1 Introduction
When giving an alternate formulation of the Ma–Trudinger–Wang regularity theory, Kim and McCann [Reference Kim and McCannKM10] defined pseudo-Riemmanian metrics with signature
$(n,n)$
on the product space
$M\times \bar {M}$
. This was followed [Reference Kim, McCann and WarrenKMW10] by the observation that the graph of the optimal transportation map is a volume maximizing n-dimensional submanifold (of codimension n), with respect to a conformal modification of the metric in [Reference Kim and McCannKM10]. In this note, we explore the idea that one can directly derive regularity theory purely from the property of being maximal. If the metric is of Kim–McCann type, a calibrated Lagrangian submanifold is either the graph of a solution to an optimal transportation problem, or, if the manifold has topology, could be the graph of a Lie solution to the optimal transportation problem [Reference WarrenWar11]. Here, to keep things simple, we restrict our attention to the case when
$n=2.$
Recall the setting of Kim–McCann [Reference Kim and McCannKM10] where metrics on the product space
$M\times \bar {M}$
are locally given by
$$ \begin{align} h=\left( \begin{array} [c]{cc} 0 & h_{i\bar{j}}\left( x,\bar{x}\right) \\ h_{i\bar{j}}^{T}\left( x,\bar{x}\right) & 0 \end{array} \right), \end{align} $$
where
$$ \begin{align} h_{i\bar{j}}(x,\bar{x})=-\frac{1}{2}\partial_{\bar{j}}\partial_{i}c(x,\bar {x}) \end{align} $$
for some cost function
$c:M\times \bar {M}\rightarrow \mathbb {R}.$
We assume here and in the sequel that c is twice differentiable and satisfies the (A2) condition on
$\mathcal {N}\subset M\times \bar {M} \backslash \mathcal {C}$
where
$\mathcal {C}$
is a measure zero set which we call the “cut locus.” In particular,
$h_{i\bar {j}}\left ( x,\bar {x}\right ) $
is non-degenerate on
$\mathcal {N}$
, where h defines an
$\left ( n,n\right ) $
signature metric.
The underlying optimal transportation problem is to find the map between the measures defined by bounded mass densities
$\rho ,\bar {\rho }$
that minimizes the total cost:
$$\begin{align*}\int_{M}c(x,T(x))\rho(x)dx \end{align*}$$
over the space of maps T satisfying
$$\begin{align*}T_{\#}\rho(x)dx=\bar{\rho}(\bar{x})d\bar{x}. \end{align*}$$
In [Reference Kim, McCann and WarrenKMW10], it was illustrated that when taking the conformal metric
$$ \begin{align} \tilde{h}=\left[ \frac{\rho(x)\bar{\rho}\left( \bar{x}\right) }{\det\left( -\partial_{\bar{j}}\partial_{i}c(x,\bar{x})\right) }\right] ^{1/n}\left( \begin{array} [c]{cc} 0 & h_{i\bar{j}}\left( x,\bar{x}\right) \\ h_{i\bar{j}}^{T}\left( x,\bar{x}\right) & 0 \end{array} \right) \end{align} $$
graphs of optimal transportation plans T are calibrated maximal submanifolds with respect to this metric [Reference Kim, McCann and WarrenKMW10, Theorem 1.1]. The conformal modification of the Kim–McCann metric can sometimes lead to technical computational issues. So here we point out that a weighted approach can be convenient: Consider instead the weighted manifold
$$ \begin{align} \left( M\times\bar{M},h,\left( \frac{\rho(x)\bar{\rho}(\bar{x})}{\det\left( -\partial_{\bar{j}}\partial_{i}c(x,\bar{x})\right) }\right) ^{1/2}\right). \end{align} $$
One can check that the volume of an n-submanifold with this weight is the same as the volume of the n-submanifold in the conformal setting defined by (1.3). Thus the minimal surfaces are the same in either setting. However, instead of the minimal surface equation occuring in the setting (1.3), in the latter setting (1.4), the manifolds will be locally defined by the weighted minimal surface equation
$$ \begin{align} \vec{H}+\left( \nabla f\right) ^{N}=0, \end{align} $$
where
$$\begin{align*}e^{-f}=\left( \frac{\rho(x)\bar{\rho}(\bar{x})}{\det\left( -\partial _{\bar{j}}\partial_{i}c(x,\bar{x})\right) }\right) ^{1/2}. \end{align*}$$
Here, we offer a proof-of-concept that the regularity theory can be derived geometrically, obtaining yet another approach for the Ma–Trudinger–Wang regularity theory. The original paper [Reference Ma, Trudinger and WangMTW05] presents a maximum principle argument applied to the Monge–Ampère equation, while the approach by Loeper [Reference LoeperLoe09] is a careful analysis of the cost function which requires less a priori regularity. The setting of Kim–McCann offers a solid pseudo-Riemannian geometric formulation of the approach offered by Loeper, where they identify the MTW condition as a cross-curvature condition on the metric
$h,$
namely
$$\begin{align*}Rm_{h}(e_{i},e_{\bar{j}},e_{\bar{j}},e_{i})>0\text{ } \end{align*}$$
whenever
$e_{i}\in T_{p}M,e_{\bar {j}}\in T_{\bar {p}}\bar {M}$
and
$h(e_{i},e_{\bar {j}})=0.$
The cross-curvature positivity is preserved under conformal changes [Reference Kim, McCann and WarrenKMW10, Remark 4.2] so if it is present in the Kim–McCann metric (1.1) it will also be present in the metric (1.3).
To significantly simplify the computation we restrict to the case of compact ambient manifolds of the form
$M\times \bar {M}$
with metric of the form (1.1) where both manifolds have dimension
$n=2.$
While we focus on the case where the metric is of the Kim–McCann form (1.2), it should be possible to loosen this to more general pseudo-Riemmanian submanifolds satisfying suitable curvature conditions. Indeed, our approach is motivated by the paper [Reference Li and SalavessaLS11] where general regularity results follow from curvature conditions.
Recently Brendle–Léger–McCann–Rankin [Reference Brendle, Léger, McCann and RankinBLMR23] have proven regularity for maximal surfaces in pseudo-Riemannian metrics with positive cross-curvature in general dimensions via a generalized method which applies to maximal surfaces in manifolds of general codimension. The method is similar but slightly different: here our strategy is to bound the Hodge dual of an alternating two form (following [Reference Li and SalavessaLS11]), whereas the approach in [Reference Brendle, Léger, McCann and RankinBLMR23] is to bound the maximum eigenvalue of a symmetric
$\left ( 0,2\right ) $
tensor field restricted to the submanifold. Naturally, both approaches rely on covariantly differentiating the
$\left ( 0,2\right ) $
tensor twice and using the positive curvature terms that arise via the Codazzi formula, together with the minimal surface equation.Footnote
1
2 Prelimaries
We consider the graph
$$\begin{align*}\Gamma=\left\{ (x,T(x):x\in M\right\} \subset M\times\bar{M} \end{align*}$$
for an optimal transport map
$$\begin{align*}T:M\rightarrow\bar{M}. \end{align*}$$
Assume that the graph
$\Gamma $
lies in a compact subset
$\mathcal {N}\subset $
$M\times \bar {M}\backslash \mathcal {C}$
where
$\mathcal {C}$
is the cut locus where c might not be smooth. Here we choose
$\mathcal {N}$
to be a compact subset staying clear of the cut locus. (An important aspect of regularity theory for optimal transport maps is the “stay-away” property, that is, there is an a priori lower bound on the distance of the graph to the cut locus. This is typically argued in the process of a regularity argument - but as we are focusing on an alternative approach to the local regularity, we may steer clear of these arguments by assuming the graph stays within a compact set avoiding the cut locus.)
