Boundary Liouville Conformal Field Theory in Four Dimensions
Adwait Gaikwad 111<EMAIL_ADDRESS>Amitay C. Kislev 222
<EMAIL_ADDRESS>Tom Levy<EMAIL_ADDRESS>and Yaron Oz
<EMAIL_ADDRESS>
School of Physics and Astronomy,
Tel-Aviv University, Tel-Aviv 69978, Israel

Higher dimensional Euclidean Liouville conformal field theories (LCFTs)
consist of a log-correlated real scalar field with a background charge and an
exponential potential. We analyse the LCFT on a four-dimensional manifold with
a boundary. We extend to the boundary, the conformally covariant GJMS operator
and the ${\cal Q}$-curvature term in the LCFT action and classify the
conformal boundary conditions. Working on a flat space with plate boundary, we
calculate the dimensions of the boundary conformal primary operators, the two-
and three-point functions of the displacement operator and the boundary
conformal anomaly coefficients.

###### Contents

  1. 1 Introduction
  2. 2 Boundary Liouville CFT in Four Dimensions
    1. 2.1 Boundary Conformal Primary Operators
    2. 2.2 Boundary Liouville Action
      1. 2.2.1 Extension of the Action to the Boundary
    3. 2.3 Spectrum and $b_{B}$
    4. 2.4 Conformal Boundary Conditions
      1. 2.4.1 Variation of the Action and Boundary Conditions
  3. 3 Boundary Conformal Anomalies
    1. 3.1 The Stress Tensor and the Displacement Operator
    2. 3.2 Green Function and Boundary Conformal Anomalies
      1. 3.2.1 Consistent Boundary Conditions
      2. 3.2.2 Boundary Anomaly Coefficients
  4. 4 Outlook
  5. 5 Acknowledgements
  6. A Ward Identities for Displacement Operator
  7. B Conformal Transformation Between the Ball and Half Euclidean Space

##  1 Introduction

Two-dimensional Liouville conformal field theory (LCFT) has been proposed as a
description of two-dimensional quantum gravity [1]. A higher dimensional
Euclidean generalization of LCFT [2] has been introduced in [3, 4] as part of
a Euclidean field theory description of steady state incompressible fluid
turbulence. In this framework, the Liouville scalar field is the Nambu-
Godstone boson associated with a spontaneous symmetry breaking (SSB) due to
the energy flux of the separate space and time scale symmetries of turbulence
in the inertial range of scales. This SSB to Kolmogorov linear scaling [3, 4]
provides an analytical formula for the universal anomalous scaling exponents
of turbulence [5, 3, 4] by a generalization of KPZ scaling in two-dimensional
quantum gravity [6, 7]. Mathematical aspects of higher dimensional LCFT have
been addressed in [8, 9, 10]. Higher dimensional LCFT appears in the study of
anomalies [11, 12], in CFT complexity [13, 14, 15], in Anderson transistion
[16] and potentially in RG flows in tensor models [17, 18]. LCFT in dimensions
higher than two is defined by a higher derivative action (see [19, 20, 21, 22]
for work on higher derivative field theories). In odd number of dimensions the
LCFT is non-local [23] (see work on non-local field theories [24, 25]). The
four-dimensional LCFT has $N=1$ and $N=2$ supeconformal generalizations
studied in [26, 27] (see [28] for higher derivative supersymmetry).

The action of LCFT on a $d$-dimensional manifold $\mathcal{M}$ reads:

$S_{L}[\phi,g]=\frac{d}{2\Omega_{d}(d-1)!}\int_{\mathcal{M}}d^{d}x\sqrt{g}\left(\phi\mathcal{P}_{g}\phi+2Q\mathcal{Q}_{g}\phi+\frac{2}{d}\Omega_{d}(d-1)!\
\mu e^{db\phi}\right)\ .$ (1)

$\phi$ is the Liouville log-correlated scalar field,
$\Omega_{d}=(2\pi)^{(d+1)/2}/\Gamma[(d+1)/2]$ is the surface volume of the
d-dimensional sphere $S^{d}$, and the dimensionless parameters are $Q$, $\mu$
and $b$. $\mathcal{P}_{g}$ is the conformally covariant GJMS operator [29]:

$\mathcal{P}_{g}=(-\Delta)^{d/2}+\cdots\ ,$ (2)

where $x^{\mu}$ are local coordinates on $\mathcal{M}$,
$\Delta=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ with $\nabla_{\mu}$ being the
covariant derivative and $\cdots$ are subleading derivative terms. The
$\mathcal{Q}$-curvature scalar [30], $\mathcal{Q}_{g}$, is:

$\mathcal{Q}_{g}=\frac{1}{(d-2)}\ (-\Delta)^{\frac{d}{2}-1}R+\cdots\ ,$ (3)

where $R$ is the Ricci scalar of $\mathcal{M}$. The integral of
$\mathcal{Q}_{g}$ is a conformal invariant of the manifold $\mathcal{M}$.

Under a $d$-dimensional Weyl transformation:

$\displaystyle g_{\mu\nu}$ $\displaystyle\rightarrow e^{2\omega}g_{\mu\nu}\,,$
(4a) $\displaystyle\phi$ $\displaystyle\rightarrow\phi-Q\sigma\,,$ (4b)
$\displaystyle\mathcal{P}_{g}$ $\displaystyle\rightarrow
e^{-d\omega}\mathcal{P}_{g}\,,$ (4c) $\displaystyle\mathcal{Q}_{g}$
$\displaystyle\rightarrow e^{-d\omega}(\mathcal{Q}_{g}+\mathcal{P}_{g}\sigma)\
,$ (4d)

the action (1) transforms as

$S_{CG}[\phi-Q\sigma,e^{2\sigma}g]=S_{CG}[\phi,g]-S_{CG}[Q\sigma,g]\ ,$ (5)

where $S_{CG}\equiv S_{L}(\mu=0)$ is the higher dimensional Coulomb gas
theory. The complete Liouville action (1) transforms as (5) when we fix
$Q=\frac{1}{b}$. Thus, the action is Weyl invariant up to a term that does not
affect the classical field equations. Semi-classically, the critical points of
the even-dimensional LCFT functional answer the problem of Uniformization,
i.e. does every compact even-dimensional manifold carry a conformal metric
with constant negative ${\cal Q}$-curvature.

We will work in four dimensions, where the GJMS (Paneitz) operator and the
${\cal Q}$-curvature read:

$\mathcal{P}_{g}=\Delta^{2}+2\nabla_{\mu}\left(R^{\mu\nu}-\frac{1}{3}g^{\mu\nu}R\right)\nabla_{\nu}\,,\quad\quad\mathcal{Q}_{g}=-\frac{1}{6}\left(\Delta
R+3R_{\mu\nu}R^{\mu\nu}-R^{2}\right)\ ,$ (6)

with $R_{\mu\nu}$ being the Ricci tensor of $\mathcal{M}$.

The aim of this work is to study the four-dimensional LCFT on a manifold with
a boundary. We will extend the LCFT action (1) to the boundary by extending
the GJMS operator and the ${\cal Q}$-curvature (6), and we will classify the
consistent conformal boundary conditions. Working on the ball and on the
conformally equivalent flat space with a plate boundary, we will calculate the
dimensions of the boundary conformal primary operators, the two- and three-
point functions of the displacement operator and the boundary conformal
anomaly coefficients. The paper is organized as follows. In section 2, we
introduce conformal geometry objects that are needed in order to extend to the
boundary the GJMS operator and the ${\cal Q}$-curvature, as well as to specify
the conformal boundary conditions. We construct the complete bulk plus
boundary action and its Weyl transformation analogous to (5). In subsection
2.3 we work on the ball, or equivalently on a flat space with a plate
boundary. We define the boundary vertex operators, derive their conformal
dimensions and analyse the restrictions on the boundary Liouville interaction.
In subsection 2.4, we classify the set of boundary conditions required for a
consistent variation of the action. In section 3, we construct the
displacement operator for the different boundary conditions, we show that only
two of the boundary conditions have the correct form consistent with the Green
function of the Liouville field and compute the B-type boundary anomaly
coefficients. In section 4 we conclude with an outlook. Details of the
displacement operator and the conformal mapping between the ball and a flat
space with a plate boundary are given in the appendices.

##  2 Boundary Liouville CFT in Four Dimensions

Let $\mathcal{M}$ be a four-dimensional manifold with a boundary
$\partial\mathcal{M}$. We will use Greek alphabet to index the bulk
coordinates $x^{\mu}$, and Roman alphabet to index the boundary coordinates
$x^{i}$. We will also denote the boundary position vector as $\mathbf{x}$. The
boundary is defined by the embedding map $x^{\mu}(\mathbf{x})$. The
coordinates are dimensionful and the metric is dimensionless. We denote the
normal vector to the boundary by $n^{\mu}$, and the normal derivative as
$\partial_{n}\equiv n^{\mu}\partial_{\mu}$. The induced metric on the boundary
(first fundamental form) is $\hat{g}_{ij}=\frac{\partial
x^{\mu}}{\partial\mathrm{x}^{i}}\frac{\partial
x^{\nu}}{\partial\mathrm{x}^{j}}\ g_{\mu\nu}$, where $g_{\mu\nu}$ is the bulk
metric. The extrinsic curvature (second fundamental form) is
$K_{ij}=\frac{\partial x^{\mu}}{\partial\mathrm{x}^{i}}\frac{\partial
x^{\nu}}{\partial\mathrm{x}^{j}}\nabla_{\mu}n_{\nu}$. We define the traceless
part of the extrinsic curvature as
$\mathring{K}_{ij}=K_{ij}-\frac{K}{3}g_{ij}$, where $K$ is its trace. For
correlation function computations, we will consider the upper half space
$\mathbb{H}\mathbb{R}^{4}$, which has a flat metric in the bulk and a plate
boundary, i.e. $y=0$, $x^{\mu}=(y,\mathbf{x})$.

###  2.1 Boundary Conformal Primary Operators

A set of conformally covariant boundary operators associated to the Paneitz
operator (GJMS operator in four dimensions) $\mathcal{P}_{g}$ with a general
background and boundary was presented in [31] (also see [32]). These boundary
operators read:

$\displaystyle\mathcal{B}_{g,0}\phi=\Phi^{(0)}$
$\displaystyle=\phi|_{\partial\mathcal{M}},$ (7a)
$\displaystyle\mathcal{B}_{g,1}\phi=\Phi^{(1)}$
$\displaystyle=\partial_{n}\phi|_{\partial\mathcal{M}},$ (7b)
$\displaystyle\mathcal{B}_{g,2}\phi=\Phi^{(2)}$
$\displaystyle=\left(-\hat{\Delta}+n^{\mu}n^{\nu}\nabla_{\mu}\partial_{\nu}+\frac{1}{3}K\partial_{n}\right)\phi\bigg{|}_{\partial\mathcal{M}},$
(7c) $\displaystyle\mathcal{B}_{g,3}\phi=\Phi^{(3)}$
$\displaystyle=\left(-\partial_{n}\Delta-2\hat{\Delta}\partial_{n}+2\mathring{K}^{ij}\hat{\nabla}_{i}\partial_{j}-\frac{2}{3}K\hat{\Delta}+\frac{2}{3}\partial^{i}K\partial_{i}+\mathcal{S}_{g,2}\partial_{n}\right)\phi\bigg{|}_{\partial\mathcal{M}}\
,$ (7d)

where

$\mathcal{S}_{g,2}=\frac{1}{2}\hat{R}+\frac{1}{6}R-R_{\mu\nu}n^{\mu}n^{\nu}-\frac{1}{3}K^{2}+\frac{1}{2}\mathring{K}^{ij}\mathring{K}_{ij}\
,$ (8)

and the quantities with hat belong purely to the boundary. In the case of a
plate boundary, it is easy to see the construction of these boundary conformal
primaries. The details are given in [22], here we describe the logic of the
derivation. Out of all the generators of the conformal group $SO(d,2)$,
$P_{n}$ (translations normal to the boundary), $K_{n}$ (boosts normal the
boundary), and $M_{\mu n}$ (rotations in a plane with one axis along and the
other transverse to the boundary), where $n$ denotes the direction normal to
the boundary, are broken. New boundary primary operators can be build by
acting on a bulk primary operator by powers of $P_{n}$ and ensuring that the
state is annihilated by all $K_{i}$ and not necessarily by $K_{n}$.
We are interested in quantities which have nice Weyl transformations for the
construction of the boundary action. The quantities which are useful to us
along with (7) are scalar invariants called T-curvatures:

$\displaystyle\mathcal{T}_{g,1}$ $\displaystyle=\frac{1}{3}K\,,$ (9a)
$\displaystyle\mathcal{T}_{g,2}$
$\displaystyle=\frac{1}{4}\hat{R}+\frac{1}{12}R-\frac{1}{2}R_{\mu\nu}n^{\mu}n^{\nu}+\frac{1}{18}K^{2}\,,$
(9b) $\displaystyle\mathcal{T}_{g,3}$
$\displaystyle=\frac{1}{6}\partial_{n}R-\frac{2}{3}\hat{\Delta}K-2K^{ij}\hat{R}_{ij}+K\hat{R}+\frac{1}{3}KK^{ij}K_{ij}-\frac{5}{27}K^{3}\
.$ (9c)

Under a Weyl transformation
$g_{\mu\nu}\to\tilde{g}_{\mu\nu}=e^{2\sigma}g_{\mu\nu}$, the T-curvatures and
the boundary operators transform as

$\mathcal{B}_{e^{2\sigma}g,k}=e^{-k\sigma}\mathcal{B}_{g,k},\quad\mathcal{T}_{e^{2\sigma}g,k}=e^{-k\sigma}\left(\mathcal{T}_{g,k}+\mathcal{B}_{g,k}\sigma\right),\quad\Phi^{(k)}\to
e^{-k\sigma}\left(\Phi^{(k)}-Q\sigma^{(k)}\right)\ \ \text{for }k=0,1,2,3\ ,$
(10)

where we denote

$\mathcal{B}_{g,k}\sigma|_{\partial\mathcal{M}}=\sigma^{(k)}\ .$ (11)

###  2.2 Boundary Liouville Action

In the presence of a boundary, we have to add boundary terms to the bulk
Liouville action in order to have a well defined variation of the action. We
would like to construct a classically Weyl invariant action such that fixing
the background metric yields a conformal field theory. Every term in the
boundary will contain a factor of $\sqrt{\hat{g}}$ coming from the volume
element. Under a Weyl transformation, we have $\sqrt{\hat{g}}\to
e^{-3\sigma}\sqrt{\hat{g}}$, hence we should choose the combinations of
$\Phi^{(k)}$ (7) and $\mathcal{T}_{g,k}$ (9) such that under Weyl
transformation it receives an overall factor of $e^{3\sigma}$. The Weyl
transformation, apart from multiplicative exponential factor, also produces
non-homogeneous terms in $\sigma$ (10,11). We have to make sure that such non-
homogeneous pieces are cancelled or vanish when boundary conditions are
imposed.

