. there exists $\eta_1>0$ such that $|v_0|\leqslant 1-\eta_1$ (reduced later; technical proof deferred to an appendix).

Geometric intuition for this uniform bound: as $|v|\to1$, $\Sigma_{(v,t)}$ becomes quantitatively close to a geodesic sphere, forcing area strictly below $2\pi^2$, so such $v$ cannot maximize. More precisely, for $t=0$, $\Sigma_v=\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, so $\mathrm{Area}(\Sigma_v)$ is close to $\mathrm{Area}(S)\le 4\pi$, and for small $|t|$ one uses
\[
\bigl|\mathrm{Area}(\Sigma_{(v,t)})-\mathrm{Area}(\Sigma_v)\bigr|\lesssim |t|\,\mathcal{W}(\Sigma_v)
\]
so the area stays $<5\pi$. For larger $|t|$, either (i) if $\Sigma_v$ stays away from the poles, an Euclidean-model argument shows $\Sigma_{(v,t)}$ is a small Lipschitz graph over a geodesic sphere hence has area $<5\pi$, or (ii) if $\Sigma_v$ lies close to a pole, comparison geometry gives $\mathrm{Area}(\Sigma_{(v,t)})<5\pi$. Thus for $|v|$ sufficiently close to $1$, $\mathrm{Area}(\Sigma_{(v,t)})<5\pi<2\pi^2$, proving $|v_0|\le 1-\eta_1$.

With this uniform control of $v$, the remaining task is to derive a quantitative stability estimate for minimal surfaces; the paper then uses the earlier area lower bound to reduce Willmore stability to minimal-surface stability, and later performs linearization/stability analysis to obtain the quantitative bounds (conformal factor, conformal structure, and $W^{2,2}$-distance to the Clifford torus), thereby proving Theorem~\ref{Thm: main}.

Inequality or Bound
Biconditional or Equivalence

Question:
Let \(\Sigma\subset \mathbb{S}^3\) be a smooth embedded surface of genus at least \(1\). For \(v\in \mathring{B}^4\), define the conformal transformation
\[
F_v(x)=(1-|v|^2)\frac{x-v}{|x-v|^2}-v.
\]
If \(\mathbb{S}^3\setminus \Sigma=A\cup A^*\) is the decomposition into connected components, set \(\Sigma_v:=F_v(\Sigma)\), \(A_v:=F_v(A)\), let \(d_v\) be the signed distance to \(\Sigma_v\), and define the canonical five-parameter family
\[
\Sigma_{(v,t)}=\partial\{x\mid d_v(x)<t\}, \qquad (v,t)\in \mathring{B}^4\times[-\pi,\pi].
\]
Which quantitative area estimate holds for this family, including the equality case?

\[\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]} \operatorname{Area}(\Sigma_{(v,t)})\ge 2\pi^2.\] Moreover, equality holds if and only if \(\Sigma\) is the Clifford torus \(\mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times \mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset \mathbb{S}^3\).

Choices:
B. \[\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]} \operatorname{Area}(\Sigma_{(v,t)})\le 2\pi^2.\] Moreover, equality holds if and only if for some \((v_0,t_0)\) the slice \(\Sigma_{(v_0,t_0)}\) is a conformal image of the Clifford torus.
C. \[\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]} \operatorname{Area}(\Sigma_{(v,t)})\ge 2\pi^2.\]
D. \[\operatorname{Area}(\Sigma_{(v,t)})\ge 2\pi^2\qquad\text{for every }(v,t)\in \mathring{B}^4\times[-\pi,\pi].\] Moreover, equality holds if and only if \(\Sigma\) is the Clifford torus.
E. \[\max_{v\in \mathring{B}^4} \operatorname{Area}(\Sigma_v)\ge 2\pi^2.\] Moreover, equality holds if and only if \(\Sigma\) is the Clifford torus \(\mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times \mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset \mathbb{S}^3\).