Recall that for graphs of optimal transport maps,
$\Gamma $
is a Lagrangian submanifold with respect to the symplectic form given by [Reference Kim and McCannKM10, (5.3)]
$$\begin{align*}h_{i\bar{j}}dx^{i}\wedge d\bar{x}^{j}, \end{align*}$$
and that there is a relation between this symplectic form and the metric given by
$$ \begin{align} h(V,W)=h_{i\bar{j}}dx^{i}\wedge d\bar{x}^{j}(KV,W), \end{align} $$
where
$$\begin{align*}K:T_{\left( p,\bar{p}\right) } \left(M\times\bar{M}\right) \rightarrow T_{\left( p,\bar {p}\right) } \left(M\times\bar{M}\right) \end{align*}$$
is the map represented as
$$\begin{align*}K_{p,\bar{p}}=I_{T_{p}M}\oplus\left( -I\right) _{T_{\bar{p}}\bar{M}} \end{align*}$$
that is, K acts as the identity on
$TM$
while performing a sign change on
$T\bar {M}.$
Since
$\Gamma $
is a Lagrangian submanifold, where (2.1) holds, K maps the tangent space to the normal space along
$\Gamma .$
Remark 2.1 In the sequel, we encounter an unfortunate collision of standard notations. When dealing with connections, we use
$\bar {\nabla }$
to denote the ambient connection (Levi–Civita with respect to ambient metric h), even while using
$\bar {M}, \bar {x}$
etc. to indicate the target space of optimal transport. Fortunately, we don’t have occasion to refer to the Levi–Civita connection on
$\bar {M}.$
Claim 2.2 K is parallel.
Proof Identifying K via musical isomorphism with the symplectic form, we can check that the symplectic form is parallel:
$$\begin{align*}\bar{\nabla}_{X}h(KV,W)=Xh(KV,W)-h(K\bar{\nabla}_{X}V,W)-h(KV,\bar{\nabla} _{X}W). \end{align*}$$
First assume that
$V,W$
are in
$T_{p}M.$
In this case, by [Reference Kim and McCannKM10, Lemma 4.1] the connection maps
$T_{p}M$
to
$T_{p}M$
, so all terms above vanish. Similarly if both
$V,W$
are in
$T_{\bar {p}}\bar {M}.$
Now consider the case that
$V\in T_{p}M,W\in T_{\bar {p}}\bar {M}.$
$$\begin{align*}\bar{\nabla}_{X}h(KV,W)=Xh(V,W)-h(\bar{\nabla}_{X}V,W)-h(V,\bar{\nabla}_{X}W) \end{align*}$$
which is just the fact that h is parallel with respect to itself. On the other hand if roles are reversed
$$\begin{align*}\bar{\nabla}_{X}h(KV,W)=Xh(-V,W)-h(-\bar{\nabla}_{X}V,W)-h(-V,\bar{\nabla} _{X}W). \end{align*}$$
Thus
$K^{\flat }$
is parallel.
Given a frame for the submanifold
$\Gamma ,$
we may define the components of the second fundamental form
$$\begin{align*}b_{ijk}=h(\bar{\nabla}_{\partial_{i}}\partial_{j},K\partial_{k}). \end{align*}$$
Claim 2.3 The expression for
$b_{ijk}$
is symmetric in all three indices.
Proof Because K is parallel,
$K^{2}=I$
and
$h(K\cdot ,K\cdot )=-h(\cdot ,\cdot )$
$$ \begin{align*} 0 & =\partial_{i}h(\partial_{j},K\partial_{k})\\ & =h(\bar{\nabla}_{\partial_{i}}\partial_{j},K\partial_{k})+h(\partial _{j},K\bar{\nabla}_{\partial_{i}}\partial_{k})\\ & =h(\bar{\nabla}_{\partial_{i}}\partial_{j},K\partial_{k})-h(K\partial _{j},\bar{\nabla}_{\partial_{i}}\partial_{k})\\ & =b_{ijk}-b_{ikj}. \end{align*} $$
This establishes symmetry in the last two. Symmetry in the first two is well-known for the second fundamental form.
The following is the Ma–Trudinger–Wang curvature condition as uncovered by Kim and McCann in the product manifold setting [Reference Kim and McCannKM10, Definition 2.3].
Definition 2.1 The metric h is strictly regular, if whenever
$\left ( x,\bar {x}\right ) \in N$
and
$$ \begin{gather*} 0\neq\partial_{i}\in T_{x}M\\ 0\neq\partial_{\bar{j}}\in T_{\bar{x}}\bar{M}\\ \text{ }h_{(x,\bar{x})}(\partial_{i},\partial_{\bar{j}})=0 \end{gather*} $$
then the curvature of the metric h satisfies
$$\begin{align*}Rm_{h}\left( \partial_{i},\partial_{\bar{j}},\partial_{\bar{j}},\partial _{i}\right)>0. \end{align*}$$
Note that the metric
$\tilde {h}$
will also satisfy this property [Reference Kim, McCann and WarrenKMW10, Remark 4.2].
3 Special coordinate charts at a point
We start by fixing arbitrary Riemannian metrics
$g_{1},g_{2}$
on
$M,\bar {M}.$
Let
$\left ( p,\bar {p}\right ) \in M\times \bar {M}.$
Given
$\left \{ e_{1},e_{2}\right \} $
an oriented
$g_{1}$
-orthonormal basis for
$T_{p}M$
, we may take exponential coordinates (w.r.t
$g_{1}$
) near p for a neighborhood of
$M,$
and then do similarly for an oriented orthonormal basis for
$T_{\bar {p}}\bar {M},$
giving us a local product neighborhood
$U\times \bar {U} .$
In this
$U\times \bar {U},$
we have
$$\begin{align*}h=\left( \begin{array} [c]{cc} 0 & h_{i\bar{j}}\left( x,\bar{x}\right) \\ h_{i\bar{j}}^{T}\left( x,\bar{x}\right) & 0 \end{array} \right). \end{align*}$$
Now take a local change of coordinates
$$\begin{align*}\varphi:\tilde{U}\rightarrow\bar{U} \end{align*}$$
so that
$$ \begin{align} \varphi(0) & =\bar{p}\end{align} $$
$$ \begin{align} D\varphi & =h_{i\bar{j}}\left( p,\bar{p}\right) ^{-1} \end{align} $$
as
$h_{ij}$
is nondegenerate. After this coordinate change, the metric at the point
$\left ( p,\bar {p}\right ) $
with respect to the basis on
$U\times \tilde {U}$
becomes
$$ \begin{align} h=\left( \begin{array} [c]{cc} 0 & I\\ I & 0 \end{array} \right). \end{align} $$
Now forgetting the old vertical basis vectors here, consider the new ones
$\left \{ \bar {e}_{1},\bar {e}_{2}\right \} $
that occur after transformation
$D\varphi .$
These are not expected to be orthonormal with respect to
$g_{2}$
but satisfy
$$\begin{align*}h(e_{i},\bar{e}_{j})=\delta_{ij} \end{align*}$$
and
$$ \begin{align} \frac{1}{C_{1}}\leq\left\Vert \bar{e}_{i}\right\Vert _{g_{2}}\leq C_{1} \end{align} $$
for some universal constant
$C_{1}$
depending on bounds on
$h_{i\bar {j} }\left ( p,\bar {p}\right ) ^{-1}$
and our choice of
$g_{2}.$
Now given a
$g_{1}$
-orthonormal
$\left \{ e_{1},e_{2}\right \} $
basis at p and a graphical manifold
$\Gamma ,$
let g be the induced metric from h on
$\Gamma ,$
that is
$$\begin{align*}g_{11}=h(e_{1},e_{1})\, \text{etc.} \end{align*}$$
This can be diagonalized by an orthogonal rotation of
$\left \{ e_{1} ,e_{2}\right \} $
so we may assume that the expression of the metric g is diagonal for our choice of
$\left \{ e_{1},e_{2}\right \} $
: There will be positive
$\lambda _{1},\lambda _{2}$
such that
$$ \begin{align} \left\Vert e_{i}\right\Vert _{g}^{2} & =\lambda_{i}^{2},\text{ } i=1,2.\\ \langle e_{1},e_{2}\rangle_{g} & =0\nonumber\\ \langle e_{i},e_{j}\rangle_{g_{1}} & =\delta_{ij}\nonumber \end{align} $$
and WLOG
$\lambda _{1}\geq \lambda _{2}.$
(Here we use
$e_{i}$
in two senses, identifying
$e_{i}$
first as an abstract tangent vector on the manifold M, which is measured by
$g_{1}$
, but also as a tangent vector to the submanifold
$\Gamma ,$
which is measured by
$h,$
restricting to the metric g, which is positive definite on
$\Gamma $
as the submanifold
$\Gamma $
is spacelike).
