\section{Introduction}

Given a closed Riemannian manifold $(M,g)$ we consider the
conformal class of the metric $g$, $[g]$. The Yamabe
constant of $[g]$,  $Y(M,[g])$, is the
infimum of the normalized total scalar curvature functional on
the conformal class. Namely,

$$Y(M,[g])= \inf_{h\in [g]}
\frac{\int {\bf s}_h \ dvol(h)}{(Vol(M,h))^{\frac{n-2}{n}}},$$

\noindent
where ${\bf s}_h$ denotes the scalar curvature of the metric $h$
and $dvol(h)$ its volume element.

If one writes metrics conformal to $g$ as $h=f^{4/(n-2)} \ g$,
one obtains the expression

$$Y(M,[g])= \inf_{f\in C^{\infty} (M)}
\frac{\int ( \  4a_n {\| \nabla f \|}_g^2 + f^2 {\bf s}_g \ )
\  dvol(g)}{{\| f\|}_{p_n}^2},$$

\noindent
where $a_n =4(n-1)/(n-2) $ and $p_n =2n/(n-2)$. It is a fundamental
result
on the subject that the infimum is actually achieved
(\cite{Yamabe, Trudinger, Aubin, Schoen}). The functions $f$ achieving
the infimum are called {\it Yamabe functions} and the corresponding metrics
$f^{4/(n-2)} \ g$ are called {\it Yamabe metrics}. Since the critical points
of the total scalar curvature functional restricted to a conformal
class of metrics are precisely the metrics of constant scalar
curvature in the conformal class, Yamabe metrics are metrics of
constant scalar curvature.

It is well known that by considering functions supported in a small
normal neighborhood of a point one can prove that
$Y(M^n,[g]) \leq Y(S^n ,[g_0 ])$, where $g_0$ is the round metric
of radius one on the sphere and $(M^n ,g)$ is any closed n-dimensional
Riemannian manifold (\cite{Aubin}).
We will use the notation $Y_n = Y(S^n ,[g_0 ])$ and
$V_n =Vol(S^n ,g_0 )$. Therefore $Y_n =n(n-1)V_n^{\frac{2}{n}}$.

Then one
defines the {\it Yamabe invariant} of a closed manifold $M$
\cite{Kobayashi, Schoen2} as

$$Y(M)=\sup_g Y(M,[g]) \leq  Y_n .$$

It follows that $Y(M)$ is positive if and only if $M$ admits a
metric of positive scalar curvature. Moreover, the sign of
$Y(M)$ determines the technical difficulties in understanding
the invariant. When the Yamabe constant of a conformal class
is non-positive there is a unique metric  (up to multiplication
by a positive constant) of constant scalar curvature in the
conformal class and if $g$ is any metric in the conformal
class, the Yamabe constant is bounded from below by
$(\inf_M {\bf s}_g  ) \ (Vol(M,g))^{2/n}$. This can be used for instance
to study the behavior of the invariant under surgery and so to
obtain information using cobordism theory \cite{Yun, Petean, Botvinnik}.
Note also that in the non-positive case the Yamabe invariant
coincides with Perelman's invariant \cite{Ishida}.
The previous estimate  is no longer true
in the positive case, but one does get a lower bound in the case of
positive Ricci curvature by a theorem of S. Ilias:
if $Ricci(g)\geq \lambda g $
($\lambda >0$) then $Y(M,[g]) \geq n \lambda (Vol(M,g))^{2/n}$
(\cite{Ilias}). Then in order to use this inequality to
find lower bounds on the Yamabe invariant of a closed
manifold $M$ one would try to maximize the volume of the manifold
under some positive lower bound of the Ricci curvature.
Namely, if one denotes ${\bf Rv} (M)= \sup \{ Vol(M,g): Ricci(g)\geq
(n-1) g \} $ then one gets $Y(M) \geq n(n-1) ({\bf Rv} (M))^{2/n}$
(one should define ${\bf Rv} (M) =0$ if $M$ does not admit
a metric of positive Ricci curvature). Very little is known
about the invariant ${\bf Rv} (M)$. Of course, Bishop's inequality
tells us that for any n-dimensional closed manifold
${\bf Rv} (M^n) \leq {\bf Rv} (S^n )$
(which is of course attained by the volume
of the metric of constant sectional curvature 1). Moreover,
G. Perelman \cite{Perelman} proved that there is a
constant $\delta =\delta_n >0$ such that if  ${\bf Rv} (M) \geq
{\bf Rv} (S^n ) -\delta_n $ then
$M$ is homeomorphic to $S^n$. Beyond this, results on
${\bf Rv} (M)$ have been obtained by computing Yamabe invariants, so
for instance ${\bf Rv} ({\bf CP}^2 )= 2 \pi^2 $
(achieved by the Fubini-Study
metric as shown by C. LeBrun \cite{Lebrun} and M. Gursky and C.
LeBrun \cite{Gursky}) and ${\bf Rv} ({\bf RP}^3) = \pi^2$ (achieved by the
metric of constant sectional curvature as shown by H. Bray and
A. Neves \cite{Bray}).