direction of the min-max area bound and misplaced equality on a slice

max over the five-parameter family replaced by pointwise lower bound for every slice

removed the signed-distance parameter t and restricted to the conformal four-parameter subfamily


\begin{theorem}\label{main thm 1}
Let $r\ge 2$ be an integer and $\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \ref{Table 1}. Let $a(r) = 2\log{2}\left(3.0758r+1.38402\log(4r)-1.5379\right)$. Then, for $q \ge \max\{10^{1145},e^{e^{a(r)}}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$

    Furthermore, as $q\rightarrow\infty$, we have a constant $D(r)$ from Table \ref{Table 1} such that
    $$|S_{\chi}(M,N)| \le (D(r)+o(1))N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

Let $q$ be a positive integer.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and~$N\ge1$, we consider the character sum
\begin{equation*}
 S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}

Burgess, in a series of papers~\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\left|S_{\chi}(M,N)\right| \ll N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+\eps},$$
for $r = 2,3$ and for any $r\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\chi}(M,N)$. In the case when $q$ is prime, Booker \cite{Boo2006} proved a result for $\chi$ quadratic, and McGown \cite{mcgown2012} and Trevi\~no \cite{ET2} made improvements for general $\chi$. The current best result for $q$ prime and $2\le r\le 10$ is due to Francis \cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \cite{ET2})
$$|S_{\chi}(M,N)| \le 2.74 N^{1-\frac{1}{r}}p^{\frac{r+1}{4r^2}}(\log{p})^{\frac{1}{r}},$$
for all integers $r\ge 2$ and for all primes $p\ge 10^7$.

These explicit results have been used to determine the class number of a quadratic field of large discriminant \cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \cite{mcgown2012}, to bound the least primitive root modulo a prime \cite{COT-2016}, to give an explicit bound for $L(1,\chi)$ in the case of quadratic characters \cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \cite{Elkies}, among other applications.

In the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \cite{JKL}.
\begin{theoremA*}
 Let $\chi$ be a primitive character mod $q$ with $q \ge e^{e^{9.594}}$. Then, for $N\le q^{\frac{5}{8}}$,
 $$\left|S_{\chi}(M,N)\right|\le 9.07 \sqrt{N} q^{\frac{3}{16}} \log^{\frac{1}{4}}(q)\left(2^{\omega(q)}\tau(q)\right)^{\frac{3}{4}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{2}},$$
 where $\tau(q)$ is the number of divisors of $q$ and $\omega(q)$ is the number of distinct prime factors of~$q$.
\end{theoremA*}

In other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\ge 3$ when $q$ is cubefree.

To state our theorem, we use the following standard labels of arithmetic functions:
let $\omega(q)$ denote the number of distinct prime factors of~$q$, let $\phi(q)$ be Euler's totient function, and let $\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\tau(n) = \tau_2(n)$ is the ordinary number-of-divisors function. Also define
\begin{equation} \label{meet Mr. Q}
m_r(q) = \min \biggl\{ \tau_{2r}(q), \biggl( \frac{\tau(q)}2 \biggr)^{2r-1} , \frac{q}{2r} \biggr\}.
\end{equation}

\label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{

To state our theorem, we use the following standard labels of arithmetic functions:
let $\omega(q)$ denote the number of distinct prime factors of~$q$, let $\phi(q)$ be Euler's totient function, and let $\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\tau(n) = \tau_2(n)$ is the ordinary number-of-divisors function. Also define
\begin{equation} \label{meet Mr. Q}
m_r(q) = \min \biggl\{ \tau_{2r}(q), \biggl( \frac{\tau(q)}2 \biggr)^{2r-1} , \frac{q}{2r} \biggr\}.
\end{equation}

