\label{theo.finite.top}
For fixed $n\,,m \in \mathbb{Z}^+\,,n \geqslant 3\,, m \geq1\,,$ and $ \Gamma \, ,\Lambda \in \mathbb{R} \,,  \Gamma\,,\Lambda \geqslant 0\,$, there exists $N = N(n,m,\Gamma,\Lambda) \in \mathbb{N}$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\mathbb{R}^{n+m}$ satisfying that  $\int_{M} \lvertA\rvert^{n}d\mu_M  \leqslant \Gamma$ and
$\vol_M ( B_R(0)) \leqslant  \Lambda R^n$ for any $R>0\,.$

\begin{theo} \label{theo.finite.top}
For fixed $n\,,m \in \ZZ^+\,,n \geq 3\,, m \geq1\,,$ and $ \Gamma \, ,\Lambda \in \RR \,,  \Gamma\,,\Lambda \geq 0\,$, there exists $N = N(n,m,\Gamma,\Lambda) \in \NN$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\RR^{n+m}$ satisfying that  $\int_{M} \abs{A}^{n}d\mu_M  \leq \Gamma$ and
$\vol_M ( B_R(0)) \leq  \Lambda R^n$ for any $R>0\,.$
\end{theo}

For minimal surfaces in $\mathbb{R}^3$, finite total curvature means that the Gaussian curvature integral is finite. Chern and Osserman \cite{Chern} proved that every minimal surface in $\mathbb{R}^3$ is conformally equivalent to a compact Riemann surface $\overline M$ punctured at a finite number of points, and the Gauss map on the surface can extend conformally to $\overline M$.

Collin \cite{collin1} proved that any properly embedded minimal
surface in $\mathbb{R}^3$ with finite topology and more than one end, has finite total curvature.
Colding and Minicozzi \cite{ColdingMincozzi1} removed the proper condition, where they proved that a complete embedded minimal surface $\Sigma$ with finite topology in $\mathbb{R}^3$ must be proper.
Meeks, Perez and Ros \cite{Meeks2} showed that the number of ends of $\Sigma$ is bounded by a constant depending on its genus.

Given an immersed minimal submanifold $M^{n}$  in $ \mathbb{R}^{n+m}\,,$ $M$ is said to have \emph{finite total  curvature} if
 $$\int_{M} \lvertA\rvert^{n}d\mu_M < \infty\,,$$
where $A$ denotes the second fundamental form of $M$ in $ \mathbb{R}^{n+m}$, and $\mu_M$ denotes the volume element of $M$.

Anderson \cite{anderson1984compactification} gave a generalization of the
Chern-Osserman theorem \cite{Chern} on minimal surfaces of finite total curvature: a complete minimal submanifold $M^n$ with finite total curvature is diffeomorphic to a compact $C^{\infty} $ manifold $\overline{M}^n$ punctured at a finite number of points $\{p_i\}_{1}^{\ell} \in \overline{M}^n$ and the Gauss map $\gamma:M^n \to G_{n,m}$ extends to a $C^{n-2}$ map $\overline \gamma: \overline  M^n \to G_{n,m} $ of the compactification(where $G_{n,m}$ denotes the Grassmann manifold of $n$-planes in Euclidean $(n+m)$-space).
In particular, $M$ has Euclidean volume growth with ratio bounded by a constant depending on $\ell$.
For complete minimal hypersurfaces in $\mathbb{R}^{n+1}$ with $3 \leqslant n \leqslant 6\,,$ Tysk \cite{Tysk} proved that finite index and Euclidean volume growth imply finite total curvature.

Chodosh, Ketover, and  Maximo \cite[Theorem 1.1]{Chodosh17} proved that
for a fixed  closed Riemannian manifold $(M^n,g)(3 \leqslant  n \leqslant 7)\,,$
 there can be at most $ N = N(M,g,\Lambda,I)$ distinct diffeomorphism types in the set of
 embedded minimal hypersurfaces $\Sigma \subset (M,g)$ with $\ind(\Sigma) \leqslant  I$ and $\vol_g(\Sigma) \leqslant  \Lambda\,.$
 In particular, for $n=3\,$, there is $r_0 = r_0(M,g,\Lambda,I)$ so that
 any embedded minimal surface $\Sigma$ in $(M^3,g)$ with $\ind(\Sigma) \leqslant I$ and $\mathrm{area}_g(\Sigma) \leqslant \Lambda$ has
 $\genus(\Sigma) \leqslant  r_0$; for $4 \leqslant n \leqslant 7$, there is $N = N(n,I,\Lambda) \in  \mathbb{N}$ so that there are at most
 $N$ mutually non-diffeomorphic complete embedded minimal hypersurfaces $\Sigma^{n-1} \subset  \mathbb{R}^n$
 with $\ind(\Sigma) \leqslant I$ and $\vol(\Sigma\cap B_R(0)) \leqslant \Lambda R^{n-1}$ for all $R > 0$ \cite[Theorem 1.2]{Chodosh17}.

Antoine Song\cite{songantoine1} showed that  given a closed embedded minimal hypersurface $\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$ with $2\leqslant n \leqslant 6\,,$  the sum of dimensions over a fixed field of the cohomology groups  of $\Sigma$ can be bounded by the area and  Morse index of $\Sigma$ by a combinatorial argument.

We get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \cite{anderson1984compactification}.

Collin \cite{collin1} proved that any properly embedded minimal
surface in $\mathbb{R}^3$ with finite topology and more than one end, has finite total curvature.
Colding and Minicozzi \cite{ColdingMincozzi1} removed the proper condition, where they proved that a complete embedded minimal surface $\Sigma$ with finite topology in $\mathbb{R}^3$ must be proper.
Meeks, Perez and Ros \cite{Meeks2} showed that the number of ends of $\Sigma$ is bounded by a constant depending on its genus.

Given an immersed minimal submanifold $M^{n}$ in $ \mathbb{R}^{n+m}\,,$ $M$ is said to have \emph{finite total curvature} if
 $$\int_{M} \lvertA\rvert^{n}d\mu_M < \infty\,,$$
where $A$ denotes the second fundamental form of $M$ in $ \mathbb{R}^{n+m}$, and $\mu_M$ denotes the volume element of $M$.

Anderson \cite{anderson1984compactification} gave a generalization of the
Chern-Osserman theorem \cite{Chern} on minimal surfaces of finite total curvature: a complete minimal submanifold $M^n$ with finite total curvature is diffeomorphic to a compact $C^{\infty} $ manifold $\overline{M}^n$ punctured at a finite number of points $\{p_i\}_{1}^{\ell} \in \overline{M}^n$ and the Gauss map $\gamma:M^n \to G_{n,m}$ extends to a $C^{n-2}$ map $\overline \gamma: \overline M^n \to G_{n,m} $ of the compactification(where $G_{n,m}$ denotes the Grassmann manifold of $n$-planes in Euclidean $(n+m)$-space).
In particular, $M$ has Euclidean volume growth with ratio bounded by a constant depending on $\ell$.
For complete minimal hypersurfaces in $\mathbb{R}^{n+1}$ with $3 \leqslant n \leqslant 6\,,$ Tysk \cite{Tysk} proved that finite index and Euclidean volume growth imply finite total curvature.

Chodosh, Ketover, and Maximo \cite[Theorem 1.1]{Chodosh17} proved that
for a fixed closed Riemannian manifold $(M^n,g)(3 \leqslant n \leqslant 7)\,,$
 there can be at most $ N = N(M,g,\Lambda,I)$ distinct diffeomorphism types in the set of
 embedded minimal hypersurfaces $\Sigma \subset (M,g)$ with $\ind(\Sigma) \leqslant I$ and $\vol_g(\Sigma) \leqslant \Lambda\,.$
 In particular, for $n=3\,$, there is $r_0 = r_0(M,g,\Lambda,I)$ so that
 any embedded minimal surface $\Sigma$ in $(M^3,g)$ with $\ind(\Sigma) \leqslant I$ and $\mathrm{area}_g(\Sigma) \leqslant \Lambda$ has
 $\genus(\Sigma) \leqslant r_0$; for $4 \leqslant n \leqslant 7$, there is $N = N(n,I,\Lambda) \in \mathbb{N}$ so that there are at most
 $N$ mutually non-diffeomorphic complete embedded minimal hypersurfaces $\Sigma^{n-1} \subset \mathbb{R}^n$
 with $\ind(\Sigma) \leqslant I$ and $\vol(\Sigma\cap B_R(0)) \leqslant \Lambda R^{n-1}$ for all $R > 0$ \cite[Theorem 1.2]{Chodosh17}.

Antoine Song\cite{songantoine1} showed that given a closed embedded minimal hypersurface $\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$ with $2\leqslant n \leqslant 6\,,$ the sum of dimensions over a fixed field of the cohomology groups of $\Sigma$ can be bounded by the area and Morse index of $\Sigma$ by a combinatorial argument.

We get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \cite{anderson1984compactification}.

We get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \cite{anderson1984compactification}.

Our proof is inspired by the ideas in \cite{Chodosh17}, but we need further research in some situations. For instance, one point of concentration is a plane in Proposition 7.1 of \cite{Chodosh17}, while in our situation it may be a non-flat minimal submanifold with finite total curvature. In Theorem \ref{prop.diffeo}, we can resolve it by an induction argument on the total curvature.

\begin{lemm}\label{lemm:curv.est}
For fixed $I \in \ZZ^+,0 <r_0<R_0< \infty\,,$ suppose that $M_j( \iota_j :M_{j} \to \RR^{n+m})$ is a sequence of $n$-dimensional complete properly immersed minimal submanifolds with nonempty boundary and $\iota_j(M_j) \subset B_{R_0}(0)$ such that $ \int_{M_j} \abs{A_{M_j}}^n d\mu_{M_j} < IK_0 , \limsup \limits_{j\to \infty}\nu_j(B_{R_0}(0)\setminus B_{\fh{r_0}{2}}(0))< K_0$ and $\vol(B_R(q)) \leq \Lambda R^n$ for any $B_R(q) \subset B_{R_0}(0)$. Then, after passing to a subsequence, we have:
\begin{enumerate}[itemsep=5pt, topsep=5pt]
 \item There exist $C>0$ and a sequence of smooth blow-up sets $\cB_{j} \subset M_{j}$ so that
\begin{equation} \label{eq:seq.cur.esti}
 |A_{M_{j}}|(x)d(\iota_j(x),\iota_j(\cB_{j} \cup \de M_j) ) \leq C\, , \ |\cB_{j}|< I\, , \ \iota_j(\cB_j) \subset B_{\fh{3}{4}r_0}(0)\, ,
\end{equation}
for all $x \in M_{j}\,.$
\item $\iota_j(\cB_j)$ converges to $ \widetilde \cB_ \infty \subset \RR^{n+m}$ in the Hausdorff distance sense and the Radon measure $\nu_j$ converges to $ \nu_\infty$ in the Radon measure sense with $\nu_\infty(p_\infty)\geq 2K_0$ for any $p_\infty \in \widetilde \cB_\infty\,.$
\end{enumerate}
\end{lemm}
\begin{proof}
Firstly, we assume the conclusion in item(1) holds for the fixed $I\,,$ and prove the conclusion in item(2) holds for the same fixed $I\, $.
 Since $\iota_j(\cB_j) \subset B_{\fh{3}{4}r_0}(0) \,,$ after passing to a subsequence, we can assume $\iota_j(\cB_j)$ converges to $\widetilde \cB_\infty \subset \RR^{n+m}$ in the Hausdorff distance sense, i.e., $d_{\cH}(\iota_j(\cB_j),\widetilde \cB_\infty) \to 0\, $. Since $\nu_j(\RR^{n+m})=\nu_j(B_{R_0}(0)) = \int_{M_j} \abs{A_{M_j}}^n d\mu_{M_j}< IK_0\,,$ after passing to a subsequence, we can assume $\nu_j \to \nu_\infty$ in the Radon measure sense.