Of course, there is no hope to apply the previous comments directly
when the fundamental group of $M$ is infinite. Nevertheless it
seems that even in this case the Yamabe invariant is
realized by conformal classes of metrics which maximize volume
with a fixed positive lower bound on the Ricci curvature
``in certain sense''. The standard example is $S^{n-1} \times
S^1$. The fact that $Y(S^n \times S^1 ) =Y_{n+1}$ is one
of the first things we learned about the Yamabe invariant
\cite{Kobayashi, Schoen2}. One way to see this is as follows:
first one notes that $\lim_{T\rightarrow \infty}
Y(S^n \times S^1 ,[g_0 + T^2 dt^2 ])=
Y(S^n \times {\mathbb R}, [g_0 + dt^2 ])$ \cite{Akutagawa}
(the Yamabe constant for a non-compact Riemannian manifold
is computed as the infimum of the Yamabe functional over
compactly supported functions).
But the Yamabe
function for $g_0 + dt^2$ is precisely the conformal factor
between $S^n \times {\mathbb R}$ and $S^{n+1} -\{ S, N \}$. Therefore
one can think of $Y(S^n \times S^1 ) =Y_{n+1}$ as realized
by the positive
Einstein metric on $S^{n+1} -\{ S, N \} $. We will see in this
article that a similar situation occurs for any closed positive
Einstein manifold $(M,g)$  (although we only get the lower
bound for the invariant).

\vspace{.3cm}

Let $(N,h)$ be a closed Riemannian manifold. An
{\it isoperimetric region}
is an open subset $U$ with boundary $\partial U$ such that
$\partial U$ minimizes area among hypersurfaces bounding a
region of volume $Vol(U)$. Given any positive number $s$,
$s<Vol(N,h)$, there exists an isoperimetric region of
volume $s$. Its boundary is a stable constant mean curvature
hypersurface with some singularities of codimension at least 7.
Of course one does not need a closed Riemannian manifold
to consider isoperimetric regions, apriori one only
needs to be able to compute volumes of open subsets and areas
of hypersurfaces. One defines the {\it isoperimetric function}
of $(N,h)$ as $I_h :(0,1) \rightarrow {\mathbb R}>0$ by

$$I_h (\beta) =\inf \{ Vol(\partial U)/Vol(N,h) :
Vol(U,h) = \beta Vol(N,h) \},$$

\noindent
where $Vol(\partial U)$ is measured with the Riemannian metric
induced by $h$ (on the non-singular part of $\partial U$).

Given a closed Riemannian manifold $(M,g)$ we will call
the {\it spherical cone} on $M$ the space $X$ obtained collapsing
$M \times \{0 \} $ and $M\times \{ \pi \}$ in
$M\times [0,\pi ]$ to points $S$ and $N$ (the vertices)
with the metric ${\bf g} =\sin^2 (t)g + dt^2$
(which is a Riemannian metric on $X-\{ S,N \}$). Now if
$Ricci(g) \geq (n-1) g$ one can see that $Ricci({\bf g})
\geq n{\bf g}$. One should compare this with the Euclidean cones
considered by F. Morgan and M. Ritor\'{e} in \cite{Morgan}:
$\hat{g} =t^2 g + dt^2$ for which $Ricci(g) \geq (n-1)g $
implies that $Ricci(\hat{g}) \geq 0$. The importance of these
spherical cones for the study of Yamabe constants is that
if one takes out the vertices the corresponding (non-complete)
Riemannian manifold is conformal to
$M\times {\mathbb R}$. But using the (warped product version) of the
Ros Product Theorem \cite[Proposition 3.6]{Ros}  (see
\cite[Section 3]{Morgan2}) and the Levy-Gromov isoperimetric
inequality \cite{Gromov} one can understand isoperimetric
regions in these spherical cones. Namely,

\begin{Theorem} Let $(M^n,g)$ be a compact manifold with
Ricci curvature $Ricci(g) \geq (n-1)g$. Let $(X,{\bf g})$ be
its spherical cone. Then geodesic balls around any of the
vertices are isoperimetric.
\end{Theorem}

But now, since the spherical cone over $(M,g)$
is conformal to $(M\times {\mathbb R} ,
g+ dt^2 )$ we can use the previous result
and symmetrization of a function with respect to the
geodesic balls centered at a vertex to prove:

\begin{Theorem} Let $(M,g)$ be a closed Riemannian manifold of
positive Ricci curvature, $Ricci(g) \geq (n-1)g$ and volume $V$.
Then

$$Y(M\times {\mathbb R} ,[g+dt^2 ]) \geq
(V/V_n )^{\frac{2}{n+1}} \  Y_{n+1} .$$
\end{Theorem}