It will be convenient to use the orthonormal basis with respect to
$g:$
$$ \begin{align*} \partial_{1} & =\frac{1}{\lambda_{1}}e_{1}\\ \partial_{2} & =\frac{1}{\lambda_{2}}e_{2} \end{align*} $$
so that
$$\begin{align*}\left\Vert \partial_{1}\right\Vert _{g}=\left\Vert \partial_{2}\right\Vert _{g}=1. \end{align*}$$
$$ \begin{align*} \partial_{1} & =(\frac{1}{\lambda_{1}},0,\lambda_{1},\ast)\\ \partial_{2} & =(0,\frac{1}{\lambda_{2}},\ast,\lambda_{2}) \end{align*} $$
(here
$*$
indicates we haven’t determined this value from (3.5) and (3.3)) and then (3.3) with the fact that
$\Gamma $
is Lagrangian gives us (still only at the point)
$$ \begin{align} \partial_{1} & =(\frac{1}{\lambda_{1}},0,\lambda_{1},0)\\ \partial_{2} & =(0,\frac{1}{\lambda_{2}},0,\lambda_{2}).\nonumber \end{align} $$
At the point
$\left ( p,\bar {p}\right ) $
, consider also the orthonormal basis for the normal space
$$ \begin{align} n_{1} & =(\frac{1}{\lambda_{1}},0,-\lambda_{1},0)\\ n_{2} & =(0,\frac{1}{\lambda_{2}},0,-\lambda_{2})\nonumber \end{align} $$
which can be extended to a normal frame via
$$\begin{align*}n_{i}:=K\partial_{i}. \end{align*}$$
Now with respect to the Euclidean systems on
$U\times \tilde {U}$
we have the representation
$$ \begin{align} DT=\left( \begin{array} [c]{cc} \lambda_{1}^{2} & 0\\ 0 & \lambda_{2}^{2} \end{array} \right). \end{align} $$
4 Setup before the covariant differentiation
Our goal is to prove the following theorem, which is a special case of general work by Ma–Trudinger–Wang and Loeper.
Theorem 4.1 Suppose that T is a smooth optimal transport plan from an oriented compact two-dimensional manifold M to an oriented compact two-dimensional manifold
$\bar {M}$
with respect to a cost function c which satisfies a positive cross-curvature condition (Definition 2.1), for measures
$\rho $
and
$\bar {\rho }$
which are given. Assume that the densities of
$\rho ,\bar {\rho }$
are smooth and bounded away from zero, and that the graph of T lies in an a priori determined compact set
$\mathcal {N}$
on which c is smooth. Then with respect to any given metrics
$g_{1}$
and
$g_{2}$
on M and
$\bar {M},$
the derivative
$DT$
satisfies an a priori bound:
$$\begin{align*}\left\Vert DT\right\Vert \leq C(g_{1},g_{2},c,\rho,\bar{\rho},\mathcal{N}), \end{align*}$$
where
$$\begin{align*}\left\Vert DT\right\Vert =\max\left\{ \left\Vert DT\left( V\right) \right\Vert _{g_{2}}\mid V\in T_{x}M,\left\Vert V\right\Vert _{g_{1} }=1\right\}. \end{align*}$$
Choose
$g_{1},g_{2}$
to be arbitrary Riemannian metrics on
$M,\bar {M}$
. These metrics will be fixed and will serve as a gauge against which to obtain estimates. We assume that M is oriented, so on any neighborhood U we can take an oriented orthonormal (w.r.t
$g_{1}$
) frame
$\left \{ e_{1} ,e_{2}\right \} $
for
$U.$
Now define
$$ \begin{align} \Omega=\left[ g_{1}\left( e_{1},\cdot\right) +h_{1}(e_{1},\cdot)\right] \wedge\left[ g_{1}\left( e_{2},\cdot\right) +h_{1}(e_{2},\cdot)\right] \end{align} $$
to be a two-form on
$U\times \bar {M}.$
One can check that this does not depend on the choice of oriented orthonormal frame
$\left \{ e_{1} ,e_{2}\right \} $
so using a partition of unity, we may extend
$\Omega $
to a well-defined form everywhere where h is defined on
$M\times \bar {M}.$
Our goal is to show that the maximum value of the Hodge dual of this two-form restricted to
$\Gamma $
is a priori bounded.
4.1 Jacobian condition
Given charts, the objects
$\rho ,\bar {\rho }$
can be represented by measure densities, so when we write down the Jacobian equation satisfied by an optimal map (or more generally, a map whose graph is calibrated as in [Reference Kim, McCann and WarrenKMW10]):
$$\begin{align*}\det DT(x)\bar{\rho}(T(x))=\rho(x) \end{align*}$$
we are implictly using coordinate systems on both sides to define
$\det DT(x)$
and each of the densities. Given a choice of metric
$g_{1}$
we can define
$\rho \left ( x\right ) $
to be a well-defined density by letting
$$\begin{align*}\rho(x)=\frac{d\rho}{dV_{g_{1}}} \end{align*}$$
and similar for
$\bar {\rho }.$
Fixing these metrics, if
$$ \begin{align*} 0 & <\frac{1}{\mu_{1}}\leq\rho\leq\mu_{1}\\ 0 & <\frac{1}{\mu_{2}}\leq\bar{\rho}\leq\mu_{2} \end{align*} $$
we may conclude that
$$ \begin{align} \frac{1}{\Lambda}\leq\det DT\leq\Lambda \end{align} $$
in any coordinates with
$dV_{g_{1}}(x)=dV_{g_{2}}\left ( T(x)\right ) =1$
. By (4.2) it follows that there exists
$\Lambda \left ( h,g_{1},g_{2},\rho ,\bar {\rho }\right ) $
(maybe slightly different from previous
$\Lambda $
) such that
$$ \begin{align} \frac{1}{\Lambda}\leq\lambda_{1}\lambda_{2}\leq\Lambda \end{align} $$
provided the measure densities are continuous and bounded away from
$0$
and
$M,\bar {M}$
are compact.