####  2.2.1 Extension of the Action to the Boundary

We would like to extend the bulk action (1) to the boundary. Consider first
the extension of the $\mathcal{Q}$-curvature, whose integral should be a
conformal invariant of a manifold with a boundary. The Gauss-Bonnet-Chern
formula for the Euler characteristic of a four-manifold with boundary reads
[33, 34]:

$\chi(\mathcal{M})=\frac{1}{8\pi^{2}}\int_{\mathcal{M}}d^{4}x\sqrt{g}\left(\mathcal{Q}_{g}+\frac{1}{4}W^{\mu\nu\sigma\rho}W_{\mu\nu\sigma\rho}\right)+\frac{1}{8\pi^{2}}\int_{\partial\mathcal{M}}d^{3}\mathrm{x}\sqrt{\hat{g}}\left(\mathcal{T}_{g,3}-\frac{2}{3}\mathring{K}_{i}^{\;j}\mathring{K}_{j}^{\;k}\mathring{K}_{k}^{\;i}\right)\
.$ (12)

Since the second term in both the bulk and boundary integrals are conformal
invariants, the integral

$\int_{\mathcal{M}}d^{4}x\sqrt{g}\
\mathcal{Q}_{g}+\int_{\partial\mathcal{M}}d^{3}y\sqrt{\hat{g}}\
\mathcal{T}_{g,3}\ ,$ (13)

is a conformal invariant of ${\cal M}$. Thus, $\mathcal{T}_{g,3}$ is a natural
extension of the $\mathcal{Q}$-curvature to the boundary.

Consider next the kinetic term in the bulk action. In [31], it was shown that

$\braket{\phi_{1},\phi_{2}}=\int_{\mathcal{M}}d^{4}x\sqrt{g}\
\phi_{1}\mathcal{P}_{g}\phi_{2}+\int_{\partial\mathcal{M}}d^{3}\mathrm{x}\sqrt{\hat{g}}\left(\Phi_{1}^{(0)}\Phi_{2}^{(3)}+\Phi_{1}^{(1)}\Phi_{2}^{(2)}\right)$
(14)

is a symmetric bilinear form and $\braket{\phi,\phi}$ is a conformal
invariant. Using above two results, we can extend the bulk Liouville action to
the boundary as:

$\displaystyle S_{L}[\phi,g]$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{\mathcal{M}}d^{4}x\sqrt{g}\left(\phi\mathcal{P}_{g}\phi+2Q\mathcal{Q}_{g}\phi+8\pi^{2}\mu
e^{4b\phi}\right)+$
$\displaystyle+\frac{1}{8\pi^{2}}\int_{\partial\mathcal{M}}d^{3}\mathrm{x}\sqrt{\hat{g}}\left(\Phi^{(0)}\Phi^{(3)}+\Phi^{(1)}\Phi^{(2)}+2Q\mathcal{T}_{g,3}\Phi^{(0)}+8\pi^{2}\mu_{B}e^{3b_{B}\Phi^{(0)}}\right),$
(15)

where $b_{B}$ and $\mu_{B}$ are dimensionless parameters, and the extension of
the Liouville potential to the boundary is trivial. The Coulomb gas action
($\mu=0,\ \mu_{B}=0$) has the Weyl transformation:

$S_{CG}[\phi-Q\sigma,e^{2\sigma}g]=S_{CG}[\phi,g]-S_{CG}[Q\sigma,g]-\frac{Q}{4\pi^{2}}\int_{\partial\mathcal{M}}dx^{3}\sqrt{\hat{g}}\
\left(\Phi^{(1)}-Q\ \sigma^{(1)}\right)\sigma^{(2)}\ .$ (16)

We will show later that the last boundary term vanishes upon imposing
consistent boundary conditions. Note, that the Liouville action is Weyl
invariant up to the classical contribution to the A-type anomaly term. If we
fix $Q=\frac{1}{b}$ and $b_{B}=b$, we find that the complete Liouville action
(2.2.1) transforms as (16). We therefore expect that in the semi-classical
limit $b\to 0$, the value of $b_{B}$ that also ensures the Weyl invariance at
the quantum level will satisfy $\lim_{b\to 0}b_{B}(b)\to b$.

###  2.3 Spectrum and $b_{B}$

In the $d$-dimensional Coulomb gas theory, the two point correlation function
of $\phi$ in the flat space reads:

$\braket{\phi(x)\phi(y)}=\frac{2}{d}\log\left(\frac{L}{|x-y|}\right)+\cdots\
,$ (17)

where $L$ is an IR regulator. The vertex operators
$V_{\alpha}(x)=e^{d\alpha\phi(x)}$ are conformal primaries with conformal
dimensions $\Delta_{\alpha}=d\alpha(Q-\alpha)$. Conformal invariance dictates
that the Liouville interaction term in (1) has the dimension $\Delta_{b}=d$,
which fixes the quantum corrected background charge to be $Q=b+\frac{1}{b}$.
Indeed, in the classical limit $b\to 0$, we have $Q\approx 1/b$.

Define boundary vertex operators as
$B_{\beta}(\mathbf{x})=e^{\beta\phi(\mathbf{x})}$. Consider the theory to be
defined on a manifold with topology of a ball555We can go from a manifold with
flat metric and a plate boundary to a 4-ball with an $\mathbb{S}^{3}$ boundary
via a singular Weyl transformation. See appendix B for which
$\chi(\mathcal{M})=1$. Thus, under a constant Weyl transformation
$\sigma(x)=\phi_{0}$, we have:

$S_{CG}[\phi+\phi_{0},e^{2\phi_{0}}g]=S_{CG}[\phi,g]+dQ\phi_{0}.$ (18)

Invariance of the two point function of $B_{\beta}$ under above shift imposes
$Q=2\beta/d$. Considering the fact that the dimension of $B_{\beta}$ is zero
when $\beta=0$, and the dimension formula should reduce to the known dimension
for analogous operator in two dimensions [35] fixes the dimension to

$\Delta_{\beta}=\beta\left(Q-\frac{2}{d}\beta\right)\ .$ (19)

Requiring that the boundary Liouville interaction term has the correct
dimension i.e. $\Delta_{(d-1)b_{B}}=d-1$ yields:

$b_{B}=\frac{1+b^{2}-\sqrt{1-b^{2}(6-8/d)+b^{4}}}{4(1-1/d)b}\ ,$ (20)

where we impose the condition that in the semi-classical limit
$b_{B}=b+O(b^{3})$. In $d=2$, formula (20) reduces to expected result
$b_{B}=b$ .

The formula (20) contains a square root whose argument is negative in a
following window

$b\in\left({\left(-\frac{2\sqrt{2}\sqrt{d^{2}-3d+2}}{d}-\frac{4}{d}+3\right)^{1/2}},\left({\frac{2\sqrt{2}\sqrt{d^{2}-3d+2}}{d}-\frac{4}{d}+3}\right)^{1/2}\right)\
.$ (21)

In this window, $b_{B}$ is complex. In $d=2$, this window vanishes. In $d=4$,
the window is
$b\in\left(\left(2-\sqrt{3}\right)^{1/2},\left(2+\sqrt{3}\right)^{1/2}\right)$
and it asymptotes to
$b\in\left(\left(3-2\sqrt{2}\right)^{1/2},\left(3+2\sqrt{2}\right)^{1/2}\right)$
at large $d$. Despite this fact, the dimension of the operator is real and is
equal to $d-1$, and hence its correlators are well defined.

###  2.4 Conformal Boundary Conditions

LCFT in $d$ dimensions is a $d$-derivative theory, where generally, one has to
define additional canonical fields and their conjugate momenta [19]. The
number of additional fields depends on the order of the derivatives in the
action, and we can have the usual Neumann and Dirichlet boundary conditions
for all of them. A four-derivative theory requires two canonical fields and
their conjugate momenta, hence we have four boundary conditions, two for each
conjugate pair.

We will study LCFT on the four-dimensional ball, which is conformally
equivalent to the upper half space $\mathbb{H}\mathbb{R}^{4}$. Performing a
Weyl transformation (see appendix B) to the upper half space results in
boundary conditions for $\phi$:

$\phi(x)=-2Q\log\left(|x|\right)+O(1),\quad|x|\to\infty\ .$ (22)

Note, that this is the boundary condition away from $y=0$ surface.

####  2.4.1 Variation of the Action and Boundary Conditions

In this subsection, we describe the consistent set of boundary conditions. For
an analysis of free theory without background charge see [22]. We denote
$\delta\Phi^{(k)}=0$ for $k=0,1$ as Dirichlet boundary conditions and
$\delta\Phi^{(3-k)}=0$ for $k=0,1$ as Neumann boundary conditions. We classify
all of them as:

$\displaystyle(D,D)$ $\displaystyle:\delta\Phi^{(0)}=\delta\Phi^{(1)}=0\qquad$
$\displaystyle(D,N)$ $\displaystyle:\delta\Phi^{(0)}=\delta\Phi^{(2)}=0$
$\displaystyle(N,D)$ $\displaystyle:\delta\Phi^{(3)}=\delta\Phi^{(1)}=0\qquad$
$\displaystyle(N,N)$ $\displaystyle:\delta\Phi^{(3)}=\delta\Phi^{(2)}=0\ .$

When we impose $\delta\Phi^{(k)}=0$ for $k>0$ then, unless the variation
problem dictates the value of $\Phi^{(k)}$, we are forced to impose
$\Phi^{(k)}=0$ since $\Phi^{(k)}$ carries a non-zero dimension. Also, in order
to preserve this boundary condition we have to restrict the set of Weyl
transformations (10) and impose
$\mathcal{B}_{g,k}\sigma|_{\partial\mathcal{M}}=\sigma^{(k)}=0$. We don’t have
to impose such a condition when $k=0$ since $\Phi^{(0)}$ is dimensionless and
$\Phi^{(0)}=C$, where $C$ is an arbitrary constant, is an allowed boundary
condition.

The boundary terms in the variation of the action (2.2.1) reads:

$\displaystyle\delta S$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{\partial\mathcal{M}}d^{3}\mathrm{x}\sqrt{\hat{g}}\left[\left(\Phi^{(3)}+Q\mathcal{T}_{g,3}+24\pi^{2}b_{B}\mu_{B}e^{3b_{B}\Phi^{(0)}}\right)\delta\Phi^{(0)}+\Phi^{(2)}\
\delta\Phi^{(1)}\right]\ .$ (23)

Using (23) we get following boundary equations for respective boundary
conditions:

$\displaystyle(D,D)$ $\displaystyle:\Phi^{(1)}=0,\quad\Phi^{(0)}=C\quad$
$\displaystyle\implies$ $\displaystyle\quad\sigma^{(1)}=0,$ (24a)
$\displaystyle(D,N)$ $\displaystyle:\Phi^{(2)}=0,\quad\Phi^{(0)}=C\quad$
$\displaystyle\implies$ $\displaystyle\quad\sigma^{(2)}=0,$ (24b)
$\displaystyle(N,D)$
$\displaystyle:\Phi^{(1)}=0,\quad\Phi^{(3)}+Q\mathcal{T}_{g,3}+24\pi^{2}b_{B}\mu_{B}e^{3b_{B}\Phi^{(0)}}=0$
$\displaystyle\implies$ $\displaystyle\quad\sigma^{(1)}=0,$ (24c)
$\displaystyle(N,N)$
$\displaystyle:\Phi^{(2)}=0,\quad\Phi^{(3)}+Q\mathcal{T}_{g,3}+24\pi^{2}b_{B}\mu_{B}e^{3b_{B}\Phi^{(0)}}=0$
$\displaystyle\implies$ $\displaystyle\quad\sigma^{(2)}=0\ .$ (24d)

After imposing boundary conditions (24), we see that (16) becomes:

$S_{CG}[\phi-Q\sigma,e^{2\sigma}g]=S_{CG}[\phi,g]-S_{CG}[Q\sigma,g]\ .$ (25)

The boundary conditions simplify when $\mu_{B}=0$ and restricting to the case
of a flat metric with a plate boundary we get:

$\displaystyle(D,D)$
$\displaystyle:\quad\partial_{n}\phi|_{\partial\mathcal{M}}=0,\quad\phi|_{\partial\mathcal{M}}=0,$
(26a) $\displaystyle(D,N)$
$\displaystyle:\quad\partial_{n}^{2}\phi|_{\partial\mathcal{M}}=0,\quad\phi|_{\partial\mathcal{M}}=0,$
(26b) $\displaystyle(N,D)$
$\displaystyle:\quad\partial_{n}\phi|_{\partial\mathcal{M}}=0,\quad\partial_{n}^{3}\phi|_{\partial\mathcal{M}}=0,$
(26c) $\displaystyle(N,N)$
$\displaystyle:\quad\left(\partial_{n}^{3}+3\partial_{n}\partial^{2}_{i}\right)\phi|_{\partial\mathcal{M}}=0,\quad\left(\partial_{n}^{2}-\partial_{i}^{2}\right)\phi|_{\partial\mathcal{M}}=0\
.$ (26d)

We will see in the next section that only the Liouville Green functions
corresponding to the boundary conditions $(D,N)$ and $(N,D)$ have the correct
conformal transformations.

##  3 Boundary Conformal Anomalies

###  3.1 The Stress Tensor and the Displacement Operator

The trace anomalies of a four-dimensional boundary conformal field theory take
the form [36]:

$\left<T^{\mu}_{\ \mu}\right>=\frac{1}{16\pi^{2}}\left(c\
W_{\mu\nu\eta\rho}^{2}-a\ E_{4}\right)+\frac{\delta(y)}{16\pi^{2}}\left(a\
E_{4}^{(\mathrm{bry})}-b_{1}\ \mathrm{tr}\mathring{K}^{3}-b_{2}\
\hat{g}^{\mu\nu}\mathring{K}^{\eta\rho}W_{\mu\nu\eta\rho}\right)\ .$ (27)

The $a$ and $c$ anomaly coefficients of the four-dimensional LCFT were
computed in [20, 2]. Our aim is to calculate the $b_{1}$ and $b_{2}$ anomaly
coefficients. The $b_{1}$ and $b_{2}$ anomaly coefficients can be read from
the two- and three-point functions of the displacement operator $D^{\nu}$ [37,
38], defined by

$\nabla_{\mu}T^{\mu\nu}(y,\mathbf{x})=\delta(y)D^{\nu}(\mathbf{x})\ .$ (28)

In the rest of this section, all the results presented are for flat space with
plate boundary, where only the normal component $D\equiv D^{n}$ of the
displacement operator is non-zero (see Appendix A). The displacement operators
in this case reads:

$T^{nn}(y,\mathbf{x})|_{y=0}=D(\mathbf{x})\ ,$ (29)

and its two- and three-point functions take the form [39]:

$\left<D(\mathbf{x})D(\mathbf{0})\right>=\frac{15\
b_{2}}{2\pi^{4}|\mathbf{x}|^{8}},\quad\left<D(\mathbf{x})D(\mathbf{x^{\prime}})D(\mathbf{0})\right>=\frac{35\
b_{1}}{2\pi^{6}|\mathbf{x}|^{4}|\mathbf{x^{\prime}}|^{4}|\mathbf{x-x^{\prime}}|^{4}}\
.$ (30)

In the higher derivative, $\Delta^{2}\phi=0$, free scalar theory without a
background charge, the stress energy tensor reads [20, 21]:

$8\pi^{2}T_{\mu\nu}|_{Q=0}=\frac{4}{3}\partial_{\mu}\partial_{\nu}\partial^{\alpha}\phi\partial_{\alpha}\phi+4\partial_{\mu}\partial_{\nu}\phi\partial^{2}\phi-\frac{8}{3}\partial_{\mu}\partial^{\alpha}\phi\partial_{\nu}\partial_{\alpha}\phi-2\partial_{\mu}\partial^{2}\phi\partial_{\nu}\phi-2\partial_{\mu}\phi\partial_{\nu}\partial^{2}\phi\hskip
113.81102pt\\\ \hskip 213.39566pt+\
\delta_{\mu\nu}\left(\frac{2}{3}\partial_{\mu}\phi\partial^{\mu}\partial^{2}\phi+\frac{2}{3}\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\partial^{\nu}\phi-\frac{1}{2}\partial^{2}\phi\partial^{2}\phi\right)\
.$