\begin{table}[h!]
\begin{center}
 \begin{tabular}{ | c | c | c|}
 \hline
 $r$ & $C(r)$ & $D(r)$ \\ \hline
 2 & 15.219 & 8.362 \\ \hline
 3 & 5.359 & 4.581\\ \hline
 4 & 3.671 & 3.396\\ \hline
 5 & 2.953 & 2.811\\ \hline
 6 & 2.549 & 2.462\\ \hline
 7 & 2.290 & 2.229\\ \hline
 8 & 2.108 & 2.063\\ \hline
 9 & 1.973 & 1.938\\ \hline
 $\ge$10 & 1.869 & 1.841 \\ \hline
 \end{tabular}
 \caption{Constants in the Burgess inequality for values of $r$.}\label{Table 1}
\end{center}
\end{table}

\begin{theorem}\label{main theorem 2}
 Let $\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \ref{Table 1}. Then, for $q \ge \max\{10^{1145}, 2^{4r-2}\}$, if $r=2$ or $q$ is cubefree, we have
 $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

Let $r\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\ge 3$ for the remainder of the paper.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and ~$N\ge1$, we consider the character sum
\begin{equation*}
 S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}
Given an integer $r\ge2$, we use induction to prove a bound of the form
\begin{equation} \label{eq desired bound}
| S_\chi(M,N) | \le E_{q,r}(N), \quad\text{where }
E_{q,r}(N) = C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q),
\end{equation}
where $C\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \eqref{meet Mr. Q}, and
\begin{equation}\label{meet T(q)}
T(q) = \begin{cases}\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}-\frac{1}{2r^2}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main thm 1},\\
\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main theorem 2}
.\end{cases}
\end{equation}

\begin{lemma}\label{lem: alpha bound}
Let $r,q\ge 2$ and $N \ge q^{\frac{1}{4}+\frac{1}{4r}} $ be integers and let $i \in \{1,2\}$. Let $A = A_i(r,q)$, $B = B_i(r,q)$, $\kappa$ be defined as in \eqref{choice of kappa}, and $t_i(r)$ be defined as in Lemma \ref{A lower bound lemma}.
Let
\begin{equation*}
 \alpha
 =
 \left[ \frac{\left(\kappa\left(\frac{\phi(q)}{q}\right)+\frac{B2^{\omega(q)-1}}{N} \right)^2}{B}
 +
 2\kappa\left(\frac{6}{\pi^2}\log{(A)}+\delta+\frac{2\log{(A)}+2}{A-1}\right)\right],
\end{equation*}
where $\delta = \frac6{\pi^2} - \frac{36\zeta'(2)}{\pi^4} \approx 0.954422$.
Then, for $q\ge \max\{10^{1145}, (10r)^{18}, t_i(q)\}$,
$$\alpha \le \frac{4}{3}\kappa \log{q}.$$
\end{lemma}

\begin{lemma}\label{lemma to set up the main theorems}
For $i\in\{1, 2\}$ and $T_i$ be defined as in \eqref{eq T1} and \eqref{eq T2}. Let $s, \alpha, u, $ and $w$ be defined as in \eqref{eq s}, \eqref{eq alpha}, \eqref{eq u}, and \eqref{w}, respectively. Then
 \begin{equation*}
\begin{split}
 \left| S_\chi(M, N) \right|
 \le
 \left(\frac{B}{B-1}\right)N^{1-1/r}q^{(r+1)/{4r^2}}(\log q)^{\frac{1}{2r}}
 \left(
 u(\alpha w)^{\frac{1}{2r}}
 +sT_i(q)
 \right).
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
 Fix $i\in \{1,2\}$ and let $q\ge \max\{10^{1145},t_i(r)\}$. From Lemma \ref{eq after Holder}, once we choose $A$ and $B$ as described in Definition \ref{def A B v}, replacing $T(q)$ with $T_i(q)$ and $C(r)$ with $C$, we have
\begin{align*}
 \left| S_\chi\left(M, N\right)\right|
 &\le
 \frac{1}{\#\cA\cdot \#\cB}\left(\sum_{x=1}^q v_\cA(x)\right)^{1-\frac{1}{r}}
 \left(\sum_{x=1}^q v^2_{\cA}(x)\right)^{\frac{1}{2r}}\left(\sum_{x=1}^q\left|\sum_{b \in \cB}\chi\left(x+b\right)\right|^{2r}\right)^\frac{1}{2r}
 \\
 &\qquad{}+
\frac{2C}{2-1/r} A^{1-1/r} \frac{(\lfloor B\rfloor+1)^{2-1/r}}{\#\cB} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T_i(q). \end{align*}