\vspace{.2cm}

As we mentioned before one of the differences between the positive
and non-positive cases in the study of the Yamabe constant is
the non-uniqueness of constant scalar curvature metrics on
a conformal class with positive Yamabe constant. And the simplest
family of examples of non-uniqueness comes from Riemannian
products. If $(M,g)$ and $(N^n ,h)$ are closed Riemannian manifolds
of constant scalar curvature and ${\bf s}_g$ is positive then
for small $\delta >0$, $\delta g + h$ is a constant scalar
curvature metric on $M \times  N$ which cannot be a Yamabe
metric. If $(M,g)$ is Einstein and $Y(M)=Y(M,[g])$ it seems
reasonable that $Y(M\times N)= \lim_{\delta \rightarrow 0}
Y(M\times N ,[ \delta g + h ])$.
Moreover as it is shown in \cite{Akutagawa}

$$ \lim Y(M\times N , [\delta g + h ]) =Y(M\times {\mathbb R}^n,[ g+ dt^2 ]).$$

The only case which is well understood is when $M=S^n$ and $N=S^1$.
Here every Yamabe function is a function of the $S^1$-factor
\cite{Schoen2} and the Yamabe function for $(S^n \times {\mathbb R} , g_0 +
dt^2 )$ is the factor which makes $S^n\times {\mathbb R}$ conformal to
$S^{n+1} -\{ S, N \}$. It seems possible that under
certain conditions on $(M,g)$ the Yamabe functions of
$(M \times {\mathbb R}^n , g+dt^2 )$ depend only on the second
variable. The best case scenario would be that this is true
if $g$ is a Yamabe metric but it seems more attainable the
case when $g$ is Einstein. It is a corollary to the previous
theorem that this is actually true in the case $n=1$. Namely,
using the notation (as in \cite{Akutagawa})
$Y_N (M\times N , g +h)$
to denote the infimum of the $(g+h)$-Yamabe functional restricted
to functions of the $N$-factor we have:

\begin{Corollary} Let $(M^n,g)$ be a closed positive Einstein manifold
with Ricci curvature $Ricci(g)=(n-1)g$. Then

$$Y(M\times {\mathbb R} , [g+ dt^2 ])=Y_{{\mathbb R}}(M\times {\mathbb R} , g+ dt^2 )=
{\left( \frac{V}{V_n} \right) }^{\frac{2}{n+1}} \ Y_{n+1}.$$

\end{Corollary}

\vspace{.3cm}

As $Y(M\times {\mathbb R} , [g+ dt^2 ]) = \lim_{T\rightarrow \infty }
Y(M\times S^1 ,[g+T dt^2 ])$ it also follows from Theorem 1.2
that:

\begin{Corollary} If $(M^n ,g)$ is a closed Einstein manifold
with $Ricci(g) = (n-1)g$ and volume $V$ then

$$Y(M\times S^1) \geq (V/V_n )^{\frac{2}{n+1}} \  Y_{n+1} .$$

\end{Corollary}

\vspace{.3cm}

So for example using the product metric we get

$$Y(S^2 \times S^2 \times S^1 )\geq  {\left(\frac{2}{3}
  \right)}^{(2/5)}  \ Y_5  $$

\noindent
and using the Fubini-Study metric we get

$$Y({\bf CP}^2 \times S^1 ) \geq {\left(\frac{3}{4} \right)}^{(2/5)}
\ Y_5 .$$

\vspace{.4cm}

{\it Acknowledgements:} The author would like to  thank
Manuel Ritor\'{e}, Kazuo Akutagawa and Frank Morgan
for several useful comments on the first drafts of
this manuscript.

\section{Isoperimetric regions in spherical cones}

As we mentioned in the introduction, the isoperimetric
problem for spherical cones (over manifolds with
Ricci curvature $\geq n-1$)  is understood using
the Levy-Gromov isoperimetric inequality
(to compare the isoperimetric functions of
$M$ and of $S^n$) and the Ros Product Theorem for  warped products
(to compare then the isoperimetric functions of
the spherical cone over $M$ to the isoperimetric function
of $S^{n+1}$).
See for example section 3 of \cite{Morgan2}
(in particular {\bf 3.2} and the remark after it). For the
reader familiar with isoperimetric problems, this should be
enough to understand Theorem 1.1. In this section, for the
convenience of the reader, we will
give a brief outline on these issues. We will mostly
discuss and follow section 3 of \cite{Ros} and ideas
in \cite{Morgan, Montiel} which we think might be useful in
dealing with other problems arising from the study of Yamabe
constants.

Let $(M^n ,g)$ be a closed Riemannian manifold
of volume $V$ and Ricci curvature $Ricci(g) \geq (n-1)g$.
We will
consider $(X^{n+1}, \bf{g}) $ where as a topological space $X$ is the
suspension of $M$ ($X=M\times [0,\pi ]$ with $M\times \{ 0 \}$ and
$M\times \{ \pi \}$ identified to points $S$ and $N$)
and $\bf{g}$ $ =\sin^2 (t) \ g \ +
dt^2$. Of course $X$ is not a manifold (except when $M$ is $S^n$) and
$\bf{g} $ is a Riemannian metric only on $X-\{ S,N \}$.

The following is a standard result in geometric measure theory.

$\bf{Theorem:}$  For any positive number $r< Vol(x)$ there exists
an isoperimetric open subset $U$ of $X$ of volume $r$. Moreover
$\partial U$ is a smooth stable constant mean curvature
hypersurface of $X$ except for a singular piece $\partial_1 U$
which consists of (possibly)
$S$, $N$, and a subset of codimension at least 7.

