# On the level set version of partial uniform ellipticity and applications

Ri-Rong Yuan School of Mathematics, South China University of Technology,
Guangzhou 510641, China
Email address<EMAIL_ADDRESS>
###### Abstract

We derive level set version of partial uniform ellipticity for symmetric
concave functions. This suggests an effective approach to investigate second
order fully nonlinear equations of elliptic and parabolic type.

##  1 Introduction

Let $f$ be a smooth symmetric function defined in an open symmetric convex
cone $\Gamma\subset\mathbb{R}^{n}$ containing the positive cone

$\Gamma_{n}=\left\\{\lambda\in\mathbb{R}^{n}:\mbox{ each
}\lambda_{i}>0\right\\}\subseteq\Gamma$

with vertex at the origin and with nonempty boundary
$\partial\Gamma\neq\emptyset$. The study of fully nonlinear equations of the
form

$\displaystyle F(D^{2}u):=f(\lambda(D^{2}u))=\psi\mbox{ in
}\Omega\subset\mathbb{R}^{n}$ (1.1)

starts form the pioneering work [1] of Caffarelli-Nirenberg-Spruck. Since then
the equations of this type have been extensively studied in real and complex
variables. The following two basic hypotheses are imposed in the literature:

$f$ is a concave function in $\Gamma$, (1.2) $\displaystyle
f_{i}(\lambda):=\frac{\partial f}{\partial\lambda_{i}}(\lambda)>0\mbox{ in
}\Gamma,\,$ $\displaystyle\forall 1\leq i\leq n.$ (1.3)

In some cases one may replace (1.3) by a weaker condition

$\displaystyle f_{i}(\lambda)\geq 0\mbox{ in }\Gamma,\quad\forall 1\leq i\leq
n.$ (1.4)

As is well known, the typical examples satisfying (1.2)-(1.3) are as follows:

$f(\lambda)=\sigma_{k}^{1/k}(\lambda)\mbox{ or
}(\sigma_{k}/\sigma_{l})^{1/(k-l)}(\lambda),\mbox{ }1\leq l<k\leq n,\mbox{
}\Gamma=\Gamma_{k}$

where $\sigma_{k}$ is the $k$-th elementary symmetric function. Here

$\Gamma_{k}=\left\\{\lambda\in\mathbb{R}^{n}:\sigma_{j}(\lambda)>0,\mbox{
}\forall 1\leq j\leq k\right\\}.$

The linearlized operator of (1.1) at $u$ is given by
$\mathfrak{L}_{u}w=\frac{\partial F(D^{2}u)}{\partial u_{ij}}\cdot w_{ij}.$
One can check that the eigenvalues of $\left(\frac{\partial F}{\partial
u_{ij}}(D^{2}u)\right)$ are precisely given by

$f_{1}(\lambda),\cdots,f_{n}(\lambda)\mbox{ for }\lambda=\lambda(D^{2}u).$

In particular, for the Poisson equation corresponding to
$f(\lambda)=\sum_{i=1}^{n}\lambda_{i}$,

$f_{i}(\lambda)\equiv 1,\quad\forall\lambda\in\mathbb{R}^{n},\quad\forall
1\leq i\leq n.$

This means that (1.1) is uniformly elliptic. However, the fully nonlinear
equations analogous to (1.1) fail to be uniformly elliptic in general, which
causes various hard difficulties in the investigation, especially in proof of
a priori (interior) estimates. Consequently, it is important to compare
$f_{i}(\lambda)$ with $\sum_{j=1}^{n}f_{j}(\lambda)$. This leads to the notion
of partial uniform ellipticity.

######  Definition 1.1 (Partial uniform ellipticity).

Let $H$ be a symmetric nonempty subset of $\Gamma$. We say that $f$ is of
$\mathrm{m}$-uniform ellipticity in $H$, if (1.4) holds and there exists a
uniform positive constant $\vartheta$ such that for any $\lambda\in H$ with
$\lambda_{1}\leq\cdots\leq\lambda_{n}$,

$\displaystyle
f_{{i}}(\lambda)\geq\vartheta\sum_{j=1}^{n}f_{j}(\lambda)>0,\quad\forall 1\leq
i\leq\mathrm{m}.$ (1.5)

In particular, $n$-uniform ellipticity is also called fully uniform
ellipticity.

The author [25]***The paper [25] is essentially extracted from
[arXiv:2011.08580] and [arXiv:2101.04947]. introduced an integer
$\kappa_{\Gamma}$ for $\Gamma$

$\displaystyle\kappa_{\Gamma}=\max\left\\{k:(-\alpha_{1},\cdots,-\alpha_{k},\alpha_{k+1},\cdots,\alpha_{n})\in\Gamma,\mbox{
where }\alpha_{j}>0,\mbox{ }\forall 1\leq j\leq n\right\\}$ (1.6)

and proved that the concave symmetric functions satisfying

$\displaystyle\lim_{t\rightarrow+\infty}f(t\lambda)>f(\mu)\mbox{ for any
}\lambda,\mbox{ }\mu\in\Gamma$ (1.7)

is exactly of $(\kappa_{\Gamma}+1)$-uniform ellipticity in $\Gamma$. More
precisely, there exists a uniform positive constant depending only on $\Gamma$
such that for any $\lambda\in\Gamma$ with
$\lambda_{1}\leq\cdots\leq\lambda_{n}$,

$\displaystyle
f_{{i}}(\lambda)\geq\vartheta_{\Gamma}\sum_{j=1}^{n}f_{j}(\lambda),\quad\forall
1\leq i\leq 1+\kappa_{\Gamma}.$ (1.8)

In the case $(f,\Gamma)=(\sigma_{k}^{1/k},\Gamma_{k})$, (1.8) was proved by
Lin-Trudinger [16]. Such a partial uniform ellipticity is relevant to various
partial differential equations of elliptic and parabolic type. A surprising
consequence of the conclusion (1.8) is that a type 2 cone means in some sense
that the corresponding equations are uniformly elliptic.

However, the condition (1.7) is not fulfilled in some situations. For
instance, it does not allow

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq-K\sum_{i=1}^{n}f_{i}(\lambda),\mbox{
for some }K\geq 0,\quad\forall\underline{A}\leq f(\lambda)\leq\overline{A}.$
(1.9)

Such a condition includes

$\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq\delta>0,\quad\forall\underline{A}\leq
f(\lambda)\leq\overline{A}$ (1.10)

as a special case. These two conditions appeared in the study of certain fully
nonlinear equations from differential geometry, see e.g. [2, 11, 17, 20, 8, 9]
and the references therein. We shall remark that assumptions (1.10) and (1.9)
hold only on the range of the given function.

Motivated by this and related topics, it would be necessary to derive the
level set version of partial uniform ellipticity.

Before stating results we introduce some notions and impose appropriate
assumptions. For $\sigma$, we denote the level set by

$\partial\Gamma^{\sigma}=\\{\lambda\in\Gamma:f(\lambda)=\sigma\\}.$

Conditions (1.2) and (1.4) imply that (see Lemma A.1)

$\sum_{i=1}^{n}f_{i}(\lambda)>0\mbox{ for }f(\lambda)<\sup_{\Gamma}f.$ (1.11)

As a result, the level set $\partial\Gamma^{\sigma}$ (when
$\partial\Gamma^{\sigma}\neq\emptyset$) is a smooth complete noncompact convex
hypersurface. Throughout this paper we assume

$\sigma<\sup_{\Gamma}f\mbox{ and }\partial\Gamma^{\sigma}\neq\emptyset.$

Let’s denote

$t_{\lambda}=\frac{\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}}{\sum_{j=1}^{n}f_{j}(\lambda)},\quad\vec{\bf
1}=(1,\cdots,1)\in\mathbb{R}^{n}.$

Geometrically, the tangent plane $T_{\lambda}\partial\Gamma^{\sigma}$ of
$\partial\Gamma^{\sigma}$ at $\lambda\in\partial\Gamma^{\sigma}$, intersects
the diagonal at $t_{\lambda}\vec{\bf 1}$, i.e.

$T_{\lambda}\partial\Gamma^{\sigma}\cap\\{t\vec{\bf
1}:t\in\mathbb{R}\\}=\\{t_{\lambda}\vec{\bf 1}\\}.$

We assume $t_{\lambda}$ has lower bound

$\displaystyle\liminf_{|\lambda|\rightarrow+\infty,\lambda\in\partial\Gamma^{\sigma}}t_{\lambda}>-\infty.$
(1.12)

And then we denote

$\tau_{\sigma}=\inf_{\lambda\in\partial\Gamma^{\sigma}}t_{\lambda}.$

Let $c_{\sigma}$ be the positive constant with $f(c_{\sigma}\vec{\bf
1})=\sigma$. By (1.2), we know

$\tau_{\sigma}\leq c_{\sigma}$

with equality holding if and only if

$f_{1}(\lambda)=f_{2}(\lambda)=\cdots=f_{n}(\lambda),\quad\forall\lambda\in\partial\Gamma^{\sigma}.$

Consequently, we assume throughout this paper that

$\displaystyle\tau_{\sigma}<c_{\sigma}.$

######  Definition 1.2.

Let $\Gamma_{\sigma,f}$ denote the cone

$\Gamma_{\sigma,f}=\left\\{t(\lambda-\tau_{\sigma}\vec{\bf
1}):\lambda\in\partial\Gamma^{\sigma},t>0\right\\}.$

Furthermore
$\Gamma_{\sigma,f}(1):=\partial\Gamma^{\sigma}-\tau_{\sigma}\vec{\bf 1}$
simply denotes a slice of $\Gamma_{\sigma,f}$. For such $\Gamma_{\sigma,f}$,
we define

$\kappa_{\Gamma_{\sigma,f}}=\max\left\\{k:(-\alpha_{1},\cdots,-\alpha_{k},\alpha_{k+1},\cdots,\alpha_{n})\in\Gamma_{\sigma,f},\mbox{
}\alpha_{i}>0\right\\}.$

Below we state the results on partial uniform ellipticity.

######  Theorem 1.3.

Assume (1.2), (1.4) and (1.12) hold. Then there exists a uniform positive
constant $\vartheta_{\Gamma_{\sigma,f}}$ depending only on $\Gamma_{\sigma,f}$
such that for each $\lambda\in\partial\Gamma^{\sigma}$ with
$\lambda_{1}\leq\cdots\leq\lambda_{n}$,

$\displaystyle
f_{{i}}(\lambda)\geq\vartheta_{\Gamma_{\sigma,f}}\sum_{j=1}^{n}f_{j}(\lambda),\quad\forall
1\leq i\leq 1+\kappa_{\Gamma_{\sigma,f}}.$

######  Remark 1.4.

When replacing (1.4) and (1.12) by (1.7), Theorem 1.3 gives back (1.8).

As a consequence of Theorem 1.3, we can confirm an important inequality.

######  Theorem 1.5.

Suppose, in addition to (1.2) and (1.4), that (1.9) holds for
$\sup_{\partial\Gamma}f<\underline{A}<\overline{A}<\sup_{\Gamma}f$. Then there
is a positive constant $\theta$ depending on $K$, $\underline{A}$,
$\overline{A}$ such that

$\displaystyle
f_{i}(\lambda)\geq\theta\left(1+\sum_{j=1}^{n}f_{j}(\lambda)\right)\,\mbox{ if
}\lambda_{i}\leq-K,\mbox{ }\forall\underline{A}\leq
f(\lambda)\leq\overline{A}.$ (1.13)

In particular, if $K=0$, i.e. $\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq 0$
for $\underline{A}\leq f(\lambda)\leq\overline{A}$, then

$\displaystyle
f_{i}(\lambda)\geq\theta\left(1+\sum_{j=1}^{n}f_{j}(\lambda)\right)\mbox{ if
}\lambda_{i}\leq 0.$ (1.14)

######  Remark 1.6.

The inequality (1.14) was imposed as a vital assumption by Li [15] and later
by many experts to study certain geometric PDEs from classical differential
geometry and conformal geometry, see e.g. [20, 10, 17, 3, 4, 6, 21, 8]. Our
results can improve related results obtained there.

The paper is organized as follows. The level set version of partial uniform
ellipticity is derived in Section 2. As applications, we derive the interior
estimates for first and second order derivatives for complex fully nonlinear
equations with Laplacian terms in Section 3, and briefly discuss real Hessian
fully nonlinear equations in Section 4. In Appendixes, we summarize and prove
some lemmas.

##  2 Partial uniform ellipticity: Level set version

###  2.1 Proof of Theorem 1.3

The concavity assumption (1.2) gives

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)(\mu_{i}-\lambda_{i})\geq
f(\mu)-f(\lambda),\quad\forall\lambda,\mbox{ }\mu\in\Gamma.$ (2.1)

First we prove the following lemma.

######  Lemma 2.1.

Suppose (1.2), (1.4) and (1.12) hold. Then

$\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geq
0,\quad\forall\lambda\in\partial\Gamma^{\sigma},\mbox{
}\forall\mu\in\Gamma_{\sigma,f}.$

###### Proof.

Given $\lambda\in\partial\Gamma^{\sigma}$. Without loss of generality, we
choose $\mu\in\Gamma_{\sigma,f}(1)$. So $\mu+\tau_{\sigma}\vec{\bf
1}\in\partial\Gamma^{\sigma}$. The inequality (2.1) simply yields

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)(\tau_{\sigma}+\mu_{i}-\lambda_{i})\geq
0.$

Thus

$\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geq\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}-\tau_{\sigma}\sum_{i=1}^{n}f_{i}(\lambda)\geq
0.$

∎

Let $\lambda_{1}\leq\cdots\leq\lambda_{n}$, the concavity and symmetry of $f$
imply

$f_{1}(\lambda)\geq\cdots\geq f_{n}(\lambda)\mbox{ and
}f_{1}(\lambda)\geq\frac{1}{n}\sum_{i=1}^{n}f_{i}(\lambda).$

If $\kappa_{\Gamma_{\sigma,f}}=0$ then Theorem 1.3 clearly follows. For
$\kappa_{\Gamma_{\sigma,f}}\geq 1$, it is a consequence of the following
proposition.