The contribution to the stress tensor from the background charge $Q$ was
computed in [2], and the full stress energy tensor reads:

$T_{\mu\nu}=T_{\mu\nu}|_{Q=0}+\frac{Q}{18\Omega_{4}}\left(4\Delta\partial_{\mu}\partial_{\nu}-\
\delta_{\mu\nu}\Delta^{2}\right)\phi\ .$ (31)

The $Q$ dependent contribution to $b_{1}$ and $b_{2}$ anomaly coefficients
(30) should be visible in the classical limit ($Q\to\infty$). The only such
term that appears in our action (2.2.1), by construction, is the A-type
anomaly one. Thus, we do not expect any $Q$ dependent term in the $b_{1}$ and
$b_{2}$ anomaly and hence we will ignore $Q$ dependent terms from the
displacement operator during computation.

let us first rewrite the normal-normal component of the above expression with
separating the derivatives with respect to normal and tangent coordinates to
the boundary ($y=0$):

$8\pi^{2}T_{yy}=\partial^{2}_{y}\phi\partial^{2}_{y}\phi-2\partial_{y}\phi\partial^{3}_{y}\phi+2\partial_{i}\partial_{y}^{2}\phi\partial_{i}\phi-\partial_{i}^{2}\phi\partial_{j}^{2}\phi+2\partial_{i}^{2}\phi\partial_{y}^{2}\phi\hskip
142.26378pt\\\ \hskip
142.26378pt-\frac{10}{3}\partial_{i}^{2}\partial_{y}\phi\partial_{y}\phi+\frac{2}{3}\partial_{i}\partial_{j}\phi\partial_{i}\partial_{j}\phi+\frac{2}{3}\partial_{i}\partial_{j}^{2}\phi\partial_{i}\phi-\frac{4}{3}\partial_{a}\partial_{y}\phi\partial_{a}\partial_{y}\phi\
.$

Taking the boundary limit of the expression above and plugging (26), we get
the form of the displacement operators for all the boundary conditions:

$\displaystyle 8\pi^{2}\ D^{DD}$
$\displaystyle=\partial^{2}_{y}\phi\partial^{2}_{y}\phi,$ (32a) $\displaystyle
8\pi^{2}\ D^{NN}$
$\displaystyle=2\partial^{2}_{y}\phi\partial^{2}_{y}\phi-\frac{8}{9}\partial^{3}_{y}\phi\partial_{y}\phi+\frac{8}{3}\partial_{i}\partial_{y}^{2}\phi\partial_{i}\phi+\frac{2}{3}\partial_{i}\partial_{j}\phi\partial_{i}\partial_{j}\phi-\frac{4}{3}\partial_{i}\partial_{y}\phi\partial_{i}\partial_{y}\phi,$
(32b) $\displaystyle 8\pi^{2}\ D^{DN}$
$\displaystyle=-2\partial_{y}\phi\partial_{y}^{3}\phi-\frac{10}{3}\partial_{i}^{2}\partial_{y}\phi\partial_{y}\phi-\frac{4}{3}\partial_{i}\partial_{y}\phi\partial_{i}\partial_{y}\phi,$
(32c) $\displaystyle 8\pi^{2}\ D^{ND}$
$\displaystyle=\partial_{y}^{2}\phi\partial_{y}^{2}\phi+2\partial_{i}\partial_{y}^{2}\phi\partial_{i}\phi-\partial_{i}^{2}\phi\partial_{j}^{2}\phi+2\partial_{i}^{2}\phi\partial_{y}^{2}\phi+\frac{2}{3}\partial_{i}\partial_{j}\phi\partial_{i}\partial_{j}\phi+\frac{2}{3}\partial_{i}\partial_{j}^{2}\phi\partial_{i}\phi\
.$ (32d)

With these expressions, we can compute the $b_{1}$ and $b_{2}$ anomaly
coefficients.

###  3.2 Green Function and Boundary Conformal Anomalies

The Liouville Green function in four dimensions is a solution to the equation:

$\Delta_{x}^{2}\ G_{0}(x,x^{\prime})=\delta^{(4)}(x-x^{\prime})\ .$ (33)

The solution is not unique as we can add to it the solutions of the
homogeneous equation, $\Delta_{x}^{2}\ F(x,x^{\prime})=0$, which are important
when we impose boundary conditions. Denote by
$r^{2}=|\textbf{x}-\textbf{x}^{\prime}|^{2}+(y-y^{\prime})^{2}$ and
$\bar{r}^{2}=|\textbf{x}-\textbf{x}^{\prime}|^{2}+(y+y^{\prime})^{2}$. The
solution to (33) is

$G_{0}(x,x^{\prime})=-\frac{1}{2}\log\left(\frac{r}{L}\right)\ ,$ (34)

where $L$ is an IR cutoff and a solution to the homogeneous equation is

$F(x,x^{\prime})=-\frac{1}{2}\log\left(\frac{\bar{r}}{L}\right)\ .$ (35)

For DN and ND boundary conditions, we have

$\braket{\phi(x)\phi(x^{\prime})}=-\frac{1}{2}\log\left(\frac{r}{L}\right)\pm\frac{1}{2}\log\left(\frac{\bar{r}}{L}\right)$
(36)

with ‘$+$’(’$-$’) for DN (ND) boundary conditions.

####  3.2.1 Consistent Boundary Conditions

The subgroup of conformal symmetry that preserves the boundary dictates that
the Green function in four dimensions should have the form [22]:

$\braket{\phi(x)\phi(x^{\prime})}=f(\zeta),\quad\text{with}\quad\zeta=\frac{r^{2}}{4yy^{\prime}}\
,$ (37)

where $\zeta$ is the conformally invariant cross ratio. The homogeneous
equation then becomes

$\zeta^{2}(1+\zeta)^{2}f^{\prime\prime\prime\prime}(\zeta)+6\
\zeta(1+\zeta)(1+2\zeta)f^{\prime\prime\prime}(\zeta)+6\
\left(1+6\zeta(1+\zeta)\right)f^{\prime\prime}(\zeta)+12\
(1+2\zeta)f^{\prime}(\zeta)=0\ ,$ (38)

where prime is a $\zeta$ derivative. This is a third order ODE and we expect
two additional homogeneous solutions apart from the one given above. These
solutions are $1/\zeta$ and $1/(1+\zeta)$. If we consider these power-law
solutions along with the log solution (35) it is possible to satisfy all the
boundary conditions. The problem with considering the power-law solutions is
that $\phi$ is not a conformal primary but the vertex operator
$e^{\alpha\phi}$ is, and we expect its two-point function to transform as a
two-point function of a conformal primary. This condition leaves the
logarithmic solutions as the only choice and prevents the power-law solutions.
Without the power-law solutions, it is not possible to satisfy the $(D,D)$ and
$(N,N)$ boundary conditions, leaving the $(N,D)$, $(D,N)$ as the only
consistent ones.

####  3.2.2 Boundary Anomaly Coefficients

Using equation (36), equation (32d), and equation (32c), we can compute the
two- and three-point functions of the displacement operator using wicks
theorem. We get:

$\braket{D^{DN(ND)}(\textbf{x})\
D^{DN(ND)}(0)}=-\frac{4}{\pi^{4}|\textbf{x}|^{8}}\quad\implies\quad
b_{2}=-\frac{8}{15}\ ,$ (39)

and

$\braket{D^{DN(ND)}(\textbf{x})\ D^{DN(ND)}(\textbf{x}^{\prime})\
D^{DN(ND)}(0)}=-\frac{52}{9\pi^{6}|\textbf{x}|^{4}|\textbf{x}^{\prime}|^{4}|\textbf{x}-\textbf{x}^{\prime}|^{4}}\quad\implies\quad
b_{1}=-\frac{104}{315}\ .$ (40)

In [40], it was observed that $b_{2}=8c$ with the choice of normalization used
in (27). It was later understood to be only valid for a free theory and that a
marginal deformation can modify this relation [37]. The computation of the $c$
anomaly coefficient was done for $\Delta^{2}\phi=0$ theory in [20] and we
indeed find that $8c=-8/15$ (after adopting it to our normalization
convention), the same as $b_{2}$ (39).

##  4 Outlook

Adding boundaries to CFTs often reveals a rather rich structure, and as in the
case of two-dimensional LCFT, it is imperative to study boundary LCFT in
higher dimensions. In this work we made first steps in the study of the four-
dimensional case. Adding boundaries requires a classification of the
consistent boundary conditions for these higher derivative theories and
appropriate extensions of the GJMS operators and the ${\cal Q}$-curvature to
manifolds with boundary, as done in this work. Judging by the two-dimensional
boundary LCFT, the extensions to the boundary of higher-dimensional LCFTs is
likely to reveal new structures such as that of the correlations functions of
boundary operators, the conformal boundary anomalies and new duality
relations. Another route to take is the study of line and surface operators
that will reveal new aspects of the higher-dimensional LCFTs. Being nonlocal
field theories, the odd-dimensional LCFTs on manifolds with boundary can add
another layer of interesting QFT structure. The stress energy tensor of these
theories is nonlocal, and thus the whole realization of the conformal algebra.
This also raises a question about the proper definition and the classification
of the anomalies.

RG monotonicity theorems state that certain numbers that characterize the UV
and IR fixed points of relativistic quantum field theories decrease along the
flow connecting them. Such numbers are the central charge $c$ of a two-
dimensional CFT [41], the A-type Weyl anomaly coefficient $a$ of four-
dimensional CFTs [42], the regularized free energy of a three-dimensional CFT
on the sphere $F$ [43] and the one-dimensional thermal free energy $g$ [44].
The proof of typical RG monotonicity theorems assumes unitarity, while LCFTs
in dimensions higher than two are non-unitary. Yet, it may still be that such
RG constraints can be generalized also to non-unitary cases, and the fact that
higher-dimensional LCFTs are tractable makes them a valuable framework to
study this question. In [22], a rich structure of boundary RG flows is
presented by deforming the boundary action with specific relevant boundary
operator for specific boundary conditions. It will be valuable to consider the
boundary RG flows of higher dimensional LCFTs.

Finally, the original motivation to introducing higher dimensional LCFTs has
been for the construction of an effective theory for incompressible fluid
turbulence [3, 4]. This proposed application of LCFTs requires the presence of
boundaries: one can think of the forcing scale as the boundary of the inertial
range of scales where a universal description of turbulence is expected. Thus,
we expect that progress in understanding higher-dimensional boundary LCFTs as
initiated in the current work, in particular in the nonlocal three-dimensional
case, is likely to provide new insights to the out of equilibrium dynamics of
statistical turbulence.

##  5 Acknowledgements

AG would like to thank S. Biswas for useful discussions. This work is
supported in part by Israeli Science Foundation excellence center, the US-
Israel Binational Science Foundation and the Israel Ministry of Science.

##  Appendix A Ward Identities for Displacement Operator

In this appendix, we will present the ward identities of the displacement
operator relevant to us (see [38, 45] for more details). We consider a
boundary defined by $y=0$ but not necessarily a flat metric and adopt the
notation, $x^{\mu}=(y,x^{i})$. In the Gaussian normal coordinates, we can
write metric near the boundary as:

$ds^{2}=dy^{2}+\left(h_{ij}-2y\ K_{ij}+\sum_{n=2}^{\infty}y^{n}\
q^{(n)}_{ij}\right)dx^{i}dx^{j}\ ,$ (41)

where $q^{(n)}_{\mu\nu}=\partial_{y}^{n}g^{\mu\nu}/\Gamma(n+1)$. Let
$e^{\mu}_{i}=\partial x^{\mu}/\partial x^{i}$,
$h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}$ be the transverse metric where
$n_{\mu}$ is the normal, and $\gamma_{ij}=h_{\mu\nu}e^{\mu}_{i}e^{\nu}_{j}$ be
the induced metric.

Define the generating functional $W=-\text{log}\
Z\left[g_{\mu\nu},x^{\mu}(x^{i})\right]$ where $x^{\mu}(x^{i})$ is general
embedding of the boundary, the variation of which reads:

$\delta W=\frac{1}{2}\int dx^{d}\ \sqrt{g}\braket{T^{\mu\nu}_{bulk}}\ \delta
g_{\mu\nu}+\frac{1}{2}\int dx^{d-1}\sqrt{h}\
\left[\braket{T_{bdy}^{\mu\nu}}\delta h_{\mu\nu}+2\braket{D_{\mu}}\delta
x^{\mu}+\braket{J^{(n)\mu\nu}}\ \delta q^{(n)}_{\mu\nu}\right]\ .$ (42)

The full stress tensor is $T^{\mu\nu}=T^{\mu\nu}_{bulk}+\delta(y)\
T_{bdy}^{\mu\nu}$.

A small reparameterization of the boundary is defined as follows:

$\delta g_{\mu\nu}=\delta x^{\mu}=0,\qquad\delta x^{i}=-\zeta^{i},\qquad\delta
x^{\mu}=\zeta^{i}\partial_{i}x^{\mu}\ .$ (43)

This gives

$\delta W=-\int d\textbf{x}^{d-1}\sqrt{h}\ \left[\
\braket{D_{\mu}}e^{\mu}_{i}\zeta^{i}+\cdots\right]\quad\implies\quad\braket{D_{\mu}}e^{\mu}_{i}=0\
,$ (44)

which means that only non-zero components of $\braket{D_{\mu}}$ are normal to
the boundary.