Chodosh proved removal singularity theorem for embedded minimal hypersurfaces with finite total curvature in Euclidean space in his note \cite{Chodosh1}. We adapt his method to study the immersed submanifolds of arbitrary codimension.
\begin{theo}[Removal singularity]
\label{theo.removal.singularity}
 Suppose that $\iota: M^{n} \to {B_2(0)}\setminus\{0\} $ is a smooth minimal immersion, i.e.\,,$$\int_M \div_MY d\mu_{M} = 0\, , \text{ for any } Y\in C^\infty_c(B_2(0)\setminus\{0\},\RR^{n+m})\,.$$ The minimal submanifold $M$ satisfies that $0 \in \bar {\iota(M)}\,,$ $\vol(\iota^{-1}(B_r(0)\setminus\{0\})) \leq \Lambda r^n$ for any $0<r<2$ and $\int_M\abs{A_M}^n d\mu_{M} <\infty\,.$
 Then there exists a smooth minimal immersion $\hat \iota: \hat M \to {B_2(0)}\, $, i.e., $$\int_{\hat M} \div_MY d\mu_{M} = 0\, , \text{ for any }Y\in C^\infty_c(B_2(0),\RR^{n+m})\,,$$ satisfying that $\hat M \setminus \hat \iota^{-1}(0) = M$ and $\hat \iota_{|M} =\iota\,.$
\end{theo}
\begin{proof}
 Let $f(x):=|\iota(x)|\,.$ We claim that after rescaling $M\,,$ we can assume that $|\hat \iota(x)|$ has no critical points in $B_2(0)$ and $M$ intersects $\de B_1(0)$ transversely.
 \begin{lemm} \label{lemm.no.critital.point}
 There exists $\delta_1$ small depending on the immersion $\iota$ such that any critical point of $f$ on $M\cap \iota^{-1}(B_{\delta_1}(0)\setminus \{0\})$ is a local minimum point.

Since $\hat M_{\infty}$ has finite total curvature and satisfies the volume growth condition of Theorem \ref{theo.removal.singularity} by the varifold convergence and monotonicity formula, the possible singularity at $\{0\}$ is removable. Hence $\hat M_{\infty}$ is an immersed minimal submanifold in $\RR^{n+m}$ with total curvature $\int_{\hat M_{\infty}} \abs{A_{\hat M_{\infty}}}^n d\mu_{\hat M_{\infty}}\leq I K_0$ and $\vol(\hat M_{\infty}\cap B_{r}(0)) \leq \Lambda r^{n}$ for any $r>0\,.$ Moreover, $\hat M_{\infty}$ has bounded number of components due to the volume bound. By Theorem \ref{eqival1}, every component of $\hat M_{\infty}$ is regular at infinity. Then we can choose $\gamma \geq 1$ large enough so that $\partial B_{\gamma}(0)$ intersects each component of $\hat M_{\infty}$ transversely, and each component of $\hat M_{\infty} \cap \partial B_{\gamma}(0)$ is diffeomorphic to $\SS^{n-1}$ with the standard smooth structure. Moreover the number of components of $\hat M_{\infty} \cap \partial B_{\gamma}(0)$ is no more than $\Lambda\,.$ By the choice of $\delta_{j}\,$, the curvature estimates \eqref{eq:prop.cur1} hold for all $x \in M_{j} \cap \left( B_{2}(0) \setminus B_{\gamma \delta_{j}}(\iota_j(p_{j}))\right)\,$. Then by applying Lemma \ref{lemm.Annular.composition}, we see that $M_{j} \cap \left( B_{2}(0) \setminus B_{\gamma \delta_{j}}(\iota_j(p_{j}))\right)$ is diffeomorphic to the union of annular regions. In particular, $M_{j}\cap B_{\gamma\delta_{j}}(\iota_j(p_{j}))$ must be connected (because we have assumed that $M_{j}$ is connected in \hyperlink{defi:beth}{$(\beth)$}). Then we only need to prove $\hat M_j \cap B_\gamma(0)$ are diffeomorphic to each other after passing to a subsequence.

\begin{proof}[Proof of Theorem \ref{theo.finite.top}]
 We will prove Theorem \ref{theo.finite.top} by contradiction. Since the volume bound and monotonicity formula imply that there exist at most finite number of components for any minimal submanifold satisfying the assumption of Theorem \ref{theo.finite.top}, without loss of generality, we can assume the minimal submanifolds satisfying the assumption of Theorem \ref{theo.finite.top} are connected. If $M_j^n$ is a sequence of pairwise non-diffeomorphic complete connected, immersed minimal submanifold in $\RR^{n+m}$ with $\vol(M_j\cap B_{R}(0)) \leq \Lambda R^{n}$ for any $R>0$ and $$\int_{M_j} \abs{A_{M_j}}^n d \mu_{M_j}\leq \Gamma < IK_0\,\,.$$ By rescaling $M_j,$ we can assume $$\int_{M_j\setminus B_{\fh{1}{j}}(0)}\abs{A_{M_j}}^n d \mu_{M_j} <\fh{1}{j}\,,$$ and $M_j $ intersects $\de B_1(0)$ transversely. By
 Theorem \ref{eqival1},
 $M_j$ is properly immersed and regular at infinity. By rescaling $M_j\,,$ we can assume $M_j \setminus B_{\fh1 2}(0) $ is the union of minimal graph and each minimal graph is defined over the exterior of a bounded region in an $n$-plane passing $0 \in \RR^{n+m}\,.$ So after passing to a subsequence, we can assume the $M_j\cap B_1(0) $ are pairwise non-diffeomorphic. By Lemma \ref{lemm:curv.est}, the sequence $M_j\cap B_2(0)$ satisfies \hyperlink{defi:beth}{$(\beth)$}. Then by Theorem \ref{prop.diffeo}, after passing to a subsequence, all of the $M_j\cap B_1(0)$ are diffeomorphic. This is a contradiction.
 \end{proof}

Our proof is inspired by the ideas in \cite{Chodosh17}, but we need further research in some situations. For instance, one point of concentration is a plane in  Proposition 7.1 of \cite{Chodosh17}, while in our situation it may be a non-flat minimal submanifold with finite total curvature. In Theorem \ref{prop.diffeo}, we can resolve it by an induction argument on the total curvature.

\subsection{Outline of the paper}
\begin{itemize}
    \item In \S \ref{sec.pre}, we state several definitions
 and curvature estimates for minimal submanifolds which are needed in the following.
 \item  In \S \ref{sec.geo}, we describe the geometry of ends of complete immersed minimal submanifolds in $\mathbb{R}^{n+m}$ with finite total curvature, enlightened by \cite{schocen} and \cite{anderson1984compactification}.
This helps us to derive  curvature estimates away from finitely many points in Lemma \ref{lemm:curv.est}.
 \item In \S \ref{sec.top}, we prove a key topological result in Lemma \ref{lemm.Annular.composition} allowing us to control the topology of the “intermediate regions”,
then combined the curvature estimates in \S \ref{sec.geo} we can prove Theorem \ref{theo.finite.top} by an induction argument on the total curvature.
\end{itemize}

The proof is “inspired by the ideas in \cite{Chodosh17},” but differs because “in our situation [a concentration point] may be a non-flat minimal submanifold with finite total curvature” (rather than a plane as in Proposition 7.1 of \cite{Chodosh17}). This issue is resolved “by an induction argument on the total curvature” (cf. Theorem \ref{prop.diffeo}).

The stated structure to prove Theorem~\ref{theo.finite.top} is:
\begin{itemize}
\item Develop “definitions and curvature estimates for minimal submanifolds” in \S\ref{sec.pre}.
\item In \S\ref{sec.geo}, “describe the geometry of ends of complete immersed minimal submanifolds in $\mathbb{R}^{n+m}$ with finite total curvature,” which “helps us to derive curvature estimates away from finitely many points in Lemma~\ref{lemm:curv.est}.”
\item In \S\ref{sec.top}, prove “a key topological result in Lemma~\ref{lemm.Annular.composition} allowing us to control the topology of the \lq intermediate regions\rq,” and then, “combined [with] the curvature estimates in \S\ref{sec.geo} we can prove Theorem~\ref{theo.finite.top} by an induction argument on the total curvature.”
\end{itemize}

The proof is “inspired by the ideas in \cite{Chodosh17},” but differs because “in our situation [a concentration point] may be a non-flat minimal submanifold with finite total curvature” (rather than a plane as in Proposition 7.1 of \cite{Chodosh17}). This issue is resolved “by an induction argument on the total curvature,” using the following theorem:

\begin{theo} \label{prop.diffeo}
Given a sequence $M_{j}$ satisfying \hyperlink{defi:beth}{$(\beth)$} and each $M_j$  intersects $\partial B_{1}(0)$ transversely. By passing to a subsequence, all of the $M_{j}\cap B_{1}(0)$ are diffeomorphic.
\end{theo}

The stated structure to prove the main theorem is:
\begin{itemize}
\item Develop “definitions and curvature estimates for minimal submanifolds” next.
\item After that, “describe the geometry of ends of complete immersed minimal submanifolds in $\mathbb{R}^{n+m}$ with finite total curvature,” which “helps us to derive curvature estimates away from finitely many points” in the following lemma.