######  Proposition 2.2.

In addition to (1.2), (1.4) and (1.12), we assume
$\kappa_{\Gamma_{\sigma,f}}\geq 1$. Let $\alpha_{1},\cdots,\alpha_{n}$ be $n$
positive constants with

$(-\alpha_{1},\cdots,-\alpha_{\kappa_{\Gamma_{\sigma,f}}},\alpha_{1+\kappa_{\Gamma_{\sigma,f}}},\cdots,\alpha_{n})\in\Gamma_{\sigma,f}.$

Assume in addition that
$\alpha_{1}\geq\cdots\geq\alpha_{\kappa_{\Gamma_{\sigma,f}}}$. Then for
$\lambda\in\partial\Gamma^{\sigma}$ with
$\lambda_{1}\leq\cdots\leq\lambda_{n}$,

$\displaystyle
f_{1+\kappa_{\Gamma_{\sigma,f}}}(\lambda)\geq\frac{\alpha_{1}}{\sum_{i=1+\kappa_{\Gamma_{\sigma,f}}}^{n}\alpha_{i}-\sum_{i=2}^{\kappa_{\Gamma_{\sigma,f}}}\alpha_{i}}f_{1}(\lambda).$
(2.2)

###### Proof.

According to Lemma 2.1, we have

$\displaystyle-\sum_{i=1}^{\kappa_{\Gamma_{\sigma,f}}}\alpha_{i}f_{i}(\lambda)+\sum_{i=1+\kappa_{\Gamma_{\sigma,f}}}^{n}\alpha_{i}f_{i}(\lambda)\geq
0.$

This simply yields
$f_{1+\kappa_{\Gamma_{\sigma,f}}}(\lambda)\geq\frac{\alpha_{1}}{\sum_{i=1+\kappa_{\Gamma_{\sigma,f}}}^{n}\alpha_{i}}f_{1}(\lambda)$.
In addition, one derives (2.2) by using iteration.

∎

######  Remark 2.3.

In the case $\kappa_{\Gamma_{\sigma,f}}\geq 1$, the constant
$\vartheta_{\Gamma_{\sigma,f}}$ in Theorem 1.3 can be achieved as

$\vartheta_{\Gamma_{\sigma,f}}=\sup_{(-\alpha_{1},\cdots,-\alpha_{\kappa_{\Gamma_{\sigma,f}}},\alpha_{1+\kappa_{\Gamma_{\sigma,f}}},\cdots,\alpha_{n})\in\Gamma_{\sigma,f}}\frac{\alpha_{1}/n}{\sum_{i=1+\kappa_{\Gamma_{\sigma,f}}}^{n}\alpha_{i}-\sum_{i=2}^{\kappa_{\Gamma_{\sigma,f}}}\alpha_{i}}.$

###  2.2 A new criterion for $f$ satisfying (1.7)

Building on Lemma B.1, we can deduce a new criterion for (1.7).

######  Lemma 2.4.

In the presence of (1.2), (1.4) and

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)>0\mbox{ in }\Gamma,$ (2.3)

condition (1.7) is equivalent to

$\displaystyle\Gamma\subseteq\Gamma_{\sigma,f},\quad\forall\sup_{\partial\Gamma}f<\sigma<\sup_{\Gamma}f.$
(2.4)

###### Proof.

$\Leftarrow$ Fix $\lambda$, $\mu\in\Gamma$, let $\sigma=f(\lambda)$. Since
$\Gamma\subseteq\Gamma_{\sigma,f}$, we have
$\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geq 0$ by Lemma 2.1. Thus (1.7) holds by
Lemma B.5.

$\Rightarrow$ For $\lambda\in\Gamma$ and
$\sup_{\partial\Gamma}f<\sigma<\sup_{\Gamma}f$, one has $f(t\lambda)>\sigma$
for some $t>0$. By Lemma B.1, $\tau_{\sigma}\geq 0$. There is $0<t_{0}<t$ such
that $f(t_{0}\lambda+\tau_{\sigma}\vec{\bf 1})=\sigma$. This yields

$\Gamma\subseteq\Gamma_{\sigma,f}.$

∎

The $(\kappa_{\Gamma}+1)$-uniform ellipticity as asserted in (1.8) follows as
a consequence of Theorem 1.3, Lemma 2.4 and Corollary B.2.

###  2.3 Confirming an inequality

######  Proposition 2.5.

Suppose (1.2), (1.4) and (1.12) hold. Then for
$\lambda\in\partial\Gamma^{\sigma}$,

$\displaystyle
f_{i}(\lambda)\geq\vartheta_{\Gamma_{\sigma,f}}\sum_{j=1}^{n}f_{j}(\lambda)\mbox{
whenever }\lambda_{i}\leq\tau_{\sigma}.$ (2.5)

In particular, replacing (1.12) by

$\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq 0\mbox{ in
}\partial\Gamma^{\sigma},$ (2.6)

then for any $\lambda\in\partial\Gamma^{\sigma}$ we get

$\displaystyle
f_{i}(\lambda)\geq\vartheta_{\Gamma_{\sigma,f}}\sum_{j=1}^{n}f_{j}(\lambda)\mbox{
if }\lambda_{i}\leq 0.$ (2.7)

###### Proof.

Given $\lambda\in\partial\Gamma^{\sigma}$ with
$\lambda_{1}\leq\cdots\leq\lambda_{n}$. Let
$\mu_{i}=\lambda_{i}-\tau_{\sigma}$, then
$\mu=(\mu_{1},\cdots,\mu_{n})\in\Gamma_{\sigma,f}$. By the definition of
$\kappa_{\Gamma_{\sigma,f}}$, $\mu_{1+\kappa_{\Gamma_{\sigma,f}}}\geq 0$, i.e.
$\lambda_{1+\kappa_{\Gamma_{\sigma,f}}}\geq\tau_{\sigma}$. For each
$\lambda_{i}\leq\tau_{\sigma}$, we have

$f_{i}(\lambda)\geq f_{1+\kappa_{\Gamma_{\sigma,f}}}(\lambda).$

Consequently, (2.5) follows from Theorem 1.3. ∎

It follows from (2.1) that if $f$ satisfies (1.12) then there is a positive
constant $\theta=\theta(\sigma)$ depending only on $\sigma$ such that

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\geq\theta(\sigma)\mbox{ in
}\partial\Gamma^{\sigma}.$ (2.8)

Theorem 1.5 then follows from (2.8) and Proposition 2.5.

Below we consider two special cases.

######  Lemma 2.6.

If $(f,\Gamma)$ satisfies (1.2), (1.3) and

$\displaystyle\lim_{t\rightarrow 0^{+}}f(t\vec{\bf 1})>-\infty,$ (2.9)

then we have $\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}>0$ and (1.14).

###### Proof.

According to Lemmas B.1 and B.6, we have
$\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}>0$. Theorem 1.5 then gives (1.14). ∎

######  Lemma 2.7.

Let $(f,\Gamma)$ satisfy (1.2) and

$\displaystyle f>0\mbox{ in }\Gamma,\,$ $\displaystyle f=0\mbox{ on
}\partial\Gamma$ (2.10)

then we have $\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq 0$, (1.4) and
(1.14).

###### Proof.

The conclusions (1.4) and $\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq 0$ are
deduced from Lemma B.3. Again, by Theorem 1.5 we have (1.14). ∎

######  Remark 2.8.

These two lemmas allows one to improve some results on Weingarten equations
obtained by Li [15] and Trudinger [20] respectively; we decide to omit the
details here.

##  3 Applications to complex fully nonlinear equations

Let $(M,\omega)$ be a compact Hermitian manifold of complex dimension $n\geq
2$ possibly with boundary. Let $\chi$ be a smooth real $(1,1)$-form, $\psi$ a
$C^{2}$-smooth function, $\Delta$ the Laplacian operator,

$Z=\frac{1}{(n-1)!}*\mathfrak{Re}(\sqrt{-1}\partial
u\wedge\overline{\partial}\omega^{n-2}),$

where $*$ is the Hodge star operator with respect to $\omega$. Recently,
Székelyhidi-Tosatti-Weinkove [18] proved Gauduchon’s conjecture, by solving
the Monge-Ampère equation for $(n-1)$-PSH functions on a closed Hermitian
manifold

$\displaystyle\left(\omega_{0}+\frac{1}{n-1}\left(\Delta
u\omega-\sqrt{-1}\partial\overline{\partial}u\right)+Z\right)^{n}=e^{(n-1)\phi}\omega^{n},\quad\omega_{0}>0.$
(3.1)

When $M$ admits a balanced metric and an astheno-Kähler metric, it closely
connects to the Form-type Calabi-Yau equation [5]. Subsequently, the author
[24]†††It is essentially extracted from the second parts of [arXiv:2001.09238]
and [arXiv:2106.14837]. solved the Dirichlet problem for (3.1), thereby
extending Székelyhidi-Tosatti-Weinkove’s results to complex manifolds with
boundary. In addition, the author [24] has investigated the Dirichlet problem
for equations of the form

$\displaystyle f\left(\lambda\left(\chi+\frac{1}{n-1}\left(\Delta
u\omega-\sqrt{-1}\partial\overline{\partial}u\right)+Z\right)\right)=\psi$
(3.2)

where $\chi$ is a smooth real $(1,1)$-form, $\lambda(A)$ are the eigenvalues
of $A$ with respect to $\omega$. More precisely, when imposing (1.7),
$\Gamma=\Gamma_{n}$ and the unbound condition

$\displaystyle\lim_{t\to+\infty}f(\lambda_{1}+t,\cdots,\lambda_{n-1}+t,\lambda_{n})=\sup_{\Gamma}f,\quad\forall\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\Gamma,$
(3.3)

the Dirichlet problem for (3.2) was solved by the author in [24, Section 7].
The assumption (3.3) allows

$f=({\sigma_{n}}/{\sigma_{k}})^{1/(n-k)},\quad 0\leq k\leq n-2.$

While $\Gamma\neq\Gamma_{n}$, even without assuming (3.3) as well with
degenerate right-hand side, the Dirichlet problem for (3.2) was completely
solved there. These results reveal that there are significant differences
between $\Gamma=\Gamma_{n}$ and $\Gamma\neq\Gamma_{n}$. Based on the partial
uniform ellipticity, we are able to figure out those differences.

######  Proposition 3.1.

Given $(f,\Gamma)$, we define

$\displaystyle\tilde{f}(\lambda)={f}(\mu),\quad\mu_{i}=\sum_{j\neq
i}\lambda_{j},\quad\tilde{\Gamma}=\\{\lambda\in\mathbb{R}^{n}:\mu\in\Gamma\\}.$
(3.4)

In the presence of (1.2) and (1.7), we have the following:

  * $\mathrm{\bf(1)}$

If $\Gamma\neq\Gamma_{n}$ holds, then $\tilde{\Gamma}$ is of type 2 cone and
$\tilde{f}$ is of fully uniform ellipticity in $\tilde{\Gamma}$.

  * $\mathrm{\bf(2)}$

If $\Gamma=\Gamma_{n}$, then $\tilde{f}$ is of $(n-1)$-uniform ellipticity in
$\tilde{\Gamma}$.

The proposition shows that in the presence of (1.2) and (1.7), equation (3.2)
is uniformly elliptic in the case $\Gamma\neq\Gamma_{n}$, while it is only of
$(n-1)$-uniform ellipticity when $\Gamma=\Gamma_{n}$. Such differences
motivate us to derive more delicate results. We apply level set version of
partial uniform ellipticity to derive interior estimates for (3.2), when
imposing proper restrictions:

$\displaystyle\Gamma\neq\Gamma_{n},$ (3.5)
$\displaystyle\lim_{t\to+\infty}f(t,\cdots,t,0)>\sup_{M}\psi,$ (3.6)
$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq 0\mbox{ in
}\Gamma^{\underline{\psi},\overline{\psi}}$ (3.7)

where

$\displaystyle\Gamma^{\underline{\psi},\overline{\psi}}=\left\\{\lambda:\inf_{M}\psi\leq
f(\lambda)\leq\sup_{M}\psi\right\\}.$ (3.8)

######  Theorem 3.2.

Let $B_{r}$ be a geodesic ball in $(M,\omega)$. Suppose (1.2), (1.4), (3.5),
(3.6) and (3.7) hold. Then for any solution $u\in C^{4}(B_{r})$ to equation
(3.2) satisfying

$\displaystyle\lambda\left(\chi+\frac{1}{n-1}\left(\Delta
u\omega-\sqrt{-1}\partial\overline{\partial}u\right)+Z\right)\in\Gamma\mbox{
in }B_{r},$ (3.9)

there is a uniform positive constant $C$ depending only on
$|u|_{C^{0}(B_{r})}$, $|\psi|_{C^{2}(B_{r})}$ and geometric quantities on
$B_{r}$, such that

$\displaystyle\sup_{B_{r/2}}(|\nabla
u|^{2}+|\partial\overline{\partial}u|)\leq\frac{C}{r^{2}}.$ (3.10)

According to Lemma B.1 and Corollary B.2 below, (1.4) and (3.7) are
simultaneously satisfied when $f$ satisfies (1.2) and (1.7). As a consequence,
we obtain

######  Corollary 3.3.

Theorem 3.2 holds when $(f,\Gamma)$ satisfies (1.2), (1.7) and (3.5).

In fact interior estimate (3.10) still holds for equations

$\displaystyle f\left(\lambda\left(\chi+\Delta
u\omega-\varrho\sqrt{-1}\partial\overline{\partial}u+\gamma
Z\right)\right)=\psi$ (3.11)

provided that $(f,\Gamma)$ satisfies (1.2) and (1.7), $\gamma$ is a
$C^{2}$-smooth function, and $\varrho$ is a $C^{2}$-smooth function satisfying

$\displaystyle\varrho<\frac{1}{1-\kappa_{\Gamma}\vartheta_{\Gamma}}\mbox{ and
}\varrho\neq 0.$ (3.12)

Here $\kappa_{\Gamma}$ and $\theta_{\Gamma}$ are the constants in (1.6) and
(1.8), respectively.

######  Theorem 3.4.

Let $B_{r}$ be a geodesic ball in $(M,\omega)$, and let $u\in C^{4}(B_{r})$ be
a solution to (3.11) in $B_{r}$ with

$\displaystyle\lambda\left(\chi+\Delta
u\omega-\varrho\sqrt{-1}\partial\overline{\partial}u+\gamma
Z\right)\in\Gamma\mbox{ in }B_{r}.$

Suppose in addition that (1.2), (1.7) and (3.12) hold. Then

$\displaystyle\sup_{B_{r/2}}(|\nabla
u|^{2}+|\partial\overline{\partial}u|)\leq\frac{C}{r^{2}}$

where $C$ depends only on $|u|_{C^{0}(B_{r})}$, $|\psi|_{C^{2}(B_{r})}$ and
geometric quantities on $B_{r}$.