Let us call $\partial_0 U$ the regular part of $\partial U$,
$\partial_0 U= \partial U - \partial_1 U$. Let
$X_t$, $t\in (-\varepsilon ,\varepsilon )$,
be a variation of  $\partial_0 U$ such that the
volume of the enclosed region $U_t$ remains constant.
Let $\lambda (t)$ be the area of $X_t$. Then $\lambda '(0) =0$
and $\lambda ''(0) \geq 0$. The first condition is satisfied
by hypersurfaces of constant mean curvature and the ones
satisfying the second condition are called ${\it stable}$.
If $N$ denotes a  normal
vector field to the hypersurface then variations are obtained
by picking a function $h$ with compact support on $\partial_0 U$ and
moving $\partial_0 U$ in the direction of  $h \ N$. Then
we have that if the mean of $h$  on $\partial_0 U$
is 0 then  $\lambda_h '(0) =0$
 $\lambda_h ''(0) \geq 0$. This last condition is written as

$$Q(h,h)=-\int_{\partial_0 U} h(\Delta h + (Ricci (N,N) +
\sigma^2 )h ) dvol(\partial_0 U) \geq 0.$$

\noindent
Here we consider $\partial_0 U$ as a Riemannian manifold
(with the induced metric) and  use the corresponding Laplacian
and volume element. $\sigma^2$ is the square of the norm of the second
fundamental form.
This was worked out by J. L. Barbosa, M. do Carmo and
J. Eschenburg in \cite{Barbosa,
doCarmo}. As we said before, the function $h$
should apriori have compact support
in $\partial_0 U$ but as shown by F. Morgan and M. Ritor\'{e}
\cite[Lemma 3.3]{Morgan} it is enough that $h$ is bounded
and $h\in L^2 (\partial_0 U)$. This is important in order to study
stable constant mean curvature surfaces on a space like $X$ because
$X$ admits what is called a ${\it conformal}$ vector field $V=
\sin (t) \partial /\partial t$ and the function $h$ one wants to
consider is $h=div (V-{\bf g}(V,N) \ N )$ where $N$ is the unit
normal to the hypersurface (and then $h$ is the divergence of
the tangencial part of $V$). This has been used for instance in
\cite{Montiel,Morgan} to classify stable constant mean curvature
hypersurfaces in Riemannian manifolds with a conformal vector field.
When the hypersurface is smooth this function $h$ has mean 0 by
the divergence theorem and one can apply the stability condition.
But when the hypersurface has singularities one would apriori need
the function $h$ to have compact support on the regular part. This
was done by   F. Morgan and M. Ritor\'{e}  in
\cite[Lemma 3.3]{Morgan}.

We want to prove that the geodesic balls around $S$ are
isoperimetric. One could try to apply the techniques of
Morgan and Ritor\'{e} in \cite{Morgan} and see that they are
the only stable constant mean
curvature hypersurfaces in $X$. This should be possible, and
actually it might be necessary to deal with isoperimetric regions
of more general singular spaces that appear naturally in the study of
Yamabe constants of Riemannian products.
But in this case we will instead
take a more direct approach using the Levy-Gromov
isoperimetric inequality \cite{Gromov} and Ros Product Theorem
\cite{Ros}.

\vspace{.3cm}

The sketch of the proof is as follows: First one has to note that
geodesic balls centered at the vertices {\it produce} the same
isoperimetric function as the one of the round sphere. Therefore
to prove that geodesic balls around the vertices are isoperimetric
is equivalent to prove that the isoperimetric function of ${\bf g}$
is bounded from below by the isoperimetric function of $g_0$. To
do this, given any open subset $U$ of $X$ one considers
its symmetrization
$U^s \subset S^{n+1}$, so the the {\it slices} of $U^s$ are geodesic
balls with the same normalized volumes as the slices of $U$.  Then
by the Levy-Gromov isoperimetric inequality we can compare the
normalized areas of the boundaries of the slices. We have to
prove that the normalized area of $\partial U^s$ is at most the
normalized area of $\partial U$.
This follows from
the warped product version of \cite[Proposition 3.6]{Ros}. We will
give an outline following Ros' proof for the Riemannian product case.
We will use the notion of Minkowski
content. This is the bulk of the proof and we will divide it into
Lemma 2.1, Lemma 2.2 and Lemma 2.3.

\vspace{.3cm}

{\it Proof of Theorem 1.1 :}
Let $U\subset X$ be a closed  subset.
For any $t\in (0,\pi )$ let

$$U_t =U \cap (M\times \{ t \} ) .$$

Fix any point $E\in S^n$ and let $(U^s )_t$ be the geodesic ball
centered at $E$ with volume

$$Vol((U^s )_t , g_0 ) = \frac{V_n}{V} \ Vol(U_t ,g).$$

\noindent
(recall that $V=Vol(M,g)$ and $V_n = Vol(S^n ,g_0 )$).
Let $U^s
\subset S^{n+1}$ be the corresponding subset (i.e. we consider
$S^{n+1} -\{ S,N \}$ as $S^n \times (0,\pi )$ and $U^s$ is
such that $U^s \cap (S^n \times \{ t \}) $ =$(U^s )_t$.
One might add
$S$ and/or $N$ to make $U^s$ closed and connected). Note
that one can write $(U^s )_t = (U_t )^s = U_t^s$ as long as there
is no confusion (or no difference) on whether  we are considering
it as a subset of $S^n$ or as a subset of $S^{n+1}$.