Invariance under bulk diffeomorphism:

$\delta x^{\mu}=-\epsilon^{\mu},\quad\delta
g_{\mu\nu}=\nabla_{\mu}\epsilon_{\nu}+\nabla_{\nu}\epsilon_{\mu},\quad\delta
n_{\mu}=n^{\alpha}\nabla_{\alpha}\epsilon^{\mu},\quad\delta{x}^{i}=0\ ,$ (45)

implies

$\delta W=-\int dx^{d}\sqrt{g}\
\epsilon_{\mu}\nabla_{\nu}\braket{T^{\mu\nu}_{bulk}}+\int
d\textbf{x}^{d-1}\sqrt{h}\left(\braket{T^{\mu\nu}_{bulk}}n_{\mu}\epsilon_{\nu}+\braket{T^{\mu\nu}_{bdy}}\frac{\delta
h_{\mu\nu}}{2}-\epsilon_{\mu}\braket{D^{\mu}}+\cdots\right)\ ,$ (46)

with $\delta W=0$. Thus, the bulk term immediately gives us

$\nabla_{\nu}\braket{T^{\mu\nu}_{bulk}}=0\ .$ (47)

Using $h_{\mu\nu}=h^{\alpha}_{\mu}h^{\beta}_{\nu}\ g_{\alpha\beta}$ and
$h^{\mu}_{\nu}=\delta^{\mu}_{\nu}+n^{\mu}n_{\nu}$, we get

$\displaystyle\braket{T^{\mu\nu}_{bdy}}\ \delta
h_{\mu\nu}=\braket{T^{\mu\nu}_{bdy}}\left(h^{\alpha}_{\mu}h^{\beta}_{\nu}\
\delta g_{\mu\nu}+\delta h^{\alpha}_{\mu}h_{\alpha\nu}+\delta
h^{\alpha}_{\nu}h_{\alpha\mu}\right)=2\braket{T^{\mu\nu}_{bdy}}\left(h_{\nu}^{\alpha}\hat{\nabla}_{\mu}\epsilon_{\alpha}+n_{\mu}h_{\alpha\nu}\nabla_{n}\epsilon^{\alpha}\right)\
.$

Making use of the above derivation, we get (we only look at the boundary
terms)

$\displaystyle\delta W_{bdy}$ $\displaystyle=\int
dy^{d-1}\sqrt{h}\left(\braket{T^{\mu\nu}_{bulk}}n_{\mu}\epsilon_{\nu}+\braket{T^{\mu\nu}_{bdy}}\left(h_{\nu}^{\alpha}\hat{\nabla}_{\mu}\epsilon_{\alpha}+n_{\mu}h_{\alpha\nu}\nabla_{n}\epsilon^{\alpha}\right)-\epsilon_{\mu}\braket{D^{\mu}}+\cdots\right)\
,$ (48)

which after integrating by parts becomes

$\delta
W_{bdy}=\int_{\partial\mathcal{M}}\left(\braket{T^{\mu\nu}_{bulk}}n_{\mu}\epsilon_{\nu}-\left(\hat{\nabla}_{\mu}h^{\alpha}_{\nu}\right)\braket{T^{\mu\nu}_{bdy}}\epsilon_{\alpha}-\left(\hat{\nabla}_{\mu}\braket{T^{\mu\nu}_{bdy}}\right)\hat{\epsilon}_{\nu}+\braket{T^{\mu\nu}_{bdy}}n_{\mu}\nabla_{n}\hat{\epsilon}_{\nu}-\epsilon_{\mu}\braket{D^{\mu}}+\cdots\right)\
.$ (49)

Here we ignored the total derivatives. Using
$h_{\mu}^{\alpha}K_{\alpha\nu}=K_{\mu\nu}$, we can write

$\hat{\nabla}_{\mu}h^{\alpha}_{\nu}=K^{\alpha}_{\mu}n_{\nu}+K_{\mu\nu}n^{\alpha}\
.$ (50)

Using this and imposing $\delta W_{bdy}=0$, we get

$\braket{T^{\mu\nu}_{bulk}}n_{\mu}\epsilon_{\nu}-\braket{T^{\mu\nu}_{bdy}}K_{\mu\nu}n^{\alpha}\epsilon_{\alpha}-\left(\hat{\nabla}_{\mu}\braket{T^{\mu\nu}_{bdy}}\right)h^{\alpha}_{\nu}{\epsilon}_{\alpha}+\braket{T^{\mu\nu}_{bdy}}n_{\nu}\left[\nabla_{n}\left(h^{\alpha}_{\mu}{\epsilon}_{\alpha}\right)+K^{\alpha}_{\mu}\epsilon_{\alpha}\right]-\epsilon_{\mu}\braket{D^{\mu}}+\cdots=0.$
(51)

This equation is valid for arbitrary $\epsilon^{\mu}$, let us choose
$\epsilon^{\mu}=n^{\mu}$. Then using $n^{\alpha}K_{\alpha\nu}=0$ and
$h^{\alpha}_{\mu}{n}_{\alpha}=0$ yields

$n_{\mu}\braket{D^{\mu}}=\braket{T^{\mu\nu}_{bulk}}n_{\mu}n_{\nu}+\braket{T^{\mu\nu}_{bdy}}\
K_{\mu\nu}+\cdots$ (52)

Now we choose $\epsilon^{\mu}=h^{\mu}_{\nu}$, which yields

$\displaystyle\hat{\nabla}_{\mu}\braket{T^{\mu\nu}_{bdy}}$
$\displaystyle=\braket{T^{\alpha\mu}_{bulk}}n_{\alpha}h_{\mu}^{\nu}\
+\braket{T^{\alpha\mu}_{bdy}}n_{\alpha}K^{\nu}_{\mu}+\cdots$ (53)

In the case of a flat space with a plate boundary, $K_{\mu\nu}=q^{(n)}=0$. In
this case we can define

$D(y^{a})=-T^{\mu\nu}_{bulk}\ n_{\mu}n_{\nu}=-T_{yy}(0,y)\ ,$ (54)

where $D=D_{x}=-D^{\mu}n_{\mu}$.

##  Appendix B Conformal Transformation Between the Ball and Half Euclidean
Space

Define a four-dimensional ball

$|w|^{2}=w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+w_{4}^{2}\leq 1,\qquad\text{with}\qquad
ds^{2}=dw_{1}^{2}+dw_{2}^{2}+dw_{3}^{2}+dw_{4}^{2}\ ,$ (55)

and let the flat four-dimensional half space be defined by coordinates

$-\infty<x^{i}<\infty,\quad y>0\qquad i=1,2,3\ .$ (56)

The coordinate transformation between these two spaces is:

$(y,\ {x}^{1},\ {x}^{2},\
{x}^{3},)=\frac{1}{w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+(w_{4}-1)^{2}}\ (1-|w|^{2},\
2w_{1},\ 2w_{2},\ 2w_{3})\ .$ (57)

Note, that in the above transformation $|w|\to 1\implies y\to 0$. The ball
metric in the $(y,x^{i})$ coordinates reads:

$\displaystyle
g_{\alpha\beta}=e^{2\Omega}\delta_{\alpha\beta},\quad\text{with}\quad\Omega$
$\displaystyle=-\
\text{log}\left(\frac{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}+(1+y)^{2}}{2}\right)=-\text{log}\left(\frac{|x|^{2}+2y+1}{2}\right)$
$\displaystyle\approx-2\ \text{log}\ |x|+O(1),\qquad\text{as }|x|\to\infty\ .$
(58)

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# A differential Harnack inequality for noncompact evolving hypersurfaces

Stephen Lynch

###### Abstract.

We prove a differential Harnack inequality for noncompact convex hypersurfaces
flowing with normal speed equal to a symmetric function of their principal
curvatures. This extends a result of Andrews for compact hypersurfaces. We
assume that the speed of motion is one-homogeneous, uniformly elliptic, and
suitably ‘uniformly’ inverse-concave as a function of the principal
curvatures. In addition, we assume the hypersurfaces satisfy pointwise
scaling-invariant gradient estimates for the second fundamental form. For many
natural flows all of these hypotheses are met by any ancient solution which
arises as a blow-up of a singularity.

##  1\. Introduction

Differential Harnack inequalities for solutions to parabolic PDE were
introduced by Li and Yau in their seminal paper [LY86]. Let
$u:M\times(0,T]\to\mathbb{R}$ denote a bounded positive solution to the heat
equation, where $(M,g)$ is a compact Riemannian manifold with nonnegative
Ricci curvature. The Li–Yau inequality asserts that

$\partial_{t}u-\frac{|\nabla u|^{2}}{u}+\frac{n}{2t}u\geq 0.$

This inequality is saturated by the Euclidean heat kernel. Upon integration
over an appropriate path in spacetime, it recovers the classical Harnack
inequality for solutions to the heat equation: for $0<t_{0}<t_{1}\leq T$ we
have

$u(x_{1},t_{1})\geq\bigg{(}\frac{t_{0}}{t_{1}}\bigg{)}^{n/2}\exp\bigg{(}-\frac{d_{g}(x_{0},x_{1})^{2}}{4(t_{1}-t_{0})}\bigg{)}\,u(x_{0},t_{0}).$

Analogues of the Li–Yau differential Harnack inequality have since been found
for many other equations, including geometric flows. Hamilton found remarkable
Harnack inequalities for the curvature of solutions to the Ricci flow with
nonnegative curvature operator [Ham93] (see also [Bre09]), and for weakly
convex solutions of the mean curvature flow [Ham95]. These inequalities play a
fundamental role in understanding singularity formation, and have therefore
had profound implications in geometry.

Chow proved the analogue of Hamilton’s Harnack inequality for compact,
strictly convex hypersurfaces flowing by powers of their Gauss curvature
[Cho91]. Andrews generalised this result to a large class of fully nonlinear
flows [And94]. Solutions to these flows move with speed equal to a general
symmetric homogeneous function of their principal curvatures. Andrews’
estimate applies, in particular, when the speed function is homogeneous of
degree one and inverse-concave.

Solutions to curvature flows which arise as dilations of singularities are
often noncompact, and it is desirable to have a Harnack inequality which
applies to these. In the present paper we extend Andrews’ differential Harnack
inequality to noncompact convex hypersurfaces, provided the speed of motion is
homogeneous of degree one, uniformly elliptic, and suitably ‘uniformly’
inverse-concave. In addition, we need to assume the hypersurfaces satisfy
pointwise scaling-invariant gradient estimates. For certain flows all of these
hypotheses are known to be satisfied by any ancient solution which arises as a
blow-up of a singularity. For example, our Harnack inequality applies to blow-
ups of compact embedded solutions to the flows introduced in [BH17] and
[Lyn22].

Let us further discuss the flow introduced in [BH17]. There Brendle and
Huisken studied domains in a Riemannian $(n+1)$-manifold whose boundaries move
inward with speed equal to

$\gamma(\lambda)=\bigg{(}\sum_{i<j}\frac{1}{\lambda_{i}+\lambda_{j}}\bigg{)}^{-1},$

where the $\lambda_{i}$ are the principal curvatures. By implementing a
surgery procedure for this flow, they were able to classify compact Riemannian
manifolds with strictly two-convex boundary and nonnegative curvature in the
sense that $R_{ikik}+R_{jkjk}\geq 0$. Namely, these spaces are all
diffeomorphic to a standard ball or 1-handlebody. In forthcoming work with
Cogo and Vičánek Martínez we completely classify the ancient solutions which
can arise as blow-ups at a singularity of this flow—these are the shrinking
round sphere $S^{n}$, the shrinking round cylinder $\mathbb{R}\times S^{n-1}$,
and the unique rotationally symmetric translating soliton. The corresponding
result for mean curvature flow is due to Brendle and Choi [BC19, BC21]. The
differential Harnack inequality proven in this paper is an important
ingredient in our proof.

###  1.1. Main results

Let $\Gamma\subset\mathbb{R}^{n}$ denote an open, symmetric (under
permutations), convex cone. We assume that $\Gamma$ contains the positive cone
$\mathbb{R}^{n}_{+}$. Fix a function $\gamma:\Gamma\to\mathbb{R}$ which is
smooth, positive, symmetric, strictly increasing in each argument, and
homogeneous of degree one.

We consider evolving immersions $F:M\times I\to\mathbb{R}^{n+1}$,
$I\subset\mathbb{R}$, which satisfy the evolution equation

(1) $\partial_{t}F(x,t)=-G(x,t)\nu(x,t),$

for every $(x,t)\in M\times I$, where $G(x,t)=\gamma(\lambda(x,t))$ and
$\nu(x,t)$ is the outward unit normal. We write $\lambda(x,t)$ for the
principal curvatures of $F$, i.e. the eigenvalues of the Weingarten map

$A(X)=D_{X}\nu,$

at $(x,t)$. These will be labeled so that
$\lambda_{1}\leq\dots\leq\lambda_{n}$. A solution to (1) is called uniformly
$k$-convex if

$\inf_{M\times I}\frac{\lambda_{1}+\dots+\lambda_{k}}{H}>0,$

where $H$ is the mean curvature. The metric induced on $M$ by the immersion
$F(\cdot,t)$ will be denoted $g=g(t)$.

The function $\gamma$ gives rise to a smooth, $O(n)$-invariant function on the
space of symmetric matrices with eigenvalues in $\Gamma$. We also denote this
function $\gamma$. We say that $\gamma$ is (strictly) inverse-concave if
$\lambda\mapsto-\gamma(\lambda^{-1})$ is (strictly) concave on
$\mathbb{R}^{n}_{+}$ or, equivalently, if $A\mapsto-\gamma(A^{-1})$ is
(strictly) concave on the space of positive-definite symmetric matrices.
Inverse-concavity is equivalent to the pointwise inequality

(2) $\bigg{(}\frac{\partial^{2}\gamma}{\partial A_{ij}\partial
A_{kl}}(A)+2\frac{\partial\gamma}{\partial
A_{ik}}(A)A^{-1}_{jl}\bigg{)}S_{ij}S_{kl}\geq 0,$

for positive-definite symmetric $A$ and symmetric $S$.

We now state the Harnack inequality. As mentioned above, this is new when $M$
is noncompact. For compact $M$, the result was proven in [And94] under more
general hypotheses—when $M$ is compact (3), (4) and (5) are unnecessary.

######  Theorem 1.1.

Suppose $\gamma$ is inverse-concave. Let $F:M\times[0,T]\to\mathbb{R}^{n+1}$
be a complete solution to (1) which satisfies $A\geq 0$. We assume there are
positive constants $C$ and $\varepsilon$ such that at each point in
$M\times[0,T]$, with respect to an orthonormal frame, we have

(3) $C^{-1}g_{ij}\leq\frac{\partial\gamma}{\partial A_{ij}}(A)\leq Cg_{ij}.$

and

(4) $\bigg{(}\frac{\partial^{2}\gamma}{\partial A_{ij}\partial
A_{kl}}(A)+2\frac{\partial\gamma}{\partial A_{ik}}(A)(A+\varepsilon
Gg)^{-1}_{jl}\bigg{)}S_{ij}S_{kl}\geq 0$

for every symmetric $S$. In addition, we assume bounded curvature

$\sup_{M\times[0,T]}G<\infty$

and the gradient estimates

(5) $\sup_{M\times[0,T]}G^{-2}|\nabla A|+G^{-3}|\nabla^{2}A|<\infty.$

Then, for every $(x,t)\in M\times(0,T]$ and $V\in T_{x}M$, we have

(6) $\partial_{t}G+2\langle\nabla G,V\rangle+A(V,V)+\frac{G}{2t}\geq 0.$

If $\gamma$ is convex then (4) is unnecessary and instead of (5) it suffices
to assume

$\sup_{M\times[0,T]}|\nabla G|+|\partial_{t}G|<\infty.$

Let us comment on the hypotheses (3), (4) and (5). The evolution of the
Harnack quantity contains terms involving the Hessian of $\gamma$, which need
to be overcome in order to establish (6) using the maximum principle. If the
solution is noncompact, we also need to be able to localise by introducing an
auxhiliary function which grows at infinity. It is (3) and (4), and the
gradient estimates (5), which let us achieve both of these things
simultaneously. We refer to Remark 4.1 for further discussion.

As stated in Theorem 1.1, if $\gamma$ is convex then the assumption (4) is
unnecessary and (5) can be weakened substantially. In fact, in this case the
terms in the evolution of the Harnack quantity depending on the Hessian of
$\gamma$ have a favourable sign. This means Hamilton’s proof for the mean
curvature flow applies almost verbatim.

We now introduce natural conditions under which (3), (4) and (5) are met. This
leads to Theorem 1.2 below.

_Pointwise gradient estimates for ancient solutions._ A solution to (1) is
called ancient if it exists for all $t\in(-\infty,T]$. Ancient solutions are
of great interest, since they arise as models for singularity formation via
rescaling.

When $\gamma$ is convex or concave, by work of Brendle and Huisken [BH17], a
pointwise gradient estimate of the form

$G^{-2}|\nabla A|+G^{-3}|\nabla^{2}A|\leq\Lambda(n,\gamma,C,\alpha)$

holds for convex ancient solutions of (1) which satisfy (3) and are also
$\alpha$-noncollapsing.