The restriction (3.12) to parameter $\varrho$ was imposed by the author [25]
to study conformal deformation of modified Schouten tensors. When
$\Gamma=\Gamma_{n}$ it reduces to

$\varrho<1\mbox{ and }\varrho\neq 0.$ (3.13)

Hence it does not cover the Monge-Ampère equation for $(n-1)$-PSH function.
The equations analogous to (3.11) were also studied in [12] under the
assumption (3.13). Notice (3.12) allows the critical case $\varrho=1$ when
$\Gamma\neq\Gamma_{n}$. Our result in Theorem 3.4 is new.

######  Remark 3.5.

The main difficulty in Székelyhidi-Tosatti-Weinkove’s proof of second estimate
is to deal with the bad terms due to the gradient terms $\partial u$,
$\overline{\partial}u$ from $Z$. Their proof depends heavily on delicate
structures of $Z$, which cannot be extended to more general cases. In contrast
with Székelyhidi-Tosatti-Weinkove’s estimates, our results assert the interior
estimates for second and first order derivatives when $\Gamma\neq\Gamma_{n}$.
In fact, such interior estimates are not true for general complex fully
nonlinear equations.

###  3.1 Interior estimates for equations of fully uniform ellipticity

In this subsection we are concerned with an equation of the form

$\displaystyle
F(\mathfrak{g}_{i\bar{j}}):=f(\lambda(\mathfrak{g}_{i\bar{j}}))=\psi$ (3.14)

where
$\mathfrak{g}_{i\bar{j}}=u_{i\bar{j}}+\chi_{i\bar{j}}+S_{i\bar{j}}^{k}u_{k}+\overline{S_{j\bar{i}}^{k}}u_{\bar{k}}$,
$\lambda(\mathfrak{g}_{i\bar{j}})$ denote the eigenvalues of
$\mathfrak{g}_{i\bar{j}}$ with respect to $g_{i\bar{j}}$. We call $u$ an
admissible function if $\lambda(\mathfrak{g}_{i\bar{j}})\in\Gamma$. In
addition, we assume that there exists a positive constant $\theta$ such that

$\displaystyle f_{i}(\lambda)\geq\theta\sum_{j=1}^{n}f_{j}(\lambda)\mbox{ in
}\Gamma^{\underline{\psi},\overline{\psi}},\quad\forall 1\leq i\leq n.$ (3.15)

We prove the following interior estimates.

######  Theorem 3.6.

Let $B_{r}$ be a geodesic ball in $(M,\omega)$. Suppose (1.2), (1.4), (3.6)
and (3.15) hold. Then for any admissible solution $u\in C^{4}(B_{r})$ to
(3.14) in $B_{r}$, we have

$\displaystyle\sup_{B_{r/2}}(|\partial\overline{\partial}u|+|\nabla
u|^{2})\leq\frac{C}{r^{2}}$

where $C$ is a uniform constant depending only on $\theta^{-1}$,
$|u|_{C^{0}(B_{r})}$, $|\psi|_{C^{2}(B_{r})}$ and geometric quantities on
$B_{r}$.

####  3.1.1 Preliminaries

The linearized operator of equation (3.14), say $\mathcal{L}$, at solution $u$
is locally given by

$\mathcal{L}v=F^{i\bar{j}}v_{i\bar{j}}+F^{i\bar{j}}S_{i\bar{j}}^{k}v_{k}+F^{i\bar{j}}\overline{S_{j\bar{i}}^{k}}v_{\bar{k}}$
(3.16)

where $F^{i\bar{j}}=\frac{\partial F}{\partial\mathfrak{g}_{i\bar{j}}}$. One
knows that the eigenvalues of $F^{i\bar{j}}$ (w.r.t. $g^{i\bar{j}}$) are
precisely

$f_{1}(\lambda),\cdots,f_{n}(\lambda),\quad\mbox{ where
}\lambda=\lambda(\mathfrak{g}).$

Moreover

$F^{i\bar{j}}\mathfrak{g}_{i\bar{j}}=\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i},\quad\sum
F^{i\bar{j}}g_{i\bar{j}}=\sum_{i=1}^{n}f_{i}(\lambda).$

From condition (3.6), the right-hand side must satisfy

$\sup_{M}\psi<\sup_{\Gamma}f.$

Combining with Lemma A.1 below,

$\displaystyle\sum_{i,j=1}^{n}F^{i\bar{j}}{g}_{i\bar{j}}=\sum_{i=1}^{n}f_{i}(\lambda)>0.$
(3.17)

######  Remark 3.7.

The condition (3.15) yields that equation (3.14) is in effect uniformly
elliptic at admissible solution $u$. That is automatically satisfied when
imposed the same conditions as that of Theorem 3.2 or Corollary 3.3.

The following formulas are standard

$\displaystyle u_{1\bar{j}k}-u_{k\bar{j}1}=$ $\displaystyle
T^{l}_{1k}u_{l\bar{j}},$ (3.18) $\displaystyle
u_{1\bar{1}i\bar{i}}-u_{i\bar{i}1\bar{1}}=R_{i\bar{i}1\bar{p}}u_{p\bar{1}}-R_{1\bar{1}i\bar{p}}u_{p\bar{i}}$
$\displaystyle+2\mathfrak{Re}\\{\bar{T}^{j}_{1i}u_{i\bar{j}1}\\}+T^{p}_{i1}\bar{T}^{q}_{i1}u_{p\bar{q}}.$

Denote

$w=|\nabla u|^{2}\mbox{ and
}Q=|\partial\overline{\partial}u|^{2}+|\partial\partial u|^{2}.$

Under local coordinates $z=(z_{1},\cdots,z_{n})$ around $z_{0}$, with
$g_{i\bar{j}}(z_{0})=\delta_{ij}$, we have by straightforward computations

$w_{i}=u_{\bar{k}}u_{ki}+u_{k}u_{i\bar{k}},$
$w_{i\bar{j}}=u_{ki}u_{\bar{k}\bar{j}}+u_{k\bar{j}}u_{i\bar{k}}+u_{\bar{k}}u_{i\bar{j}k}+u_{k}u_{i\bar{j}\bar{k}}+R_{i\bar{j}k\bar{l}}u_{\bar{k}}u_{l}-T^{l}_{ik}u_{l\bar{j}}u_{\bar{k}}-\overline{T^{l}_{jk}}u_{i\bar{l}}u_{k}.$

One then obtains

######  Lemma 3.8.

We have

$F^{i\bar{j}}w_{i}w_{\bar{j}}\leq 2wQ\sum F^{i\bar{i}};$

and there exists $C>0$ such that

$\displaystyle\mathcal{L}(w)\geq\frac{3\theta Q}{4}\sum F^{ii}-Cw\sum
F^{i\bar{i}}-C|\nabla\psi|\sqrt{w}.$

####  3.1.2 Interior estimate for first derivative

Let’s consider the quantity

$m_{0}=\max_{\bar{M}}\eta|\nabla u|^{2}e^{\phi}$

where $\eta\geq 0$ and $\phi=\phi(z,u)$ are functions to be determined.
Suppose that $m_{0}$ is attained at an interior point $z_{0}\in M$. We choose
local coordinates $(z_{1},\ldots,z_{n})$ such that $g_{i\bar{j}}=\delta_{ij}$
at $z_{0}$. As above we denote $w=|\nabla u|^{2}$. Without loss of generality,
$w\geq 1$ at $z_{0}$. From above, the function $\log\eta+\log w+\phi$ achieves
a maximum at $z_{0}$ and therefore,

$\frac{\eta_{i}}{\eta}+\frac{w_{i}}{w}+\phi_{i}=0,\;\;\frac{\eta_{\bar{i}}}{\eta}+\frac{w_{\bar{i}}}{w}+\phi_{\bar{i}}=0,$
(3.19) $\mathcal{L}(\log\eta+\log w+\phi)\leq 0.$ (3.20)

Combining (3.19) with Cauchy-Schwarz inequality, we derive

$\frac{1}{w^{2}}F^{i\bar{j}}w_{i}w_{\bar{j}}\leq\frac{1+\epsilon}{\epsilon\eta^{2}}F^{i\bar{j}}\eta_{i}\eta_{\bar{j}}+(1+\epsilon)F^{i\bar{j}}\phi_{i}\phi_{\bar{j}}.$
(3.21)

As a result, combining with Lemma 3.8 and let $8\epsilon\leq\theta$, we derive
at $z_{0}$

$\displaystyle\mathcal{L}\log w\geq$
$\displaystyle\frac{(3\theta-8\epsilon)Q}{4w}\sum F^{i\bar{i}}-C\sum
F^{i\bar{i}}-\frac{C|\nabla\psi|}{\sqrt{w}}$ (3.22)
$\displaystyle-\frac{1-\epsilon^{2}}{\epsilon\eta^{2}}F^{i\bar{j}}\eta_{i}\eta_{\bar{j}}-(1-\epsilon^{2})F^{i\bar{j}}\phi_{i}\phi_{\bar{j}}$
$\displaystyle\geq$ $\displaystyle\frac{\theta Q}{2w}\sum F^{i\bar{i}}-C\sum
F^{i\bar{i}}-\frac{C|\nabla\psi|}{\sqrt{w}}-\frac{1}{\epsilon\eta^{2}}F^{i\bar{j}}\eta_{i}\eta_{\bar{j}}-F^{i\bar{j}}\phi_{i}\phi_{\bar{j}}.$

On the other hand,

$\displaystyle\mathcal{L}\log\eta=\frac{F^{i\bar{j}}\eta_{i\bar{j}}}{\eta}+F^{i\bar{j}}S_{i\bar{j}}^{k}\frac{\eta_{k}}{\eta}+F^{i\bar{j}}\overline{S_{j\bar{i}}^{k}}\frac{\eta_{\bar{k}}}{\eta}-F^{i\bar{j}}\frac{\eta_{i}\eta_{\bar{j}}}{\eta^{2}}.$
(3.23)

To derive the interior estimate, following [13] (see also [12]) we take $\eta$
to be a smooth function with compact support in $B_{r}\subset M$ satisfying

$0\leq\eta\leq 1,~{}~{}\eta|_{B_{\frac{r}{2}}}\equiv
1,~{}~{}|\nabla\eta|\leq\frac{C\sqrt{\eta}}{r},~{}~{}|\nabla^{2}\eta|\leq\frac{C}{r^{2}}.$
(3.24)

Thus

$\displaystyle\frac{1+\epsilon}{\epsilon\eta^{2}}F^{i\bar{j}}\eta_{i}\eta_{\bar{j}}+F^{i\bar{j}}S_{i\bar{j}}^{k}\frac{\eta_{k}}{\eta}+F^{i\bar{j}}\overline{S_{j\bar{i}}^{k}}\frac{\eta_{\bar{k}}}{\eta}-\frac{1}{\eta}F^{i\bar{j}}\eta_{i\bar{j}}\leq\frac{C}{\epsilon
r^{2}\eta}\sum F^{i\bar{i}}.$ (3.25)

As in [7], let $\phi=v^{-N}$ where $v=u-\inf_{B_{r}}u+2$ ($v\geq 2$ in
$B_{r}$) and $N\geq 1$ is an integer that is chosen later. By direct
computation

$\displaystyle\phi_{i}=-Nv^{-N-1}u_{i},\quad\phi_{\bar{i}}=-Nv^{-N-1}u_{\bar{i}}$
$\displaystyle\phi_{i\bar{j}}=N(N+1)v^{-N-2}u_{i}u_{\bar{j}}-Nv^{-N-1}u_{i\bar{j}}.$

So

$\displaystyle
F^{i\bar{i}}\phi_{i}\phi_{\bar{j}}=N^{2}v^{-2N-2}F^{i\bar{j}}u_{i}u_{\bar{j}}$
(3.26)

and

$\displaystyle\mathcal{L}\phi=$ $\displaystyle
N(N+1)v^{-N-2}F^{i\bar{j}}u_{i}u_{\bar{j}}-Nv^{-N-1}F^{i\bar{j}}u_{i\bar{j}}-Nv^{-N-1}(F^{ij}S_{i\bar{j}}^{k}u_{k}+F^{ij}\overline{S_{j\bar{i}}^{k}}u_{\bar{k}})$
(3.27) $\displaystyle=$ $\displaystyle
N(N+1)v^{-N-2}F^{i\bar{j}}u_{i}u_{\bar{j}}-Nv^{-N-1}F^{i\bar{j}}(\mathfrak{g}_{i\bar{j}}-\chi_{i\bar{j}}).$

The concavity of $f$ implies that there is a positive constant $C_{1}$ such
that

$F^{i\bar{j}}\mathfrak{g}_{i\bar{j}}=\sum_{i=1}^{n}f_{i}\lambda_{i}\leq
C_{1}\sum_{i=1}^{n}f_{i}.$ (3.28)

We choose $N\gg 1$ so that $Nv^{-N}<1$, then

$N(N+1)v^{-N-2}-N^{2}v^{-2N-2}\geq N^{2}v^{-N-2}.$ (3.29)

Plugging (3.22)-(3.23) and (3.25)-(3.29) into (3.20), we obtain

$\displaystyle\theta wN^{2}v^{-N-2}\sum F^{i\bar{i}}+\frac{\theta Q}{2w}\sum
F^{i\bar{i}}\leq CNv^{-N-1}\sum F^{i\bar{i}}+\frac{C}{r^{2}\eta}\sum
F^{i\bar{i}}+\frac{C}{\sqrt{w}}.$

Note

$\frac{\theta wN^{2}v^{-N-2}}{2}\sum F^{i\bar{i}}+\frac{\theta Q}{2w}\sum
F^{i\bar{i}}\geq\theta Nv^{-\frac{N}{2}-1}\sqrt{Q}\sum F^{i\bar{i}}$

and that there exists $R_{0}>0$ such that for any $\lambda$ with
$|\lambda|\geq R_{0}$

$|\lambda|\sum_{i=1}^{n}f_{i}(\lambda)\geq\frac{f(|\lambda|\vec{\bf
1})-f(\lambda)}{2}\geq\frac{f(R_{0}\vec{\bf 1})-\psi}{2}>0.$

As a result, we obtain Theorem 3.6.

####  3.1.3 Interior estimate for second order derivatives

We derive second order interior estimate.