Similar to the proof of Theorem \ref{main thm 1}, $10^{1145}\ge (10r)^{18}$ for $r\le 10^{50}$ while $t_2(r) \ge (10r)^{18}$ for $r\ge 10^{50}$, thus from \eqref{roughly u} and \eqref{eq alpha upper bound},
\begin{equation*}
 u \le \frac{2^{1/r}}{q^{\frac{1}{2r^2}}}\left(\frac{N}{A\phi^*}\right)^{1/r},
\end{equation*}
and
\[
\alpha \le \frac{4}{3}\kappa \log{q}.
\]
Furthermore,
\begin{equation}\label{f2}
f_2 = (4r)^{\omega(q)}m_r(q)(r-1)+1 \le r(4r)^{\omega(q)}m_r(q),
\end{equation}
and \begin{equation}\label{beta B def 2}\beta = \left(\frac{r-1}{(2r)r!}\right)^{\frac{1}{r}},
\end{equation}
therefore we bound $w$ in \eqref{w} as
\begin{equation}\label{bounding w 2}
 w \le \frac{2^{1+1/r}(r\cdot r!)^{1/r}}{(r-1)^{1+1/r}\log q} \left(4r\right)^{\omega(q)}m_r(q).
\end{equation}
Using \eqref{beta B def 2}, $B = r^2q^{\frac{1}{2r}} \beta$, and $AB = \kappa N$, we get
\begin{equation}\label{bounding N over A 2}
\frac{N}{A q^{\frac{1}{2r}}} = \frac{r^2}{\kappa}\left(\frac{r-1}{r!(2r)}\right)^{\frac{1}{r}}.
\end{equation}
Therefore, combining \eqref{eq s}, \eqref{roughly u}, \eqref{eq alpha upper bound}, \eqref{bounding w 2}, and \eqref{bounding N over A 2}, we get
\begin{align*}
 &u(\alpha w)^{\frac{1}{2r}}+sT_2(q)\\
 &\le 2^{\frac{1}{r}}\left(\frac{N}{Aq^{\frac{1}{2r}}\phi^*}\right)^{\frac{1}{r}}\left(\frac{4}{3}\kappa \log{q}\right)^{\frac{1}{2r}}\left(\frac{2^{1+\frac{1}{r}}(r\cdot r!)^{\frac{1}{r}}r^{\frac{1}{r}}}{(r-1)^{1+\frac{1}{r}}\log{q}}\right)^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\\ &+\frac{2C\kappa^{1-\frac{1}{r}}}{2-\frac{1}{r}}\left(\frac{B+1}{B}\right)^{2-\frac{1}{r}}T_2(q)\\
 &= \left(2^{\frac{5}{2r}-\frac{1}{2r^2}}3^{-\frac{1}{2r}}(r!)^{-\frac{1}{2r^2}}(r-1)^{\frac{1}{2r^2}-\frac{1}{2r}}r^{\frac{2}{r}-\frac{1}{2r^2}}\kappa^{-\frac{1}{2r}} +\frac{2C\kappa^{1-\frac{1}{r}}}{2-\frac{1}{r}}\left(\frac{B+1}{B}\right)^{2-\frac{1}{r}}\right)T_2(q).
\end{align*}
This produces the same conditions on $C$ and $D$ as in the proof of Theorem \ref{main thm 1}, thus we obtain the constants in Table \ref{Table 1}.
\end{proof}