4.2 Maximum principle argument
With the metric g on
$\Gamma ,$
we may consider the scalar function
$$\begin{align*}w:=\ast\Omega|_{\Gamma} \end{align*}$$
that is, the ratio
$$\begin{align*}w=\frac{\Omega|_{\Gamma}\left( \partial_{1},\partial_{2}\right) } {dV_{g}\left( \partial_{1},\partial_{2}\right) } \end{align*}$$
for any tangent frame
$\partial _{1},\partial _{2}.$
This function attains a maximum value at some point
$\left ( x_{\max },T(x_{\max })\right ) \in \Gamma .$
Claim 4.2 Theorem 4.1 follows from an a priori bound on the function
$w.$
Proof Suppose
$w(x)\leq \bar {C}.$
Then for any
$x,$
choosing special coordinates (3.5) (3.6), (3.7), and (3.8) we get (see Claim 6.2)
$$ \begin{align*} w(x) & =\frac{\Omega(\partial_{1},\partial_{2})}{1}\\ & =\frac{1}{\lambda_{1}\lambda_{2}}\left( 1+\lambda_{1}^{2}\right) (1+\lambda_{2}^{2})\leq\bar{C}. \end{align*} $$
$$\begin{align*}\left\Vert DT(e_{i})\right\Vert _{g_{2}}=\left\Vert \lambda_{i}^{2}\bar{e} _{i}\right\Vert _{g_{2}}\leq\lambda_{i}^{2}C_{1} \end{align*}$$
thus
$$\begin{align*}\max\left\{ \left\Vert DT\left( V\right) \right\Vert _{g_{2}}\mid V\in T_{x}M,\left\Vert V\right\Vert _{g_{1}}=1\right\} \leq \Lambda \bar{C} C_{1}.\\[-37pt] \end{align*}$$
Theorem 4.1 will be proved as follows: Go to
$\left ( x_{\max },T(x_{\max })\right ) $
and apply the maximum principle. This requires an expression for the covariant derivative of
$\Omega $
computed in the next section.
5 The covariant differentiation
Lemma 5.1 Suppose that
$\Gamma =\left ( x,T(x)\right ) $
is a submanifold with mean curvature vector
$\vec {H}$
and
$\Omega $
is as described in (4.1). At
$\left ( x_{\max },\bar {x}_{\max }\right ) \in \Gamma ,$
with frames defined by (3.6) and (3.7) we have
$$ \begin{align*} &\sum_{k}\nabla_{\partial_{k}}\nabla_{\partial_{k}}\Omega(\partial_{1} ,\partial_{2})- \sum_{k}\bar{\nabla}_{\partial_{k}}\bar{\nabla} _{\partial_{k}}\Omega(\partial_{1},\partial_{2})\\& \quad =\left\Vert B\right\Vert ^{2}\Omega(\partial_{1},\partial_{2})+2\sum _{k}\Omega(B(\partial_{k},\partial_{1}),B(\partial_{k},\partial_{2}))\\& \qquad +2\sum_{k}\left\{ \bar{\nabla}_{\partial_{k}}\Omega(B(\partial _{k},\partial_{1}),\partial_{2})+\bar{\nabla}_{\partial_{k}}\Omega (\partial_{1},B(\partial_{k},\partial_{2}))\right\} +\bar{\nabla}_{\vec{H} }\Omega(\partial_{1},\partial_{2})\\& \qquad -\sum_{p}\left\{ Rm_{h}\left( \partial_{2},\partial_{1},\partial _{2},n_{p}\right) \Omega(n_{p},\partial_{2})+Rm_{h}\left( \partial _{1},\partial_{2},\partial_{1},n_{p}\right) \Omega(\partial_{1,} n_{p})\right\} \\& \qquad -\sum_{s}\left\{ \left( \bar{\nabla}_{\partial_{1}}\vec{H}\cdot n_{p}\right) \Omega(n_{p},\partial_{2})+\left( \bar{\nabla}_{\partial_{2} }\vec{H}\cdot n_{p}\right) \Omega(\partial_{1},n_{p})\right\}. \end{align*} $$
Here,
$\left \Vert B\right \Vert ^{2}$
is the positive norm of the second fundamental form B for the submanifold
$\Gamma $
, and,
$Rm_{h}$
is the curvature tensor of h.
Proof To begin, extend tangent vectors
$\left \{ \partial _{1},\partial _{2}\right \} $
by taking these to be the coordinate derivatives with respect to normal coordinates on g. Then we differentiate covariantly with respect to
$g:$
$$\begin{align*}\nabla_{\partial_{k}}\Omega(\partial_{1},\partial_{2})=\partial_{k} \Omega(\partial_{1},\partial_{2})-\Omega(\nabla_{\partial_{k}}\partial _{1},\partial_{2})-\Omega(\partial_{1},\nabla_{\partial_{k}}\partial_{2}) \end{align*}$$
and again
$$ \begin{align*} \nabla_{\partial_{k}}\nabla_{\partial_{k}}\Omega(\partial_{1},\partial_{2}) & =\partial_{k}\left\{ \partial_{k}\Omega(\partial_{1},\partial_{2} )-\Omega(\nabla_{\partial_{k}}\partial_{1},\partial_{2})-\Omega(\partial _{1},\nabla_{\partial_{k}}\partial_{2})\right\} \\& \quad -\left\{ \begin{array} [c]{@{}c@{}} \nabla_{\partial_{k}}\partial_{k}\Omega(\partial_{1},\partial_{2} )-\Omega(\nabla_{\nabla_{\partial_{k}}\partial_{k}}\partial_{1},\partial _{2})-\Omega(\partial_{1},\nabla_{\nabla_{\partial_{k}}\partial_{k}} \partial_{2})\\ \partial_{k}\Omega(\nabla_{\partial_{k}}\partial_{1},\partial_{2} )-\Omega(\nabla_{\partial_{k}}\nabla_{\partial_{k}}\partial_{1},\partial _{2})-\Omega(\nabla_{\partial_{k}}\partial_{1},\nabla_{\partial_{k}} \partial_{2})\\ \partial_{k}\Omega(\partial_{1},\nabla_{\partial_{k}}\partial_{2} )-\Omega(\nabla_{\partial_{k}}\partial_{1},\nabla_{\partial_{k}}\partial _{2})-\Omega(\partial_{1},\nabla_{\partial_{k}}\nabla_{\partial_{k}} \partial_{2}) \end{array} \right\}\\& =\partial_{k}\left\{ \partial_{k}\Omega(\partial_{1},\partial_{2} )-\Omega(\nabla_{\partial_{k}}\partial_{1},\partial_{2})-\Omega(\partial _{1},\nabla_{\partial_{k}}\partial_{2})\right\} \\& \quad -\left\{ \begin{array} [c]{@{}c@{}} \partial_{k}\Omega(\nabla_{\partial_{k}}\partial_{1},\partial_{2})-\Omega (\bar{\nabla}_{\partial_{k}}\left( \nabla_{\partial_{k}}\partial_{1}\right) ^{T},\partial_{2})\\ \partial_{k}\Omega(\partial_{1},\nabla_{\partial_{k}}\partial_{2} )-\Omega(\partial_{1},\left( \nabla_{\partial_{k}}\left( \nabla _{\partial_{k}}\partial_{2}\right) \right) ^{T}) \end{array} \right\} \end{align*} $$