The (interior) $\alpha$-noncollapsing property for solutions to (1) is a form
of quantitative embeddedness. Let $F(M,t)=M_{t}$ and suppose
$M_{t}=\partial\Omega_{t}$, where $\Omega_{t}$ is an open subset of
$\mathbb{R}^{n+1}$. We say the solution $M_{t}$ is $\alpha$-noncollapsing if
there is a time-independent constant $\alpha>0$ such that $\Omega_{t}$ admits
an inscribed ball of radius $\alpha G(x,t)^{-1}$ for each $x\in M_{t}$. In
[ALM13] it was shown that for compact solutions and $\gamma$ concave, the
noncollapsing property is preserved forward in time. When we do not need to
refer to $\alpha$ we simply say that $M_{t}$ is noncollapsing.

_Uniform inverse-concavity._ For each integer $0\leq m\leq n$, we define

$\Gamma_{+}^{m}=\\{\sigma(\lambda)\in\mathbb{R}^{n}:\lambda=(0,\lambda_{1},\dots,\lambda_{m}),\;\min_{i}\lambda_{i}>0,\;\sigma\in
P_{n}\\},$

where $P_{n}$ is the group of permutations on $n$ elements. For $1\leq m\leq
n-1$, $\Gamma_{+}^{m}$ is the union of all of the $m$-dimensional facets of
$\partial\Gamma_{+}^{n}$. Each connected component of $\Gamma_{+}^{m}$ can be
identified with $\mathbb{R}_{+}^{m}$.

Since $\Gamma$ is open, convex and contains $\Gamma_{+}^{n}$, for each $1\leq
m\leq n$ we either have $\Gamma_{+}^{m}\subset\Gamma$ or else
$\Gamma_{+}^{m}\cap\Gamma=\emptyset$. Let $m_{*}$ denote the least integer
such that $\Gamma_{+}^{m}\subset\Gamma$. For each $m_{*}\leq m\leq n$ we write
$\gamma_{m}:\mathbb{R}_{+}^{m}\to\mathbb{R}$ for the function

$\gamma_{m}(\lambda_{1},\dots,\lambda_{m}):=\gamma(0,\lambda_{1},\dots,\lambda_{m}).$

When $\gamma$ is strictly inverse-concave we define $m_{\operatorname{IC}}\geq
m_{*}$ to be the least integer such that $\gamma_{m}$ is strictly inverse-
concave for every $m_{\operatorname{IC}}\leq m\leq n$.

In Section 2 we demonstrate that if the eigenvalues of $A\geq 0$ satisfy

$\min_{0\leq
m<m_{\operatorname{IC}}}\operatorname{dist}(\tfrac{\lambda}{|\lambda|},\Gamma_{+}^{m})\geq\delta,$

then (3) and (4) hold for some positive $C=C(n,\gamma,\delta)$ and
$\varepsilon=\varepsilon(n,\gamma,\delta)$. When $A\geq 0$, this is equivalent
to assuming that $\lambda$ is uniformly $k$-positive with
$k=n-m_{\operatorname{IC}}+1$. Therefore, (3) and (4) hold on a solution to
(1) which is uniformly $k$-convex with $k\leq n-m_{\operatorname{IC}}+1$.

As a result of all of this discussion, we have the following consequence of
Theorem 1.1.

######  Theorem 1.2.

Suppose $\gamma$ is convex, or concave and strictly inverse-concave. Let
$F:M\times(-\infty,0]\to\mathbb{R}^{n+1}$ be an ancient solution to (1). We
assume the hypersurfaces $M_{t}=F(M,t)$ each bound an open convex subset
$\Omega_{t}$. In addition, for each $T<\infty$, we assume $M_{t}$ is
noncollapsing and uniformly $k$-convex on the time interval $[-T,0]$, where
$k\leq n-m_{\operatorname{IC}}+1$. Finally, we assume that

$\sup_{M\times[-T,0]}G<\infty$

for each $T<\infty$. Then, for every $(x,t)\in M\times(-\infty,0]$ and $V\in
T_{x}M$, we have

(7) $\partial_{t}G+2\langle\nabla G,V\rangle+A(V,V)\geq 0.$

###  1.2. Examples

Consider the concave speeds (cf. [BH17]) given by

$\gamma(\lambda)=\bigg{(}\sum_{i_{1}<\dots<i_{k}}\frac{1}{\lambda_{i_{1}}+\dots+\lambda_{i_{k}}}\bigg{)}^{-1},$

for $k\geq 2$ and $n\geq k+1$. In this case we may take

$\Gamma=\Big{\\{}\lambda:\min_{i_{1}<\dots<i_{k}}\lambda_{i_{1}}+\dots+\lambda_{i_{k}}>0\Big{\\}}.$

We then have $m_{\operatorname{IC}}=m_{*}=n-k+1$, so Theorem 1.2 applies to
convex ancient solutions of (1) which are noncollapsing and uniformly
$k$-convex.

Consider the ratios of elementary symmetric polynomials
$\gamma=\sigma_{k}/\sigma_{k-1}$ for $k\geq 2$ and $n\geq k+1$. We may take
$\Gamma$ to be the cone where $\sigma_{k}>0$, in which case $m_{*}=k$ and
$m_{\operatorname{IC}}=k+1$ (the function $\gamma_{k}$ is the harmonic mean,
which is inverse-concave but not strictly inverse-concave). So Theorem 1.2
applies to convex ancient solutions which are noncollapsing and uniformly
$(n-k)$-convex.

If $\gamma:\Gamma\to\mathbb{R}$ is strictly inverse-concave,
$\beta:\Gamma\to\mathbb{R}$ is inverse-concave, and
$h:\mathbb{R}_{+}^{2}\to\mathbb{R}$ is inverse-concave, then the composition

$\lambda\mapsto h(\gamma(\lambda),\beta(\lambda))$

is strictly inverse-concave. Using this observation, one finds that Theorem
1.2 applies to convex, noncollapsing, uniformly $k$-convex ancient solutions
to the flows introduced in [Lyn22]. This class includes all blow-up limits at
a singularity of a compact embedded solution.

###  1.3. Translating solitons

Harnack inequalities are closely related to solitons. A solution to (1) is
called a translating soliton if there is a constant vector $\xi$ on
$\mathbb{R}^{n+1}$ such that the hypersurfaces $M_{t}=F(M,t)$ satisfy
$M_{t}=M_{0}+t\xi$. Translating solitons are characterised by the identity

$G=-\langle\xi,\nu\rangle.$

Notice that when $A>0$ we have

$2\langle\nabla G,V\rangle+A(V,V)\geq-A^{-1}(\nabla G,\nabla G),$

with equality for $V=-A^{-1}(\nabla G)$. In this case (7) becomes

$\partial_{t}G-A^{-1}(\nabla G,\nabla G)\geq 0.$

If equality is attained here, and $\gamma$ is strictly inverse-concave, then
the solution is a translating soliton.

######  Corollary 1.3.

Suppose $\gamma$ is strictly inverse-concave. Let
$F:(-\infty,0]\to\mathbb{R}^{n+1}$ be an ancient solution to (1) such that
$A>0$. Suppose the Harnack inequality

$\partial_{t}G-A^{-1}(\nabla G,\nabla G)\geq 0$

holds at each point in spacetime. In addition, we assume bounded curvature,

$\sup_{M\times[-T,0]}G<\infty,$

and the uniform ellipticity condition (3) on $[-T,0]$ for each $T<\infty$. If
there is a point in spacetime at which

$\partial_{t}G-A^{-1}(\nabla G,\nabla G)=0,$

then $F$ is a translating soliton.

###  1.4. Pointwise Harnack estimate

When $A>0$, (6) implies a pointwise estimate comparing $G$ at different points
in spacetime. Indeed, assuming $A>0$, the inequality (6) may be restated as

$\partial_{t}G-A^{-1}(\nabla G,\nabla G)+\frac{G}{2t}\geq 0.$

Given times $0<t_{0}<t_{1}\leq T$, integration of this inequality yields

$G(x_{1},t_{1})\geq\bigg{(}\frac{t_{0}}{t_{1}}\bigg{)}^{\frac{1}{2}}\exp\bigg{(}-\frac{1}{4}\inf_{\gamma}\int_{t_{0}}^{t_{1}}\frac{A(\dot{\gamma},\dot{\gamma})}{G}\,dt\bigg{)}\,G(x_{0},t_{0}),$

where the infimum is over smooth paths $\gamma:[t_{0},t_{1}]\to M$ satisfying
$\gamma(t_{0})=x_{0}$ and $\gamma(t_{1})=x_{1}$.

###  1.5. Acknowledgements

The author is grateful to M. Langford for his valuable comments.

##  2\. Uniform inverse-concavity

In this section we establish conditions under which (3) and (4) hold. We begin
by introducing some notation.

We write $\dot{\gamma}^{i}(\lambda)$ and $\ddot{\gamma}^{ij}(\lambda)$ for
derivatives with respect to eigenvalues, so that

$\frac{d}{ds}\bigg{|}_{s=0}\gamma(\lambda+s\mu)=\dot{\gamma}^{i}(\lambda)\mu_{i},\qquad\frac{d}{ds}\bigg{|}_{s=0}\dot{\gamma}^{i}(\lambda+s\mu)=\ddot{\gamma}^{ij}(\lambda)\mu_{j},$

and write $\dot{\gamma}^{ij}(A)$ and $\ddot{\gamma}^{ij,kl}(A)$ for
derivatives with respect to matrix entries, so that

$\frac{d}{ds}\bigg{|}_{s=0}\gamma(A+sB)=\dot{\gamma}^{ij}(A)B_{ij},\qquad\frac{d}{ds}\bigg{|}_{s=0}\dot{\gamma}^{ij}(A+sB)=\ddot{\gamma}^{ij,kl}(A)B_{kl}.$

When $A$ is a diagonal matrix with entries $\lambda$, $\dot{\gamma}^{ij}(A)$
is also diagonal, with entries $\dot{\gamma}^{i}(\lambda)$.

Let $\operatorname{Sym}(n)$ denote the space of symmetric $n\times
n$-matrices. We assume that $\gamma$ is strictly inverse-concave. That is, the
function $A\mapsto-\gamma(A^{-1})$ is strictly concave for positive-definite
$A\in\operatorname{Sym}(n)$. In terms of derivatives this means that

$(\ddot{\gamma}^{ij,kl}(A)+2\dot{\gamma}^{ik}(A)A^{-1}_{jl})S_{ij}S_{kl}>0$

for every positive-definite $A\in\operatorname{Sym}(n)$ and every nonzero
$S\in\operatorname{Sym}(n)$. Since $\gamma$ is homogeneous of degree one, its
strict inverse-concavity is also equivalent to strict concavity of the
function $A\mapsto\gamma(A^{-1})^{-1}$ in non-radial directions. In terms of
derivatives,

$(\ddot{\gamma}^{ij,kl}(A)+2\dot{\gamma}^{ik}(A)A^{-1}_{jl}-2\gamma(A)^{-1}\dot{\gamma}^{ij}(A)\dot{\gamma}^{kl}(A))S_{ij}S_{kl}>0$

for every positive-definite $A\in\operatorname{Sym}(n)$ and every
$S\in\operatorname{Sym}(n)$ which is not a multiple of $A$.

Let $\Gamma^{\prime}$ be a closed, symmetric, convex cone which is contained
in the closure of $\Gamma_{+}^{n}$. We are interested in consequences of the
property

(8) $\min_{0\leq
m<m_{\operatorname{IC}}}\bigg{(}\inf_{\lambda\in\Gamma^{\prime}}\operatorname{dist}(\tfrac{\lambda}{|\lambda|},\Gamma_{+}^{m})\bigg{)}>0.$

######  Lemma 2.1.

Suppose $\Gamma^{\prime}$ satisfies (8). We then have

$\inf_{\lambda\in\Gamma^{\prime}}\operatorname{dist}(\tfrac{\lambda}{|\lambda|},\partial\Gamma)>0.$

###### Proof.

By definition, $\Gamma_{+}^{m}\cap\Gamma=\emptyset$ for $m<m_{*}$, and
$\Gamma_{+}^{m}\subset\Gamma$ for $m\geq m_{*}$. It follows that

(9) $\partial\Gamma\cap\bar{\Gamma}_{+}^{n}=\bigcup_{0\leq
m<m_{*}}\Gamma_{+}^{m}.$

The property (8) implies

(10) $\min_{0\leq
m<m_{*}}\bigg{(}\inf_{\lambda\in\Gamma^{\prime}}\operatorname{dist}(\tfrac{\lambda}{|\lambda|},\Gamma_{+}^{m})\bigg{)}>0.$

Indeed, we have $m_{*}\leq m_{\operatorname{IC}}$ by definition. Define

$\delta:=\inf_{\lambda\in\Gamma^{\prime}}\operatorname{dist}(\tfrac{\lambda}{|\lambda|},\partial\Gamma).$

The claim is that $\delta>0$. There is a sequence
$\lambda^{(k)}\in\Gamma^{\prime}$ such that $|\lambda^{(k)}|=1$ and
$\operatorname{dist}(\lambda^{(k)},\partial\Gamma)\to\delta$ as $k\to\infty$.
Since $\Gamma^{\prime}$ is closed, we may assume $\lambda^{(k)}$ converges to
some $\lambda\in\Gamma^{\prime}$. Consider the possibility that
$\lambda\in\partial\Gamma$. In this case (9) implies
$\lambda\in\Gamma_{+}^{m}$ for some $1\leq m<m_{*}$. But due to (10) this is
impossible. So we must have $\lambda\in\Gamma$, and hence $\delta>0$. ∎

Given a symmetric matrix $A$, we define $\lambda(A)$ to be the eigenvalues of
$A$. We define $\lambda^{-1}(\Gamma^{\prime})$ to be the space of symmetric
matrices with eigenvalues in $\Gamma^{\prime}$. In addition,
$\lambda^{-1}(\Gamma^{\prime}\cap\Gamma_{+}^{n})$ will denote the space of
positive-definite symmetric matrices with eigenvalues in $\Gamma^{\prime}$.

Lemma 2.1 shows that if (8) holds then
$\\{\lambda\in\Gamma^{\prime}:|\lambda|=1\\}$ is a compact subset of $\Gamma$.
Since $\gamma$ is homogeneous of degree one we conclude that if (8) holds then
there is a constant $C=C(n,\gamma,\Gamma^{\prime})$ such that

$C^{-1}|\xi|^{2}\leq\dot{\gamma}^{ij}(A)\xi_{i}\xi_{j}\leq
C|\xi|^{2},\qquad\gamma(A)\ddot{\gamma}^{ij,kl}(A)S_{ij}S_{kl}\geq-C|S|^{2}$

for every $A\in\lambda^{-1}(\Gamma^{\prime})$, $\xi\in\mathbb{R}^{n}$ and
$S\in\operatorname{Sym}(n)$.

######  Lemma 2.2.

Suppose (8) holds. There is then a positive constant
$\kappa=\kappa(n,\gamma,\Gamma^{\prime})$ such that

(11)
$\gamma(A)(\ddot{\gamma}^{ij,kl}(A)+2\dot{\gamma}^{ik}(A)A^{-1}_{jl})S_{ij}S_{kl}\geq\kappa|S|^{2}$

for every $A\in\lambda^{-1}(\Gamma^{\prime}\cap\Gamma_{+}^{n})$ and
$S\in\operatorname{Sym}(n)$.

###### Proof.

Let us define

$\kappa:=\inf_{A\in\lambda^{-1}(\Gamma^{\prime}\cap\Gamma_{+}^{n}),\,S\in\operatorname{Sym}(n)}|S|^{-2}\gamma(A)(\ddot{\gamma}^{ij,pq}(A)+2\dot{\gamma}^{ip}(A)A^{-1}_{jq})S_{ij}S_{pq}.$

We claim that $\kappa$ is positive.