###### Proof.

As in [12] we consider the following quantity

$\displaystyle P:=\sup_{z\in M}\max_{\xi\in
T^{1,0}_{z}M}\;e^{2\phi}\mathfrak{g}_{p\bar{q}}\xi_{p}\bar{\xi_{q}}\sqrt{g^{k\bar{l}}\mathfrak{g}_{i\bar{l}}\mathfrak{g}_{k\bar{j}}\xi_{i}\bar{\xi_{j}}}/|\xi|^{3}$

where $\phi$ is a function depending on $z$ and $|\nabla u|$. Assume that it
is achieved at an interior point $p_{0}\in M$ for some $\xi\in
T^{1,0}_{p_{0}}M$. The quantity $P$ is inspired by [19]. We choose local
coordinates $z=(z_{1},\cdots,z_{n})$ around $p_{0}$, such that at $p_{0}$

$g_{i\bar{j}}=\delta_{ij},\quad\mathfrak{g}_{i\bar{j}}=\delta_{ij}\lambda_{i}\mbox{
and }F^{i\bar{j}}=\delta_{ij}f_{i}.$

The maximum $P$ is achieved for $\xi=\partial_{1}$ at $p_{0}$. We assume
$\mathfrak{g}_{1\bar{1}}\geq 1$; otherwise we are done.

In what follows the computations are given at $p_{0}$. Similar to the
computations in [12] one has

$\mathfrak{g}_{1\bar{1}i}+\mathfrak{g}_{1\bar{1}}\phi_{i}=0,\;\;\mathfrak{g}_{1\bar{1}\bar{i}}+\mathfrak{g}_{1\bar{1}}\phi_{\bar{i}}=0,$
(3.30) $\displaystyle
0\geq\frac{F^{i\bar{i}}\mathfrak{g}_{1\bar{1}i\bar{i}}}{\mathfrak{g}_{1\bar{1}}}+F^{i\bar{i}}(\phi_{i\bar{i}}-\phi_{i}\phi_{\bar{i}})+\frac{1}{8\mathfrak{g}_{1\bar{1}}^{2}}\sum_{k>1}F^{i\bar{i}}\mathfrak{g}_{1\bar{k}i}\mathfrak{g}_{k\bar{1}\bar{i}}-C\sum
F^{i\bar{i}}.$ (3.31)

Combining with the standard formula (3.18), together with straightforward
computation, we can derive

$\displaystyle\mathfrak{g}_{1\bar{1}i\bar{i}}\geq$
$\displaystyle\mathfrak{g}_{i\bar{i}1\bar{1}}+2\mathfrak{Re}(\bar{T}^{j}_{1i}\mathfrak{g}_{1\bar{j}i})+2\mathfrak{Re}(S_{1\bar{1}}^{l}\mathfrak{g}_{i\bar{i}l}-S_{i\bar{i}}^{l}\mathfrak{g}_{1\bar{1}l})-C\sqrt{Q}$
(3.32)

where as above one denotes

$Q=|\partial\overline{\partial}u|^{2}+|\partial\partial u|^{2}.$

Differentiating equation (3.14) twice (using covariant derivative), we obtain

$\displaystyle F^{i\bar{i}}\mathfrak{g}_{i\bar{i}l}$ $\displaystyle=\psi_{l},$
(3.33) $\displaystyle
F^{i\bar{i}}\mathfrak{g}_{i\bar{i}1\bar{1}}=\psi_{1\bar{1}}-F^{i\bar{j},l\bar{m}}\mathfrak{g}_{i\bar{j}1}\mathfrak{g}_{l\bar{m}\bar{1}}.$
(3.34)

Then we have

$\displaystyle F^{i\bar{i}}\mathfrak{g}_{1\bar{1}i\bar{i}}\geq$ $\displaystyle
2\mathfrak{Re}(F^{i\bar{i}}\bar{T}^{j}_{1i}\mathfrak{g}_{1\bar{j}i})-2\mathfrak{Re}F^{i\bar{i}}S_{i\bar{i}}^{l}\mathfrak{g}_{1\bar{1}l}-C\sqrt{Q}\sum
F^{i\bar{i}}.$ (3.35)

Putting the above inequalities into (3.31) we get

$\displaystyle 0\geq$
$\displaystyle\mathfrak{g}_{1\bar{1}}F^{i\bar{i}}(\phi_{i\bar{i}}-\phi_{i}\phi_{\bar{i}})-2\mathfrak{Re}(F^{i\bar{i}}S_{i\bar{i}}^{l}\mathfrak{g}_{1\bar{1}l})+2\mathfrak{Re}(F^{i\bar{i}}\bar{T}^{1}_{1i}\mathfrak{g}_{1\bar{1}i})-C\sqrt{Q}\sum
F^{i\bar{i}}$ $\displaystyle=$
$\displaystyle\mathfrak{g}_{1\bar{1}}\mathcal{L}\phi-\mathfrak{g}_{1\bar{1}}F^{i\bar{i}}\phi_{i}\phi_{\bar{i}}-2\mathfrak{g}_{1\bar{1}}\mathfrak{Re}(F^{i\bar{i}}\bar{T}^{1}_{1i}\phi_{i})-C\sqrt{Q}\sum
F^{i\bar{i}}.$

Let $\phi=\log\eta+\varphi(w)$, where as above $w=|\nabla u|^{2}$, and $\eta$
is the cutoff function given by (3.24). Then

$\displaystyle\mathcal{L}\phi=$
$\displaystyle\frac{\mathcal{L}\eta}{\eta}-F^{i\bar{i}}\frac{|\eta_{i}|^{2}}{\eta^{2}}+\varphi^{\prime}\mathcal{L}w+\varphi^{\prime\prime}F^{i\bar{i}}|w_{i}|^{2},$
(3.36) $\displaystyle
F^{i\bar{i}}|\phi_{i}|^{2}+2\mathfrak{Re}F^{i\bar{i}}\bar{T}^{1}_{1i}\phi_{i}\leq\frac{4}{3}F^{i\bar{i}}|\phi_{i}|^{2}+C\sum
F^{i\bar{i}}$ (3.37)

and

$\displaystyle
F^{i\bar{i}}|\phi_{i}|^{2}\leq\frac{3}{2}F^{i\bar{i}}|\varphi_{i}|^{2}+3F^{i\bar{i}}\frac{|\eta_{i}|^{2}}{\eta^{2}}.$

As in [7] we set

$\varphi=\varphi(w)=\left(1-\frac{w}{2N}\right)^{-\frac{1}{2}}\mbox{ where
$N=\sup_{\\{\eta>0\\}}|\nabla u|^{2}$}.$

One can check $\varphi^{\prime}=\frac{\varphi^{3}}{4N},$
$\varphi^{\prime\prime}=\frac{3\varphi^{5}}{16N^{2}}$ and
$1\leq\varphi\leq\sqrt{2}$. And so

$\displaystyle\varphi^{\prime\prime}-2\varphi^{\prime
2}=\frac{\varphi^{5}}{16N^{2}}(3-2\varphi)>\frac{\varphi^{5}}{96N^{2}}.$
(3.38)

By Lemma 3.8 we have

$\displaystyle\mathcal{L}(w)\geq\frac{3\theta Q}{4}\sum F^{ii}-C(1+\sum
F^{i\bar{i}}).$ (3.39)

By (3.24) we obtain

$\displaystyle 0\leq\eta\leq 1,~{}~{}\eta|_{B_{\frac{r}{2}}}\equiv
1,~{}~{}F^{i\bar{i}}\frac{|\eta_{i}|^{2}}{\eta^{2}}\leq\frac{C}{r^{2}\eta},~{}~{}\frac{\mathcal{L}\eta}{\eta}\leq\frac{C}{r^{2}\eta}\sum
F^{i\bar{i}}.$ (3.40)

In conclusion we finally derive

$\displaystyle 0\geq\frac{9\theta Q}{16N}\sum
F^{i\bar{i}}-\frac{C}{r^{2}\eta}\sum
F^{i\bar{i}}-C\frac{\sqrt{Q}}{\mathfrak{g}_{1\bar{1}}}\sum F^{i\bar{i}}.$
(3.41)

This gives

$\eta\mathfrak{g}_{1\bar{1}}\leq\frac{C}{r^{2}}.$

∎

###  3.2 Completion the proof of Theorem 3.2

First, Theorem 1.3 implies the following result.

######  Theorem 3.9.

Assume (1.2), (1.4) and (2.6) hold. Then for fixed $\sigma$,

  * $\mathrm{\bf(1)}$

If $\partial\Gamma^{\sigma}\setminus\bar{\Gamma}_{n}\neq\emptyset,$ then
$\kappa_{\Gamma_{\sigma,f}}\geq 1$.

  * $\mathrm{\bf(2)}$

If $\Gamma$ is of type 2 cone and

$\displaystyle f(0,\cdots,0,t)>\sigma\mbox{ for some }t>0,$ (3.42)

then $\kappa_{\Gamma_{\sigma,f}}=n-1$ and $f$ is of fully uniform ellipticity
when restricted to $\partial\Gamma^{\sigma}$.

###### Proof.

Prove $\mathrm{\bf(1)}$: By (2.6), $\tau_{\sigma}\geq 0$. Let
$\lambda\in\partial\Gamma^{\sigma}\setminus\bar{\Gamma}_{n}$, and we assume
$\lambda_{n}<0$. Note that $\lambda-\tau_{\sigma}\vec{\bf
1}\in\Gamma_{\sigma,f}$ and $\lambda_{n}-\tau_{\sigma}\leq\lambda_{n}<0$. Thus
$\kappa_{\Gamma_{\sigma,f}}\geq 1.$

Prove $\mathrm{\bf(2)}$: From (3.42), there are some positive constants
$\epsilon_{0}$, $t_{0}$ such that
$f(-\epsilon_{0},\cdots,-\epsilon_{0},t_{0})>\sigma$. And then there is
$0<\beta<1$ such that $f(-\beta\epsilon_{0},\cdots,-\beta\epsilon_{0},\beta
t_{0})=\sigma.$ Thus
$(-\beta\epsilon_{0}-\tau_{\sigma},\cdots,-\beta\epsilon_{0}-\tau_{\sigma},\beta
t_{0}-\tau_{\sigma})\in\Gamma_{\sigma,f}$ as required. ∎

Next we present functions being of fully uniform ellipticity.

######  Corollary 3.10.

Let $(\tilde{f},\tilde{\Gamma})$ be as defined in (3.4). Let
$\sup_{\partial\Gamma}f<\sigma<\sup_{\Gamma}f$. Suppose that $(f,\Gamma)$
satisfies (1.2), (1.4), (2.6), $\Gamma\neq\Gamma_{n}$ and

$\displaystyle f(t,\cdots,t,0)>\sigma\mbox{ for some }t>0.$ (3.43)

Then $\tilde{f}$ is of fully uniform ellipticity when restricted to
$\\{\lambda\in\tilde{\Gamma}:\tilde{f}(\lambda)=\sigma\\}$.

Corollary 3.10 immediately implies the following:

######  Proposition 3.11.

If (1.2), (1.4), (3.5), (3.6) and (3.7) hold, then equation (3.2) is of fully
uniform ellipticity at any solution satisfying (3.9).

This proposition confirms all the assumptions imposed in Theorem 3.6, thereby
obtaining Theorem 3.2.

##  4 Applications to Hessian equations

Let $(M,g)$ be a $n$-dimensional compact Riemannian manifold, possibly with
boundary $\partial M$, $\bar{M}=M\cup\partial M$. We consider the Hessian
equations

$\displaystyle f(\lambda(\nabla^{2}u+A))=\psi.$ (4.1)

where $\psi$ is a smooth function and $A$ is a smooth symmetric $(0,2)$-type
tensor. The second order estimate for (4.1) on curved Riemannian manifolds was
studied by Guan [8, Section 3], extending previous results in literature, see
e.g. [14, 21]. The gradient estimate is the remaining task to the study of
Hessian equations. However, it is rather hard to prove gradient estimate for
Hessian equations on curved Riemannian manifolds. The gradient estimate was
obtained in [14] under assumptions (2.9),
$\lim_{|\lambda|\rightarrow+\infty}\sum_{i=1}^{n}f_{i}(\lambda)=+\infty$ and
that the Riemannian manifold admits nonnegative sectional curvature, and later
extended by Urbas [21] with replacing such restrictions by (1.14) and (2.10).
One may use Lemmas 2.6 and 2.7 to improve their results.

In fact, Theorem 1.5 and Proposition 2.5 allow us to derive the gradient
estimate for more general equations when

$\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq-
K_{0}\sum_{i=1}^{n}f_{i}(\lambda)\mbox{ for some }K_{0}\geq 0.$ (4.2)

If (4.2) holds for any $\lambda\in\Gamma^{\underline{\psi},\overline{\psi}}$,
then according to Proposition 2.5 and (2.8), we obtain a more general
inequality than (1.14)

$\displaystyle
f_{i}(\lambda)\geq\theta+\theta\sum_{j=1}^{n}f_{i}(\lambda)\quad\mbox{if
}\lambda_{i}\leq-K_{0},\mbox{
}\forall\lambda\in\Gamma^{\underline{\psi},\overline{\psi}}.$ (4.3)

As an application, one can follow a strategy, analogous to that used in [21],
to derive gradient bound for solutions to (4.1) under the
$\mathcal{C}$-subsolution assumption

$\displaystyle\lim_{t\rightarrow+\infty}f(\lambda(\nabla^{2}\underline{u}+A)+t\mathrm{e}_{i})>\psi\mbox{
in }\bar{M},\quad\forall 1\leq i\leq n$ (4.4)

where $\mathrm{e}_{i}$ is the $i$-$\mathrm{th}$ standard basis vector of
$\mathbb{R}^{n}$.

######  Proposition 4.1.

In addition to (1.2), (1.4) and (4.2), we assume there is a $C^{2}$-smooth
$\mathcal{C}$-subsolution $\underline{u}$. Let $u\in C^{3}(M)\cap
C^{1}(\bar{M})$ be a solution to (4.1) with $\lambda(\nabla^{2}u+A)\in\Gamma$,
then

$\displaystyle\sup_{M}|\nabla u|\leq C(1+\sup_{\partial M}|\nabla u|),$

where $C$ depends on $|\psi|_{C^{1}(\bar{M})}$,
$|\underline{u}|_{C^{2}(\bar{M})}$ and other known data.

With gradient estimate at hand, as in [8, 23], we can prove the following:

######  Theorem 4.2.

Let $(M^{n},g)$ be a compact Riemannian manifold with smooth smooth boundary.
Let $\varphi\in C^{\infty}(\partial M)$ and $\psi\in C^{\infty}(\bar{M})$ be a
function satisfying $\inf_{M}\psi>\sup_{\partial\Gamma}f$. Suppose that there
is an admissible function $\underline{u}\in C^{3,1}(\bar{M})$ satisfying

$f(\lambda(\nabla^{2}\underline{u}+A))\geq\psi,\quad\underline{u}|_{\partial
M}=\varphi.$

In addition to (1.3), (1.2), we assume (4.2) holds for

$\inf_{M}\psi\leq
f(\lambda)\leq\sup_{M}f(\lambda(\nabla^{2}\underline{u}+A)).$

Then there exists a unique smooth admissible function $u$ solving (4.1) with
$u|_{\partial M}=\varphi$.