\begin{lemm}\label{lemm:curv.est}
For fixed $I \in \ZZ^+,0 <r_0<R_0< \infty\,,$ suppose that $M_j( \iota_j :M_{j} \to  \RR^{n+m})$ is a sequence of $n$-dimensional complete properly immersed minimal submanifolds with nonempty boundary and $\iota_j(M_j) \subset B_{R_0}(0)$ such that $ \int_{M_j} \abs{A_{M_j}}^n d\mu_{M_j} < IK_0 , \limsup \limits_{j\to \infty}\nu_j(B_{R_0}(0)\setminus B_{\fh{r_0}{2}}(0))< K_0$ and $\vol(B_R(q)) \leq \Lambda R^n$ for any $B_R(q) \subset B_{R_0}(0)$. Then, after passing to a subsequence, we have:
\begin{enumerate}[itemsep=5pt, topsep=5pt]
    \item There exist $C>0$ and a sequence of smooth blow-up sets $\cB_{j} \subset M_{j}$  so that
\begin{equation} \label{eq:seq.cur.esti}
    |A_{M_{j}}|(x)d(\iota_j(x),\iota_j(\cB_{j} \cup \de M_j) ) \leq C\, , \ |\cB_{j}|< I\, , \ \iota_j(\cB_j) \subset B_{\fh{3}{4}r_0}(0)\, ,
\end{equation}
for all $x \in M_{j}\,.$
\item  $\iota_j(\cB_j)$ converges to $ \widetilde  \cB_ \infty \subset \RR^{n+m}$ in the Hausdorff distance sense and  the Radon measure $\nu_j$ converges to $ \nu_\infty$ in the Radon measure sense with $\nu_\infty(p_\infty)\geq 2K_0$ for any $p_\infty \in \widetilde \cB_\infty\,.$
\end{enumerate}
\end{lemm}

\item Next, prove “a key topological result” in the following lemma, allowing control of the topology of the \lq intermediate regions\rq.

\begin{lemm}[Annular decomposition] \label{lemm.Annular.composition}
For fixed $n\,,m\in \ZZ^+\,,n\geq 2\,,$ there is a $0 < \sigma_0 <\fh{1}{2} $ only depending on $n,m$ with the following property. Suppose that $M^n(\iota : M^n \to \bar{B_2(0)} \subset\RR^{n+m})$ is a complete properly immersed submanifold  with $\iota(\de M)  \subset \de B_2(0)\, . $ Assume that for some $\sigma \leq \sigma_0$ and $p \in B_{\sigma_0}(0)\,,$ we have:
\begin{enumerate}[itemsep=5pt, topsep=5pt]
\item For each component $M'$ of $M\,,$ $M' \cap B_{\sigma}(p) \neq \emptyset \,$.
\item The immersed submanifold $M$ intersects $\de B_{\sigma}(p)$ transversely, and $M \cap \de B_{\sigma}(p)$ has $k$ components. Moreover, each component of   $M \cap \de B_{\sigma}(p)$ is diffeomorphic to $\SS^{n-1}$ with the standard smooth structure.
\item The second fundamental form of $M$ satisfies $|A|(x)|\iota(x)-p| \leq \fh{1}{4} $ for all $x \in M \cap \left( \overline{B_{1}(0)} \setminus B_{\sigma}(p) \right)\,.$
\end{enumerate}
Then, $M$ intersects $\de B_{1}(0)$ transversely. Both  $M \cap \left(\overline{B_{1}(0)}\setminus {B_{\sigma}(p)}\right)$ and  $M \cap \de B_{1}(0)$  have $k$ components. Moreover, each component of   $M \cap \de B_{1}(0)$ is diffeomorphic to $\SS^{n-1}$ with the standard smooth structure and
 each component of $M \cap \left(\overline{B_{1}(0)}\setminus {B_{\sigma}(p)}\right)$ is diffeomorphic to $\SS^{n-1}\times [0,1]$ with the standard smooth structure.
\end{lemm}

Combined with the curvature estimates proved earlier, this yields an induction argument on the total curvature, completing the proof of the main theorem.
\end{itemize}

\label{theo.finite.top}
For fixed $n\,,m \in \mathbb{Z}^+\,,n \geqslant 3\,, m \geq1\,,$ and $ \Gamma \, ,\Lambda \in \mathbb{R} \, ,  \Gamma\,,\Lambda \geqslant 0\,$, there exists $N = N(n,m,\Gamma,\Lambda) \in \mathbb{N}$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\mathbb{R}^{n+m}$ satisfying that  $\int_{M} \lvertA\rvert^{n}d\mu_M  \leqslant \Gamma$ and
$\vol_M ( B_R(0)) \leqslant  \Lambda R^n$ for any $R>0\,.$

Existential–Universal
Inequality or Bound

Question:
Let n,m be positive integers with n \ge 3 and m \ge 1, and let \Gamma,\Lambda \ge 0. For a complete immersed minimal n-submanifold M \subset \mathbb{R}^{n+m}, write A for its second fundamental form, d\mu_M for its volume element, and \operatorname{vol}_M(B_R(0)) for the n-dimensional volume of M\cap B_R(0), where B_R(0) is the Euclidean ball of radius R centered at 0. Which statement holds for the class of all such M satisfying
\[
\int_M |A|^n\,d\mu_M \le \Gamma
\quad\text{and}\quad
\operatorname{vol}_M(B_R(0))\le \Lambda R^n\ \,\text{for every }R>0?
\]

There exists a number N=N(n,m,\Gamma,\Lambda)\in\mathbb{N} such that among complete immersed minimal n-submanifolds M\subset \mathbb{R}^{n+m} satisfying
\[
\int_M |A|^n\,d\mu_M \le \Gamma
\quad\text{and}\quad
\operatorname{vol}_M(B_R(0))\le \Lambda R^n\ \,\text{for all }R>0,
\]
there are at most N mutually non-diffeomorphic examples.

Choices:
B.
There exists a number N=N(n,m,\Gamma,\Lambda)\in\mathbb{N} such that among complete immersed minimal n-submanifolds M\subset \mathbb{R}^{n+m} satisfying
\[
\int_M |A|^n\,d\mu_M \le \Gamma
\quad\text{and}\quad
\operatorname{vol}_M(B_R(0))\le \Lambda R^n\ \,\text{for all sufficiently large }R,
\]
there are at most N mutually non-diffeomorphic examples.
C.
There exists a number N=N(n,m,\Gamma,\Lambda)\in\mathbb{N} such that among complete immersed minimal n-submanifolds M\subset \mathbb{R}^{n+m} satisfying
\[
\int_M |A|^n\,d\mu_M \le \Gamma
\quad\text{and}\quad
\operatorname{vol}_M(B_R(0))\le \Lambda R^n\ \,\text{for all }R>0,
\]
there are at most N mutually non-homeomorphic examples.
D.
There exists a number N=N(n,m,\Gamma)\in\mathbb{N} such that among complete immersed minimal n-submanifolds M\subset \mathbb{R}^{n+m} satisfying
\[
\int_M |A|^n\,d\mu_M \le \Gamma,
\]
there are at most N mutually non-diffeomorphic examples.
E.
For every complete immersed minimal n-submanifold M\subset \mathbb{R}^{n+m} satisfying
\[
\int_M |A|^n\,d\mu_M \le \Gamma
\quad\text{and}\quad
\operatorname{vol}_M(B_R(0))\le \Lambda R^n\ \,\text{for all }R>0,
\]
there exists a compact smooth manifold \overline M and a finite set of points \{p_i\}_{i=1}^{\ell}\subset \overline M, with \ell\le N(n,m,\Gamma,\Lambda), such that M\text{ is diffeomorphic to }\overline M\setminus \{p_i\}_{i=1}^{\ell}.

diffeomorphism conclusion weakened to homeomorphism

dependence on Euclidean volume growth parameter \Lambda

upgrading finiteness of diffeomorphism types to uniform bound on number of punctures in compactifications


[Quantitative stability for parabolic equations]\label{thm:parabolic}
		Let $T>0$ and $\Omega$ be a bounded domain in ${\mathbb R}^n$. For $\varepsilon\geq 0$, assume that $A_\varepsilon: {\mathbb R}^n_0\to \S_+^n$ and ${H_\varepsilon: \Omega_T\times {\mathbb R}^n\to {\mathbb R}}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\varepsilon\in C(\partial_p \Om_T)$ and $u_\varepsilon\in C(\overline{\Omega}\times [0, T))$ be bounded ${\mathcal F}$-solutions of  \eqref{general parabolic approx eq}\eqref{parabolic approx bdry-cond}. Assume in addition that $u_\varepsilon$ are equi-H\"older continuous in $\overline{\Om_T}$, that is, there exist $L>0$ and $\theta\in (0, 1]$ such that,
		\begin{equation}\label{equi-holder}
			|u_\varepsilon(x_1, t_1)-u_\varepsilon(x_2, t_2)|\leq L(|x_1-x_2|^\theta +|t_1-t_2|^\theta)
		\end{equation}
		for all $(x_1, t_1), (x_2, t_2)\in \overline{\Omega}\times [0, T)$ and $\varepsilon\geq 0$ small.
		Then, for all $\varepsilon>0$ small,
		\begin{equation}\label{error-est parabolic}
			\sup_{\overline{\Omega}\times [0, T)}|u_\varepsilon-u_0|\leq \sup_{\partial_p \Om_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}  +c_H T\varepsilon^\gamma,
		\end{equation}
		holds with
		\begin{equation}\label{nu-exponent}
			\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta, 0\}},
		\end{equation}
		where $C>0$ depends on $T, n, k, L_A, L_H, \beta, c_A$ as well as the H\"older exponent $\theta$ and the constant $L$. Here, $k>2$, $L_A\geq0$, $L_H\geq0$ appear in (A1)(A2), and $\alpha, \beta, \gamma, c_A, c_H\in {\mathbb R}$ are given in (C1)(C2).