Let $A^{(k)}\in\lambda^{-1}(\Gamma^{\prime}\cap\Gamma_{+}^{n})$ and
$S^{(k)}\in\operatorname{Sym}(n)$ be sequences such that

$|S^{(k)}|^{-2}\gamma(A^{(k)})(\ddot{\gamma}^{ij,pq}(A^{(k)})+2\dot{\gamma}^{ip}(A^{(k)})(A^{(k)})^{-1}_{jq})S^{(k)}_{ij}S^{(k)}_{pq}\to\kappa$

as $k\to\infty$. Since $\gamma$ is homogeneous of degree one, we may assume
without loss of generality that $|A^{(k)}|=1$ and $|S^{(k)}|=1$. Since
$\gamma$ is $O(n)$-invariant we may also assume $A^{(k)}$ is diagonal. By
passing to a subsequence we can arrange that $A^{(k)}$ converges to some
$\tilde{A}$, and that $S^{(k)}$ converges to some $\tilde{S}$. If
$\lambda(\tilde{A})\in\Gamma_{+}^{n}$ then we are done—since $\gamma$ is
strictly inverse-concave, it then follows that $\kappa>0$. Suppose instead
that $\lambda(A)\in\Gamma_{+}^{m}$ for some $1\leq m\leq n-1$. Since $A^{(k)}$
in $\lambda^{-1}(\Gamma^{\prime})$, (8) implies $m\geq m_{\operatorname{IC}}$.
Therefore, the function $\gamma_{m}$ is strictly inverse-concave.

To ease notation, let us write $A=A^{(k)}$ and $S=S^{(k)}$. Let $X$ be the
$n\times n$-matrix whose entries $X_{ij}=S_{ij}$ when $i,j\leq n-m$ and vanish
otherwise. Let $Z$ be the $n\times n$-matrix whose entries $Z_{ij}=S_{ij}$
when $i,j>n-m$ and vanish otherwise. We then define $Y:=S-X-Z$. By Lemma 2.1
there is a constant $C$ depending only on $n$, $\gamma$ and $\Gamma^{\prime}$
such that

$\displaystyle\ddot{\gamma}^{ij,kl}(A)S_{ij}S_{pq}\geq-C.$

In addition, writing $\lambda=\lambda(A)$, we have

$\displaystyle 2\gamma^{ip}(A)A^{-1}_{jq}S_{ij}S_{pq}$
$\displaystyle=2\sum_{i,\,j}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}}|S_{ij}|^{2}$
$\displaystyle=2\sum_{i,\,j\leq
n-m}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}}|X_{ij}|^{2}+2\sum_{i\leq
n-m,\,j>n-m}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}}|Y_{ij}|^{2}$
$\displaystyle+2\sum_{i>n-m,\,j\leq
n-m}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}}|Y_{ij}|^{2}+2\sum_{i,\,j>n-m}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}}|Z_{ij}|^{2}.$

Since $\lambda_{i}\to 0$ for every $1\leq i\leq n-m$, and $\kappa<\infty$, we
conclude that $|X|\to 0$ and $|Y|\to 0$ as $k\to\infty$. In particular,
$\tilde{S}_{ij}=0$ unless $i,j>n-m$.

To finish we combine the inequalities

$\ddot{\gamma}^{ij,pq}(A)S_{ij}S_{pq}\geq\ddot{\gamma}^{ij,pq}(A)Z_{ij}Z_{pq}-C(|X|^{2}+|Y|^{2}+|X||Y|+|X||Z|+|Y||Z|)$

and

$2\gamma^{ip}(A)A^{-1}_{jq}S_{ij}S_{pq}\geq
2\sum_{i,\,j>n-m}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}}|Z_{ij}|^{2}=2\dot{\gamma}^{ip}(A)A^{-1}_{jq}Z_{ij}Z_{pq}$

in order to obtain

$\displaystyle(\ddot{\gamma}^{ij,pq}(A)+2\gamma^{ip}(A)A^{-1}_{jq})$
$\displaystyle S_{ij}S_{pq}$
$\displaystyle\geq(\ddot{\gamma}^{ij,pq}(A)+2\gamma^{ip}(A)A^{-1}_{jq})Z_{ij}Z_{pq}$
$\displaystyle-C(|X|^{2}+|Y|^{2}+|X||Y|+|X||Z|+C|Y||Z|).$

Let us write $\hat{A}$ and $\hat{S}$ for the $m\times m$-matrices which
coincide with the lower-right $m\times m$-blocks of $\tilde{A}$ and
$\tilde{S}$, respectively. Sending $k\to\infty$ in the last inequality then
yields

$\kappa=(\ddot{\gamma}^{ij,pq}_{m}(\hat{A})+2\gamma^{ip}_{m}(\hat{A})\hat{A}^{-1}_{jq})\hat{S}_{ij}\hat{S}_{pq}.$

Since $\gamma_{m}$ is strictly inverse-concave and $|\hat{S}|=1$ we conclude
that $\kappa>0$. ∎

Next we use Lemma 2.2 to show that (8) implies (4).

######  Lemma 2.3.

Suppose (8) holds. There is then a positive constant
$\varepsilon=\varepsilon(n,\gamma,\Gamma^{\prime})$ such that

(12)
$\gamma(A)(\ddot{\gamma}^{ij,kl}(A)+2\dot{\gamma}^{ik}(A)(A+\varepsilon\gamma(A)I)^{-1}_{jl})S_{ij}S_{kl}\geq
0$

for every $A\in\lambda^{-1}(\Gamma^{\prime})$ and symmetric matrix $S$.

###### Proof.

Consider an arbitrary $A\in\lambda^{-1}(\Gamma^{\prime})$. Set
$\lambda=\lambda(A)$. Let $\theta$ be a small positive constant whose value
will be fixed later, and denote by $\ell$ the integer such that
$\lambda_{i}\leq\theta\gamma(\lambda)$ for $1\leq i\leq\ell$ and
$\lambda_{i}>\theta\gamma(\lambda)$ for $\ell+1\leq i\leq n$.

Fix an arbitrary $S\in\operatorname{Sym}(n)$ and write $S=X+Y+Z$, where
$X_{ij}=S_{ij}$ for $i,j\leq\ell$ and $X_{ij}=0$ otherwise, and
$Z_{ij}=S_{ij}$ for $i,j>\ell$ and $Z_{ij}=0$ otherwise.

We may assume without loss of generality that $A$ is diagonal and
$\gamma(A)=1$. We then have

$\displaystyle 2\dot{\gamma}^{ip}(A)(A+\varepsilon$ $\displaystyle
I)^{-1}_{jq}S_{ij}S_{pq}$
$\displaystyle=2\sum_{i,j}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}+\varepsilon}|S_{ij}|^{2}$
$\displaystyle=\sum_{i,j\leq\ell}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}+\varepsilon}|X_{ij}|^{2}+2\sum_{i\leq\ell,\,j>\ell}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}+\varepsilon}|Y_{ij}|^{2}$
$\displaystyle+2\sum_{i>\ell,\,j\leq\ell}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}+\varepsilon}|Y_{ij}|^{2}+2\sum_{i,j>\ell}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}+\varepsilon}|Z_{ij}|^{2}.$

By Lemma 2.1 we have $\dot{\gamma}^{i}(\lambda)\geq C^{-1}$ for some positive
$C=C(n,\gamma,\Gamma^{\prime})$. Therefore,

$\displaystyle
2\sum_{i,j>\ell}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}+\varepsilon}|Z_{ij}|^{2}$
$\displaystyle=2\sum_{i,j>\ell}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}}|Z_{ij}|^{2}-2\varepsilon\sum_{i,j>\ell}\frac{\dot{\gamma}^{i}(\lambda)}{\lambda_{j}(\lambda_{j}+\varepsilon)}|Z_{ij}|^{2}$
$\displaystyle\geq
2\dot{\gamma}^{ip}(A)(A)^{-1}_{jq}Z_{ij}Z_{pq}-C\varepsilon\theta^{-1}(\theta+\varepsilon)^{-1}|Z|^{2},$

and hence

$\displaystyle 2\dot{\gamma}^{ip}(A)$ $\displaystyle(A+\varepsilon
I)^{-1}_{jq}S_{ij}S_{pq}$ $\displaystyle\geq
2\dot{\gamma}^{ip}(A)(A)^{-1}_{jq}Z_{ij}Z_{pq}+C^{-1}(\theta+\varepsilon)^{-1}(|X|^{2}+|Y|^{2})$
$\displaystyle-C\varepsilon\theta^{-1}(\theta+\varepsilon)^{-1}|Z|^{2}.$

Combining this inequality with

$\ddot{\gamma}^{ij,pq}(A)S_{ij}S_{pq}\geq\ddot{\gamma}^{ij,pq}(A)Z_{ij}Z_{pq}-C(|X|^{2}+|Y|^{2}),$

we obtain

$\displaystyle(\ddot{\gamma}^{ij,pq}(A)+2$
$\displaystyle\dot{\gamma}^{ip}(A)(A+\varepsilon I)^{-1}_{jq})S_{ij}S_{pq}$
$\displaystyle\geq(\ddot{\gamma}^{ij,pq}(A)+2\dot{\gamma}^{ip}(A)(A)^{-1}_{jq})Z_{ij}Z_{pq}-C\varepsilon\theta^{-1}(\theta+\varepsilon)^{-1}|Z|^{2}$
$\displaystyle+(C^{-1}(\theta+\varepsilon)^{-1}-C)(|X|^{2}+|Y|^{2}).$

Lemma 2.2 now gives

$\displaystyle(\ddot{\gamma}^{ij,pq}$
$\displaystyle(A)+2\dot{\gamma}^{ip}(A)(A+\varepsilon
I)^{-1}_{jq})S_{ij}S_{pq}$
$\displaystyle\geq(\kappa-C\varepsilon\theta^{-1}(\theta+\varepsilon)^{-1})|Z|^{2}+(C^{-1}(\theta+\varepsilon)^{-1}-C)(|X|^{2}+|Y|^{2}).$

The right-hand side is nonnegative for $\theta<C^{-2}/2$ and
$\varepsilon<\min\\{C^{-1}\kappa\theta^{2},\theta\\}$. ∎

To conclude this section we show that for $k=n-m_{\operatorname{IC}}+1$,
uniform $k$-positivity implies (8). The converse is also true. We write

$\operatorname{tr}(\lambda)=\lambda_{1}+\dots+\lambda_{n}.$

######  Lemma 2.4.

Let $k=n-m_{\operatorname{IC}}+1$. The condition (8) holds if and only if

(13)
$\inf_{\lambda\in\Gamma^{\prime}}\bigg{(}\min_{i_{1}<\dots<i_{k}}\frac{\lambda_{i_{1}}+\dots+\lambda_{i_{k}}}{\operatorname{tr}(\lambda)}\bigg{)}>0.$

###### Proof.

Suppose first that (13) holds. Let us define

$\delta=\inf_{0\leq
m<m_{\operatorname{IC}}}\bigg{(}\inf_{\lambda\in\Gamma^{\prime}}\operatorname{dist}(\tfrac{\lambda}{|\lambda|},\Gamma_{+}^{m})\bigg{)}.$

The claim is that $\delta>0$. If, to the contrary, $\delta=0$, then there is a
point $\lambda\in\Gamma^{\prime}$ such that $|\lambda|=1$ and
$\lambda\in\Gamma_{+}^{m}$ for some $1\leq m<m_{\operatorname{IC}}$. We may
assume $\lambda_{1}\leq\dots\leq\lambda_{n}$, in which case
$\lambda\in\Gamma_{+}^{m}$ implies

$0=\lambda_{1}=\dots=\lambda_{n-m},\qquad
0<\lambda_{n-m+1}\leq\dots\leq\lambda_{n}.$

On the other hand, by (13), we have

$\lambda_{1}+\dots+\lambda_{k}>0.$

Since $k=n-m_{\operatorname{IC}}+1<n-m+1$, this is a contradiction. Therefore,
we must have $\delta>0$.

Now suppose (8) holds. Let us define

$\rho=\inf_{\lambda\in\Gamma^{\prime}}\bigg{(}\min_{i_{1}<\dots<i_{k}}\frac{\lambda_{i_{1}}+\dots+\lambda_{i_{k}}}{\operatorname{tr}(\lambda)}\bigg{)}.$

The claim is that $\rho>0$. Assume to the contrary $\rho=0$. There is then a
point $\lambda\in\Gamma^{\prime}$ such that $\operatorname{tr}(\lambda)=1$,
$\lambda_{1}\leq\dots\leq\lambda_{n}$, and

$0=\lambda_{1}=\dots=\lambda_{k}.$

Using $k=n-m_{\operatorname{IC}}+1$ we conclude that
$\lambda\in\Gamma_{+}^{m}$ for some $m<m_{\operatorname{IC}}$. This
contradicts (8), so we must have $\rho>0$. ∎

##  3\. Evolution of the Harnack quantity

Let $F:M\times(0,T]\to\mathbb{R}^{n+1}$ a solution to (1). Let $g=g(t)$ denote
the metric on $M$ induced by $F(\cdot,t)$. We define a time derivative acting
on vector fields by

$\nabla_{t}X^{i}:=\partial_{t}X^{i}-GA^{i}_{j}X^{j},$

and extend to tensors via the usual Leibniz rule. This time derivative has the
property that $\nabla_{t}g=0$.