Now

$$Vol(U)=\int_0^{\pi} \sin^n (t) \ Vol(U_t ,g) \ dt $$

$$= \frac{V}{V_n} \int_0^{\pi} \sin^n (t) \ Vol((U^s )_t ,g_0 ) \ dt
= \frac{V}{V_n} Vol(U^s ,g_0 ).$$

Also if $B(r) =M\times [0,r]$ (the geodesic ball of radius
$r$ centered at the vertex at 0) then

$$Vol(B(r))=\int_0^r \sin^n (t) V dt = \frac{V}{V_n}
\int_0^r \sin^n (t) V_n dt = \frac{V}{V_n} Vol (B_0 (r))  \ \ (1)$$

\noindent
where $B_0 (r)$ is the geodesic ball of radius $r$ in the
round sphere. And

$$Vol(\partial B(r))=\sin^n (r) V =\frac{V}{V_n}
Vol(\partial B_0 (r)) \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2).$$

Formulas (1) and (2) tell us that the geodesic balls around the
vertices in $X$ produce the same isoperimetric function as
the round metric $g_0$. Therefore given any open subset $U \subset
X$ we want to compare the area of $\partial U$ with the area
of the boundary
of the geodesic ball in $S^{n+1}$ with the same normalized volume
as $U$.

\vspace{.3cm}

Given a closed set $W$ let $B(W,r)$ be the set of points at distance
at most $r$ from $W$. Then one considers the {\it Minkowski content}
of $W$,

$$\mu^+ (W) = \liminf \frac{Vol (B(W,r) ) -Vol(W)}{r}.$$

\noindent
If $W$ is a smooth submanifold with boundary then
$\mu^+ (W) = Vol (\partial W)$. And this is still true if the
boundary has singularities of codimension $\geq 2$ (and finite
codimension 1 Hausdorff measure).

The Riemannian measure on $(S^n ,g_0 )$, normalized to be a
probability measure is what is called a {\it model measure}:
if $D^t$, $t\in (0,1)$ is the family of geodesic balls
(with volume $Vol(D^t )=t$) centered at some fixed point then
they are
isoperimetric regions which are ordered by volume and such
that for any $t$, $ B(D^t ,r) =D^{t'}$ for some $t'$.
See \cite[Section 3.2]{Ros}. The following result follows
directly from the
Levy-Gromov isoperimetric inequality \cite[Appendix C]{Gromov}
and \cite[Proposition 3.5]{Ros} (see the lemma in
\cite[page 77]{Morgan3} for a more elementary proof and point of view
on \cite[Proposition 3.5]{Ros}).

\begin{Lemma}: Let $(M,g)$ be a closed Riemannian manifold
of volume $V$ and Ricci curvature $Ricci(g) \geq (n-1) g$.
For any nonempty closed subset $\Omega \subset M$ and
any $r\geq 0$ if $B_{\Omega}$ is a geodesic ball in
$(S^n , g_0 )$ with volume $Vol(B_{\Omega})=(V_n /V)
Vol(\Omega )$ then $Vol(B(B_{\Omega} ,r))  \leq
(V_n /V) Vol(B(\Omega ,r))$.
\end{Lemma}

\begin{proof} Given any closed Riemannian manifold $(M,g)$,
dividing the
Riemannian measure by the volume one obtains a probability
measure which we will denote $\mu_g$.
As we said before, the round metric on the sphere
gives a model measure $\mu_{g_0}$. On the other hand the Levy-Gromov
isoperimetric inequality \cite{Gromov}
says that $I_{\mu_g} \geq I_{\mu_{g_0}}$.
The definition of $B_{\Omega}$ says that $\mu_g (\Omega )=\mu_{g_0}
(B_{\Omega})$ and what we want to prove is that $\mu_g (B(\Omega ,r))
\geq \mu_{g_0}
(B(B_{\Omega} ,r)  )$ .
Therefore the
statement of the lemma is precisely \cite[Proposition 3.5]{Ros}.

\end{proof}

Fix a positive constant $\lambda$. Note that the previous lemma
remains unchanged if we replace $g$ and $g_0$ by $\lambda g$
and $\lambda g_0$: the correspondence $\Omega \rightarrow
B_{\Omega}$ is the same and $\mu_{\lambda g} = \mu_g$.

\begin{Lemma} For any $t_0 \in (0,\pi )$
$B((U^s )_{t_0} ,r) \subset (B(U_{t_0} ,r ))^s $.
\end{Lemma}

\begin{proof}  First note that the distance from a point
$(x,t) \in X$ to a vertex depends only on $t$ and not on $x$
(or even on $X$). Therefore if $r$ is greater than the
distance $\delta$ between $t_0$ and $0$ or $\pi$
then both sets in the lemma
will contain a geodesic ball of radius $r-\delta$ around the
corresponding vertex.

Also observe that the distance between points $(x,t_0 )$ and
$(y,t)$ depends only on the distance between $x$ and $y$
(and $t$, $t_0$, and the function in the warped product,
which in this case is $\sin$) but not on $x, y$ or $X$.
In particular for any $t$ so that $|t-t_0 |<r$,
$(B((U^s )_{t_0} ,r) )_t$ is a geodesic ball.