\begin{equation}\label{np-parabolic}
	\partial_t u-\Delta^N_p u+H(x, u, \nabla u)=0 \quad \text{in $\Om_T:=\Omega\times (0, T)$}
\end{equation}

\begin{thm}[Quantitative stability for parabolic equations]\label{thm:parabolic}
		Let $T>0$ and $\Omega$ be a bounded domain in $\R^n$. For $\vep\geq 0$, assume that $A_\vep: \R^n_0\to \S_+^n$ and ${H_\vep: \Omega_T\times \R^n\to \R}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\vep\in C(\partial_p \Om_T)$ and $u_\vep\in C(\Oba\times [0, T))$ be bounded $\F$-solutions of  \eqref{general parabolic approx eq}\eqref{parabolic approx bdry-cond}. Assume in addition that $u_\vep$ are equi-H\"older continuous in $\ol{\Om_T}$, that is, there exist $L>0$ and $\theta\in (0, 1]$ such that,
		\begin{equation}\label{equi-holder}
			|u_\vep(x_1, t_1)-u_\vep(x_2, t_2)|\leq L(|x_1-x_2|^\theta +|t_1-t_2|^\theta)
		\end{equation}
		for all $(x_1, t_1), (x_2, t_2)\in \Oba\times [0, T)$ and $\vep\geq 0$ small.
		Then, for all $\vep>0$ small,
		\begin{equation}\label{error-est parabolic}
			\sup_{\Oba\times [0, T)}|u_\vep-u_0|\leq \sup_{\partial_p \Om_T}|g_\vep-g_0|+C\vep^{\nu}  +c_H T\vep^\gamma,
		\end{equation}
		holds with
		\begin{equation}\label{nu-exponent}
			\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta, 0\}},
		\end{equation}
		where $C>0$ depends on $T, n, k, L_A, L_H, \beta, c_A$ as well as the H\"older exponent $\theta$ and the constant $L$. Here, $k>2$, $L_A\geq0$, $L_H\geq0$ appear in (A1)(A2), and $\alpha, \beta, \gamma, c_A, c_H\in \R$ are given in (C1)(C2).
	\end{thm}

\begin{equation}\label{regularized eq}
		\partial_t u-\left(|\nabla u|^2+\vep^2\right)^{\frac{p'-p}{2}}\div\left(\left(|\nabla u|^2+\vep^2\right)^{\frac{p-2}{2}}\nabla u\right)=0 \quad \text{in $\Omega_T$,}
	\end{equation}

\begin{rmk}\label{rmk:periodic}
		Our discussions on the Cauchy-Dirichlet problem in Theorem~\ref{thm:parabolic} and Theorem~\ref{thm:parabolic2} extend readily to the case of spatially periodic settings.
		Instead of considering \eqref{general parabolic lim eq} in a bounded domain $\Omega$, we may obtain the same stability results as in Theorem~\ref{thm:parabolic} and Theorem~\ref{thm:parabolic2} for the whole space $\R^n$, provided that the initial value $g$ and $H$ are both periodic with respect to $x$. Note that the periodic setting essentially plays the same role as a bounded domain (without boundary) in our comparison arguments and therefore does not substantially affect the proofs. For clarity of presentation, here we choose not to repeat the statements for this case.
	\end{rmk}

\begin{equation}\label{mono-0}
		\rho\mapsto H_\vep(x, \rho, \xi)-\lambda \rho \quad \text{is nondecreasing in $\R$. }
	\end{equation}

\subsection{Motivation}
	In this paper we study the stability of a class of quasilinear parabolic partial differential equations, which takes the form
	\begin{equation}\label{general parabolic lim eq}
		\partial_t u-\operatorname{tr}(A(\nabla u)\nabla^2 u)+H(x, t, \nabla u)=0 \quad \text{in $\Omega\times (0, T)$,}
	\end{equation}
	where $\Omega$ is a bounded domain in ${\mathbb R}^n$, $T>0$ is fixed, and
	\[
	A: {\mathbb R}^n\setminus \{0\}\to {\mathcal S}^n_+, \quad H: \Omega\times (0, T)\times {\mathbb R}^n\to {\mathbb R},
	\]
	are given functions satisfying appropriate assumptions to be introduced later. Here, ${\mathcal S}^n_+$ represents the set of all nonnegative symmetric $n\times n$ matrices. A key feature of this general setting is that the elliptic operator may exhibit singularity at vanishing gradients.

	One typical example of applicable equations we have in mind is the following parabolic normalized $p$-Laplace type equation (with $1<p<\infty$)
	\begin{equation}\label{np-parabolic0}
		\partial_t u-\Delta^N_p u=0 \quad \text{in $\Om_T:=\Omega\times (0, T)$,}
	\end{equation}
	where
	\[
	\Delta^N_p u:= |\nabla u|^{2-p}\operatorname{div}(|\nabla u|^{p-2}\nabla u).
	\]
	Another example is the parabolic equation with the variational $p$-Laplacian:
	\begin{equation}\label{vp-parabolic0}
		\partial_t u-\Delta_p u=0 \quad \text{in $\Om_T$,}
	\end{equation}
	where
	\[
	\Delta_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u).
	\]
	We consider the associated Cauchy-Dirichlet boundary value problem for these parabolic equations. The parabolic boundary condition is given by
	\begin{equation}\label{parabolic bdry-cond}
		u=g \quad \text{on $\partial_p \Om_T$}
	\end{equation}
	for
	\[
	\partial_p \Om_T=(\Omega\times \{0\})\cup (\partial \Omega\times [0, T)),
	\]
	where $g: \partial \Omega\to {\mathbb R}$ is assumed to be continuous.

	One of the main questions addressed in this paper concerns the stability of $p$-Laplace type equations under perturbations of the parameter $p$. Assume that $H\equiv 0$. For a fixed parabolic boundary value $g$, it is known that, as $p\to q$ for some $q\in(1,\infty)$, the unique solution $u_p$ of the Cauchy-Dirichlet problem \eqref{np-parabolic0} or \eqref{vp-parabolic0} converges uniformly to the solution $u_q$ of the corresponding problem with parameter $q$. In the framework of viscosity solutions, consult \cite[Theorem~6.1]{OS} and \cite[Theorem~2.2.1]{Gbook} for such qualitative stability results on the variational and normalized $p$-Laplace equations of parabolic type. See also \cite{KP, LP} for related analysis with different approaches. However, these works do not provide an explicit rate of convergence. Our goal is therefore to further quantify the uniform convergence 	and to discuss the optimality of the resulting estimates.

	On the other hand, when studying the solutions $u$ of singular parabolic equations such as \eqref{np-parabolic} and \eqref{vp-parabolic0}, it is common to approximate them with regularized equations in the form
	\begin{equation}\label{regularized eq}
		\partial_t u-\left(|\nabla u|^2+\varepsilon^2\right)^{\frac{p'-p}{2}}\operatorname{div}\left(\left(|\nabla u|^2+\varepsilon^2\right)^{\frac{p-2}{2}}\nabla u\right)=0 \quad \text{in $\Omega_T$,}
	\end{equation}
	where $\varepsilon>0$ is small. Here, we take $p'=2$ and $p'=p$ to obtain regularized approximations of \eqref{np-parabolic} and \eqref{vp-parabolic0}, respectively.  Under the comparison principle for the limit singular equations, sending  $\varepsilon\to 0$ yields the uniform convergence of the solution $u_\varepsilon$ of \eqref{regularized eq} to the solution $u$ of the corresponding singular equation. It is then natural to ask about the rate of this convergence in applications. The case with $p'=2$ and $p=1$ corresponds to the level-set mean curvature flow equation, for which such a regularization was introduced in \cite{ES}. The convergence rate in this case was essentially established by Deckelnick \cite{De}, and later proved by Mitake \cite{Mi} using a different method based on comparison arguments. Addressing convergence rates for the general approximation \eqref{regularized eq} is another main objective of the present paper.

	The method employed in this paper builds on the arguments developed by Mitake in \cite{Mi}, as well as earlier works \cite{CGL, JK1}, which established general continuous dependence estimates for viscosity solutions of nonlinear parabolic equations. Further related results include extensions to nonlocal PDEs in \cite{JK2} and to general Neumann-type boundary value problems in \cite{JG}. While it may be possible to adapt these methods to study quantitative stability for $p$-Laplace type equations in regimes without vanishing-gradient singularities, precise estimates do not appear to have been explicitly given, to the best of our knowledge. Also, the singular case, such as \eqref{vp-parabolic0} with $1<p<2$, seems unexplored and constitutes one of the focuses of the present work.

	A recent paper of Bungert \cite{Bun} studies the convergence rate of $p$-harmonic functions to infinity-harmonic functions, using partly the variational structure of the $p$-Laplace operator. In contrast, based on the viscosity solution theory, we here provide a purely PDE approach to show the convergence rate for a general class of related parabolic stability problems. Our method develops the techniques in \cite{CGL, JK1, Mi} and further extends them to treat convergence rates arising from more general stability and regularization problems.

	As explained through the examples above, the equations of interest may exhibit singularity at vanishing gradients. To handle such singularities, we employ a generalized notion of viscosity solutions, known as $\mathcal{F}$-solutions. A brief overview of this approach is given in Section~\ref{sec:preli}; for a detailed introduction, we again refer the reader to \cite{OS, Gbook}.

	\subsection{Main results}
	Let us now introduce more details about our setting. Let $T>0$ and $\Omega\subset\mathbb{R}^n$ be a bounded domain. In the sequel, we set $\Om_T=\Omega\times (0, T)$ and $\Rn_0={\mathbb R}^n\setminus \{0\}$. Consider a family of parabolic equations
	\begin{equation}\label{general parabolic approx eq}
		\partial_t u-\operatorname{tr}(A_\varepsilon(\nabla u)\nabla^2 u)+H_\varepsilon(x, t, \nabla u)=0 \quad \text{in $\Om_T$}
	\end{equation}
	for $\varepsilon\geq 0$, where $A_\varepsilon: \Rn_0\to {\mathcal S}^n_+$ and $H_\varepsilon:\Omega_T \times {\mathbb R}^{n}\mapsto {\mathbb R}$  are continuous functions satisfying appropriate assumptions to be introduced below.

	Let $\partial_p \Om_T$ denote the parabolic boundary of $\Omega\times (0, T)$,  defined by
	\[
	\partial_p \Om_T=(\Omega\times \{0\})\cup (\partial \Omega\times [0, T)).
	\]
	We impose the initial and boundary condition 	\begin{equation}\label{parabolic approx bdry-cond}
		u =g_\varepsilon \quad \text{on $\partial_p \Om_T$}
	\end{equation}
	for \eqref{general parabolic approx eq}, where $g_\varepsilon\in C(\partial \Om_T)$ are given for $\varepsilon\geq 0$.

	Suppose that $u_\varepsilon$ (with $\varepsilon\geq 0$) is a unique viscosity solution of \eqref{general parabolic approx eq}\eqref{parabolic approx bdry-cond}. Our main result quantifies the convergence of $u_\varepsilon\to u_0$ in terms of $\varepsilon>0$ under certain assumptions on the convergence rate of $A_\varepsilon\to A_0$, $H_\varepsilon\to H_0$, and $g_\varepsilon \to g_0$ as $\varepsilon\to 0$.