\begin{table}[h!]
\begin{center}
  \begin{tabular}{ | c | c | c|}
    \hline
    $r$ & $C(r)$ & $D(r)$ \\ \hline
    2 & 15.219 & 8.362 \\ \hline
    3 & 5.359 & 4.581\\ \hline
    4 & 3.671 & 3.396\\ \hline
    5 & 2.953 & 2.811\\ \hline
    6 & 2.549 & 2.462\\ \hline
    7 & 2.290 & 2.229\\ \hline
    8 & 2.108 & 2.063\\ \hline
    9 & 1.973 & 1.938\\ \hline
    $\ge$10 & 1.869 & 1.841 \\ \hline
  \end{tabular}
  \caption{Constants in the Burgess inequality for values of $r$.}\label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{theorem}\label{main theorem 2}
    Let $\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \ref{Table 1}. Then, for $q \ge \max\{10^{1145}, 2^{4r-2}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

To prove Theorems \ref{main thm 1} and \ref{main theorem 2} we need the following explicit Weil-type inequality, which we consider of independent interest.

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

The paper is organized as follows. We prove Theorem \ref{weil} in Section~\ref{sec: weil}. In Section~\ref{General setup} we describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \cite[Theorem 12.6]{IK2004}. We prove other technical lemmas in Section~\ref{sec: technical}, some important bounds in Section~\ref{section asymptotics}, and we prove our main theorems in Section~\ref{sec: main theorems}.

\subsection*{Notation}
Most of the notation in the paper is standard, for example, $\phi(n)$ is Euler's totient function, for a finite set $S$, $\#S$ is the number of elements of $S$, $\lfloor x\rfloor$ is the floor of $x$, for a pair of integers $a,b$, $(a,b) = \gcd(a,b)$. We also use Landau's $O$ notation, i.e., $f(x) = O(g(x))$ if there exists a positive constant $C$ such that $|f(x)| \le C|g(x)|$ for large enough $x$. A notation we use that is less well-known is $O^*$. We say that $f(x) = O^*(g(x))$ if for all $x\ge x_0$, $|f(x)| \le |g(x)|$.

To prove Theorems \ref{main thm 1} and \ref{main theorem 2} the authors say they "need the following explicit Weil-type inequality" (Theorem \ref{weil}). They then describe the structure of the proof: they prove Theorem \ref{weil} in Section~\ref{sec: weil}, in Section~\ref{General setup} they "describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \cite[Theorem 12.6]{IK2004}," they prove "other technical lemmas" in Section~\ref{sec: technical} and obtain "some important bounds" in Section~\ref{section asymptotics}, and finally they "prove our main theorems" in Section~\ref{sec: main theorems}.

To prove the main theorem and Theorem \ref{main theorem 2} the authors say they “need the following explicit Weil-type inequality”. We first record this Weil-type inequality.

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

They then describe the structure of the proof as follows: they prove Theorem \ref{weil} earlier in the paper; next, in

\begin{align}\label{eq bad tuples}
    \sum_{(b_1,b_2,\ldots,b_{2r})\in \cB_2}\sum_{x\bmod{q}}\chi(f_1(x)f_2^{(\phi(q)-1)}(x)) \le r^{2r} \binom{\lfloor B\rfloor}{r} q \le \frac{r^{2r} \lfloor B\rfloor^r}{r!}q.
\end{align}
Combining the bounds in \eqref{eq good tuples} and \eqref{eq bad tuples}, we get the desired upper bound in the Weil-type inequality.
\end{proof}

\section{General setup}\label{General setup}

Let $r\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\ge 3$ for the remainder of the paper.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and ~$N\ge1$, we consider the character sum
\begin{equation*}
    S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}
Given an integer $r\ge2$, we use induction to prove a bound of the form
\begin{equation} \label{eq desired bound}
| S_\chi(M,N) | \le E_{q,r}(N), \quad\text{where }
E_{q,r}(N) = C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q),
\end{equation}
where $C\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \eqref{meet Mr. Q}, and
\begin{equation}\label{meet T(q)}
T(q) = \begin{cases}\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}-\frac{1}{2r^2}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main thm 1},\\
\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main theorem 2}
.\end{cases}
\end{equation}