We have to prove that for any $t$

$$(B((U^s )_{t_0} ,r) )_t \subset ((B(U_{t_0} ,r ))^s )_t.$$

\noindent
But since they are both geodesic balls centered at the same point
it is enough to prove that the volume of the subset on the left is
less than or equal to the volume of the subset on the right.
By the definition of symmetrization the normalized volume of
$ ((B(U_{t_0} ,r ))^s )_t$ is equal to the normalized volume of
$(B(U_{t_0} ,r ))_t$. But from the previous comment there exist
$\rho >0$ such that, considered as subsets of $M$,

$$(B(U_{t_0} ,r ))_t = B(U_{t_0} ,\rho )$$

\noindent
and, as subsets of $S^n$,

$$(B((U^s )_{t_0} ,r) )_t =B(U^s_{t_0} ,\rho ).$$

The lemma then follows from Lemma 2.1 (and the comments after it).

\end{proof}

Now for any closed subset $U\subset X$ let $B_U$ be a
geodesic ball in $(S^{n+1} ,g_0 )$ with volume
$Vol(B_U ,g_0 )= (V_n /V)
Vol(U,{\bf g})$. Since geodesic balls in round spheres are isoperimetric
(and $Vol(B_U ,g_0 )=Vol(U^s ,g_0 )$)
it follows that $Vol(\partial B_U )\leq \mu^+ (U^s )$.

\begin{Lemma} Given any closed set $U\subset X$, $\mu^+(U)
\geq (V/V_n ) Vol(\partial B_U )$.
\end{Lemma}

\begin{proof}
Since   $(B(U,r) )^s$ is closed
and  $B(U^s ,r)$ is the closure of
$\cup_{t\in (0,\pi )} \ B(U_t ^s ,r)$
we have from the previous lemma that

$$B(U^s ,r) \subset (B(U,r ) )^s .$$

 Then

$$Vol(\partial B_U )\leq \mu^+ (U^s )
=\liminf \frac{Vol(B(U^s ,r) ) - Vol (U^s )}{r}$$

$$\leq \liminf \frac{Vol((B(U ,r))^s  ) - Vol (U^s )}{r}$$

$$=(V_n /V)\liminf \frac{Vol(B(U,r) ) - Vol (U)}{r}
 =(V_n /V) \mu^+ (U) $$

\noindent
and the lemma follows.

\end{proof}

Now if we let $B_U^M$ be a geodesic ball around a vertex in $X$
with volume

$$Vol(B_U^M ,{\bf g}) = Vol(U,{\bf g} ) =
\frac{V}{V_n} Vol(B_U, g_0 )$$

\noindent
then it follows from (1) and (2) in the beginning of the proof that

$$Vol(\partial B_U^M ,{\bf g}) = \frac{V}{V_n} Vol(\partial B_U ,g_0 ).$$

\noindent
and so by Lemma 2.3

$$Vol(\partial B_U^M ,{\bf g}) \leq \mu^+ (U)$$

\noindent
and Theorem 1.1 is proved.

{\hfill$\Box$\medskip}

\section{The Yamabe constant of $M\times {\mathbb R}$}

Now assume that $g$ is a metric of positive Ricci curvature,
$Ricci(g) \geq (n-1)g$ on $M$ and consider as before the
spherical cone $(X,{\bf g})$ with ${\bf g} =\sin^2 (t) g + dt^2$.
By a direct
computation the sectional curvature of ${\bf g}$ is given by:

$$K_{{\bf g}} (v_i ,v_j )=\frac{K_g (v_i ,v_j )-\cos^2 (t)}{\sin^2 (t)}$$

$$K_{\bf g} (v_i ,\partial /\partial t)=1,$$

\noindent
for a $g$-orthonormal basis $\{ v_1 ,...,v_n \}$. And the Ricci
curvature is given by:

$$Ricci({\bf g}) (v_i ,\partial /\partial t )=0$$

$$Ricci({\bf g}) (v_i ,v_j )= Ricci(g) (v_i ,v_j ) - (n-1)\cos^2 (t)\delta_i^j
+\sin^2 (t) \delta_i^j$$

$$Ricci({\bf g}) (\partial_t ,\partial_t )=n.$$

Therefore  by picking $\{ v_1 ,...,v_n \}$ which diagonalizes $Ricci(g)$ one
easily sees that if $Ricci(g)\geq (n-1)g$ then $Ricci({\bf g})\geq n
{\bf g}$. Moreover, if $g$ is an Einstein metric with Einstein
constant $n-1$ the ${\bf g}$ is Einstein with Einstein constant $n$.

Let us recall that for non-compact
Riemannian manifolds one defines
the Yamabe constant of a metric as the infimum of the Yamabe
functional of the metric
over smooth compactly supported functions (or functions
in $L_1^2$, of course). So for instance if $g$ is a Riemannian metric
on the closed manifold $M$ then

$$Y(M\times {\mathbb R} ,[g+dt^2 ]) =\inf_{f \in C^{\infty}_0 (M\times {\mathbb R} )}
\frac{\int_{M\times {\mathbb R} } \left( \ a_{n+1} {\| \nabla f \|}^2 +
{\bf s}_g  \ f^2 \ \right)
dvol(g+dt^2)}{
{\| f \|}_{p_{n+1}}^2 } .$$

\vspace{.2cm}

{\it Proof of Theorem 1.2 :}
We have a closed Riemannian manifold $(M^n ,g)$
such that $Ricci(g) \geq (n-1) g$. Let $f_0 (t)= \cosh^{-2} (t)$
and consider the diffeomorphism

$$H: M \times (0, \pi )  \rightarrow M \times {\mathbb R}  $$

\noindent
given by $H(x,t)=(x,h_0 (t))$, where $h_0 :(0,\pi ) \rightarrow {\mathbb R} $
is the diffeomorphism defined by $h_0 (t) =cosh^{-1} ( (\sin
(t))^{-1})$
on $[\pi /2, \pi )$ and $h_0 (t)=-h_0 (\pi /2 -t)$ if
$t\in(0,\pi /2 )$.