Given an orthonormal frame $\\{e_{i}\\}$ of tangent vectors to $M$, we write

$\Delta_{\gamma}=\dot{\gamma}^{ij}\nabla_{i}\nabla_{j},\qquad|A|^{2}_{\gamma}=\dot{\gamma}^{ij}A^{2}_{ij}=\dot{\gamma}^{ij}A_{ik}A_{kj}.$

The first variation formula for the second fundamental form asserts that

$\nabla_{t}A_{kl}=\nabla_{k}\nabla_{l}G+GA^{2}_{kl}.$

Simons’ identity then yields the parabolic equation

$(\nabla_{t}-\Delta_{\gamma})A_{kl}=|A|^{2}_{\gamma}A_{kl}+\ddot{\gamma}^{ij,pq}\nabla_{k}A_{ij}\nabla_{l}A_{pq}.$

Since $\gamma$ is homogeneous of degree one, tracing this formula with respect
to $\dot{\gamma}^{kl}$ gives

$(\partial_{t}-\Delta_{\gamma})G=|A|^{2}_{\gamma}G.$

The above evolution equations, together with the Gauss and Codazzi relations,
imply the following commutation identities

$\displaystyle\Delta_{\gamma}\nabla_{k}f$
$\displaystyle=\nabla_{k}\Delta_{\gamma}f+GA_{kl}\nabla^{l}f-\dot{\gamma}^{ij}A_{ik}A_{jl}\nabla_{l}f-\ddot{\gamma}^{ij,pq}\nabla_{i}\nabla_{j}f\nabla_{k}A_{pq}$

and

$\displaystyle\partial_{t}\Delta_{\gamma}f$
$\displaystyle=\Delta_{\gamma}\partial_{t}f+2G\dot{\gamma}^{ij}A_{ik}\nabla_{k}\nabla_{j}f+2\dot{\gamma}^{ij}A_{jk}\nabla_{i}G\nabla_{k}f$
$\displaystyle+\ddot{\gamma}^{ij,kl}\nabla_{i}\nabla_{j}f\nabla_{t}A_{kl}.$

Using these, straightforward computations yield

$\displaystyle(\partial_{t}-\Delta_{\gamma})\partial_{t}G$
$\displaystyle=|A|^{2}_{\gamma}\partial_{t}G+2G\dot{\gamma}^{ij}A_{ik}(2\nabla_{t}A_{kj}-GA^{2}_{jk})$
(14)
$\displaystyle+2\dot{\gamma}^{ij}A_{jk}\nabla_{i}G\nabla_{k}G+\ddot{\gamma}^{ij,pq}\nabla_{t}A_{ij}\nabla_{t}A_{pq}$

and

$\displaystyle(\nabla_{t}-\Delta_{\gamma})\nabla_{k}G$
$\displaystyle=|A|^{2}_{\gamma}\nabla_{k}G+\dot{\gamma}^{ij}A_{ik}A_{jl}\nabla_{l}G+2\dot{\gamma}^{ij}A_{il}\nabla_{k}A_{lj}G$
(15) $\displaystyle+\ddot{\gamma}^{ij,pq}\nabla_{t}A_{ij}\nabla_{k}A_{pq}.$

In case $A>0$, one finds that

$\displaystyle(\nabla_{t}-\Delta_{\gamma})A^{-1}_{kl}$
$\displaystyle=-|A|^{2}_{\gamma}A^{-1}_{kl}$
$\displaystyle-(\ddot{\gamma}^{ij,pq}+2\dot{\gamma}^{ip}A^{-1}_{jq})A^{-1}(e_{k},\nabla
A_{ij})A^{-1}(e_{l},\nabla A_{pq}).$

We combine this formula with (3) and the identity

$\displaystyle-4\dot{\gamma}^{ij}\nabla_{i}A^{-1}(\nabla G,\nabla_{j}\nabla
G)$ $\displaystyle=4\dot{\gamma}^{ip}A^{-1}_{jq}\nabla_{t}A_{ij}A^{-1}(\nabla
G,\nabla A_{pq})$ $\displaystyle-4G\dot{\gamma}^{ij}A_{il}A^{-1}(\nabla
G,\nabla A_{lj}),$

in order to derive

$\displaystyle(\partial_{t}$ $\displaystyle-\Delta_{\gamma})(A^{-1}(\nabla
G,\nabla G))$ $\displaystyle=|A|^{2}_{\gamma}A^{-1}(\nabla G,\nabla
G)-2\dot{\gamma}^{ij}A^{-1}(\nabla_{i}\nabla G,\nabla_{j}\nabla
G)+2\dot{\gamma}^{ij}A_{jk}\nabla_{i}G\nabla_{k}G$
$\displaystyle+2(\ddot{\gamma}^{ij,pq}+2\dot{\gamma}^{ip}A^{-1}_{jq})\nabla_{t}A_{ij}A^{-1}(\nabla
G,\nabla A_{pq})$
$\displaystyle-(\ddot{\gamma}^{ij,pq}+2\dot{\gamma}^{ip}A^{-1}_{jq})A^{-1}(\nabla
G,\nabla A_{ij})A^{-1}(\nabla G,\nabla A_{pq}).$

Combining this with (3) and simplifying, we obtain

$\displaystyle(\partial_{t}-\Delta_{\gamma})$
$\displaystyle(\partial_{t}G-A^{-1}(\nabla G,\nabla G))$
$\displaystyle=|A|^{2}_{\gamma}(\partial_{t}G-A^{-1}(\nabla G,\nabla G))$
$\displaystyle+(\ddot{\gamma}^{ij,pq}+2\dot{\gamma}^{ip}A^{-1}_{jq})(\nabla_{t}A_{ij}-A^{-1}(\nabla
G,\nabla A_{ij}))(\nabla_{t}A_{pq}-A^{-1}(\nabla G,\nabla A_{pq})).$

Consequently,

$\displaystyle(\partial_{t}-\Delta_{\gamma})$
$\displaystyle\bigg{(}\partial_{t}G-A^{-1}(\nabla G,\nabla
G)+\frac{1}{2t}G\bigg{)}$
$\displaystyle=\bigg{(}|A|^{2}_{\gamma}-\frac{2}{t}\bigg{)}\bigg{(}\partial_{t}G-A^{-1}(\nabla
G,\nabla G)+\frac{1}{2t}G\bigg{)}$
$\displaystyle+\frac{2}{t}(\partial_{t}G-A^{-1}(\nabla G,\nabla
G))+\frac{1}{2t^{2}}G$
$\displaystyle+(\ddot{\gamma}^{ij,pq}+2\dot{\gamma}^{ip}A^{-1}_{jq})(\nabla_{t}A_{ij}-A^{-1}(\nabla
G,\nabla A_{ij}))(\nabla_{t}A_{pq}-A^{-1}(\nabla G,\nabla A_{pq})).$

In case $A>0$ and $\gamma$ is inverse-concave, we have

$\displaystyle(\ddot{\gamma}^{ij,pq}+2$
$\displaystyle\dot{\gamma}^{ip}A^{-1}_{jq})(\nabla_{t}A_{ij}-A^{-1}(\nabla
G,\nabla A_{ij}))(\nabla_{t}A_{pq}-A^{-1}(\nabla G,\nabla A_{pq}))$
$\displaystyle\geq\frac{2}{G}(\partial_{t}G-A^{-1}(\nabla G,\nabla G))^{2}.$

Inserting this above yields

$\displaystyle(\partial_{t}-\Delta_{\gamma})$
$\displaystyle\bigg{(}\partial_{t}G-A^{-1}(\nabla G,\nabla
G)+\frac{1}{2t}G\bigg{)}$
$\displaystyle\geq\bigg{(}|A|^{2}_{\gamma}-\frac{2}{t}\bigg{)}\bigg{(}\partial_{t}G-A^{-1}(\nabla
G,\nabla G)+\frac{1}{2t}G\bigg{)}$ (16)
$\displaystyle+\frac{2}{G}\bigg{(}\partial_{t}G-A^{-1}(\nabla G,\nabla
G)+\frac{1}{2t}G\bigg{)}^{2}.$

Andrews found this inequality in [And94] by working in the Gauss map
parameterization (this simplifies the computation considerably). By applying
the parabolic maximum to (3), Andrews could prove the following Harnack
estimate for compact, strictly convex solutions of (1): at each point in
spacetime one has

$\partial_{t}G-A^{-1}(\nabla G,\nabla G)+\frac{1}{2t}G\geq 0.$

This implies (6).

The inequality (3) does not seem to be suitable for proving a Harnack
inequality for noncompact solutions—various difficulties arise from the fact
that $A^{-1}(\nabla G,\nabla G)$ might be unbounded. To prove Theorem 1.1, we
modify Hamilton’s approach to the Harnack estimate for mean curvature flow
[Ham95]. That is, we define a form $Q$ acting on tangent vectors by

$Q(V):=\partial_{t}G+2\langle\nabla G,V\rangle+A(V,V)+\frac{1}{2t}G,$

and use the maximum principle to conclude that $Q$ is nonnegative. The key
advantage of this is that at a spacetime minimum of $Q$, given any minimizing
vector $V$, there is some freedom in how we extend $V$ to a neighbourhood.
This freedom can be exploited to introduce terms which are favourable, but
suitably controlled—see Remark 4.1.

The rest of the computations in this section apply to a general solution of
(1).

Let $V$ be a time-dependent field of tangent vectors on $M$. Following
[Ham95], we introduce tensors

$\displaystyle X_{i}$ $\displaystyle=\nabla_{i}G+A_{ij}V^{j}$ $\displaystyle
Y_{ij}$ $\displaystyle=\nabla_{i}V_{j}-GA_{ij}-\frac{1}{2t}g_{ij}$
$\displaystyle W_{ij}$
$\displaystyle=\nabla_{t}A_{ij}+V^{k}\nabla_{k}A_{ij}+\frac{1}{2t}A_{ij}$
$\displaystyle U^{i}$
$\displaystyle=(\nabla_{t}-\Delta_{\gamma})V^{i}+\dot{\gamma}^{ij}A_{jk}\nabla_{k}G+\frac{1}{t}V^{i}.$

We compute at a point $(x_{0},t_{0})$ in spacetime with respect to a local
orthonormal frame for which $\nabla_{i}e_{j}=0$ and $\nabla_{t}e_{i}=0$ at
$(x_{0},t_{0})$.

######  Proposition 3.1.

At the point $(x_{0},t_{0})$ we have

$\displaystyle(\partial_{t}$ $\displaystyle-\Delta_{\gamma})(Q(V))$
$\displaystyle=\bigg{(}|A|^{2}_{\gamma}-\frac{2}{t}\bigg{)}Q(V)+2X_{i}U_{i}-4\dot{\gamma}^{ij}Y_{ik}W_{jk}-2\dot{\gamma}^{ij}Y_{ik}A_{kl}Y_{jl}$
(17)
$\displaystyle+\ddot{\gamma}^{ij,pq}(\nabla_{t}A_{ij}+V_{k}\nabla_{k}A_{ij})(\nabla_{t}A_{pq}+V_{l}\nabla_{l}A_{pq}).$

###### Proof.

A nice account of Hamilton’s computation for the mean curvature flow is given
in [ACGL22, Proposition 10.7]. We follow the exposition there.

Note that

$Q(V)=\dot{\gamma}^{ij}W_{ij}+X_{i}V_{i}.$

Using (3) and the first variation formula

$\nabla_{t}A_{ij}=\nabla_{i}\nabla_{j}G+GA^{2}_{ij},$

we compute

$\displaystyle(\partial_{t}-\Delta_{\gamma})(X_{k}V_{k})$
$\displaystyle=|A|^{2}_{\gamma}X_{k}V_{k}+(X_{k}+A_{ik}V_{i})(\nabla_{t}-\Delta_{\gamma})V_{k}+2G\dot{\gamma}^{ij}A_{il}\nabla_{k}A_{lj}V_{k}$
$\displaystyle+\dot{\gamma}^{ij}A_{il}A_{jk}V_{k}\nabla_{l}G-2\dot{\gamma}^{ij}(\nabla_{i}X_{k}+V_{l}\nabla_{i}A_{kl})\nabla_{j}V_{k}$
$\displaystyle+\ddot{\gamma}^{ij,pq}V_{k}\nabla_{k}A_{ij}V_{l}\nabla_{l}A_{pq}+\ddot{\gamma}^{ij,pq}\nabla_{t}A_{ij}V_{k}\nabla_{k}A_{pq}.$

Next we use (3) to derive

$\displaystyle(\partial_{t}-\Delta_{\gamma})(\dot{\gamma}^{ij}W_{ij})$
$\displaystyle=|A|^{2}_{\gamma}\dot{\gamma}^{ij}W_{ij}+\nabla_{i}G(\nabla_{t}-\Delta_{\gamma})V_{i}+\dot{\gamma}^{ij}A_{il}A_{jk}\nabla_{l}GV_{k}$
$\displaystyle+2G\dot{\gamma}^{ij}A_{ik}(2\nabla_{t}A_{jk}-GA^{2}_{jk}+V_{l}\nabla_{l}A_{jk})$
$\displaystyle-2\dot{\gamma}^{ij}\nabla_{i}V_{k}\nabla_{j}\nabla_{k}G+2\dot{\gamma}^{ij}A_{jk}\nabla_{i}G\nabla_{k}G-\frac{1}{2t^{2}}G$
$\displaystyle+\ddot{\gamma}^{ij,pq}\nabla_{t}A_{ij}\nabla_{t}A_{pq}+\ddot{\gamma}^{ij,pq}V_{k}\nabla_{k}A_{ij}\nabla_{t}A_{pq}.$

Summing these two formulae yields

$\displaystyle(\partial_{t}-\Delta_{\gamma})(Q(V))$
$\displaystyle=|A|^{2}_{\gamma}Q(V)+2X_{i}(\nabla_{t}-\Delta_{\gamma})V_{i}$
$\displaystyle+2G\dot{\gamma}^{ij}A_{ik}(2\nabla_{t}A_{jk}-GA^{2}_{jk}+2V_{l}\nabla_{l}A_{jk})$
$\displaystyle-2\dot{\gamma}^{ij}\nabla_{i}V_{k}(2\nabla_{t}A_{jk}-2GA^{2}_{jk}+2\nabla_{k}A_{jl}V_{l}+A_{kl}\nabla_{j}V_{l})$
$\displaystyle+2\dot{\gamma}^{ij}A_{il}A_{jk}V_{k}\nabla_{l}G+2\dot{\gamma}^{ij}A_{jk}\nabla_{i}G\nabla_{k}G-\frac{1}{2t^{2}}G$
$\displaystyle+\ddot{\gamma}^{ij,pq}(\nabla_{t}A_{ij}+V_{k}\nabla_{k}A_{ij})(\nabla_{t}A_{pq}+V_{l}\nabla_{l}A_{pq}).$

We may rewrite this as

$\displaystyle(\partial_{t}$ $\displaystyle-\Delta_{\gamma})(Q(V))$
$\displaystyle=\bigg{(}|A|^{2}_{\gamma}-\frac{2}{t}\bigg{)}Q(V)+2X_{i}U_{i}-4\dot{\gamma}^{ij}Y_{ik}W_{jk}-2\dot{\gamma}^{ij}\nabla_{i}V_{k}A_{kl}Y_{jl}$
$\displaystyle-2G\dot{\gamma}^{ij}\bigg{(}\frac{1}{2t}A_{jk}+GA^{2}_{jk}\bigg{)}+2\dot{\gamma}^{ij}\nabla_{i}V_{k}\bigg{(}\frac{1}{2t}A_{jk}+GA^{2}_{jk}\bigg{)}$
$\displaystyle+2\dot{\gamma}^{ij}A_{ik}\nabla_{k}G(A_{jl}V_{l}+\nabla_{j}G-X_{j})-\frac{1}{t}\dot{\gamma}^{ij}\bigg{(}\frac{1}{2t}A_{ij}+GA^{2}_{ij}\bigg{)}$
$\displaystyle+\ddot{\gamma}^{ij,pq}(\nabla_{t}A_{ij}+V_{k}\nabla_{k}A_{ij})(\nabla_{t}A_{pq}+V_{l}\nabla_{l}A_{pq}).$

The first term on the penultimate line is zero. Collecting the remaining
terms, the claim follows. ∎

Notice that if $A>0$ then (3) is recovered when we set $V=-A^{-1}(\nabla G)$
in (3.1).

##  4\. Application of the maximum principle

We now prove Theorem 1.1. The function $\gamma$ is assumed to be inverse-
concave (when $\gamma$ is convex the proof is analogous but much simpler). Let
$F:M\times[0,T]\to\mathbb{R}^{n+1}$ be a complete solution to (1) such that
$A\geq 0$. We assume that (3) and (4) hold on $M\times[0,T]$ for some positive
constants $C$ and $\varepsilon$. That is, with respect to any orthonormal
frame,

$C^{-1}g_{ij}\leq\dot{\gamma}^{ij}\leq Cg_{ij},$

and

$(\ddot{\gamma}^{ij,kl}+2\dot{\gamma}^{ik}(A+\varepsilon
Gg)^{-1}_{jl})S_{ij}S_{kl}\geq 0$

for every symmetric $S$. We also assume bounded curvature,

$\Xi:=\sup_{M\times[0,T]}G<\infty,$

and pointwise gradient estimates of the form (5),

$\Lambda:=\sup_{M\times[0,T]}G^{-2}|\nabla A|+G^{-3}|\nabla^{2}A|<\infty.$

We define

$\hat{Q}(V)=Q(V)+\varphi+\psi|V|^{2}$

on $M\times(0,T]$, where $\varphi(x,t)$ and $\psi(t)$ are functions to be
chosen later. For now we only assume that $\varphi$ and $\psi$ are bounded
from below by positive constants, and that $\varphi$ grows at spatial infinity
in the following sense: for some $p_{0}\in M$,

(18) $\lim_{R\to\infty}\inf_{(x,t)\in M\setminus
B_{g(0)}(p_{0},R)\times(0,T]}\varphi(x,t)=\infty.$

Since $A\geq 0$ and the remaining terms in $Q$ are bounded, these assumptions
ensure that $\hat{Q}>0$ on $M$ at times close to zero.