We take $1\le N\le q^{\frac{1}{4}+\frac{1}{4r}}$ as the base case. In this case, the inequality~\eqref{eq desired bound} follows from the trivial bound $|S_\chi(M,N)| \le N$, since
\begin{align*}
&N \le C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q) \\
\implies &N^{1/r} \le C q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q) \\
\implies &N \le q^{\frac{r+1}{4r}} \bigl( C (\log q)^{\frac{1}{2r}} T(q) \bigr)^r,
\end{align*}
when $N \le q^{\frac{1}{4}+\frac{1}{4r}}$ (because $C(\log q)^{\frac{1}{2r}} T(q) \ge 1$).
\\
Therefore, in the induction step, we assume
\begin{align}\label{inductive hypothesis}
\text{for $N > q^{\frac{1}{4}+\frac{1}{4r}}$, the bound in \eqref{eq desired bound} holds for all smaller positive integers.}
\end{align}

they “describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in E. Kowalski and H. Iwaniec, \cite[Theorem 12.6]{IK2004}.” They then prove other technical lemmas later, obtain some important bounds later, and finally establish the main theorem together with Theorem \ref{main theorem 2}.

\label{main thm 1}
Let $r\ge 2$ be an integer and $\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{. Let $a(r) = 2\log{2}\left(3.0758r+1.38402\log(4r)-1.5379\right)$. Then, for $q \ge \max\{10^{1145},e^{e^{a(r)}}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$

    Furthermore, as $q\rightarrow\infty$, we have a constant $D(r)$ from Table \label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{ such that
    $$|S_{\chi}(M,N)| \le (D(r)+o(1))N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$,

Question:
Let $q$ be a positive integer, let $\chi$ be a primitive Dirichlet character modulo $q$, and for integers $M$ and $N\ge 1$ define
$$S_{\chi}(M,N)=\sum_{M<n\le M+N}\chi(n).$$
For arithmetic functions, let $\omega(q)$ be the number of distinct prime factors of $q$, let $\phi(q)$ be Euler’s totient function, let $\tau_k(q)$ be the number of ordered factorizations of $q$ into $k$ positive integers (so $\tau(q)=\tau_2(q)$), and define
$$m_r(q)=\min\left\{\tau_{2r}(q),\left(\frac{\tau(q)}{2}\right)^{2r-1},\frac{q}{2r}\right\}.$$
Fix an integer $r\ge 2$, and set
$$a(r)=2\log 2\,\bigl(3.0758r+1.38402\log(4r)-1.5379\bigr).$$
Let $C(r)$ and $D(r)$ be the constants given by
$$(C(r),D(r))=\begin{cases}
(15.219,8.362),&r=2,\\
(5.359,4.581),&r=3,\\
(3.671,3.396),&r=4,\\
(2.953,2.811),&r=5,\\
(2.549,2.462),&r=6,\\
(2.290,2.229),&r=7,\\
(2.108,2.063),&r=8,\\
(1.973,1.938),&r=9,\\
(1.869,1.841),&r\ge 10.
\end{cases}$$
Assume that either $r=2$ or $q$ is cubefree. Which quantitative estimate holds for $S_\chi(M,N)$?

For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
Furthermore, as $q\to\infty$, under the same hypothesis $r=2$ or $q$ cubefree,
$$|S_{\chi}(M,N)|\le (D(r)+o(1))N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$

Choices:
B.
For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
Furthermore, as $q\to\infty$, under the same hypothesis $r=2$ or $q$ cubefree,
$$|S_{\chi}(M,N)|\le (D(r)+o(1))N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
C.
For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
D.
For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
Furthermore, as $q\to\infty$, if $q$ is cubefree then
$$|S_{\chi}(M,N)|\le (D(r)+o(1))N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
E.
For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1r}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
Furthermore, as $q\to\infty$, under the same hypothesis $r=2$ or $q$ cubefree,
$$|S_{\chi}(M,N)|\le (D(r)+o(1))N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1r}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$