######  Remark 4.3.

Condition (4.2) for $K_{0}=0$ was also used by [8] to derive boundary estimate
for Dirichlet problem of (4.1), and later by [9] (for $K_{0}\geq 0$) to study
first initial boundary problems. Our results indicate that the technique
assumptions $\mathrm{\bf(i)}$-$\mathrm{\bf(iii)}$ imposed in [8, Theorem 1.10]
as well as assumptions $\mathrm{\bf(i)}$-$\mathrm{\bf(iv)}$ in [9, Theorem
1.9] can be removed.

### Acknowledgements

The author was supported by the National Natural Science Foundation of China
under grant 11801587.

##  Appendix A Some standard lemmas

In this appendix we summarize some standard lemmas.

######  Lemma A.1.

Let $f$ satisfy (1.2) and (1.4), then (1.11) holds.

The above lemma has been used in [12]. Below we present another one.

######  Lemma A.2.

Suppose $f$ satisfies (1.2) and (1.4). Then for any $\sigma$ with
$\sigma<\sup_{\Gamma}f$ and $\partial\Gamma^{\sigma}\neq\emptyset$, there
exists $c_{\sigma}\vec{\bf 1}\in\partial\Gamma^{\sigma}.$

###### Proof.

For $\sigma<\sup_{\Gamma}f$, the level set $\partial\Gamma^{\sigma}$ (if
$\partial\Gamma^{\sigma}\neq\emptyset$) is a convex noncompact hypersurface
contained in $\Gamma$. Moreover, $\partial\Gamma^{\sigma}$ is symmetric with
respect to the diagonal.

Let $\lambda^{0}\in\partial\Gamma^{\sigma}$ be the closest point to the
origin. (Such a point exists, since $\partial\Gamma^{\sigma}$ is a closed
set). The idea is to prove $\lambda^{0}$ is the one what we look for.

Assume $\lambda^{0}$ is not in the diagonal. Then by the Implicit Function
Theorem, and the convexity and symmetry of $\partial\Gamma^{\sigma}$, one can
choose $\lambda\in\partial\Gamma^{\sigma}$ which has strictly less distance
than that of $\lambda^{0}$. It is a contradiction.

∎

##  Appendix B Criterion for $f$ satisfying (1.7)

We summarize characterizations of $f$ when it satisfies (1.2) and (1.7). The
following lemma was first proposed by [22]‡‡‡The paper is extracted from
[arXiv:2203.03439] and the first parts of [arXiv:2001.09238;
arXiv:2106.14837]. and further reformulated in [25].

######  Lemma B.1 ([22, 25]).

For $f$ satisfying (1.2), the following statements are equivalent.

  * •

$f$ satisfies (1.7).

  * •

$\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}>0,\mbox{ }\forall\lambda,\mbox{
}\mu\in\Gamma.$

  * •

$f(\lambda+\mu)>f(\lambda),\mbox{ }\forall\lambda,\mbox{ }\mu\in\Gamma.$

######  Corollary B.2 ([25]).

Assume (1.2) and (1.7) hold. Then we have (1.4) and
$\sum_{i=1}^{n}f_{i}(\lambda)>0$.

Inspired by the following key observation derived from (2.1)

$\displaystyle\mbox{For any }\lambda,\mbox{ }\mu\in\Gamma,\mbox{
}\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geq\limsup_{t\rightarrow+\infty}f(t\mu)/t$

the author [22] introduced the following two conditions:

$\displaystyle\mbox{For any }\lambda\in\Gamma,\mbox{
}\lim_{t\rightarrow+\infty}f(t\lambda)>-\infty,$ (B.1) $\displaystyle\mbox{For
any }\lambda\in\Gamma,\mbox{ }\limsup_{t\rightarrow+\infty}f(t\lambda)/t\geq
0.$ (B.2)

Obviously, it leads to

######  Lemma B.3 ([22]).

Suppose $f$ satisfies (1.2) and (B.2). Then

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geq 0\mbox{ for any
}\lambda,\mbox{ }\mu\in\Gamma.$ (B.3)

In addition, $f_{i}(\lambda)\geq 0$ in $\Gamma$ for all $1\leq i\leq n$. In
particular it satisfies

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq
0,\quad\forall\lambda\in\Gamma.$ (B.4)

We have criteria for concave symmetric functions.

######  Lemma B.4.

In the presence of (1.2), the following statements are equivalent.

  * •

$f$ satisfies (B.1).

  * •

$f$ satisfies (B.2).

  * •

$f$ satisfies (B.3).

  * •

$f$ satisfies (B.4).

We can deduce the following lemma when (2.3) holds.

######  Lemma B.5.

Suppose that (1.2) and (2.3) hold. Then the following statements are
equivalent to each other.

  * •

$f$ satisfies (1.7).

  * •

$f$ satisfies (B.1).

  * •

$f$ satisfies (B.2).

  * •

$f$ satisfies (B.3).

  * •

$f$ satisfies (B.4).

###### Proof.

From Lemmas B.1 and B.4, it requires only to prove

$\eqref{addistruc-5}\Rightarrow\eqref{addistruc}.$

Fix $\lambda$, $\mu\in\Gamma$. Note that $t\mu-\lambda-\vec{\bf 1}\in\Gamma$
for $t>t_{\mu,(\lambda+{\vec{\bf 1}})}>0$. Together with (2.1) and (2.3), we
derive

$f(t\mu)\geq f(\lambda)+\sum_{i=1}^{n}f_{i}(t\mu)>f(\lambda)\mbox{ for
}t>t_{\mu,(\lambda+{\vec{\bf 1}})}.$

∎

######  Lemma B.6 ([22]).

If $f$ satisfies (1.2), (1.3) and (2.9), then it obeys (1.7).

######  Remark B.7.

Lemma B.5 was proved in [22] when $f$ satisfies (1.2)-(1.3).

## References

  * [1] L. Caffarelli, L. Nirenberg and J. Spruck, _The Dirichlet problem for nonlinear second-order elliptic equations III: Functions of eigenvalues of the Hessians_ , Acta Math. 155 (1985), 261–301.
  * [2] L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces, Comm. Pure Appl. Math. 41 (1988), 47–70.
  * [3] S.-Y. S. Chen, Boundary value problems for some fully nonlinear elliptic equations, Calc. Var. PDE. 30 (2007), 1–15.
  * [4] S.-Y. S. Chen, Conformal deformation on manifolds with boundary, Geom. Funct. Anal. 19 (2009), 1029–1064.
  * [5] J.-X. Fu, Z.-Z. Wang and D.-M. Wu, _Form-type Calabi-Yau equations_ , Math. Res. Lett. 17 (2010), 887–903.
  * [6] B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. PDE. 8 (1999), 45–69.
  * [7] B. Guan, Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 105, 25 pp.
  * [8] B. Guan, _Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds_ , Duke Math. J. 163 (2014), 1491–1524.
  * [9] B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE. 8 (2015), 1145–1164.
  * [10] B. Guan and J. Spruck, Interior gradient estimates for solutions of prescribed curvature equations of parabolic type, Indiana Univ. Math. J. 40 (1991), 1471–1481.
  * [11] B. Guan and J. Spruck, Locally convex hypersurfaces of constant curvature with boundary, Comm. Pure Appl. Math. 57 (2004), 1311–1331.
  * [12] B. Guan, C.-H. Qiu and R.-R. Yuan, Fully nonlinear elliptic equations for conformal deformation of Chern-Ricci forms, Adv. Math. 343 (2019), 538–566.
  * [13] P.-F. Guan and G.-F. Wang, Local estimates for a class of fully nonlinear equations arising from conformal geometry, Int. Math. Res. Not. 2003, 1413–1432.
  * [14] Y.-Y. Li, _Some existence results of fully nonlinear elliptic equations of Monge-Ampère type_ , Comm. Pure Appl. Math. 43 (1990), 233–271.
  * [15] Y.-Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Differential Equations 90 (1991), 172–185.
  * [16] M. Lin and N. Trudinger, On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc. 50 (1994), 317–326.
  * [17] W.-M. Sheng, J. Urbas and X.-J. Wang, Interior curvature bounds for a class of curvature equations, Duke Math. J. 123 (2004), 235–264.
  * [18] G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form, Acta Math. 219 (2017), 181–211.
  * [19] V. Tosatti and B. Weinkove, Hermitian metrics, $(n-1,n-1)$ forms and Monge-Ampère equations, J. Reine Angew. Math. 755 (2019), 67–101.
  * [20] N. Trudinger, _The Dirichlet problem for the prescribed curvature equations_ , Arch. Rational Mech. Anal. 111 (1990), 153–179.
  * [21] J. Urbas, Hessian equations on compact Riemannian manifolds. Nonlinear problems in mathematical physics and related topics, II, vol. 2 of Int. Math. Ser. (N. Y.), Kluwer/Plenum, New York, 2002, pp. 367–377.
  * [22] R.-R. Yuan, On the regularity of Dirichlet problem for fully non-linear elliptic equations on Hermitian manifolds, arXiv:2203.04898.
  * [23] R.-R. Yuan, On the Dirichlet problem for a class of fully nonlinear elliptic equations, Calc. Var. PDE. 60 (2021), no. 5, Paper No. 162, 20 pp.
  * [24] R.-R. Yuan, The Dirichlet problem for Monge-Ampère equation for $(n-1)$-PSH functions on Hermitian manifolds, arXiv:2203.07232.
  * [25] R.-R. Yuan, The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics, arXiv:2203.13212.


# The origin of lopsided satellite galaxy distribution around isolated systems
in MillenniumTNG

Yikai Liu1,2, Peng Wang1,3, Hong Guo1, Volker Springel4, Sownak Bose5, Rüdiger
Pakmor4, Lars Hernquist6
1Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai
200030, China
2University of Chinese Academy of Sciences, Beijing 100049, China
3Astronomical Research Center, Shanghai Science & Technology Museum, Shanghai,
201306, China
4Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748
Garching, Germany
5Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK
6Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA
02138, USA
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>0009-0007-2848-7454
0000-0003-2504-3835 0000-0003-4936-8247 0000-0001-5976-4599
0000-0002-0974-5266 0000-0003-3308-2420

###### Abstract

Dwarf satellites in galaxy groups are distributed in an anisotropic and
asymmetric manner, which is called the “lopsided satellite distribution”. This
lopsided signal has been observed not only in galaxy pairs but also in
isolated systems. However, the physical origin of the lopsided signal in
isolated systems is still unknown. In this work, we investigate this in the
state-of-the-art hydrodynamical simulation of the MillenniumTNG Project by
tracing each system back to high redshift. We find that the lopsided signal is
dominated by satellites located in the outer regions of the halo and is also
dominated by recently accreted satellites. The lopsided signal originates from
the anisotropic accretion of galaxies from the surrounding large-scale
structure and that, after accretion, the nonlinear evolution of satellites
inside the dark-matter halo weakens the lopsidedness. The signal decreases as
cosmic time passes because of a competition between anisotropic accretion and
internal evolution within dark matter halos. Our findings provide a useful
perspective for the study of galaxy evolution, especially for the origin of
the spatial satellite galaxy distributions.

###### keywords:

galaxies: formation – large-scale structure of Universe – methods: numerical

††pagerange: The origin of lopsided satellite galaxy distribution around
isolated systems in MillenniumTNG–LABEL:lastpage

##  1 Introduction

The matter distribution across the Universe is homogeneous and isotropic on
large scales, which is in agreement with the predictions of the $\Lambda$CDM
cosmological model. However, the anisotropy and inhomogeneity of the matter
distribution increases in localised regions. On scales of a few to hundreds of
megaparsecs, the distribution of matter is characterised by a web-like
structure known as the cosmic web, which is made up of knots, filaments,
sheets, and voids (Zel’dovich, 1970; Bond et al., 1991). Within galaxy groups,
the phase-space distribution of dwarf satellite galaxies around their central
galaxies is anisotropic (Zaritsky et al., 1997; Zentner et al., 2005;
Brainerd, 2005).

In recent decades, numerous studies have been conducted on the phase-space
distribution of satellites (Knebe et al., 2004; Wang et al., 2005; Knebe et
al., 2008; Libeskind et al., 2011; Pawlowski, 2021b). After a long period of
debate (Sastry, 1968; Holmberg, 1969), it is now widely accepted that
satellites are preferentially located around the main axis of their central
galaxies (Brainerd, 2005; Libeskind et al., 2005; Agustsson & Brainerd, 2010;
Li et al., 2013; Welker et al., 2018; Wang et al., 2019; Tang et al., 2020).
This alignment signal depends on the colour of the galaxy. The red satellites
around the red centrals show the most significant alignment, while the blue
satellites around the blue centrals are more randomly distributed (Yang et
al., 2006; Azzaro et al., 2007; Kang et al., 2007). However, another issue of
satellite phase-space distribution, the so-called “satellite plane problem”,
is still not well understood and controversially discussed (Shaya & Tully,
2013; Gillet et al., 2015; Maji et al., 2017; Pawlowski, 2018, 2021a; Sawala
et al., 2023; Pham et al., 2023). In the Local Universe, the brightest
satellites of the Milky Way (Kroupa et al., 2005; Metz et al., 2008; Pawlowski
& Kroupa, 2013) and Andromeda (Ibata et al., 2013; Conn et al., 2013; Wan et
al., 2020) are found to be located in a thin corotating plane, whereas such a
satellite plane is rare (Ibata et al., 2014) in cosmological simulations.
Recently, a similar satellite plane was also discovered around the central
galaxy Centaurus A (Tully et al., 2015; Müller et al., 2018).

In the Local Group, the Milky Way and M31 are considered as a galaxy pair, and
there are 25 satellites located in the area between them. By using galaxy
pairs taken from SDSS DR10 (York et al., 2000; Ahn et al., 2014), Libeskind et
al. (2016) discovered that 8% more satellites than would be expected from a
uniform distribution are situated in the region between two hosts of a galaxy
pair similar to the MW-M31 pair, which is known as the problem of the
“lopsided satellite distribution” (LSD). Following these observations,
Pawlowski et al. (2017) identified a comparable signal of LSD in a galaxy pair
in a $\Lambda$CDM cosmological N-body simulation.