	Now let us state our assumptions on $A_\varepsilon$ and $H_\varepsilon$ with $\varepsilon\geq 0$.
	Note first that for a matrix $M\in {\mathcal S}^n_+$, one can find its square root, denoted by $M^{1/2}$, which is also nonnegative. There thus exists the matrix valued function ${A_\varepsilon}^{1/2}: \Rn_0\to {\mathcal S}^n_+$ for $\varepsilon\geq 0$. We impose the following assumption on $A_\varepsilon$ and $H_\varepsilon$.
	\begin{enumerate}
		\item[(A1)] There exists $k>2$ such that for any $\varepsilon\geq 0$ small, $f(x)=|x|^k$ satisfies
		\begin{equation}\label{singular-cancel}
			\operatorname{tr} \left(A_\varepsilon(\nabla f(x)) \nabla^2 f(x)\right)\to 0 \quad \text{as $x\to 0$.}
		\end{equation}
	\end{enumerate}
	\begin{enumerate}
		\item[(A2)] $H_\varepsilon: \Om_T\times {\mathbb R}^n\to {\mathbb R}$ is continuous, and there exists $L_H\geq0$ such that
		\[
		|H_\varepsilon(x_1, t_1, \xi)-H_\varepsilon(x_2, t_2, \xi)|\leq L_H(1+|\xi|)\left(|x_1-x_2|+\left|t_1-t_2\right|\right)
		\]
		for all $(x_1, t_1),\ (x_2, t_2)\in \Omega_T$,  $\xi\in {\mathbb R}^n$ and all $\varepsilon\geq 0$.

	\end{enumerate}
	These assumptions guarantee the existence and uniqueness of ${\mathcal F}$-solutions to the associated Cauchy-Dirichlet problem. The existence of $k>2$ required in (A1) is related to the possible singularity of the equation at vanishing gradients. A typical example of singular operators we will consider in our later study of the stability is the case of the $p$-Laplacian for $p$ near a fixed $q\in (1, 2)$, where $A_\varepsilon$ is taken to be
	\begin{equation}\label{vp-operator}
		A_\varepsilon(\xi)= |\xi|^{q+\varepsilon-2} \left(I+(q+\varepsilon-2) \frac{\xi\otimes \xi}{|\xi|^2}\right)
	\end{equation}
	for $\varepsilon\geq 0$. In this case, one can choose  $k>q/(q-1)$ for $\varepsilon\geq 0$ sufficiently small.

	For general $A_\varepsilon$ and $H_\varepsilon$, we next impose assumptions on the convergence of $A_\varepsilon\to A_0$, $H_\varepsilon\to H_0$ as $\varepsilon\to 0$:
	\begin{enumerate}
		\item[(C1)] There exist $c_A>0$ and $\alpha>0, \beta>\frac{2-k}{2(k-1)}$ such that
		\begin{equation}\label{eq:c1}
			\|A_\varepsilon(\xi)^{1/2} -A_0(\xi)^{1/2}\|\leq c_A\varepsilon^{\alpha} (1+|\xi|^\beta)
		\end{equation}
		holds for all $\varepsilon>0$ small and $\xi\in \Rn_0$. Here, $k$ is the constant given in (A1).
		\item[(C2)] There exist $c_H>0$ and $\gamma>0$  such that
		\[
		|H_\varepsilon(x, t, \xi)-H_0(x, t, \xi)|\leq c_H \varepsilon^\gamma
		\]
		holds for all $\varepsilon>0$ small and $(x, t)\in \Om_T,  \xi\in {\mathbb R}^n$.
	\end{enumerate}
	Note that by the condition $\beta>\frac{2-k}{2(k-1)}$ in (C1), we have $\beta>-1/2$ for $k>2$. It is worth remarking that one can generalize (C1) by assuming the existence of $\Sigma_\varepsilon: {\mathbb R}^n_0\to {\mathbb R}^{m\times n}$ with $m\in {\mathbb N}$ for all $\varepsilon\geq 0$ small such that $\Sigma_\varepsilon^T\Sigma_\varepsilon=A_\varepsilon$ in ${\mathbb R}^n_0$ and \eqref{eq:c1} holds for $\Sigma_\varepsilon$ in place of $A_\varepsilon^{1/2}$. For clarity and simplicity, we restrict our attention to the case $\Sigma_\varepsilon=A_\varepsilon^{1/2}$, which already covers the applications of interest.

	Let us present our main theorem.

\subsection{Motivation}
 In this paper we study the stability of a class of quasilinear parabolic partial differential equations, which takes the form
 \begin{equation}\label{general parabolic lim eq}
 \partial_t u-\operatorname{tr}(A(\nabla u)\nabla^2 u)+H(x, t, \nabla u)=0 \quad \text{in $\Omega\times (0, T)$,}
 \end{equation}
 where $\Omega$ is a bounded domain in ${\mathbb R}^n$, $T>0$ is fixed, and
 \[
 A: {\mathbb R}^n\setminus \{0\}\to {\mathcal S}^n_+, \quad H: \Omega\times (0, T)\times {\mathbb R}^n\to {\mathbb R},
 \]
 are given functions satisfying appropriate assumptions to be introduced later. Here, ${\mathcal S}^n_+$ represents the set of all nonnegative symmetric $n\times n$ matrices. A key feature of this general setting is that the elliptic operator may exhibit singularity at vanishing gradients.

\subsection{Main results}
 Let us now introduce more details about our setting. Let $T>0$ and $\Omega\subset\mathbb{R}^n$ be a bounded domain. In the sequel, we set $\Om_T=\Omega\times (0, T)$ and $\Rn_0={\mathbb R}^n\setminus \{0\}$. Consider a family of parabolic equations
 \begin{equation}\label{general parabolic approx eq}
 \partial_t u-\operatorname{tr}(A_\varepsilon(\nabla u)\nabla^2 u)+H_\varepsilon(x, t, \nabla u)=0 \quad \text{in $\Om_T$}
 \end{equation}
 for $\varepsilon\geq 0$, where $A_\varepsilon: \Rn_0\to {\mathcal S}^n_+$ and $H_\varepsilon:\Omega_T \times {\mathbb R}^{n}\mapsto {\mathbb R}$ are continuous functions satisfying appropriate assumptions to be introduced below.

Let $\partial_p \Om_T$ denote the parabolic boundary of $\Omega\times (0, T)$, defined by
 \[
 \partial_p \Om_T=(\Omega\times \{0\})\cup (\partial \Omega\times [0, T)).
 \]
 We impose the initial and boundary condition \begin{equation}\label{parabolic approx bdry-cond}
 u =g_\varepsilon \quad \text{on $\partial_p \Om_T$}
 \end{equation}
 for \eqref{general parabolic approx eq}, where $g_\varepsilon\in C(\partial \Om_T)$ are given for $\varepsilon\geq 0$.

Now let us state our assumptions on $A_\varepsilon$ and $H_\varepsilon$ with $\varepsilon\geq 0$.
 Note first that for a matrix $M\in {\mathcal S}^n_+$, one can find its square root, denoted by $M^{1/2}$, which is also nonnegative. There thus exists the matrix valued function ${A_\varepsilon}^{1/2}: \Rn_0\to {\mathcal S}^n_+$ for $\varepsilon\geq 0$. We impose the following assumption on $A_\varepsilon$ and $H_\varepsilon$.
 \begin{enumerate}
 \item[(A1)] There exists $k>2$ such that for any $\varepsilon\geq 0$ small, $f(x)=|x|^k$ satisfies
 \begin{equation}\label{singular-cancel}
 \operatorname{tr} \left(A_\varepsilon(\nabla f(x)) \nabla^2 f(x)\right)\to 0 \quad \text{as $x\to 0$.}
 \end{equation}
 \end{enumerate}
 \begin{enumerate}
 \item[(A2)] $H_\varepsilon: \Om_T\times {\mathbb R}^n\to {\mathbb R}$ is continuous, and there exists $L_H\geq0$ such that
 \[
 |H_\varepsilon(x_1, t_1, \xi)-H_\varepsilon(x_2, t_2, \xi)|\leq L_H(1+|\xi|)\left(|x_1-x_2|+\left|t_1-t_2\right|\right)
 \]
 for all $(x_1, t_1),\ (x_2, t_2)\in \Omega_T$, $\xi\in {\mathbb R}^n$ and all $\varepsilon\geq 0$.

For general $A_\varepsilon$ and $H_\varepsilon$, we next impose assumptions on the convergence of $A_\varepsilon\to A_0$, $H_\varepsilon\to H_0$ as $\varepsilon\to 0$:
 \begin{enumerate}
 \item[(C1)] There exist $c_A>0$ and $\alpha>0, \beta>\frac{2-k}{2(k-1)}$ such that
 \begin{equation}\label{eq:c1}
 \|A_\varepsilon(\xi)^{1/2} -A_0(\xi)^{1/2}\|\leq c_A\varepsilon^{\alpha} (1+|\xi|^\beta)
 \end{equation}
 holds for all $\varepsilon>0$ small and $\xi\in \Rn_0$. Here, $k$ is the constant given in (A1).
 \item[(C2)] There exist $c_H>0$ and $\gamma>0$ such that
 \[
 |H_\varepsilon(x, t, \xi)-H_0(x, t, \xi)|\leq c_H \varepsilon^\gamma
 \]
 holds for all $\varepsilon>0$ small and $(x, t)\in \Om_T, \xi\in {\mathbb R}^n$.
 \end{enumerate}
 Note that by the condition $\beta>\frac{2-k}{2(k-1)}$ in (C1), we have $\beta>-1/2$ for $k>2$. It is worth remarking that one can generalize (C1) by assuming the existence of $\Sigma_\varepsilon: {\mathbb R}^n_0\to {\mathbb R}^{m\times n}$ with $m\in {\mathbb N}$ for all $\varepsilon\geq 0$ small such that $\Sigma_\varepsilon^T\Sigma_\varepsilon=A_\varepsilon$ in ${\mathbb R}^n_0$ and \eqref{eq:c1} holds for $\Sigma_\varepsilon$ in place of $A_\varepsilon^{1/2}$. For clarity and simplicity, we restrict our attention to the case $\Sigma_\varepsilon=A_\varepsilon^{1/2}$, which already covers the applications of interest.

Let us present our main theorem.

\begin{equation}\label{eq:c1}
			\|A_\vep(\xi)^{1/2} -A_0(\xi)^{1/2}\|\leq c_A\vep^{\alpha} (1+|\xi|^\beta)
		\end{equation}

\begin{equation}\label{general parabolic approx eq}
		\partial_t u-\tr(A_\vep(\nabla u)\nabla^2 u)+H_\vep(x, t, \nabla u)=0 \quad \text{in $\Om_T$}
	\end{equation}

\begin{equation}\label{parabolic approx bdry-cond}
		u =g_\vep \quad \text{on $\partial_p \Om_T$}
	\end{equation}

Let us present our main theorem.