Suppose, with the aim of deriving a contradiction, that $\hat{Q}$ fails to be
positive on $M\times[0,T]$. Then there exists a spacetime point
$(x_{0},t_{0})$ such that $\hat{Q}>0$ for $0<t<t_{0}$ and $\hat{Q}(V)=0$ at
$(x_{0},t_{0})$ for some $V\in T_{x_{0}}M$. For any extension of $V$,
Proposition 3.1 shows that at $(x_{0},t_{0})$ we have

$\displaystyle 0$
$\displaystyle\geq-4\dot{\gamma}^{ij}Y_{ik}W_{jk}-2\dot{\gamma}^{ij}Y_{ik}A_{kl}Y_{jl}$
$\displaystyle+\ddot{\gamma}^{ij,pq}(\nabla_{t}A_{ij}+V_{k}\nabla_{k}A_{ij})(\nabla_{t}A_{pq}+V_{l}\nabla_{l}A_{pq})$
$\displaystyle+(\partial_{t}-\Delta_{\gamma})\varphi+\partial_{t}\psi|V|^{2}+2\psi
V_{i}(\nabla_{t}-\Delta_{\gamma})V_{i}$
$\displaystyle-2\psi\dot{\gamma}^{ij}\nabla_{i}V_{k}\nabla_{j}V_{k}+\bigg{(}\frac{2}{t}-|A|^{2}_{\gamma}\bigg{)}(\varphi+\psi|V|^{2}).$

######  Remark 4.1.

To derive a contradiction we would like to choose $\varphi$ and $\psi$, and an
extension of $V$, so that the right-hand side of the last inequality is
positive. Given any matrix $B$, we are free to extend $V$ so that, at the
point $(x_{0},t_{0})$,

$Y_{ij}=B_{ij},\qquad U=0.$

In case $\gamma$ is convex it is sufficient to choose $B=0$. If $A>0$ and
$\gamma$ is inverse-concave, choosing $B_{ij}=-W_{ik}A^{-1}_{kj}$ results in a
positive term which overcomes the term involving second derivatives of
$\gamma$. But when the solution is noncompact, $A^{-1}$ is unbounded, and this
introduces errors which cannot be absorbed using $\varphi$ and $\psi$. This is
why we assume (3), (4) and (5). We extend $V$ so that

(19) $Y_{ij}=-W_{ik}(A+\varepsilon Gg)^{-1}_{kj}$

at $(x_{0},t_{0})$, where $\varepsilon$ is the constant in (4). This is
sufficient to overcome the term involving second derivatives of $\gamma$, and
only introduces errors which are bounded in terms of $C$, $\Xi$, $\Lambda$ and
$\varepsilon$.

In what follows $\Theta$ is a large constant which may depend on $n$,
$\gamma$, $C$, $\Xi$ and $\Lambda$. The value of $\Theta$ may change from line
to line.

Inserting (19), we obtain

$\displaystyle-4\dot{\gamma}^{ij}Y_{ik}W_{jk}-2\dot{\gamma}^{ij}Y_{ik}A_{kl}Y_{jl}=2\dot{\gamma}^{ik}(A+\varepsilon
Gg)^{-1}_{jl}W_{ij}W_{kl}+2\varepsilon G\dot{\gamma}^{ij}Y_{ik}Y_{jk}.$

Since $\gamma$ is inverse-concave and homogeneous of degree one, if $A$ is
positive-definite at $(x_{0},t_{0})$ then we have

$\ddot{\gamma}^{ij,pq}A_{ij}S_{pq}=-(2\dot{\gamma}^{ip}A^{-1}_{jq}-2G^{-1}\dot{\gamma}^{ij}\dot{\gamma}^{pq})A_{ij}S_{pq}=0$

for every symmetric matrix $S$. Actually the same holds even if $A$ is only
nonnegative—to see this, approximate $A_{ij}$ by a sequence of positive-
definite symmetric matrices and pass to the limit. It follows that

$\displaystyle\ddot{\gamma}^{ij,pq}$
$\displaystyle(\nabla_{t}A_{ij}+V_{k}\nabla_{k}A_{ij})(\nabla_{t}A_{pq}+V_{l}\nabla_{l}A_{pq})=\ddot{\gamma}^{ij,pq}W_{ij}W_{pq},$

and hence

$\displaystyle\ddot{\gamma}^{ij,pq}$
$\displaystyle(\nabla_{t}A_{ij}+V_{k}\nabla_{k}A_{ij})(\nabla_{t}A_{pq}+V_{l}\nabla_{l}A_{pq})-4\dot{\gamma}^{ij}Y_{ik}W_{jk}-2\dot{\gamma}^{ij}A_{jl}Y_{ki}Y_{kl}$
$\displaystyle=(\ddot{\gamma}^{ij,pq}+2\dot{\gamma}^{ip}(A+\varepsilon
Gg)^{-1}_{jq})W_{ij}W_{pq}+2\varepsilon G\dot{\gamma}^{ij}Y_{ik}Y_{jk}.$

The right-hand side is nonnegative by hypothesis. Moreover, if we extend $V$
so that $U=0$ at $(x_{0},t_{0})$, then

$2\psi
V_{i}(\nabla_{t}-\Delta_{\gamma})V_{i}\geq-\Theta\psi|V|-\frac{2}{t}\psi|V|^{2}.$

Combining these facts we find that

$\displaystyle 0$
$\displaystyle\geq(\partial_{t}-\Delta_{\gamma})\varphi+(\partial_{t}\psi-\Theta\psi)|V|^{2}-2\psi\dot{\gamma}^{ij}\nabla_{i}V_{k}\nabla_{j}V_{k}+\bigg{(}\frac{2}{t}-|A|^{2}_{\gamma}\bigg{)}\varphi-\Theta\psi$

at $(x_{0},t_{0})$.

We may assume without loss of generality $\varepsilon<1$. Since

$G|(A+\varepsilon Gg)^{-1}|\leq\Theta\varepsilon^{-1},$

we have

$\displaystyle\dot{\gamma}^{ij}\nabla_{i}V_{k}$ $\displaystyle\nabla_{j}V_{k}$
$\displaystyle\leq\Theta(|Y|^{2}+G^{2}|A|^{2}+t^{-2})$
$\displaystyle\leq\Theta\varepsilon^{-2}G^{-2}|W|^{2}+\Theta(t^{-2}+1)$
$\displaystyle\leq\Theta\varepsilon^{-2}(G^{-2}|\nabla^{2}A|^{2}+G^{-2}|\nabla
A|^{2}|V|^{2}+G^{-2}|A|^{6}+t^{-2}G^{-2}|A|^{2})$
$\displaystyle+\Theta(t^{-2}+1)$
$\displaystyle\leq\Theta\varepsilon^{-2}(G^{-6}|\nabla^{2}A|^{2}+G^{-4}|\nabla
A|^{2}|V|^{2}+t^{-2}+1).$

As a consequence of the pointwise gradient estimates,

$-2\psi\dot{\gamma}^{ij}\nabla_{i}V_{k}\nabla_{j}V_{k}\geq-\Theta\varepsilon^{-2}\psi|V|^{2}-\Theta\varepsilon^{-2}\bigg{(}1+\frac{1}{t^{2}}\bigg{)}\psi.$

Therefore, at $(x_{0},t_{0})$ we have

$\displaystyle 0$
$\displaystyle\geq(\partial_{t}-\Delta_{\gamma})\varphi+(\partial_{t}\psi-\Theta\varepsilon^{-2}\psi)|V|^{2}+\bigg{(}\frac{2}{t}-\Theta\bigg{)}\varphi-\Theta\varepsilon^{-2}\bigg{(}1+\frac{1}{t^{2}}\bigg{)}\psi.$

Let $b$, $K$ and $L$ be positive constants. For $t\in(0,T]$ set

$\psi(t):=be^{Kt},\qquad\varphi(x,t):=bt^{-3/2}e^{Lt}f(x,t),$

where $f(x,t)\geq 1$ is such that (18) holds and we have

$(\partial_{t}-\Delta_{\gamma})f\geq 2Lf.$

The construction of a function $f$ with these properties is standard—see eg.
[CCG+08, Lemma 12.7]. Inserting these definitions gives

$\displaystyle 0$
$\displaystyle\geq\bigg{(}\frac{1}{2t}+2L-\Theta\bigg{)}\frac{be^{Lt}}{t^{3/2}}f-\Theta\varepsilon^{-2}\bigg{(}1+\frac{1}{t^{2}}\bigg{)}be^{Kt}+(K-\Theta\varepsilon^{-2})\psi|V|^{2}$

at $(x_{0},t_{0})$. For $K>\Theta\varepsilon^{-2}$ and
$L>\max\\{\Theta/2,K\\}$ we conclude that

$\displaystyle 0$
$\displaystyle\geq\bigg{(}\frac{1}{2t}+2L-\Theta\bigg{)}\frac{1}{t^{3/2}}-\Theta\varepsilon^{-2}\bigg{(}1+\frac{1}{t^{2}}\bigg{)}.$

at $t=t_{0}$. But if $L$ is sufficiently large the quantity on the right is
positive for every $t\in(0,T]$. This is a contradiction.

We thus conclude that $\hat{Q}(V)>0$ for every point $(x,t)\in M\times(0,T]$
and tangent vector $V\in T_{x}M$. Sending $b\to 0$, we obtain $Q(V)\geq 0$.
With this the proof of Theorem 1.1 is complete.

Theorem 1.2 is proven as follows. Here we are assuming $\gamma$ is convex, or
concave and strictly inverse-concave. Suppose $F$ is a convex ancient solution
satisfying the hypotheses of the theorem. Fix a $T<\infty$. Since $F$ is
assumed to be uniformly $k$-convex on $[-T,0]$, with $k\leq
n-m_{\operatorname{IC}}+1$, Lemma 2.1 implies there is a constant $C$ such
that (3) holds for $t\in[-T,0]$. Moreover, Lemma 2.3 implies there is a
constant $\varepsilon$ such that (4) holds for $t\in[-T,0]$. Since we are
assuming convexity and noncollapsing, Corollary 5.2 in [BH17] (that result is
for a specific flow, but the proof generalises—see [Lyn20, Theorem 4.14] for
the general case) yields the pointwise gradient estimates

$G^{-2}|\nabla A|+G^{-3}|\nabla^{2}A|\leq\Lambda$

for some $\Lambda<\infty$. This $\Lambda$ depends only on $n$, $\gamma$, the
ellipticity constant $C$ and the noncollapsing constant. Therefore, $F$
satisfies all of the hypotheses of Theorem 1.1 on $[-T,0]$. Consequently, we
have

$\partial_{t}G+2\langle\nabla G,V\rangle+A(V,V)+\frac{G}{2(t+T)}\geq 0$

at every point $(x,t)\in M\times(-T,0]$ and for every $V\in T_{x}M$. Since
this is true for every $T<\infty$, we may send $T\to\infty$ to obtain

$\partial_{t}G+2\langle\nabla G,V\rangle+A(V,V)\geq 0.$

This completes the proof of Theorem 1.2.

##  5\. The equality case

In this section we prove Theorem 1.3. Suppose $\gamma$ is strictly inverse-
concave. Let $F:M\times(-\infty,0]\to\mathbb{R}^{n+1}$ be an ancient solution
to (1) which satisfies $A>0$ and the Harnack inequality

$\partial_{t}G-A^{-1}(\nabla G,\nabla G)\geq 0$

at every point in spacetime. In addition, we assume bounded curvature

$\sup_{M\times[-T,0]}G<\infty$

and the uniform ellipticity condition (3) in $[-T,0]$ for each $T<\infty$.

Suppose there is some $(x_{0},t_{0})\in M\times(-\infty,0]$ at which

$\partial_{t}G-A^{-1}(\nabla G,\nabla G)=0.$

The computations at the beginning of Section 3 show that

$\displaystyle(\partial_{t}$
$\displaystyle-\Delta_{\gamma})(\partial_{t}G-A^{-1}(\nabla G,\nabla G))$
$\displaystyle\geq|A|^{2}_{\gamma}(\partial_{t}G-A^{-1}(\nabla G,\nabla G))$
$\displaystyle+(\ddot{\gamma}^{ij,kl}+2\dot{\gamma}^{ik}A^{-1}_{jl})(\nabla_{t}A_{ij}-A^{-1}(\nabla
G,\nabla A_{ij}))(\nabla_{t}A_{kl}-A^{-1}(\nabla G,\nabla A_{kl}))$

on $M\times(-\infty,0]$. The right-hand side is nonnegative, so the strong
maximum principle implies

$\partial_{t}G-A^{-1}(\nabla G,\nabla G)=0$

for $t\leq t_{0}$. Since $\gamma$ is assumed to be strictly inverse-concave,
we conclude that

$\nabla_{t}A_{ij}-A^{-1}(\nabla G,\nabla A_{ij})=0,$

for $t\leq t_{0}$.

Define $\xi=-G\nu-A^{-1}(\nabla G)$. Using the Codazzi equations we compute

$\displaystyle\langle D_{i}\xi,e_{j}\rangle$
$\displaystyle=A^{-1}_{jk}A^{-1}(\nabla G,\nabla
A_{ik})-GA_{ij}-A_{jk}^{-1}\nabla_{i}\nabla_{k}G$
$\displaystyle=A^{-1}_{jk}A^{-1}(\nabla G,\nabla
A_{ik})-A_{jk}^{-1}\nabla_{t}A_{ik}$ $\displaystyle=0$

for $t\leq t_{0}$, where $D$ is the Euclidean connection on
$\mathbb{R}^{n+1}$. It follows that $\xi$ can be extended to a constant vector
field $\xi$ on $\mathbb{R}^{n+1}$ and, by definition,

$G=-\langle\xi,\nu\rangle.$

That is, $F$ is a translating soliton with velocity $\xi$ for $t\leq t_{0}$.

It is now straightforward to show that $F$ is a translating soliton with
velocity $\xi$ for $t\leq 0$. Since $\xi$ is constant on $\mathbb{R}^{n+1}$,
the quantity $u:=G+\langle\xi,\nu\rangle$ satisfies

$\partial_{t}u=\dot{\gamma}^{ij}\nabla_{i}\nabla_{j}u+\dot{\gamma}^{ij}A_{ik}A_{kj}u.$

Moreover, because we have bounded curvature and uniform ellipticity, the
function $h(x,t)=e^{Kt}(|x|^{2}+1)$ satisfies

$\partial_{t}h>\dot{\gamma}^{ij}\nabla_{i}\nabla_{j}h+\dot{\gamma}^{ij}A_{ik}A_{kj}h$

for $t\geq t_{0}$ if the constant $K$ is sufficiently large. By the maximum
principle, $\sup_{M_{t}}\tfrac{|u|}{h}$ is nonincreasing for $t_{0}\leq t\leq
0$. Since $u$ vanishes at $t=t_{0}$, we have $G=-\langle\xi,\nu\rangle$ for
all $t\leq 0$.

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