discarding terms that vanish when taking normal coordinates. (Implicity
$\Omega $
is identified with its restriction to
$\Gamma $
; normal covariant differentiation is via the connection on normal bundle.) Now repeat with respect to the ambient connection:
$$ \begin{align*} \bar{\nabla}_{\partial_{k}}\bar{\nabla}_{\partial_{k}}\Omega(\partial _{1},\partial_{2}) & =\partial_{k}\left\{ \partial_{k}\Omega(\partial _{1},\partial_{2})-\Omega(\bar{\nabla}_{\partial_{k}}\partial_{1},\partial _{2})-\Omega(\partial_{1},\bar{\nabla}_{\partial_{k}}\partial_{2})\right\} \\& \quad -\left\{ \begin{array} [c]{@{}c@{}} \bar{\nabla}_{\partial_{k}}\partial_{k}\Omega(\partial_{1},\partial _{2})-\Omega(\bar{\nabla}_{\bar{\nabla}_{\partial_{k}}\partial_{k}} \partial_{1},\partial_{2})-\Omega(\partial_{1},\bar{\nabla}_{\bar{\nabla }_{\partial_{k}}\partial_{k}}\partial_{2})\\ \partial_{k}\Omega(\bar{\nabla}_{\partial_{k}}\partial_{1},\partial _{2})-\Omega(\bar{\nabla}_{\partial_{k}}\bar{\nabla}_{\partial_{k}} \partial_{1},\partial_{2})-\Omega(\bar{\nabla}_{\partial_{k}}\partial_{1} ,\bar{\nabla}_{\partial_{k}}\partial_{2})\\ \partial_{k}\Omega(\partial_{1},\bar{\nabla}_{\partial_{k}}\partial _{2})-\Omega(\bar{\nabla}_{\partial_{k}}\partial_{1},\bar{\nabla} _{\partial_{k}}\partial_{2})-\Omega(\partial_{1},\bar{\nabla}_{\partial_{k} }\bar{\nabla}_{\partial_{k}}\partial_{2}) \end{array} \right\}. \end{align*} $$
Computing the difference, using definition of second fundamental form
$$\begin{align*}B(\partial_{k},\partial_{j})=\bar{\nabla}_{\partial_{k}}\partial_{j} -\nabla_{\partial_{k}}\partial_{j} \end{align*}$$
we get
$$ \begin{align} \nabla_{\partial_{k}}\nabla_{\partial_{k}}\Omega(\partial_{1},\partial _{2})-\bar{\nabla}_{\partial_{k}}\bar{\nabla}_{\partial_{k}}\Omega (\partial_{1},\partial_{2}) & =2\partial_{k}\left\{ \Omega(B(\partial _{k},\partial_{1}),\partial_{2})+\Omega(\partial_{1},B(\partial_{k} ,\partial_{2}))\right\} \\& \quad +\Omega(\bar{\nabla}_{\partial_{k}}\left( \nabla_{\partial_{k}} \partial_{1}\right) ^{T}-\bar{\nabla}_{\partial_{k}}\bar{\nabla} _{\partial_{k}}\partial_{1}-\bar{\nabla}_{\bar{\nabla}_{\partial_{k}} \partial_{k}}\partial_{1},\partial_{2})\nonumber\\& \quad +\Omega(\partial_{1},\left( \nabla_{\partial_{k}}\left( \nabla _{\partial_{k}}\partial_{2}\right) \right) ^{T}-\bar{\nabla}_{\partial_{k} }\bar{\nabla}_{\partial_{k}}\partial_{2}-\bar{\nabla}_{\bar{\nabla} _{\partial_{k}}\partial_{k}}\partial_{2})\nonumber\\& \quad -2\Omega\left( B(\partial_{k},\partial_{1}),B(\partial_{k},\partial _{2})\right) +B(\partial_{k},\partial_{k})\Omega(\partial_{1},\partial _{2}).\nonumber \end{align} $$
Summing over
$k,$
recalling that we have chosen an orthonormal basis at the point
$$\begin{align*}\vec{H}=B(\partial_{k},\partial_{k}). \end{align*}$$
Noting also that in normal coordinates (introducing
$B_{k1}:=B(\partial _{k},\partial _{1})$
etc., as shorthand)
$$ \begin{align} & \bar{\nabla}_{\partial_{k}}\bar{\nabla}_{\partial_{k}}\partial_{1} -\nabla_{\partial_{k}}\nabla_{\partial_{k}}\partial_{1}=\bar{\nabla} _{\partial_{k}}\left\{ \nabla_{\partial_{k}}\partial_{1}+B_{k1}\right\} -\nabla_{\partial_{k}}\nabla_{\partial_{k}}\partial_{1}\nonumber\\& \quad =\bar{\nabla}_{\partial_{k}}\left\{ \Gamma_{k1}^{l}\partial_{l} +B_{k1}\right\} -\nabla_{\partial_{k}}\left( \Gamma_{k1}^{l}\partial _{l}\right) \nonumber\\& \quad =\left\{ \partial_{k}\Gamma_{k1}^{l}\partial_{l}+\Gamma_{k1}^{l}\left( \nabla_{\partial_{k}}\partial_{l}+B(k,l)\right) +\bar{\nabla}_{\partial_{k} }B_{k1}\right\} -\left( \partial_{k}\Gamma_{k1}^{l}\partial_{l}+\Gamma _{k1}^{l}\nabla_{k}\partial_{l}\right) \nonumber\\& \quad =\bar{\nabla}_{\partial_{k}}B_{k1}. \end{align} $$
Now in the first line of (5.1) we may use
$$\begin{align*}\partial_{k}\Omega(B_{k1},\partial_{2})=\bar{\nabla}_{\partial_{k}} \Omega(B_{k1},\partial_{2})+\Omega\left( \bar{\nabla}_{\partial_{k}} B_{k1},\partial_{2}\right) +\Omega\left( B_{k1},\bar{\nabla}_{\partial_{k} }\partial_{2}\right) \end{align*}$$
together with (5.2) to get
$$ \begin{align} \nabla_{\partial_{k}}\nabla_{\partial_{k}}\Omega(\partial_{1},\partial _{2})-\bar{\nabla}_{\partial_{k}}\bar{\nabla}_{\partial_{k}}\Omega (\partial_{1},\partial_{2}) & =2\bar{\nabla}_{\partial_{k}}\Omega (B_{k1},\partial_{2})+2\bar{\nabla}_{\partial_{k}}\Omega(\partial_{1} ,B_{k2})\\& \quad +\bar{\nabla}_{\vec{H}}\Omega(\partial_{1},\partial_{2})+2\Omega\left( B_{k1},\bar{\nabla}_{\partial_{k}}\partial_{2}\right) \nonumber\\& \quad +\Omega\left( \bar{\nabla}_{\partial_{k}}B_{k1},\partial_{2}\right) +\Omega(\partial_{1},\bar{\nabla}_{\partial_{k}}B_{k2}).\nonumber \end{align} $$
Now using the alternating nature of
$\Omega $
and tangential and normal decomposition, we have
$$ \begin{align*} \Omega(\bar{\nabla}_{\partial_{k}}B_{k1},\partial_{2}) & =\Omega\left( \left( \bar{\nabla}_{\partial_{k}}B_{k1}\right) ^{T}+\left( \bar{\nabla }_{\partial_{k}}B_{k1}\right) ^{N},\partial_{2}\right) \\ & =\Omega\left( \left( \left( \bar{\nabla}_{\partial_{k}}B_{k1}\right) \cdot\partial_{1}\right) \partial_{1},\partial_{2}\right) +\Omega\left( \left( \bar{\nabla}_{\partial_{k}}B_{k1}\right) ^{N},\partial_{2}\right) \\ & =-B_{k1}\cdot B_{k1}\Omega(\partial_{1},\partial_{2})+\Omega\left( \left( \bar{\nabla}_{\partial_{k}}B_{k1}\right) ^{N},\partial_{2}\right). \end{align*} $$