Our proof of Theorem~\ref{thm:parabolic} develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\"older regularity of solutions into the estimates. Without effecting these estimates, this result can be readily adapted to the spatially periodic setting by simply taking $\Omega$ to be a torus; see Remark~\ref{rmk:periodic} for more details. Moreover, the assumption of H\"older continuity in time in \eqref{equi-holder} can be removed when vanishing gradients induce only a mild singularity of the equation. Such an improvement follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.

Our quantitative stability result for the elliptic equation is as below. For the sake of completeness, we also include a proof, which to a large extent resembles that of Theorem~\ref{thm:parabolic}.
 \newpage
 \begin{thm}[Quantitative stability for elliptic equations]\label{thm:elliptic}
 Let $\Omega$ be a bounded domain in $\R^n$. For $\vep\geq 0$, assume that $A_\vep: \R^n_0\to \S_+^n$ and $H_\vep: \Omega\times \R\times \R^n\to \R$ are continuous functions satisfying (A1)(A2')(A3) and (C1)(C2'). Let $g_\vep\in C(\partial\Omega)$ and $u_\vep\in C(\Oba)$ be $\F$-solutions of \eqref{general approx eq}\eqref{approx bdry-cond}. Assume in addition that $u_\vep$ are equi-H\"older continuous in $\Oba$, that is, there exist $L>0$ and $0<\theta\leq 1$ such that \[
 |u_\vep(x_1)-u_\vep(x_2)|\leq L|x_1-x_2|^\theta
 \]
 for all $x_1, x_2\in \Oba$ and $\vep\geq 0$.
 Then, for all $\vep>0$ small,
 \begin{equation}\label{error-est}
 \max_{\Oba} |u_\vep-u_0|\leq \max_{\partial \Omega} |g_\vep-g|+\frac{1}{\lambda}\left(C_0\vep^{\nu}+c_H \vep^\gamma \right)
 \end{equation}
 holds with $\nu$ given as in \eqref{nu-exponent}, where $C_0>0$ is a constant depending on
 $k>2$,$L_A>0$, $L_H>0$, $\lambda>0$ in (A1)(A2')(A3) and $\alpha, \beta, c_A, \in \R$ given in (C1). The constants $\gamma, c_H>0$ are given as in (C2').
 \end{thm}

\begin{thm}\label{thm:p-lap1par}
 Let $\Omega\subset \R^n$ be a bounded domain and $T>0$. Assume that $H: \Omega\times (0,T)\times \R^n\to \R$ satisfies (A2). Fix $q\in [1, \infty)$ and take $p\in [1, \infty)$ close to $q$. For any such $p$, let $u_p\in C(\Om_T)$ be a solution to \eqref{np-parabolic} satisfying \eqref{parabolic bdry-cond} with with boundary data $g=g_p\in C(\partial_p \Om_T)$. Assume in addition that there exist $L>0$ and $\theta\in (0, 1]$ such that
 \[
 |u_p(x_1, t)-u_p(x_2, t)|\leq L|x_1-x_2|^\theta
 \]
 for all $x_1, x_2\in \overline{\Omega}$, $0\leq t<T$ and all $p$ near $q$. Then, there exists $C>0$ independent of $p$, such that
 \begin{equation}\label{rate:p-lap1par}
 \sup_{\overline{\Omega}\times[0,T)}|u_p-u_q|\leq \sup_{\partial_p \Om_T}|g_p-g_q| +C|p-q|^{\theta}.
 \end{equation}
\end{thm}
\begin{proof}
 Note that for the normalized $p$-Laplacian, (A1) holds with $k>0$ large for all $q$ near $p$.
 Moreover, by a straightforward computation, we have
 \[
 A_\eps(\xi)^{1/2}=I+(\sqrt{p-1}-1) \frac{\xi\otimes \xi}{|\xi|^{2}}, \quad A_0(\xi)^{1/2}=I+(\sqrt{q-1}-1) \frac{\xi\otimes \xi}{|\xi|^{2}}
 \]
 with $p, q\in (1, \infty)$ satisfying $\vep=|p-q|$.
 This yields the existence of $c>0$ such that
 \[
 \norm{A_\vep(\xi)^{1/2} -A_0(\xi)^{1/2}}\leq |\sqrt{p-1}-\sqrt{q-1}|\leq c|p-q|
 \]
 for all $q$ near $p$. This amounts to saying that (C1) holds with $\alpha=1$ and $\beta=0$. We thus complete the proof of inequality \eqref{rate:p-lap1par} by applying Theorem~\ref{thm:parabolic2} with $c_H=0$ and $\nu=\theta$.
\end{proof}

\begin{thm}\label{thm:p-lap2par}
 Let $\Omega\subset \R^n$ be a bounded domain and $T>0$. Assume that $H: \Omega\times (0,T)\times \R^n\to \R$ satisfies (A2). Fix $q\in (1, \infty)$ and take $p\in (1, \infty)$ close to $q$. For any such $p$, let $u_p\in C(\Om_T)$ be a solution to \eqref{vp-parabolic} satisfying \eqref{parabolic bdry-cond} with $g=g_p\in C(\partial_p \Om_T)$. If for all $p$ close to $q$, $u_p$ are equi-H\"older continuous in $\overline{\Om_T}$ with exponent $\theta\in (0, 1]$, then the following results hold:
 \begin{enumerate}
 \item If $p<2$ and $1<q\leq 2$, then for any $-1/2<\beta<p/2-1$, there exists $C>0$ independent of $p$ near $q$ such that
 \begin{equation}\label{rate:p-lap2par-new}
 \sup_{\overline{\Omega}\times[0,T)}|u_p-u_q|\leq \sup_{\partial_p\Omega_T}|g_p-g_q| +C|p-q|^{\theta}.
 \end{equation}
 \item If $p>2$ and $q\geq 2$, then for any $\beta>q/2-1$, there exists $C>0$ independent of $q$ near $p$ such that
 \begin{equation}\label{rate:p-lap2par}
 \sup_{\overline{\Omega}\times[0,T)}|u_p-u_q|\leq \sup_{\partial_p\Omega_T}|g_p-g_q| +C|p-q|^{\frac{\theta}{1+(1-\theta)\beta}}.
 \end{equation}
 \end{enumerate}
\end{thm}
As explained in Remark~\ref{rmk holder}, for the case (2), one may relax the regularity of $u_\vep$, assuming that they are equi-H\"older continuous only in $x$.

\begin{thm}\label{thm:pq-lap1par}
 Let $\Omega\subset \R^n$ be a bounded domain and $T>0$. Fix $(q, q')\in (1, \infty)^2$ and take $(p, p')$ close to $(q, q')$. For any such $(p, p')$, let $u_{p,p'}\in C(\Om_T)$ be a solution to \eqref{eq:pq-parabolic} satisfying \eqref{parabolic bdry-cond} with $g=g_{p,p'}\in C(\partial_p \Om_T)$. Assume in addition that for all such $(p, p')$, $u_{p,p'}$ are equi-H\"older continuous in $\overline{\Om_T}$ with exponent $\theta\in (0, 1]$. Let $\beta>-1/2$ be an arbitrary value satisfying \eqref{beta-choice}. Then the following results hold:
 \begin{enumerate}
 \item For $p'> 2$ and $q'\geq 2$, there exists $C>0$ independent of $(p, p')$ such that
 \begin{equation}\label{rate:pq-lap1par}
 \sup_{\ol{\Omega}\times[0,T)}|u_{p,p'}-u_{q,q'}|\leq \sup_{\partial_p\Omega_T}|g_{p,p'}-g_{q,q'}| +C(|p-q|+|p'-q'|)^{\theta}.
 \end{equation}
 \item For $1<p'< 2$ and $1<q'\leq 2$, there exists $C>0$ independent of $(p, p')$ such that
 \begin{equation}\label{rate:pq-lap1par2}
 \sup_{\ol{\Omega}\times[0,T)}|u_{p,p'}-u_{q,q'}|\leq \sup_{\partial_p\Omega_T}|g_{p,p'}-g_{q,q'}| +C(|p-q|+|p'-q'|)^{\frac{\theta}{1+(1-\theta)\beta}}.
 \end{equation}
 \item In the special case that $p'= q'\in (1, \infty)$, there exists $C>0$ independent of $q$ such that
 \begin{equation}\label{rate:pq-lap2par}
 \sup_{\ol{\Omega}\times[0,T)}|u_{p,p'}-u_{q,p'}|\leq \sup_{\partial_p\Omega_T}|g_{p,p'}-g_{q,p'}| +C|p-q|^{\nu},
 \end{equation}
 holds with
 \begin{equation}\label{nu-pq}
 \nu=\frac{2\theta}{2\theta+(1-\theta){q'}}.
 \end{equation}
 \end{enumerate}
\end{thm}
In view of Remark \ref{rmk holder} and Theorem \ref{thm:parabolic2}, in the case $p'> 2$, $q'\geq 2$, we can drop the equi-H\"older continuity of $u_{p, p'}$ with respect to the time variable and only keep it for the space variable.

As for (C2), we have
\begin{equation*}
 \abs{H_\vep(x, t, \xi)-H_0(x, t, \xi)}=\abs{a\abs{\xi} -a\sqrt{\abs{\xi}^2+\vep_2^2}}\leq \abs{a}\vep_2,
\end{equation*}
so it holds with $\gamma=1$. We can now use Theorem \ref{thm:parabolic} to obtain the following result.
\begin{thm}
 Let $\Omega\subset \R^n$ be a bounded domain and $T>0$. For $\eps=(\vep_1, \vep_2)$ with $\vep_1, \vep_2\geq0$, let $u_{\eps}\in C(\Om_T)$ be a solution to \eqref{eq:biasedreg} with boundary value $u_\vep=g_{\eps}\in C(\partial_p \Om_T)$. Let $u_0\in C(\Om_T)$ be a solution to \eqref{eq:biased} satisfying $u_0=g_0\in C(\partial_p \Om_T)$. Suppose that $u_{\eps}$ are equi-H\"older continuous in $\overline{\Om_T}$ with exponent $\theta\in (0, 1]$ for all $\vep_1, \vep_2$ small. Then, there exists $C>0$ such that
 \begin{equation*}
 \sup_{\ol{\Om}\times[0,T)}|u_{\eps}-u_0|\leq \sup_{\partial_p\Omega_T}|g_{\eps}-g_0| +C\eps_1^{\alpha\theta}+C\vep_2
 \end{equation*}
 for any $\alpha\in (0, 1/2)$.
\end{thm}

This result applies to important quasilinear parabolic equations with vanishing gradient singularities, for which typically (C1) holds with $\beta<0$. While the parameter $\beta<0$ represents the singularity strength, the term $\max\{\beta, 0\}$ in \eqref{nu-exponent} indicates that our estimate \eqref{error-est parabolic} does not capture any influence of this singularity. Moreover, for equi-Lipschitz solutions, corresponding to the case $\theta=1$, we always have $\nu=\alpha$, and the behavior of $A_\varepsilon(\xi)$ with respect to the gradient variable $\xi$ essentially plays no decisive role in determining the stability estimate.