Examining isolated systems such as the Milky Way (MW) and M31, it has been
observed that 7 of the 11 brightest MW satellites and 21 of the 27 M31
satellites are located in the same hemisphere. Studies of other MW-like
galaxies in the Local Volume, such as M101, have revealed that 7 of the 8
brightest satellites are located in one hemisphere (Merritt et al., 2014;
Bennet et al., 2019, 2020). Brainerd & Samuels (2020) investigated the
satellite distribution around isolated galaxies in the NASA-Sloan Atlas
catalogue and found that the probability distribution for the polar angles of
the satellites differs significantly from a random distribution. Wang et al.
(2021) further found that the lopsided signal depends on the mass, colour, and
large-scale environment, with satellites that reside in the large radius of
low-mass blue hosts exhibiting the most lopsided signal. Samuels & Brainerd
(2023) also studied the lopsided satellite distribution by selecting isolated
host galaxies and their satellites from mock redshift surveys of an N-body
simulation and obtained results similar to those of Brainerd & Samuels (2020).

Observations and simulations have both demonstrated that the lopsided
distribution of satellites is not exclusive to galaxy pairs, but also applies
to isolated galaxies. The magnitude of the lopsidedness depends on both the
mass and the colour. The physical source of LSD has been explored, with Bowden
et al. (2014) suggesting that the lopsidedness of M31 is more likely due to a
dwarf association falling into its halo than the tidal field of the Milky Way.
Gong et al. (2019) found that the lopsided signal is generated from the first
satellite approach and weakens over time due to interactions between satellite
and central pairs. However, the origin of LSD in isolated systems remains a
mystery. It is worth investigating whether the origins of LSD in isolated
systems are similar to or completely different from those in galaxy pair
systems, given the unique environment of isolated systems.

To understand the source of LSD in isolated systems, we investigate the
history of these systems and study the development of the LSD signal in the
state-of-the-art MilleniumTNG hydrodynamic simulation (see Section 2 and the
references therein for more details). The MilleniumTNG simulation is based on
the IllustrisTNG (Springel et al., 2018; Marinacci et al., 2018; Naiman et
al., 2018; Nelson et al., 2018; Pillepich et al., 2018a, b; Weinberger et al.,
2017) physics model but features a much larger volume at still high
resolution, making it possible to obtain a substantial galaxy group sample to
statistically explore the origin of LSD.

Our paper is structured as follows. Section 2 describes the simulation we use
and how we measure the intensity of the lopsided signal. In Section 3, we look
back at the evolution of the lopsided signal and try to identify its origin by
comparing lopsided and nonlopsided systems in both the dark-matter halo and
the satellites themselves. Section 4 provides a summary and a discussion.

##  2 Data and Methodology

Figure 1: A schematic diagram illustrating the calculation of the lopsided
angle $\theta_{\rm lop}$ (left) and lopsided fraction $f_{\rm lop}$ (right).
The central galaxies are shown with cloud-like symbols located in the centre,
and small circles indicate satellite positions around the central galaxies. In
the left pane, $\rm\theta_{ij}$ is the angle between satellite pairs $i$ and
$j$. In the right panel, $\rm N_{sat}$ is the total number of satellites in a
given system, and $\rm N_{max}$ is the satellite number with the semicircle
that contains the most satellite galaxies.

We used the galaxy catalogue from the MTNG740 simulation of the MillenniumTNG
Project (hereafter MTNG; Hernández-Aguayo et al., 2023; Pakmor et al., 2023).
This simulation was carried out in a box size of $\rm(500\ cMpc/h)^{3}$ or
$(\rm 738.1\ cMpc)^{3}$ with cosmological parameters taken from Planck (Planck
Collaboration et al., 2016), i.e., $\Omega_{m}=0.3089$,
$\Omega_{\Lambda}=0.6911$, $\Omega_{b}=0.0486$ and $h=0.6774$. It contains
$4320^{3}$ dark matter particles and $4320^{3}$ initial gas cells, with mass
resolutions of $1.7\times 10^{8}\,{\rm M_{\odot}}$ and $3.1\times 10^{7}\,{\rm
M_{\odot}}$, respectively. Detailed information on the MTNG project refer to
papers of Hernández-Aguayo et al. (2023); Pakmor et al. (2023). More
introductory papers of MTNG includes past lightcones (Barrera et al., 2023),
large-scale clustering (Bose et al., 2023), cosmological parameters analysis
(Contreras et al., 2023), intrinsic alignments of galaxies and haloes (Delgado
et al., 2023), weak lensing (Ferlito et al., 2023), halo occupation models
(HODs) (Hadzhiyska et al., 2023a), satellite population model (Hadzhiyska et
al., 2023b) and galaxy population at high redshift (Kannan et al., 2023).

Dark matter halos are identified by the Friend-of-Friend(FoF) algorithm and
substructures are identified by the SUBFIND-HBT algorithm (Springel et al.,
2021). The GADGET-4 code (Springel et al., 2021) was used to construct the
halo merger trees. The merger trees are self-contained, meaning that all
subhalos are in the same tree as their progenitors and descendants, allowing
us to easily trace the history of galaxy groups.

In this study, we selected central galaxies with halo viral mass $M_{200}$
greater than $10^{11}{\rm M}_{\sun}/h$ and at least two satellites with a
total mass in their subhalos greater than $10^{10}{\rm M}_{\sun}/h$. We use
the isolation criteria of Wang et al. (2021) to identify our target groups as
isolated groups, which are groups whose distance from the nearest massive
group (i.e. $M_{200}$ $\textgreater$ $10^{11}{\rm M}_{\sun}/h$) is greater
than 2$\,{\rm{cMpc/h}}$. We note that two primary parameters determine the
sample selection process: the mass of satellites and the distance used to
define isolated systems. Wang et al. (2021) has studied a variety of
combinations of these two parameters and found a weak dependence on them. For
further information, we refer the reader to Figure 4 in Wang et al. (2021). In
the end, 35579 groups and 110346 satellites were selected for the following
analysis.

Figure 2: Left panel: Relation between the angle of lopsidedness and the
fraction of lopsidedness. Each black dot represents a galaxy group in MTNG.
The blue dashed vertical line is the lopsided angle expected for a random
distribution, that is, 90∘. Right panel: lopsided signal $\rm\theta_{lop}$ as
a function of the number of satellites per galaxy group. Shaded regions with
different colours show random distributions from 1$\sigma$ to 3$\sigma$, as
labelled in the legend. The black dots show the galaxy systems in MTNG.

To quantify the lopsidedness of a given isolated system, we define the
lopsided angle, $\theta_{\rm lop}$, and the lopsided fraction $f_{\rm lop}$,
as in Figure 1. To calculate $\rm\theta_{lop}$, we project all the satellite
galaxies onto the $xy$ plane and place the central galaxy at the origin. As
mentioned in Wang et al. (2021), the viewing angle (that is, the choice of
projection plane) does have an effect on the lopsided angle for a given
system, but there is a strong linear correlation between the calculated
lopsided signal projected in the 2D and 3D space. Thus, an arbitrarily chosen
projection of the $xy$-plane is suitable and does not affect the statistical
results. We calculate the lopsided angle as the average of angles between
central-satellite galaxy pairs $\theta_{ij}$, i.e.,

$\theta_{\rm lop}=\left<\theta_{ij}\right>,$ (1)

where the subscript $ij$ refers to satellite galaxies $i$ and $j$. The
expected $\theta_{\rm lop}$ for a random distribution is close to
$90^{\circ}$. However, in a lopsided system, satellite galaxies are usually
situated on one side of the central, so the expected $\theta_{\rm lop}$ would
be less than $90^{\circ}$. For example, the lopsided angle is $\theta_{\rm
lop}=77.56^{\circ}$ for the MW. It is important to remember that the
lopsidedness of two systems cannot be compared simply by looking at their
$\theta_{\rm lop}$. For galaxy groups with only two satellites, the
$\theta_{\rm lop}$ for random distributions can vary greatly. On the other
hand, the dispersion for galaxy groups with a large number of satellites is
very small. Thus, the same $\theta_{\rm lop}$ can signify different levels of
lopsidedness depending on the number of satellites.

The second quantity is the lopsided fraction, $f_{\rm lop}$. To calculate
$f_{\rm lop}$, it is necessary to draw a line through the centre of the galaxy
to divide the projected plane into two sections, with one of them containing
the maximum number of satellites. For a group with $N_{\rm sat}$ satellite
galaxies, we can fit a straight line passing the centrals with the most
satellites ($N_{\rm max}$) on one side in the $xy$-plane. Then the lopsided
fraction is defined as

$f_{\rm lop}=\frac{N_{\rm max}}{N_{\rm sat}}.$ (2)

In the local universe, for example, the lopsided fraction is $f_{\rm
lop}=0.82$ for MW, which shows an obvious lopsided signal of its satellite
distribution.

The correlation between lopsided angles and lopsided fractions is demonstrated
in Figure 2. This correlation is particularly evident in a large number of
satellite galaxies. When the number of satellites is high, a larger $f_{\rm
lop}$ implies a more pronounced lopsided signal. However, this is not the case
for fewer satellites, as all $N_{\rm sat}=2$ groups have a $f_{\rm lop}$ of 1.
Further details can be found in the appendix of Wang et al. (2021).

The significance of a given system depends on the number of satellites,
$N_{\rm sat}$. To determine whether a lopsided signal is inconsistent with a
random distribution, we calculate the significance of $\theta_{\rm lop}$. To
do this, for a system with a given $N_{\rm sat}$, we keep the distance between
each satellite galaxy and the central galaxy fixed but randomise their angular
distributions in the projected plane. We then repeat this randomisation 10,000
times and save the corresponding $\theta_{\rm lop,Ran}$ each time. We can then
estimate the mean lopsided angle $\langle\theta_{\rm lop,Ran}\rangle$ for the
random process and the corresponding $\sigma(\theta_{\rm lop,Ran})$ for each
system with the given $N_{\rm sat}$. The significance is then calculated as

$\rm{significance}=\frac{\left|\theta_{\rm lop}-\langle\theta_{\rm
lop,Ran}\rangle\right|}{\sigma(\theta_{\rm lop,Ran})}.$ (3)

The right panel of Figure 2 shows three different coloured shadows that
signify 1$\sigma$, 2$\sigma$, and 3$\sigma$ of the lopsided angle of the
random trials. Groups with different numbers of satellites show different
levels of significance. A considerable number of systems ($\sim$ 5% of the
total) are located beyond 3$\sigma$, which demonstrates a clear lopsided
signal. The findings in Figure 2 agree with those of Wang et al. (2021).

##  3 Results

###  3.1 General evolution of the lopsided signal

Figure 3: The evolution of the satellite distribution of three galaxy groups
is depicted from top to bottom. The three columns from left to right represent
three different snapshots corresponding to redshifts $z=0.25$, $z=0.1$, and
$z=0$. The red star in the centre of each snapshot represents the central
galaxy. The dashed circles indicate the viral radius at the corresponding
snapshot time. The orange dots indicate satellites that were not present in
the previous snapshot, while the blue dots indicate satellites that were
already present in the previous snapshot. We note that the relative size of
the circle does not indicate the radius size. It is used to make sure that all
satellites show in the figure.

Inspection of some selected examples can help to intuitively elucidate the
origin of the lopsided signal and its evolutionary trajectory. In Figure 3, we
show the evolution of satellite distributions for three distinct groups in
three different snapshots, which we denote as snap n-X, snap n and snap n+X,
corresponding to redshifts $z=0.25$, $z=0.1$ and $z=0$, respectively. The
central galaxy in each group is represented by a red star. The blue dots
represent the satellites that were present in the previous snapshot, while the
orange dots represent satellites that have recently been accreted from the
large-scale environment. The lopsided angle $\,{\rm\theta_{lop}}$ changes from
snapshot n-X to snapshot n, showing the effect of satellite accretion from the
large-scale environment and from snapshot n to snapshot n+X, corresponding to
the internal evolution of satellites within the halo.

For Group 1 (top panels), the lopsided angle $\,{\rm\theta_{lop}}$ remains
unchanged throughout the three snapshots, since the system had a large number
of satellites at snapshot n-X, and the relatively small number of satellites
that were accreted at snapshot n was insufficient to modify the original
lopsided signal.

For Group 2 (middle panels), the satellite distribution was lopsided with
$\,{\rm\theta_{lop}}\sim 42^{\circ}$ in the snapshot n-X. The host accreted
only six satellites (orange dots) in snapshot n from the large-scale
environment, but the internal evolution of the existing satellites (blue dots)
caused the lopsidedness to increase to $\sim 61^{\circ}$. From snapshot n to
snapshot n+X, satellites became almost randomly distributed
($\,{\rm\theta_{lop}}\sim 92^{\circ}$), mainly due to the effects of internal
evolution.

In snapshot n-X, Group 3 (bottom panels) had a relatively small number of
satellite galaxies, and their distribution was close to random. By snapshot n,
the host had accreted more satellites from the large-scale environment,
although the original distribution of the existing satellites was closer to
the inner region and more randomly distributed. Most newly accreted satellites
cluster in the same direction in the outer region, thus increasing the
lopsided signal. At snapshot n+X, more satellites moved to the centre, causing
the angle of lopsidedness to slightly increase.

We can deduce that the lopsidedness of the satellite distribution is mainly
caused by two factors. The first is the satellites that fall into the dark-
matter halo from the external environment, which is especially important,
since most of them are accreted from a single direction, thus intensifying the
signal. The second is the internal evolution of the satellites within the host
dark matter halo, which causes them to move closer to the centre and randomise
their distribution, thus weakening the signal.

Investigating the evolution of LSD signals is a direct way to explore their
origin. Wang et al. (2021) demonstrated an slightly increase of the lopsided
angle $\theta_{\rm lop}$ (a decrease in the lopsidedness, which can be seen in
the orange line of Figure 4) as the universe evolves by studying isolated
systems and calculating the lopsided angles at various redshifts without
tracing the systems back in time. In this work, we trace the main progenitors
of all target systems at $z=0$ back to high redshifts until the satellite
number is less than 2. At $z=0$, the sample is divided into two subsamples
with a halo mass cutoff of $10^{12}{\rm M}_{\sun}/h$.

Figure 4: Relation between the lopsided angle $\theta_{\rm lop}$ and redshift.
The solid black line with circles is for the whole sample. The blue and red
lines represent the low-mass and high-mass subsamples. The LSD shows the same
weakening trend in both subsamples. The sudden decrease of the lopsided angle
from $z=0.5$ to $z=0$ can be attributed to the rapidly increasing contribution
of the low-mass subsample. The orange line shows result from Wang et al.
(2021). The error bars are determined using bootstrap method.