Here we have used
$$ \begin{align*} 0 & =\bar{\nabla}_{\partial_{k}}\left( B_{k1}\cdot\partial_{j}\right) \\ & =\bar{\nabla}_{\partial_{k}}B_{k1}\cdot\partial_{j}+\left( B_{k1}\cdot \bar{\nabla}_{\partial_{k}}\partial_{j}\right) \\ & =\bar{\nabla}_{\partial_{k}}B_{k1}\cdot\partial_{j}+B_{k1}\cdot B_{kj}. \end{align*} $$
Now let
$$\begin{align*}\left\Vert B\right\Vert ^{2}=-\left( B_{k1}\cdot B_{k1}+B_{k2}\cdot B_{k2}\right), \end{align*}$$
where the positivity suggested is appropriate as
$B_{kj}$
are time-like vectors, and (5.3) becomes (also repeating the computation in the above lines for the last term in (5.3))
$$ \begin{align} & 2\bar{\nabla}_{\partial_{k}}\Omega(B_{k1},\partial_{2})+2\bar{\nabla }_{\partial_{k}}\Omega(\partial_{1},B_{k2})+\bar{\nabla}_{\vec{H}} \Omega(\partial_{1},\partial_{2})+2\Omega\left( B_{k1},\bar{\nabla} _{\partial_{k}}\partial_{2}\right) \\& \quad +\left\Vert B\right\Vert ^{2}\Omega(\partial_{1},\partial_{2} )+\Omega\left( \left( \bar{\nabla}_{\partial_{k}}B_{k1}\right) ^{N},\partial_{2}\right) +\Omega\left( \partial_{1},\left( \bar{\nabla }_{\partial_{k}}B_{k2}\right) ^{N}\right) .\nonumber \end{align} $$
The Codazzi equation holds in pseudo-Riemannian manifolds: If
$X,Y,Z$
are vectors on
$\Gamma $
and and
$\eta $
a normal vector, then
$$\begin{align*}\bar{\nabla}_{X}\left( B(Y,Z)\right) \cdot\eta-\bar{\nabla}_{Y}\left( B(X,Z)\right) \cdot\eta=Rm_{h}\left( X,Y,Z,\eta\right). \end{align*}$$
In our case, this gives at
$\left ( x_{\max },\bar {x}_{\max }\right ) $
$$\begin{align*}\left( \bar{\nabla}_{\partial_{k}}B_{1k}\right) \cdot\eta=\left( \bar{\nabla}_{\partial_{1}}B_{kk}\right) \cdot\eta+Rm_{h}\left( \partial _{k},\partial_{1},\partial_{k},\eta\right) \end{align*}$$
thus
$$\begin{align*}\left( \bar{\nabla}_{\partial_{k}}B_{1k}\right) ^{N}=\left( \bar{\nabla }_{\partial_{1}}\vec{H}\cdot n_{p}+Rm_{h}\left( \partial_{2},\partial_{1} ,\partial_{2},n_{p}\right) \right) n^{pq}n_{q}, \end{align*}$$
where we are using the negative definite
$n_{pq}:=n_{p}\cdot n_{q}$
(which is just
$\left ( -\delta _{pq}\right ) $
at
$\left ( x_{\max },\bar {x}_{\max }\right ) )$
. So now we have
$$ \begin{align} \Omega\left( \left( \bar{\nabla}_{k}B_{k1}\right) ^{N},\partial_{2}\right) =\Omega\left( \left( \bar{\nabla}_{\partial_{1}}\vec{H}\cdot n_{p}+Rm_{h}\left( \partial_{2},\partial_{1},\partial_{2},n_{p}\right) \right) n^{pq} n_{q},\partial_{2}\right). \end{align} $$
Substituting this into (5.4) (also with appropriate expressions for swapped indices
$1$
and
$2$
) using the fact that
$n^{pq}$
is negative definite provides the expression in the statement of the Lemma.
6 Curvature and other computations with eigenvalues
First we need:
Claim 6.1 At
$\left ( x_{\max },\bar {x}_{\max }\right ) \in \Gamma ,$
with frames defined by (3.6) and (3.7) we have
$$ \begin{align*} Rm_{h}(\partial_{2},\partial_{1},\partial_{2},n_{1}) & =-Rm_{h}(\partial _{1},\partial_{2},\partial_{1},n_{2})\\ & =R_{\bar{1}22\bar{1}}\left( \frac{\lambda_{1}}{\lambda_{2}}\right) ^{2}-R_{1\bar{2}\bar{2}1}\left( \frac{\lambda_{2}}{\lambda_{1}}\right) ^{2}, \end{align*} $$
where
$$\begin{align*}R_{\bar{1}22\bar{1}}=Rm_{h}(E_{\bar{1}},E_{2},E_{2},E_{\bar{1}})\text{, etc. } \end{align*}$$
for
$E_{i}\in T_{p}M$
and
$E_{\bar {j}}\in T_{p}\bar {M}.$
Proof Note that in the Kim–McCann metric [Reference Kim and McCannKM10, Lemma 4.1], these computations become massively simplified by the fact that the only curvature terms that do not vanish are those with two barred and two unbarred indices, so there will be at most six nontrivial terms in the expression for
$Rm(\partial _{2},\partial _{1},\partial _{2},n_{1})$
. Straightforward computations using the symmetries of the curvature tensor and (3.6), (3.7) yield the result.
Claim 6.2 At the point
$\left ( x_{\max },\bar {x}_{\max }\right ) $
we have
$$ \begin{align} \Omega\left( \partial_{1},n_{1}\right) & =0\nonumber\\ \Omega\left( \partial_{1},n_{2}\right) & =\frac{1}{\lambda_{1}\lambda_{2} }\left( 1+\lambda_{1}^{2}\right) (1-\lambda_{2}^{2})\nonumber\\ \Omega\left( \partial_{2},n_{1}\right) & =-\frac{1}{\lambda_{1}\lambda _{2}}\left( 1-\lambda_{1}^{2}\right) (1+\lambda_{2}^{2})\nonumber\\ \Omega\left( \partial_{2},n_{2}\right) & =0\nonumber\\ \Omega\left( \partial_{1},\partial_{2}\right) & =\frac{1}{\lambda _{1}\lambda_{2}}\left( 1+\lambda_{1}^{2}\right) (1+\lambda_{2} ^{2})\\ \Omega\left( n_{1},n_{2}\right) & =\frac{1}{\lambda_{1}\lambda_{2}}\left( 1-\lambda_{1}^{2}\right) (1-\lambda_{2}^{2}).\nonumber \end{align} $$
Proof Straightforward calculation noting that at the point
$$ \begin{align*} \left( dx^{i}+dy^{i}\right) \partial_{j} & =\frac{1}{\lambda_{i}} \delta_{ij}\left( 1+\lambda_{i}^{2}\right) \\\left( dx^{i}+dy^{i}\right) n_{j} & =\frac{1}{\lambda_{i}}\delta _{ij}\left( 1-\lambda_{i}^{2}\right).\\[-41pt] \end{align*} $$
Now combing the above two lemmas, we get the following.