	Our proof of Theorem~\ref{thm:parabolic} develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\"older regularity of solutions into the estimates. Without effecting these estimates, this result can be readily adapted to the spatially periodic setting by simply taking $\Omega$ to be a torus; see Remark~\ref{rmk:periodic} for more details. Moreover, the assumption of H\"older continuity in time in \eqref{equi-holder} can be removed when vanishing gradients induce only a mild singularity of the equation. Such an improvement follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.

	We are also interested in the elliptic variant of such stability problems. Consider now
	\begin{equation}\label{general approx eq}
		-\operatorname{tr}(A_\varepsilon(\nabla u)\nabla^2 u)+H_\varepsilon(x, u, \nabla u)=0 \quad \text{in $\Omega$}
	\end{equation}
	with the Dirichlet boundary condition
	\begin{equation}\label{approx bdry-cond}
		u =g_\varepsilon \quad \text{on $\partial \Omega$}
	\end{equation}
	for given $g_\varepsilon\in C(\partial \Omega)$.
	While our approach to parabolic problems can be readily extended to the elliptic case under similar assumptions on $A_\varepsilon$ and $H_\varepsilon$, we need to additionally assume a monotonicity condition on $H_\varepsilon$ with respect to $u$ for all $\varepsilon\geq 0$. More precisely, we assume that there exists $\lambda>0$ such that
	\begin{equation}\label{mono-0}
		\rho\mapsto H_\varepsilon(x, \rho, \xi)-\lambda \rho \quad \text{is nondecreasing in ${\mathbb R}$. }
	\end{equation}
	We obtain the same convergence rate as in Theorem \ref{thm:parabolic}; see Theorem \ref{thm:elliptic}. However, it is not clear to us how to drop or relax the monotonicity assumption \eqref{mono-0}.

	\subsection{Applications}

	Our general framework in Theorem~\ref{thm:parabolic} yields stability estimates for a broad class of quasilinear parabolic equations including the normalized and variational $p$-Laplace equations and their further generalizations, as well as regularizing approximations for degenerate $p$-Laplace equations ($1\leq p\leq \infty$). Below we summarize our major convergence rate results obtained for these equations. See Section~\ref{sec:app} for precise statements, detailed analysis, concrete examples, and further discussions.

	\begin{itemize}
		\item (Normalized $p$-Laplacian) Let $u_p$ be a solution of \eqref{np-parabolic0}\eqref{parabolic bdry-cond} for $1\leq p<\infty$. For any fixed $1\leq q<\infty$, if $u_p$ are spatially equi-H\"older continuous with exponent $\theta\in (0, 1]$ for all $p$ near $q$, then $u_p\to u_q$ uniformly as $p\to q$ with convergence rate $O(|p-q|^\theta)$.
		\item (Variational $p$-Laplacian) Let $u_p$ be a solution of \eqref{vp-parabolic0}\eqref{parabolic bdry-cond} for $1<p<\infty$. For any fixed $1\leq q<\infty$, if $u_p$ are equi-H\"older continuous with exponent $\theta\in (0, 1]$ for all $p$ near $q$, then, depending on $p$, $u_p\to u_q$ uniformly as $p\to q$ with convergence rate
		\begin{enumerate}
			\item[(i)] $O(|p-q|^\theta)$ when $p<2$ (and hence $q\leq 2$);
			\item[(ii)] $O(|p-q|^\nu)$ with any
			\[
			0<\nu< \frac{2\theta}{(1-\theta)q+2\theta}
			\]
			when $p>2$ (and hence $q\geq 2$).
		\end{enumerate}
		\item (Regularization for generalized $p$-Laplacian) Let $p\geq 1$, $p'\geq 2$. For all $\varepsilon>0$ small, let $u_\varepsilon$ be a solution of \eqref{regularized eq}
		and satisfy $u_\varepsilon=g$ on $\partial_p \Omega_T$ for $g\in C(\partial_p\Omega_T)$. As $\varepsilon\to 0$, $u_\varepsilon\to u$ uniformly, where $u$ is the unique solution of
		\begin{equation*}
			\partial_t u-\left|\nabla u\right|^{p'-p}\operatorname{div}\left(\left|\nabla u\right|^{p-2}\nabla u\right)=0
		\end{equation*}
		satisfying the same boundary condition. If $u_\varepsilon$ is spatially equi-H\"older continuous with exponent $\theta\in (0, 1]$ for $\varepsilon>0$ small, then we obtain the rate $O(\varepsilon^\nu)$ for the uniform convergence $u_\varepsilon\to u$, where $\nu>0$ can be chosen as follows:
		\begin{enumerate}
			\item[(i)] If $p'=2$, then $0<\nu< \theta/2$;
			\item[(ii)] If $2<p'< 3$, then $\nu=(p'-2)\theta$;
			\item[(iii)] If $3\leq p'\leq 4$, then $0<\nu<\theta$;
			\item[(iv)] If $p'>4$, then
			\[
			0<\nu< \frac{\theta}{1+(1-\theta)(p'-4)}.
			\]
		\end{enumerate}
	\end{itemize}
	Note that the Hölder regularity of solutions assumed in the results above is not the focus of this paper. We refer interested readers to \cite{Do11, JS17, AP18, IJS19, Lee2025}, among many others, for related regularity estimates about these parabolic equations.

	In addition to the applications above, our results can also be applied to study parabolic quantitative stability for general $p$-Laplace type equations that cover both the normalized and variational cases, as well as a regularizing approximation for biased infinity-Laplace equations arising in the stochastic tug-of-war games. Several examples for quasilinear elliptic equations are also discussed in Section~\ref{sec:app}.

	Our main theorem applies to the vanishing viscosity limit for Hamilton-Jacobi equations as well. For $\varepsilon\geq 0$ small, consider the solution $u_\varepsilon$ to the initial boundary value problem for the Hamilton-Jacobi equation
	\[
	\partial_t u -\varepsilon \Delta u+H(x, t, \nabla u)=0 \quad \text{in $\Omega_T$,}
	\]
	where the Hamiltonian $H$ is assumed to satisfy the same regularity assumption as in (A2). Then, for fixed initial and boundary data independent of $\varepsilon$, Theorem~\ref{thm:parabolic} immediately yields the convergence rate $\|u_\varepsilon-u_0\|_{L^\infty(\Omega_T)}=O(\varepsilon^{1/2})$ for equi-Lipschitz solutions $u_\varepsilon$ and the limit solution $u_0$ of the corresponding inviscid Hamilton-Jacobi equation. This recovers the classical result of \cite{Fl}, originally established using an approach based on differential games, and later obtained in \cite{CL1, CL2} via viscosity solution theory and in \cite{Tr} using the nonlinear adjoint method. As this application is not the main focus of the present paper, we do not pursue it further, but instead refer the reader to the aforementioned references, as well as \cite{QSTY, CD, CG} for more recent progress on improved rates in the convex Hamiltonian case.

	In this work, we do not seek general optimality of the convergence rates. We only discuss the sharpness of our estimates through some examples, including in particular the stability of spatially periodic solutions to normalized $p$-Laplace equations under perturbations of $p$. The above convergence rates for the regularization of generalized $p$-Laplacian are consistent with the estimate obtained in \cite{Mi} for Lipschitz solutions of regularized level set curvature flow equations, which corresponds to the case $p=1$, $p'=2$ in our general setting.

	\subsection*{Acknowledgments}

	The authors would like to thank Hiroyoshi Mitake and F\'elix del~Teso for helpful discussions. The work of QL was supported by JSPS Grant-in-Aid for Scientific Research, No.~22K03396.

Our proof of Theorem~\ref{thm:parabolic} \emph{“develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\"older regularity of solutions into the estimates.”} The authors also note that, when vanishing gradients induce only a mild singularity, the \emph{time} H\"older continuity in \eqref{equi-holder} can be removed: this improvement \emph{“follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.”}

Our proof of Theorem~\ref{thm:parabolic} \emph{“develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\"older regularity of solutions into the estimates.”} The authors also note that, when vanishing gradients induce only a mild singularity, the \emph{time} H\"older continuity in
\begin{equation}\label{equi-holder}
			|u_\vep(x_1, t_1)-u_\vep(x_2, t_2)|\leq L(|x_1-x_2|^\theta +|t_1-t_2|^\theta)
		\end{equation}
can be removed: this improvement \emph{“follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.”}

[Quantitative stability for parabolic equations]\label{thm:parabolic}
		Let $T>0$ and $\Omega$ be a bounded domain in ${\mathbb R}^n$. For $\varepsilon\geq 0$, assume that $A_\varepsilon: {\mathbb R}^n_0\to \S_+^n$ and ${H_\varepsilon: \Omega_T\times {\mathbb R}^n\to {\mathbb R}}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\varepsilon\in C(\partial_p \Om_T)$ and $u_\varepsilon\in C(\overline{\Omega}\times [0, T))$ be bounded ${\mathcal F}$-solutions of
		\begin{equation}\label{general parabolic approx eq}
		\partial_t u-\tr(A_\vep(\nabla u)\nabla^2 u)+H_\vep(x, t, \nabla u)=0 \quad \text{in $\Om_T$}
	\end{equation}
		and
		\begin{equation}\label{parabolic approx bdry-cond}
		u =g_\vep \quad \text{on $\partial_p \Om_T$}
	\end{equation}
		Assume in addition that $u_\varepsilon$ are equi-H\"older continuous in $\overline{\Om_T}$, that is, there exist $L>0$ and $\theta\in (0, 1]$ such that,
		\begin{equation}\label{equi-holder}
			|u_\varepsilon(x_1, t_1)-u_\varepsilon(x_2, t_2)|\leq L(|x_1-x_2|^\theta +|t_1-t_2|^\theta)
		\end{equation}
		for all $(x_1, t_1), (x_2, t_2)\in \overline{\Omega}\times [0, T)$ and $\varepsilon\geq 0$ small.
		Then, for all $\varepsilon>0$ small,
		\begin{equation}\label{error-est parabolic}
			\sup_{\overline{\Omega}\times [0, T)}|u_\varepsilon-u_0|\leq \sup_{\partial_p \Om_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}  +c_H T\varepsilon^\gamma,
		\end{equation}
		holds with
		\begin{equation}\label{nu-exponent}
			\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta, 0\}},
		\end{equation}
		where $C>0$ depends on $T, n, k, L_A, L_H, \beta, c_A$ as well as the H\"older exponent $\theta$ and the constant $L$. Here, $k>2$, $L_A\geq0$, $L_H\geq0$ appear in (A1)(A2), and $\alpha, \beta, \gamma, c_A, c_H\in {\mathbb R}$ are given in (C1)(C2).