By a direct computation  $H^* ( f_0 (g+dt^2))=
{\bf g}= \sin^2 (t) g +dt^2$ on $M\times (0,\pi )$.

Therefore by conformal invariance if we call $g_{f_0} = f_0 (g+dt^2)$

$$Y(M\times {\mathbb R} , [g+dt^2  ] )
=\inf_{ f \in C^{\infty}_0 (M\times {\mathbb R} )}
\frac{\int_{M\times {\mathbb R} } \left( \ a_{n+1} {\| \nabla f \|}_{g+dt^2}^2 +
{\bf s}_g f^2 \right) \ dvol(g+dt^2)}{
{\| f \|}_{p_{n+1}}^2 } $$

$$=\inf_{f \in C^{\infty}_0 (M\times {\mathbb R} )}
\frac{\int_{M\times {\mathbb R} } \left( \ a_{n+1}  {\| \nabla f \|}^2_{g_{f_0}}
+ {\bf s}_{g_{f_0}} f^2 \ \right) \
  dvol(g_{f_0} )}{
{\| f \|}_{p_{n+1}}^2 } $$

$$=\inf_{f \in C^{\infty}_0 (M\times (0,\pi ))}
\frac{\int_{M\times (0,\pi ) } \ \left( a_{n+1} {\| \nabla f \|}^2_{\bf g}
+ {\bf s}_{\bf g}
f^2 \ \right) \ dvol({\bf g})}{
{\| f \|}_{p_{n+1}}^2 } =Y(M\times (0,\pi ),[{\bf g}]).$$

Now, as we showed in the previous section, $Ricci({\bf g})
\geq n$. Therefore ${\bf s}_{\bf g} \geq n(n+1)$. So we get

$$Y(M\times {\mathbb R} , [g+dt^2 ]) \geq
\inf_{f \in C^{\infty}_0 (M\times (0,\pi ))}
\frac{\int_{M\times (0,\pi ) } \ \left( a_{n+1} {\| \nabla f \|}^2_{\bf g}
+ n(n+1)
f^2 \ \right) \ dvol({\bf g})}{
{\| f \|}_{p_{n+1}}^2 }.$$

To compute the infimum one needs to consider only non-negative
functions.
Now for any non-negative function
$f \in C^{\infty}_0 (M\times (0,\pi ) \  )$ consider its symmetrization
$f_* :X \rightarrow {\mathbb R}_{\geq 0}$ defined by $f_* (S) =\sup f$ and
$f_* (x,t) =s$ if and only if $Vol(B(S,t), {\bf g} )=
Vol(\{ f > s \} ,{\bf g})$ (i.e. $f_*$ is a
non-increasing function of $t$
and $Vol(\{ f_* > s \})=Vol(\{ f > s \}) $ for any $s$).
It is inmediate that the $L^q$-norms of $f_*$ and $f$ are the
same for any $q$. Also, by the coarea formula

$$\int
 \| \nabla f \|_{\bf g}^2 = \int_0^{\infty}
\left( \int_{f^{-1}(t)} \| \nabla f \|_{\bf g} d\sigma_t \right) dt.$$

$$ \geq  \int_0^{\infty} (\mu (f^{-1} (t)))^2
{\left( \int_{f^{-1}(t)} \| \nabla f \|_{\bf g}^{-1} d\sigma_t
\right)}^{-1} \ dt$$

\noindent
by H\"{o}lder's inequality, where $d\sigma_t$ is the measure induced
by ${\bf g}$ on $\{ f^{-1} (t) \}$. But

$$\int_{f^{-1}(t)} \| \nabla f \|_{\bf g}^{-1} d\sigma_t
=-\frac{d}{dt} (\mu\{ f>t \})$$

$$=-\frac{d}{dt} (\mu\{ f_* >t \}) =
\int_{f_*^{-1}(t)} \| \nabla f_* \|_{\bf g}^{-1} d\sigma_t $$

\noindent
and since $f^{-1} (t) =\partial \{ f>t \}$ by Theorem 1.1
we have $\mu (f^{-1} (t))\geq \mu (f_*^{-1} (t))$. Therefore

$$ \int_0^{\infty} (\mu (f^{-1} (t)))^2
{\left( \int_{f^{-1}(t)} \| \nabla f \|_{\bf g}^{-1} d\sigma_t
\right)}^{-1} \ dt$$

$$ \geq  \int_0^{\infty} (\mu (f_*^{-1} (t)))^2
{\left( \int_{f_*^{-1}(t)} \| \nabla f_* \|_{\bf g}^{-1} d\sigma_t
\right)}^{-1} \ dt $$

\noindent
(and since  $\| \nabla f_* \|_{\bf g}$ is constant along
$f_*^{-1}(t)$ )

$$=\int_0^{\infty}\mu (f_*^{-1} (t)) \| \nabla f_* \|_{\bf g} \ dt$$

$$= \int_0^{\infty}
\left( \int_{f_* ^{-1}(t)} \| \nabla f_* \|_{\bf g} d\sigma_t \right)
dt =\int
 \| \nabla f_*  \|_{\bf g}^2 .$$

Considering $S^{n+1}$ as the spherical cone over $S^n$ we have
the function $f^0_* : S^{n+1} \rightarrow {\mathbb R}_{\geq 0}$ which
corresponds to $f_*$.