Corollary 6.3 As above,
$$ \begin{align*} & \Omega(Rm_{h}(\partial_{2},\partial_{1},\partial_{2},n_{p})n_{p} ,\partial_{2})+\Omega(\partial_{1},Rm_{h}(\partial_{1},\partial_{2} ,\partial_{1},n_{p})n_{p})\\& \quad =2\frac{1}{\lambda_{1}\lambda_{2}}\left( \lambda_{2}^{2}-\lambda_{1} ^{2}\right) \left\{ R_{\bar{1}22\bar{1}}\left( \frac{\lambda_{1}} {\lambda_{2}}\right) ^{2}-R_{1\bar{2}\bar{2}1}\left( \frac{\lambda_{2} }{\lambda_{1}}\right) ^{2}\right\}. \end{align*} $$
Proof As
$\Omega \left ( \partial _{i},n_{i}\right ) =0$
we have
$$ \begin{align*} & \Omega\left( Rm_{h}(\partial_{2},\partial_{1},\partial_{2},n_{p} )n_{p},\partial_{2}\right) +\Omega\left( \partial_{1},Rm_{h}(\partial _{1},\partial_{2},\partial_{1},n_{p})n_{p}\right) \\&\quad =Rm_{h}(\partial_{2},\partial_{1},\partial_{2},n_{1})\frac{1}{\lambda _{1}\lambda_{2}}\left( 1-\lambda_{1}^{2}\right) (1+\lambda_{2}^{2})\\&\qquad +Rm_{h}(\partial_{1},\partial_{2},\partial_{1},n_{2})\frac{1}{\lambda _{1}\lambda_{2}}\left( 1+\lambda_{1}^{2}\right) (1-\lambda_{2}^{2}). \end{align*} $$
From Claim 6.1 the expression becomes
$$ \begin{align*} & =\frac{1}{\lambda_{1}\lambda_{2}}\left[ R_{\bar{1}22\bar{1}}\left( \frac{\lambda_{1}}{\lambda_{2}}\right) ^{2}-R_{1\bar{2}\bar{2}1}\left( \frac{\lambda_{2}}{\lambda_{1}}\right) ^{2}\right] \left\{ \begin{array} [c]{@{}c@{}} \left( 1-\lambda_{1}^{2}\right) (1+\lambda_{2}^{2})\\ -\left( 1+\lambda_{1}^{2}\right) (1-\lambda_{2}^{2}) \end{array} \right\} \\& =2\frac{1}{\lambda_{1}\lambda_{2}}\left( \lambda_{2}^{2}-\lambda_{1} ^{2}\right) \left\{ R_{\bar{1}22\bar{1}}\left( \frac{\lambda_{1}} {\lambda_{2}}\right) ^{2}-R_{1\bar{2}\bar{2}1}\left( \frac{\lambda_{2} }{\lambda_{1}}\right) ^{2}\right\}.\\[-41pt] \end{align*} $$
Claim 6.4 Suppose that
$\lambda _{2}\leq 1 \leq \lambda _{1}.$
Then for some constant
$C_{2}$
depending on
$\left \Vert Df\right \Vert _{g_{1}\times g_{2}}$
,
$g_{1},g_{2}$
and
$\Lambda $
$$\begin{align*}2\Omega\left( B(\partial_{k},\partial_{1}),B(\partial_{k},\partial _{2})\right) \geq-\frac{\left\Vert B\right\Vert ^{2}}{3}\frac{(1+\lambda _{1}^{2})(1+\lambda_{1}^{2})}{\lambda_{1}\lambda_{2}}-C_{2}(1+\lambda_{1}^{4}). \end{align*}$$
Proof The minimal surface equation
$$\begin{align*}\vec{H}=-\left( \nabla f\right) ^{N} \end{align*}$$
can be expressed in normal coordinates as the following
$$ \begin{align*} b_{111}+b_{221} & =-\nabla f\cdot n_{1}\\ b_{112}+b_{222} & =-\nabla f\cdot n_{2}, \end{align*} $$
By Claim 2.3 we then have
$$\begin{align*}b_{k22}=-\nabla f\cdot n_{k}-b_{k11}. \end{align*}$$
Now
$$ \begin{align} 2\Omega\left( B(\partial_{k},\partial_{1}),B(\partial_{k},\partial _{2})\right) & =\sum_{k,s,p}2b_{k1s}b_{k2p}\Omega(n_{s},n_{p})\nonumber\\& =\sum_{k}2b_{k11}b_{k22}\Omega(n_{1},n_{2})+\sum_{k}2b_{k12}b_{k21} \Omega(n_{2},n_{1})\nonumber\\& =\sum_{k}2\left( b_{k11}b_{k22}-b_{k12}b_{k21}\right) \frac{1} {\lambda_{1}\lambda_{2}}(\lambda_{1}^{2}-1)(\lambda_{2}^{2}-1)\nonumber\\& =\sum_{k}2\left( b_{k11}\left( -\nabla f\cdot n_{k}-b_{k11}\right) -b_{k12}^{2}\right) \frac{1}{\lambda_{1}\lambda_{2}}(\lambda_{1} ^{2}-1)(\lambda_{2}^{2}-1)\nonumber\\& =2\sum_{k}b_{k11}\left( -\nabla f\right) \cdot n_{k}\frac{1}{\lambda _{1}\lambda_{2}}(\lambda_{1}^{2}-1)(\lambda_{2}^{2}-1)\nonumber\\& \quad -2\sum_{k}\left( b_{k11}^{2}+b_{k12}^{2}\right) \frac{1}{\lambda _{1}\lambda_{2}}(\lambda_{1}^{2}-1)(\lambda_{2}^{2}-1)\nonumber\\& \geq-\left( \frac{1}{\beta}\sum_{k}\left\Vert B\right\Vert ^{2}\left\vert \nabla f\cdot n_{k}\right\vert ^{2}+\frac{\beta}{4}\right) .\frac{1} {\lambda_{1}\lambda_{2}}(\lambda_{1}^{2}+1)(\lambda_{2}^{2}+1), \end{align} $$
where we may drop the term in the second-to-last line using
$(\lambda _{2} ^{2}-1) \leq 0.$
Next, recalling that in our product system coordinates (with respect to Euclidean metric)
$$\begin{align*}\left\Vert \partial_{k}\right\Vert _{g_{1}\times g_{2}}=\left\Vert n_{k}\right\Vert _{g_{1}\times g_{2}}\leq\sqrt{\frac{1}{\lambda_{k}^{2}} +C_{1}^{2}\lambda_{k}^{2}} \end{align*}$$
we may use (4.3) to conclude that
$$ \begin{align*} \left\vert \partial_{1}\right\vert ,\left\vert n_{1}\right\vert & \leq \sqrt{\Lambda+C_{1}^{2}\lambda_{1}^{2}}\\ \left\vert \partial_{2}\right\vert ,\left\vert n_{2}\right\vert & \leq \sqrt{\Lambda\lambda_{1}^{2}+C_{1}^{2}\Lambda.} \end{align*} $$
or
$$ \begin{align} \left\vert \partial_{k}\right\vert \leq C_{0}\sqrt{1+\lambda_{1}^{2}. } \end{align} $$
Then evaluating
$$\begin{align*}\nabla f\cdot n_{k}=Df(n_{k})\leq C_{0}\sqrt{1+\lambda_{1}^{2}}\left\Vert Df\right\Vert _{g_{1}\times g_{2}}\leq C_{2}^{\prime}(1+\lambda_{1}^{2})^{1/2} \end{align*}$$
for some
$C_{2}^{\prime }$
that depends on
$\left \Vert Df\right \Vert _{g_{1}\times g_{2}}$
,
$C_{0}$
and
$\Lambda .$
Thus letting
$\beta =3\left ( C_{2}^{\prime }\right ) ^{2} (1+\lambda _{1}^{2})$
in (6.2) gives the result.
Claim 6.5 For some constants
$C_{3},C_{4}$
depending on
$\left \Vert D^{2}\Omega \right \Vert _{g_{1}\times g_{2}}$
and
$\Lambda .$
$$\begin{align*}2\left( \bar{\nabla}_{\partial_{k}}\Omega(B_{k1},\partial_{2})+\bar{\nabla }_{\partial_{k}}\Omega(\partial_{1},B_{k2})\right) \geq-\frac{\left\Vert B\right\Vert ^{2}}{3}\frac{(1+\lambda_{1}^{2})(1+\lambda_{2}^{2})}{\lambda _{1}\lambda_{2}}-C_{3}\left( 1+\lambda_{1}^{4}\right) \end{align*}$$
and
$$\begin{align*}\sum_{k}\bar{\nabla}_{\partial_{k}}\bar{\nabla}_{\partial_{k}}\Omega (\partial_{1},\partial_{2})\geq-C_{4}(1+\lambda_{1}^{4}). \end{align*}$$
Proof Recalling (6.3)
$$ \begin{align*} \left\vert \bar{\nabla}_{\partial_{k}}\Omega(B_{k1},\partial_{2})\right\vert & \leq\left\Vert D\Omega\right\Vert _{g_{1}\times g_{2}}\left\Vert \partial_{k}\right\Vert _{g_{1}\times g_{2}}\left\vert b_{k1s}\right\vert \left\Vert n_{s}\right\Vert _{g_{1}\times g_{2}}\left\Vert \partial _{2}\right\Vert _{g_{1}\times g_{2}}\\ & \leq\left\Vert D\Omega\right\Vert _{g_{1}\times g_{2}}\sum_{k,s}\left\vert b_{k1s}\right\vert \left( C_{0}\left( 1+\lambda_{1}^{2}\right) ^{1/2}\right) ^{3}\\ & \leq\frac{\left\Vert B\right\Vert ^{2}}{12}\frac{\left( 1+\lambda_{1} ^{2}\right) (1+\lambda_{2}^{2})}{\lambda_{1}\lambda_{2}}+48\left\Vert D\Omega\right\Vert{}^2 _{g_{1}\times g_{2}}C_{0}^{6} \frac{\lambda_{1}\lambda_{2}}{(1+\lambda_{2}^{2})}\left( 1+\lambda_{1} ^{2}\right) ^{2}. \end{align*} $$
Noting that
$\frac {\lambda _{1}\lambda _{2}}{(1+\lambda _{2}^{2})}$
is controlled, and repeating the same computation for
$\vert \bar {\nabla }_{\partial _{k}}\Omega (\partial _{1}, B_{k2}) \vert $
gives the first inequality.