Question:
Let \(T>0\) and let \(\Omega\subset \mathbb{R}^n\) be a bounded domain. Write \(\Omega_T=\Omega\times(0,T)\), \(\mathbb{R}^n_0=\mathbb{R}^n\setminus\{0\}\), and \(\partial_p\Omega_T=(\Omega\times\{0\})\cup(\partial\Omega\times[0,T))\). For each \(\varepsilon\ge 0\), let \(A_\varepsilon:\mathbb{R}^n_0\to \mathcal S_+^n\) and \(H_\varepsilon:\Omega_T\times\mathbb{R}^n\to\mathbb{R}\) be continuous, where \(\mathcal S_+^n\) is the set of nonnegative symmetric \(n\times n\) matrices, and assume the following: (i) there exists \(k>2\) such that, for \(f(x)=|x|^k\), one has \(\operatorname{tr}(A_\varepsilon(\nabla f(x))\nabla^2 f(x))\to 0\) as \(x\to 0\) for all sufficiently small \(\varepsilon\ge 0\); (ii) there exists \(L_H\ge 0\) such that \(|H_\varepsilon(x_1,t_1,\xi)-H_\varepsilon(x_2,t_2,\xi)|\le L_H(1+|\xi|)(|x_1-x_2|+|t_1-t_2|)\) for all \((x_1,t_1),(x_2,t_2)\in\Omega_T\), \(\xi\in\mathbb{R}^n\), and all \(\varepsilon\ge 0\); (iii) there exist \(c_A>0\), \(\alpha>0\), and \(\beta>\frac{2-k}{2(k-1)}\) such that, with \(A_\varepsilon(\xi)^{1/2}\) the nonnegative square root of \(A_\varepsilon(\xi)\), \(\|A_\varepsilon(\xi)^{1/2}-A_0(\xi)^{1/2}\|\le c_A\varepsilon^\alpha(1+|\xi|^\beta)\) for all \(\xi\in\mathbb{R}^n_0\) and all sufficiently small \(\varepsilon>0\); and (iv) there exist \(c_H>0\) and \(\gamma>0\) such that \(|H_\varepsilon(x,t,\xi)-H_0(x,t,\xi)|\le c_H\varepsilon^\gamma\) for all \((x,t)\in\Omega_T\), \(\xi\in\mathbb{R}^n\), and all sufficiently small \(\varepsilon>0\). For each \(\varepsilon\ge 0\), let \(g_\varepsilon\in C(\partial_p\Omega_T)\), and let \(u_\varepsilon\in C(\overline\Omega\times[0,T))\) be a bounded \(\mathcal F\)-solution of
\[
\partial_tu-\operatorname{tr}(A_\varepsilon(\nabla u)\nabla^2u)+H_\varepsilon(x,t,\nabla u)=0\quad\text{in }\Omega_T,
\]
with boundary condition \(u=g_\varepsilon\) on \(\partial_p\Omega_T\). Assume also that the family \(\{u_\varepsilon\}\) is equi-H\"older continuous on \(\overline\Omega\times[0,T)\): there exist \(L>0\) and \(\theta\in(0,1]\) such that
\[
|u_\varepsilon(x_1,t_1)-u_\varepsilon(x_2,t_2)|\le L\big(|x_1-x_2|^\theta+|t_1-t_2|^\theta\big)
\]
for all \((x_1,t_1),(x_2,t_2)\in\overline\Omega\times[0,T)\) and all sufficiently small \(\varepsilon\ge 0\). Which statement holds for every such family?

For all sufficiently small \(\varepsilon>0\), there exists a constant \(C>0\) depending only on the data in the assumptions such that
\[
\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}+c_HT\varepsilon^\gamma,
\]
where
\[
\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta,0\}}.
\]

Choices:
B.
For all sufficiently small \(\varepsilon>0\), there exists a constant \(C>0\) depending only on the data in the assumptions such that
\[
\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}+c_HT\varepsilon^\gamma,
\]
where
\[
\nu=\frac{\alpha\theta}{1+(1-\theta)\beta}.
\]
C.
For all sufficiently small \(\varepsilon>0\), there exists a constant \(C>0\) depending only on the data in the assumptions such that
\[
\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}+c_HT\varepsilon^\gamma
\]
for some exponent \(\nu>0\) depending only on \(\alpha,\beta,\theta\).
D.
For all sufficiently small \(\varepsilon>0\), there exists a constant \(C=C(\varepsilon)>0\) such that
\[
\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}+c_HT\varepsilon^\gamma,
\]
where
\[
\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta,0\}}.
\]
E.
For all sufficiently small \(\varepsilon>0\), there exists a constant \(C>0\) depending only on the data in the assumptions such that
\[
\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu+	heta}+c_HT\varepsilon^\gamma,
\]
where
\[
\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta,0\}}.
\]

positive-part treatment of beta in the exponent

explicit sharp formula for the rate exponent

uniform independence of the constant from epsilon

sharp power of epsilon produced by the comparison estimate


Question: Let $T>0$ and let $\Omega\subset \mathbb{R}^n$ be bounded. Write $\Omega_T=\Omega\times(0,T)$, $\partial_p\Omega_T=(\Omega\times\{0\})\cup(\partial\Omega\times[0,T))$, and $\mathbb{R}^n_0=\mathbb{R}^n\setminus\{0\}$. For each $\varepsilon\ge 0$, let $A_\varepsilon:\mathbb{R}^n_0\to \mathcal S_+^n$ and $H_\varepsilon:\Omega_T\times\mathbb{R}^n\to\mathbb{R}$ be continuous, and assume the following: (1) there exists $k>2$ such that, for $f(x)=|x|^k$, one has $\operatorname{tr}(A_\varepsilon(\nabla f(x))\nabla^2 f(x))\to 0$ as $x\to 0$ for all sufficiently small $\varepsilon\ge 0$; (2) there exists $L_H\ge 0$ such that for all $(x_1,t_1),(x_2,t_2)\in\Omega_T$, $\xi\in\mathbb{R}^n$, and all $\varepsilon\ge 0$, $$|H_\varepsilon(x_1,t_1,\xi)-H_\varepsilon(x_2,t_2,\xi)|\le L_H(1+|\xi|)\bigl(|x_1-x_2|+|t_1-t_2|\bigr);$$ (3) there exist $c_A>0$, $\alpha>0$, and $\beta>\frac{2-k}{2(k-1)}$ such that, for all sufficiently small $\varepsilon>0$ and all $\xi\in\mathbb{R}^n_0$, $$\|A_\varepsilon(\xi)^{1/2}-A_0(\xi)^{1/2}\|\le c_A\varepsilon^{\alpha}(1+|\xi|^{\beta}),$$ where $A_\varepsilon(\xi)^{1/2}$ denotes the nonnegative square root of $A_\varepsilon(\xi)$; and (4) there exist $c_H>0$ and $\gamma>0$ such that, for all sufficiently small $\varepsilon>0$, all $(x,t)\in\Omega_T$, and all $\xi\in\mathbb{R}^n$, $$|H_\varepsilon(x,t,\xi)-H_0(x,t,\xi)|\le c_H\varepsilon^{\gamma}.$$ Let $g_\varepsilon\in C(\partial_p\Omega_T)$, and let $u_\varepsilon\in C(\overline\Omega\times[0,T))$ be bounded $\mathcal F$-solutions of $$\partial_t u-\operatorname{tr}(A_\varepsilon(\nabla u)\nabla^2u)+H_\varepsilon(x,t,\nabla u)=0\quad\text{in }\Omega_T,$$ with boundary data $u=g_\varepsilon$ on $\partial_p\Omega_T$. Assume also that the family $(u_\varepsilon)$ is equi-H\"older on $\overline\Omega\times[0,T)$: there exist $L>0$ and $\theta\in(0,1]$ such that for all $(x_1,t_1),(x_2,t_2)\in\overline\Omega\times[0,T)$ and all sufficiently small $\varepsilon\ge 0$, $$|u_\varepsilon(x_1,t_1)-u_\varepsilon(x_2,t_2)|\le L\bigl(|x_1-x_2|^{\theta}+|t_1-t_2|^{\theta}\bigr).$$ Under these hypotheses, which quantitative estimate holds for all sufficiently small $\varepsilon>0$?

For all sufficiently small $\varepsilon>0$, $$\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}+c_H T\varepsilon^{\gamma},$$ where $$\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta,0\}},$$ and $C>0$ depends on $T,n,k,L_A,L_H,\beta,c_A,\theta,$ and $L$.

Choices:
B. For all sufficiently small $\varepsilon>0$, $$\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}+c_H T\varepsilon^{\gamma},$$ where $$\nu=\frac{\alpha\theta}{1+(1-\theta)\beta},$$ and $C>0$ depends on $T,n,k,L_A,L_H,\beta,c_A,\theta,$ and $L$.
C. For all sufficiently small $\varepsilon>0$, $$\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}+c_H T\varepsilon^{\gamma},$$ for some exponent $\nu>0$ and some constant $C>0$ depending only on the structural data and the equi-H\"older constants.
D. For all sufficiently small $\varepsilon>0$, $$\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu}+c_H T\varepsilon^{\gamma},$$ where $$\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta,0\}},$$ and one may choose $C>0$ independently of the equi-H\"older constants $L,\theta$.
E. For all sufficiently small $\varepsilon>0$, $$\sup_{\overline\Omega\times[0,T)}|u_\varepsilon-u_0|\le \sup_{\partial_p\Omega_T}|g_\varepsilon-g_0|+C\varepsilon^{\nu+\gamma},$$ where $$\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta,0\}},$$ and $C>0$ depends on $T,n,k,L_A,L_H,\beta,c_A,\theta,$ and $L$.

positive-part truncation in the denominator via $\max\{\beta,0\}$

explicit exponent formula and detailed parameter dependence of the constant

dependence of the comparison constant on the equi-H\"older modulus

separate additive contribution of the $H_\varepsilon-H_0$ error term