In Figure 4, we show the evolution of the lopsided angle $\theta_{\rm lop}$ of
the main progenitors. The lopsided angle is given by the median value of all
systems in each bin, and the bootstrap error is provided. For the entire
sample, represented by the circles with the black line, we find that the
lopsided angle $\theta_{\rm lop}$ increases from $z=4$ to $z=0$, indicating
that the strength of the lopsidedness decreases from high to low redshifts.
This evolutionary trend is in agreement with Wang et al. (2021), although the
value of $\theta_{\rm lop}$ at a given redshift is slightly different. The
difference can be explained as follows. The criteria used by Wang et al.
(2021) to select isolated systems remain unchanged regardless of redshift,
leading to discrepancies between their sample and ours. At higher redshifts,
particularly for $z>3.5$, the sample size is drastically reduced (as seen in
Figure 7-f of Wang et al. (2021)) and the lopsided signal is driven mainly by
massive groups, thus weakening the lopsided signal (with larger $\theta_{\rm
lop}$). The red and blue lines represent the low- and high-mass subsamples,
respectively. The evolutionary trend of the low-mass and high-mass subsamples
is comparable to that of the entire sample. However, the $\theta_{\rm lop}$ of
the low-mass subsample is consistently lower (with more lopsidedness) than
that of the high-mass subsample. For the entire sample, there is a sudden
decrease in $\theta_{\rm lop}$ from $z=0.5$ to $z=0$. We note that this does
not mean a physical increase in LSD. This is caused by the high fraction
contributed by the low-mass subsample. Considering the mass limit of the
satellites at $z=0$ and the fact that each progenitor must contain at least
two satellites, the fraction of satellites by number decreases more rapidly
than the high-mass groups.

Figure 5: A comparison is conducted between the lopsided angle, $\rm\theta_{lop}$, of lopsided and nonlopsided systems in terms of their satellite infall time (upper panel) and radial distribution (lower panel). The two systems are distinguished by their significance, with lopsided systems having a significance greater than $3\sigma$ and nonlopsided systems having a significance less than $1\sigma$. The error bars are determined by adding in quadrature the errors using the bootstrap method and the uncertainties from 1000 random projection directions.  Table 1: The number of groups, the lopsided angle and the corresponding error for each subsample shown in Figure 5. |  | nonlopsided |  | lopsided
---|---|---|---|---
|  | early | late | inner | outer |  | early | late | inner | outer
Num |  | 3597 | 3106 | 3340 | 2433 |  | 567 | 960 | 458 | 1079
$\rm\theta_{lop}$ |  | 92.62 | 85.22 | 93.41 | 76.29 |  | 41.58 | 18.19 | 42.45 | 16.20
error |  | 0.64 | 1.13 | 0.80 | 1.23 |  | 2.52 | 0.83 | 2.94 | 0.65

Figure 6: The left panel displays the trend of the rescaled number density
with respect to the radial distance from the halo centre. The middle panel
shows the cumulative distribution of the time when the satellites fell into
the host. The right panel illustrates the distribution of the time interval
$\Delta t$ of satellites that have been accreted by the host. The solid red
and blue lines in each panel represent the lopsided and nonlopsided systems,
respectively, while the black dashed line represents the entire sample.

###  3.2 Lopsided signal with radial distribution and infall time

Following our previous conjecture that the LSD is dominated by external
accretion from the large-scale environment and internal evolution within the
halo, the spatial distribution and infall time of the satellites are therefore
essential for understanding the evolution of lopsided signals. The satellite
infall time, $t_{\rm infall}$, is defined as follows: if at snapshot $n$ the
satellite was an independent FOF halo, while in the next snapshot $n+1$ it was
a part of a larger halo but still a subhalo within the host, we use the
corresponding epoch of snapshot $n$ as the infall time. The high time
resolution of the outputs of MTNG makes this definition feasible. We divide
the entire sample into two subsamples according to significance (as defined in
Equation 3). Systems with a significance less than $1\sigma$ ($\sim 40\%$ of
the total) are labelled as _not lopsided_ and systems with a significance
greater than $3\sigma$ ($\sim 5\%$ of the total) are labelled as _lopsided_
subsample. Figure 5 shows a direct comparison of the lopsided signal,
$\theta_{\rm lop}$, between the subsamples considering the infall time (upper
panel) and the radial distribution (lower panel) of the satellite galaxies.
The corresponding lopisded angles and errors are listed in Table 1.

In the upper panel, we investigate the effect of the satellite infall time on
lopsidedness by selecting the 30% earliest and 30% latest accreted satellites
as early-infall and late-infall satellites, respectively, from both lopsided
and non-lopsided systems. We compare the median lopsided angles for different
subsamples. Two types of errors are considered for each subsample. The first
type is the measurement error caused by the limited sample size, which is also
listed in Table 1. This error is estimated using the bootstrap method, similar
to Figure 4. The second source of error arises from the our selected
projection direction. Therefore, we generate 1000 random sightlines and
estimate the error from the different median $\theta_{\rm lop}$ measurements
of the 1000 sightlines (the typical error is only about $0.3^{\circ}$). The
two types of errors are combined in quadrature to obtain the final errors. The
lopsided angle of the early-infall satellites in lopsided systems is
$41.58^{\circ}$ (red star), while the late-infall satellites have a lopsided
angle of $18.18^{\circ}$ (blue star). For _non-lopsided_ systems, the lopsided
angles of early and late infall satellites are $92.62^{\circ}$ (red triangle)
and $85.22^{\circ}$ (blue triangle), respectively. The median value reveals
apparent differences between early and late infall satellites, especially in
lopsided systems.

In the lower panel of Figure 5, we divide the satellites into two groups based
on their distances to the central galaxies, inner and outer satellites. We
select satellites with the smallest 30% (inner) and largest 30% (outer)
distances to their central galaxies. We then calculate the lopsided angles of
these two groups for both lopsided and non-lopsided systems. For lopsided
systems, the inner and outer satellites have $\theta_{\rm lop}=42.45^{\circ}$
(green star) and $\theta_{\rm lop}=16.21^{\circ}$ (yellow star), respectively.
For non-lopsided systems, the corresponding values of $\theta_{\rm lop}$ are
$93.41^{\circ}$ (green triangle) for the inner regions and $76.30^{\circ}$
(yellow triangle) for the outer regions. When comparing the median values
between inner satellite samples and outer satellite samples, we find that the
early and inner satellites have larger lopsided angles (i.e. less
lopsidedness) than the late and outer counterparts. The hierarchical galaxy
formation model suggests that early-accreted satellites are mainly located in
the inner part of the dark-matter halo, whereas late-accreted satellites are
usually situated in the outer regions.

We show in the left panel of Figure 6 the radial number density distributions
which are rescaled by the number density $\rho$ at $R_{200}$ of satellite
galaxies for lopsided (red line) and nonlopsided (blue line) systems, as well
as the entire sample (black line) as a function of the relative distance to
the group centres ($r/R_{200}$), where $R_{200}$ is the halo virial radius.
Most of the satellite galaxies are located within $2R_{200}$. The two
subsamples show considerable differences. Within $R_{200}$, the satellite
number density of the _nonlopsided_ subsample is relatively high, while for a
radius larger than $R_{200}$, the satellite number density of the _lopsided_
subsample is relatively high.

In the middle panel, we show the distribution of the infall time $t_{\rm
infall}$ (defined in Section 3.2) of the entire sample and the two subsamples.
The right panel shows the time interval $\Delta t$ between the infall of each
satellite and the previous one. We found that for _lopsided_ systems, nearly
half of the satellites are accreted within the same time ($\Delta t=0$ but not
in the same single snapshot) and 80% within 1 Gyr (solid red line in the
middle panel). For _nonlopsided_ systems, this fraction of $\Delta t=0$ is
reduced to around 20% (solid blue line in the middle panel). In the case of
_lopsided_ systems, 80% of the satellites are successively accreted in a short
period of 1 Gyr (red solid line in the middle panel); however, _nonlopsided_
systems need 2.5 Gyr (blue solid line in the right panel) to reach that
fraction.

In summary, we found that for _lopsided_ systems, the majority of their
satellites were accreted recently over a relatively short time frame and are
located in the outer parts of halos. Conversely, for _nonlopsided_ systems,
the majority of their satellites were obtained in the past over a relatively
extended period and they are situated in the inner areas of halos.

The two types of system appear to be discrepant, which results in different
strengths of the lopsided signal. This could imply that the lopsided signal
originates from large scales when satellites are accreted from the environment
(Kang & Wang, 2015; Libeskind et al., 2015). They bring anisotropies from the
large scale structure, but it usually decreases when moving towards the centre
of the halo due to non-linear evolution within the halos. One may wonder how
long this kind of memory lasts after a satellite has been accreted. To
understand this, we examine the relationship between the survival time
$\Delta{t}$ of satellite galaxies after accretion and the strength of their
lopsided signal $\rm\theta_{lop}$ at a given redshift $z$. Survival time is
defined as $\Delta t=t_{\rm infall}-t_{z}$, where $t_{\rm infall}$ is the
infall time of a given satellite and $t_{z}$ is the time of a given redshift.
Large $\Delta{t}$ indicates that the satellites were accreted early and
evolved for a long time within the host halo.

Figure 7: The correlation between satellite survival time $\Delta t$ and the
strength of their lopsided signal $\rm\theta_{lop}$. Satellite survival time
$\Delta t$ is defined as $\Delta t=t_{\rm infall}-t_{z}$, where $t_{\rm
infall}$ is the infall time of a given satellite and $t_{z}$ is the time of a
given redshift $z$ (labelled). The black dashed line shows the lopsided angle
of randomly distributed satellites. The error bars are determined using the
bootstrap method.

Figure 7 shows the relation between $\Delta t$ and $\theta_{\rm lop}$. We can
see a clear trend that $\theta_{\rm lop}$ increases first with growing
$\rm\Delta t$ and then becomes independent of $\Delta t$. We find that at each
$z$ (except for $z=0$ shown in the solid red line), the lopsided angle is
close to $\rm 90^{\circ}$ for systems with $\Delta t\textgreater 0.3\,{\rm
Gyr}$, which implies that the lopsided signal brought from infalling
satellites does not survive for a long time. This trend suggests a gradual
weakening of the lopsided signal introduced by infalling satellite subhalos,
which eventually leads to its near disappearance after a certain period. The
profiles of $z=1$, $2$, and $3$ are similar, while the profile of $z=0$ is
obviously lower. This can be attributed to the large number of low-mass
systems that we mentioned in Section 3.1. The lopsided angle approaches $\rm
90^{\circ}$ at $\Delta t>1.0\,{\rm Gyr}$ and tends to become a flat profile
for larger $\Delta t$, which is not shown here.

###  3.3 Satellite accretion along large scale structure

In Section 3.2, we show that the lopsided signal is mainly due to satellites
that have recently been accreted and to those that are located in the outer
regions of halos. It is also observed that for lopsided systems most
satellites are accreted for a short period. However, if the satellites were
not accreted from a particular direction, i.e. uniform accretion, the expected
satellite distribution would still be random. Shao et al. (2018) proposed that
satellites of MW-like galaxies were obtained in the group from the closest
abundant filament. It has also been suggested that the subhalos were accreted
along the filament direction (Kang & Wang, 2015; Shi et al., 2015; Libeskind
et al., 2015; Tempel et al., 2015). Combining the fact that the lopsided
signal comes from short-time satellite galaxy accretion events and that
satellites fell preferentially from the filaments, we can conclude that the
lopsided signal should be strongly correlated with the large-scale structure.

We can verify this by looking at the pattern of satellite accretion in
relation to the large-scale structure. To do this, we first use the Cloud-in-
Cell algorithm with $1024^{3}$ cells to generate the matter density field
$\rho$. We then solve the Poisson equation ($\nabla^{2}\phi=4\pi G\rho$) to
obtain the gravitational potential $\phi$. The tidal tensor is derived from
the Hessian matrix of $\phi$ as $T_{ij}=\partial_{i}\partial_{j}\phi$ at the
position of the targeted system (Wang et al., 2020). The three eigenvalues
(ordered as $\lambda_{1}\textgreater\lambda_{2}\textgreater\lambda_{3}$) of
the tidal tensor indicate the strength of the gravitational potential, and the
three eigenvectors ($e_{1}$, $e_{2}$, $e_{3}$) show the direction of each
eigenvalue. The sign of the eigenvalue can be used to determine whether the
gravitational force of the eigenvector is inward (positive) or outward
(negative). This allows us to distinguish the environment of each halo through
the number of positive and negative eigenvalues. For example, the eigenvector
$e_{3}$ associated with the smallest eigenvalue points to the direction of the
strongest expansion. Therefore, the eigenvector of a halo located in a
filament (which has two positive eigenvalues and one negative eigenvalue) is
aligned with the direction of the filament.

We investigate the relationship between satellite lopsided distribution and
large-scale structure directions by calculating the cosine of angles between
the central-satellite direction $\bm{e_{i}}$ and the eigenvector $\bm{e_{3}}$
of the smallest eigenvalue, denoted
$\cos(\theta_{e})=\bm{e_{i}}\cdot\bm{e_{3}}$. A random distribution would have
an expectation value $\left<\cos(\theta_{e})\right>=0.5$. If
$\cos(\theta_{e})$ is greater than 0.5, the satellites are more likely to be
aligned with the large-scale direction, and if it is less than 0.5, the
satellites are more likely to be perpendicular to the large-scale direction.
We examine $\cos(\theta_{e})$ both at $z=0$ and at the infall time $z=z_{\rm
infall}$. For $z=z_{\rm infall}$, an alignment trend indicates that satellites
are more likely to be accreted along the filament direction.

Figure 8: The cumulative distribution function of the cosine of the angle
between the satellite-central vector and the primary large-scale structure
direction. The black dotted line correponds to a uniform (isotropic) satellite
accretion relative to the large-scale structure. The black solid line is for
the satellites’ position at the present time ($z=0$), while the blue dash-
dotted line represents the satellites’ position at the time of accretion.

In Figure 8, we show the cumulative distribution of the cosine of the angle
between the satellite-central vector and the direction of the large-scale
structure. It is clear that satellites at $z=0$ (solid black line) or
$z=z_{\rm infall}$ (blue dashed dotted line) are more likely to be orientated
in the direction of the large-scale structure. Moreover, there is a noticeable
gap between the distributions for $z=0$ and $z=z_{\rm infall}$, indicating
that the alignment at the time of accretion is stronger than at $z=0$. This
implies that the satellites were accreted along the direction of the large-
scale structure, which contributes to the generation of lopsided signals.
Subsequently, the internal evolution effect within the host dark matter halos
causes the satellites to lose their correlation with the directions of the
large-scale structure, resulting in a weakening of the alignment.