For the next inequality we may directly compute
$$\begin{align*}\left\vert \sum_{k}\bar{\nabla}_{\partial_{k}}\bar{\nabla}_{\partial_{k} }\Omega(\partial_{1},\partial_{2})\right\vert \leq2\left\Vert D^{2} \Omega\right\Vert _{g_{1}\times g_{2}}\left[ C_{0}\sqrt{1+\lambda_{1}^{2} }\right] ^{4}.\\[-41pt] \end{align*}$$
Finally, we bound terms involving the mean curvature.
Claim 6.6 For some constants
$C_{5},C_{6}$
$$\begin{align*}\nabla_{\vec{H}}\Omega(\partial_{1},\partial_{2})\geq-C_{5}\left( 1+\lambda_{1}^{4}\right) \end{align*}$$
$$\begin{align*}&-\sum_{s}\left\{ \left( \bar{\nabla}_{\partial_{1}}\vec{H}\cdot n_{1}\right) \Omega(n_{1},\partial_{2})+\left( \bar{\nabla}_{\partial_{2} }\vec{H}\cdot n_{2}\right) \Omega(\partial_{1},n_{2})\right\}\\& \quad \geq -\frac{\left\Vert B\right\Vert ^{2}}{3}\frac{(1+\lambda_{1}^{2})(1+\lambda _{2}^{2})}{\lambda_{1}\lambda_{2}}-C_{6}\left( 1+\lambda_{1}^{4}\right). \end{align*}$$
Proof Note that
$$ \begin{align*} \left( \nabla f\right) ^{N} & =\left( \left( \nabla f\right) \cdot n_{s}\right) n^{sp}n_{p}\\ & =\left( df(n_{s})\right) n^{sp}n_{p}, \end{align*} $$
thus we have (recall (6.3))
$$ \begin{align*} \left\Vert \vec{H}\right\Vert _{g_{1}\times g_{2}} & =\left\Vert \left( \nabla f\right) ^{N}\right\Vert _{g_{1}\times g_{2}}\\ & \leq\left\vert df(n_{s})\right\vert \left\Vert n_{p}\right\Vert _{g_{1}\times g_{2}}\\ & \leq\left\Vert Df\right\Vert _{g_{1}\times g_{2}}C_{0}^{2}\left( 1+\lambda_{1}^{2}\right). \end{align*} $$
Then
$$\begin{align*}\left\vert \nabla_{\vec{H}}\Omega(\partial_{1},\partial_{2})\right\vert \leq\left\Vert Df\right\Vert _{g_{1}\times g_{2}}\left\Vert D\Omega\right\Vert _{g_{1}\times g_{2}}C_{0}^{4}\left( 1+\lambda_{1}^{2}\right) ^{2}. \end{align*}$$
Next, note that
$$ \begin{align*} \left( \bar{\nabla}_{\partial_{1}}\vec{H}\cdot n_{1}\right) & =\bar {\nabla}_{\partial_{1}}\left( \bar{\nabla}f\right) ^{N}\cdot n_{1}\\ & =\left( \bar{\nabla}_{\partial_{1}}\left( \bar{\nabla}f\right) -\bar{\nabla}_{\partial_{1}}\left( \bar{\nabla}f\right) ^{T}\right) \cdot n_{1}\\ & =\bar{\nabla}^{2}f\left( \partial_{1},n_{1}\right) -\sum_{k}\left( \partial_{k}f\right) b_{1k1}\\ & \geq-\left\Vert D^{2}f\right\Vert _{g_{1}\times g_{2}}C_{0}^{2}\left( 1+\lambda_{1}^{2}\right) -\frac{\left\Vert B\right\Vert ^{2}}{6}-2\left\Vert Df\right\Vert _{g_{1}\times g_{2}}^{2}C_{0}^{2}\left( 1+\lambda_{1} ^{2}\right). \end{align*} $$
The inequality follows from adding the terms and using
$$ \begin{align*} \left\vert \Omega\left( \partial_{1},n_{2}\right) \right\vert ,\left\vert \Omega\left( n_{1},\partial_{2}\right) \right\vert \leq\frac{1}{\lambda _{1}\lambda_{2}}\left( 1+\lambda_{1}^{2}\right) (1+\lambda_{2} ^{2}).\\[-37pt] \end{align*} $$
7 Finish It
We return to the maximum principle argument.
Proof of Theorem 4.1
Pick an
$x_{\max }$
in
$\Gamma $
where the function w is maximized. The scalar function
$w=\frac {\Omega }{dV_{g}}$
is simply the Hodge dual of
$\Omega |_{\Gamma }$
. Covariant differentiation commutes with the Hodge operator, so
$$\begin{align*}\frac{\nabla_{\partial_{k}}\nabla_{\partial_{k}}\Omega\left( \partial _{1},\partial_{2}\right) }{dV_{g}\left( \partial_{1},\partial_{2}\right) }=\left( \frac{\Omega}{dV_{g}}\right) _{kk}\leq0. \end{align*}$$
Assume that
$\lambda _{2}\leq 1;$
If not, then
$\lambda _{1}\leq \Lambda $
and we immediately have a bound on
$w.$
Applying Lemma 5.1, and all the claims in the previous section, we have
$$ \begin{align*} 0 & \geq\left( \frac{\Omega}{dV_{g}}\right) _{kk}\geq-\left( C_{2} +C_{3}+C_{4}+C_{5}+C_{6}\right) (1+\lambda_{1}^{4})\\ & +2\frac{1}{\lambda_{1}\lambda_{2}}\left( \lambda_{1}^{2}-\lambda_{2} ^{2}\right) \left\{ R_{\bar{1}22\bar{1}}\left( \frac{\lambda_{1}} {\lambda_{2}}\right) ^{2}-R_{1\bar{2}\bar{2}1}\left( \frac{\lambda_{2} }{\lambda_{1}}\right) ^{2}\right\} \\ & \geq-C_{10}\left( 1+\lambda_{1}^{4}\right) +2\frac{1}{\Lambda}\left( \lambda_{1}^{2}-\Lambda\right) \left( R_{\bar{1}22\bar{1}}\left( \frac{\lambda_{1}^{2}}{\Lambda}\right) ^{2}-R_{1\bar{2}\bar{2}1}\right). \end{align*} $$
Now the components
$R_{\bar {1}22\bar {1}}$
are with respect to the vectors that are uniformly bounded and bounded away from zero in our gauge metrics, so
$R_{\bar {1}22\bar {1}}$
has a known lower bound. We may conclude that
$$\begin{align*}\lambda_{1}\leq C_{20} \end{align*}$$
and thus
$$\begin{align*}w(x_{\max})=\frac{\left( 1+\lambda_{1}^{2}\right) \left( 1+\lambda_{2} ^{2}\right) }{\lambda_{1}\lambda_{2}}\leq\bar{C}=2\Lambda(1+C_{20}^{2}). \end{align*}$$