##  4 Summary and Discussion

Observations and simulations have both revealed an asymmetric distribution of
satellites around paired galaxies (Libeskind et al., 2016; Pawlowski et al.,
2017) and isolated systems (Brainerd & Samuels, 2020; Wang et al., 2021).
While the origin of this lopsided distribution has been explored in paired
galaxies Gong et al. (2019), the physical origin of the same phenomenon in
isolated systems is still unknown. This paper investigates the origin of the
lopsided satellite distribution around isolated central galaxies, using the
state-of-the-art hydrodynamical $\Lambda$CDM simulation MillenniumTNG. Our
findings can be summarised as follows.

  * •

We have re-examined the lopsided signal of the isolated system selected at
$z=0$. Our results agree with Wang et al. (2021) (see Figure 2).

  * •

The lopsided signal decreases with cosmic time, and this trend is mass-
dependent. High-mass systems experience slower fading, whereas the reduction
is more rapid for low-mass systems (see Figure 4).

  * •

The lopsided signal at $z=0$ is dominated by satellites located at the outer
parts of halos and is also dominated by recently accreted satellites (see
Figure 5).

  * •

The lopsided signal originates from the anisotropic accretion of satellite
galaxies from the large-scale structure (see Figure 8), and after accretion,
the non-linear evolution of satellites inside the dark matter halos weakens
the lopsidedness (see Figures 7 and 8).

Our findings for the lopsided distribution of satellite galaxies at $z=0$ are
in general agreement with the results of Brainerd & Samuels (2020); Wang et
al. (2021); Samuels & Brainerd (2023). In addition, we have explored the
origin of this lopsidedness in isolated systems. We studied whether the
lopsided satellite distribution is caused by the external large-scale
structure or the internal evolution of dark matter halos. By tracing systems
back to the early Universe, we examined the satellite accretion pattern with
respect to the large-scale structure and satellite infall time, as well as the
radial satellite distribution. We identified the generation and evolution of
lopsided signals from satellites.

The large-scale structure has a significant impact on the formation of
lopsided satellite distributions, as it is strongly associated with the
pattern of satellite accretion, which is characterised by a preferential
accretion of satellites along certain large-scale structure directions. This
is in agreement with the findings of Gong et al. (2019), who suggest that the
lopsidedness of both central pairs and isolated systems is the result of the
specific direction of satellite fall from the environment (Libeskind et al.,
2014; Kang & Wang, 2015). The more and the later satellites that fall into a
galaxy group, the more pronounced the lopsided signal becomes. Late-infalling
satellites, which are more likely to be located in the outer parts of dark
matter halos, are particularly responsible for the lopsidedness.

Once the satellites are accreted, the internal evolution within the dark
matter halo and the external origin of the satellites begin to compete. The
relaxation of the dark matter halo causes the satellites to become more
isotropically distributed with time. The closer a satellite galaxy is to the
inner regions of a halo, the longer it has experienced the influence of
relaxation, making it lose more of the memory it had from the large-scale
structure.

Despite residing in the inner regions of the host halo, such satellites
nevertheless still exhibit some weak lopsidedness. What other factors could be
suppressing the relaxation of this signal, preventing it from dissipating
completely? How long does it take for the lopsidedness caused by the large
scales to be completely relaxed? The source of the lopsidedness from the
large-scale structure at early epochs has yet to be explored. This is why
satellites are accreted into dark matter halos in groups and aligned with the
large-scale structure. Further research is required to construct a full model
of the generation of anisotropy of the satellite distribution at high
redshifts.

In addition, the lopsided distribution of satellites that we have investigated
in this work, as well as in previous studies, is based on the definitions of
central and satellite galaxies provided by the FOF and SUBFIND-HBT algorithms.
However, bridge-like configurations sometimes caused by the FOF algorithm may
affect the lopsided signal by biasing the definition of the centre of the
system. For example, in the case of two merging systems with a very similar
mass of the central galaxies, the algorithm may identify them as a single
system. In such a double-centred system, the algorithm will designate only one
galaxy as central, and the others as satellite galaxies, potentially creating
a spurious lopsided signal. This issue will be explored further in our future
work.

## Acknowledgements

We thank the anonymous reviewr for the helpful comments that significantly
improved the presentation of this paper. We acknowledge the stimulating
discussions with Prof. Noam I. Libeskind and Prof. Xi Kang. This work is
supported by the National SKA Program of China (grant No. 2020SKA0110100). PW
is sponsored by Shanghai Pujiang Program(No. 22PJ1415100). HG is supported by
the CAS Project for Young Scientists in Basic Research (No. YSBR-092) and the
science research grants from the China Manned Space Project with NO. CMS-
CSST-2021-A02. SB is supported by the UK Research and Innovation (UKRI) Future
Leaders Fellowship [grant number MR/V023381/1]. We acknowledge the use of the
High Performance Computing Resource in the Core Facility for Advanced Research
Computing at the Shanghai Astronomical Observatory.

## Data availability

The MillenniumTNG simulations will be publicly available on https://www.mtng-
project.org in the future. The data we use in this article will be shared upon
reasonable request to the corresponding author.

## References

  * Agustsson & Brainerd (2010) Agustsson I., Brainerd T. G., 2010, ApJ, 709, 1321
  * Ahn et al. (2014) Ahn C. P., et al., 2014, ApJS, 211, 17
  * Azzaro et al. (2007) Azzaro M., Patiri S. G., Prada F., Zentner A. R., 2007, MNRAS, 376, L43
  * Barrera et al. (2023) Barrera M., et al., 2023, MNRAS,
  * Bennet et al. (2019) Bennet P., Sand D. J., Crnojević D., Spekkens K., Karunakaran A., Zaritsky D., Mutlu-Pakdil B., 2019, ApJ, 885, 153
  * Bennet et al. (2020) Bennet P., Sand D. J., Crnojević D., Spekkens K., Karunakaran A., Zaritsky D., Mutlu-Pakdil B., 2020, ApJ, 893, L9
  * Bond et al. (1991) Bond J. R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379, 440
  * Bose et al. (2023) Bose S., et al., 2023, MNRAS, 524, 2579
  * Bowden et al. (2014) Bowden A., Evans N. W., Belokurov V., 2014, ApJ, 793, L42
  * Brainerd (2005) Brainerd T. G., 2005, ApJ, 628, L101
  * Brainerd & Samuels (2020) Brainerd T. G., Samuels A., 2020, ApJ, 898, L15
  * Conn et al. (2013) Conn A. R., et al., 2013, ApJ, 766, 120
  * Contreras et al. (2023) Contreras S., et al., 2023, MNRAS, 524, 2489
  * Delgado et al. (2023) Delgado A. M., et al., 2023, MNRAS, 523, 5899
  * Ferlito et al. (2023) Ferlito F., et al., 2023, MNRAS, 524, 5591
  * Gillet et al. (2015) Gillet N., Ocvirk P., Aubert D., Knebe A., Libeskind N., Yepes G., Gottlöber S., Hoffman Y., 2015, ApJ, 800, 34
  * Gong et al. (2019) Gong C. C., et al., 2019, MNRAS, 488, 3100
  * Hadzhiyska et al. (2023a) Hadzhiyska B., et al., 2023a, MNRAS, 524, 2507
  * Hadzhiyska et al. (2023b) Hadzhiyska B., et al., 2023b, MNRAS, 524, 2524
  * Hernández-Aguayo et al. (2023) Hernández-Aguayo C., et al., 2023, MNRAS, 524, 2556
  * Holmberg (1969) Holmberg E., 1969, Arkiv for Astronomi, 5, 305
  * Ibata et al. (2013) Ibata R. A., et al., 2013, Nature, 493, 62
  * Ibata et al. (2014) Ibata R. A., Ibata N. G., Lewis G. F., Martin N. F., Conn A., Elahi P., Arias V., Fernando N., 2014, ApJ, 784, L6
  * Kang & Wang (2015) Kang X., Wang P., 2015, ApJ, 813, 6
  * Kang et al. (2007) Kang X., van den Bosch F. C., Yang X., Mao S., Mo H. J., Li C., Jing Y. P., 2007, MNRAS, 378, 1531
  * Kannan et al. (2023) Kannan R., et al., 2023, MNRAS, 524, 2594
  * Knebe et al. (2004) Knebe A., Gill S. P. D., Gibson B. K., Lewis G. F., Ibata R. A., Dopita M. A., 2004, ApJ, 603, 7
  * Knebe et al. (2008) Knebe A., Draganova N., Power C., Yepes G., Hoffman Y., Gottlöber S., Gibson B. K., 2008, MNRAS, 386, L52
  * Kroupa et al. (2005) Kroupa P., Theis C., Boily C. M., 2005, A&A, 431, 517
  * Li et al. (2013) Li Z., Wang Y., Yang X., Chen X., Xie L., Wang X., 2013, ApJ, 768, 20
  * Libeskind et al. (2005) Libeskind N. I., Frenk C. S., Cole S., Helly J. C., Jenkins A., Navarro J. F., Power C., 2005, MNRAS, 363, 146
  * Libeskind et al. (2011) Libeskind N. I., Knebe A., Hoffman Y., Gottlöber S., Yepes G., Steinmetz M., 2011, MNRAS, 411, 1525
  * Libeskind et al. (2014) Libeskind N. I., Knebe A., Hoffman Y., Gottlöber S., 2014, MNRAS, 443, 1274
  * Libeskind et al. (2015) Libeskind N. I., Hoffman Y., Tully R. B., Courtois H. M., Pomarède D., Gottlöber S., Steinmetz M., 2015, MNRAS, 452, 1052
  * Libeskind et al. (2016) Libeskind N. I., Guo Q., Tempel E., Ibata R., 2016, ApJ, 830, 121
  * Maji et al. (2017) Maji M., Zhu Q., Marinacci F., Li Y., 2017, ApJ, 843, 62
  * Marinacci et al. (2018) Marinacci F., et al., 2018, MNRAS, 480, 5113
  * Merritt et al. (2014) Merritt A., van Dokkum P., Abraham R., 2014, ApJ, 787, L37
  * Metz et al. (2008) Metz M., Kroupa P., Libeskind N. I., 2008, ApJ, 680, 287
  * Müller et al. (2018) Müller O., Pawlowski M. S., Jerjen H., Lelli F., 2018, Science, 359, 534
  * Naiman et al. (2018) Naiman J. P., et al., 2018, MNRAS, 477, 1206
  * Nelson et al. (2018) Nelson D., et al., 2018, MNRAS, 475, 624
  * Pakmor et al. (2023) Pakmor R., et al., 2023, MNRAS, 524, 2539
  * Pawlowski (2018) Pawlowski M. S., 2018, Modern Physics Letters A, 33, 1830004
  * Pawlowski (2021a) Pawlowski M. S., 2021a, Nature Astronomy, 5, 1185
  * Pawlowski (2021b) Pawlowski M. S., 2021b, Galaxies, 9, 66
  * Pawlowski & Kroupa (2013) Pawlowski M. S., Kroupa P., 2013, MNRAS, 435, 2116
  * Pawlowski et al. (2017) Pawlowski M. S., Ibata R. A., Bullock J. S., 2017, ApJ, 850, 132
  * Pham et al. (2023) Pham K., Kravtsov A., Manwadkar V., 2023, MNRAS, 520, 3937
  * Pillepich et al. (2018a) Pillepich A., et al., 2018a, MNRAS, 473, 4077
  * Pillepich et al. (2018b) Pillepich A., et al., 2018b, MNRAS, 475, 648
  * Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 594, A13
  * Samuels & Brainerd (2023) Samuels A., Brainerd T. G., 2023, ApJ, 947, 56
  * Sastry (1968) Sastry G. N., 1968, PASP, 80, 252
  * Sawala et al. (2023) Sawala T., et al., 2023, Nature Astronomy, 7, 481
  * Shao et al. (2018) Shao S., Cautun M., Frenk C. S., Grand R. J. J., Gómez F. A., Marinacci F., Simpson C. M., 2018, MNRAS, 476, 1796
  * Shaya & Tully (2013) Shaya E. J., Tully R. B., 2013, MNRAS, 436, 2096
  * Shi et al. (2015) Shi J., Wang H., Mo H. J., 2015, ApJ, 807, 37
  * Springel et al. (2018) Springel V., et al., 2018, MNRAS, 475, 676
  * Springel et al. (2021) Springel V., Pakmor R., Zier O., Reinecke M., 2021, MNRAS, 506, 2871
  * Tang et al. (2020) Tang L., Lin W., Wang Y., 2020, ApJ, 893, 87
  * Tempel et al. (2015) Tempel E., Guo Q., Kipper R., Libeskind N. I., 2015, MNRAS, 450, 2727
  * Tully et al. (2015) Tully R. B., Libeskind N. I., Karachentsev I. D., Karachentseva V. E., Rizzi L., Shaya E. J., 2015, ApJ, 802, L25
  * Wan et al. (2020) Wan Z., Oliver W. H., Lewis G. F., Read J. I., Collins M. L. M., 2020, MNRAS, 492, 456
  * Wang et al. (2005) Wang H. Y., Jing Y. P., Mao S., Kang X., 2005, MNRAS, 364, 424
  * Wang et al. (2019) Wang P., Guo Q., Libeskind N. I., Tempel E., Wei C., Kang X., 2019, MNRAS, 484, 4325
  * Wang et al. (2020) Wang P., Kang X., Libeskind N. I., Guo Q., Gottlöber S., Wang W., 2020, New Astron., 80, 101405
  * Wang et al. (2021) Wang P., Libeskind N. I., Pawlowski M. S., Kang X., Wang W., Guo Q., Tempel E., 2021, ApJ, 914, 78
  * Weinberger et al. (2017) Weinberger R., et al., 2017, MNRAS, 465, 3291
  * Welker et al. (2018) Welker C., Dubois Y., Pichon C., Devriendt J., Chisari N. E., 2018, A&A, 613, A4
  * Yang et al. (2006) Yang X., van den Bosch F. C., Mo H. J., Mao S., Kang X., Weinmann S. M., Guo Y., Jing Y. P., 2006, MNRAS, 369, 1293
  * York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579
  * Zaritsky et al. (1997) Zaritsky D., Smith R., Frenk C. S., White S. D. M., 1997, ApJ, 478, L53
  * Zel’dovich (1970) Zel’dovich Y. B., 1970, A&A, 5, 84
  * Zentner et al. (2005) Zentner A. R., Kravtsov A. V., Gnedin O. Y., Klypin A. A., 2005, ApJ, 629, 219