Then for all $s$

$$Vol (\{ f_*^0 >s \} ) =
\left( \frac{V_n}{V} \right) \  Vol( \{ f_* >s \},$$

\noindent
and so for any $q$,

$$\int (f^0_*)^q dvol(g_0 ) = \left( \frac{V_n}{V} \right)
\int (f_* )^q dvol({\bf g}).$$

Also for any $s\in (0,\pi )$

$$\mu ( (f_*^0 )^{-1} (s)) = \frac{V_n}{V} \mu (f_*^{-1} (s)),$$

\noindent
and since ${\| \nabla f_*^0 \|}_{g_0} = {\| \nabla f_* \| }_{\bf g}$
we have

$$  \int
 \| \nabla f^0_*  \|_{g_0}^2 = \frac{V_n}{V} \int
 \| \nabla f_*  \|_{\bf g}^2 .$$

We obtain

$$Y(M\times {\mathbb R} , [g+dt^2 ]) \geq
\inf_{f \in C^{\infty}_0 (M\times (0,\pi ))}
\frac{\int_{M\times (0,\pi ) } a_{n+1} {\| \nabla f \|}^2_{\bf g}
+ n(n+1)
f^2 \ dvol({\bf g})}{
{\| f \|}_{p_{n+1}} ^2 }$$

$$\geq \inf_{f \in C^{\infty}_0 (M\times (0,\pi ))}
\frac{\int_{M\times (0,\pi ) } a_{n+1} {\| \nabla f_* \|}^2_{\bf g}
+ n(n+1)
f_*^2 \ dvol({\bf g})}{
{\| f_* \|}_{p_{n+1}}^2 }$$

$$={\left( \frac{V}{V_n} \right)}^{1-(2/p_{n+1})}
\inf_{f \in C^{\infty}_0 (M\times (0,\pi ))}
\frac{\int_{M\times (0,\pi ) } a_{n+1} {\| \nabla f^0_* \|}^2_{g_0}
+ n(n+1)
{f^0_*}^2 dvol({g_0})}{
{\| f^0_* \|}_{p_{n+1}}^2 }$$

$$ \geq
{\left( \frac{V}{V_n} \right)}^{2/(n+1)} \  Y_{n+1}$$

This finishes the proof of Theorem 1.2.

{\hfill$\Box$\medskip}

{\it Proof of Corollary 1.3 :} Note that if
${\bf s}_g$ is constant $Y_{{\mathbb R}} (M \times {\mathbb R} , g +
dt^2)$
only depends on ${\bf s}_g$ and $V=Vol(M,g)$ Actually,

$$Y_{{\mathbb R}} (M\times {\mathbb R} ,g +dt^2 )=
\inf_{f\in C_0^{\infty} ( {\mathbb R} )} \frac{\int_{{\mathbb R}} \  a_{n+1} {\|\nabla f
    \|}^2_{dt^2} V
+ {\bf s}_g V f^2 \ dt^2}{(\int_{{\mathbb R}} f^p )^{2/p} \  V^{2/p}}$$

$$=V^{1-(2/p)}
\inf_{f\in C_0^{\infty} ( {\mathbb R} )} \frac{\int_{{\mathbb R}} \  a_{n+1} {\|\nabla f
    \|}^2_{dt^2}
+ {\bf s}_g  f^2 \ dt^2}{(\int_{{\mathbb R}} f^p )^{2/p}}.$$

But as we said

$$\inf_{f\in C_0^{\infty} ( {\mathbb R} )} \frac{\int_{{\mathbb R}} \  a_{n+1} {\|\nabla f
    \|}^2_{dt^2}
+ {\bf s}_g  f^2 \ dt^2}{(\int_{{\mathbb R}} f^p )^{2/p}}$$

\noindent
is independent of $(M,g)$ and it is known to be equal to
$Y_{n+1} V_n^{-2/(n+1)}$.  Corollary 1.3 then follows
directly from Theorem 1.2.

{\hfill$\Box$\medskip}


\section{Introduction}
The averaged quantities can be obtained in two different ways in
magnetohydrodynamics. The first way is to solve 3D MHD equations
and then average the results. The second way is to solve some
system of equations on averages. Combination of numerical
simulations and averaged theory brings phenomenology that can
describe observations or experimental data.

The problem of spherically symmetric accretion takes its origin
from Bondi's work \citep{bondi}. He presented idealized
hydrodynamic solution with accretion rate $\dot{M}_B.$ However,
magnetic field $\vec{B}$ always exists in the real systems. Even
small seed $\vec{B}$ amplifies in spherical infall and becomes
dynamically important \citep{schwa}.

Magnetic field inhibits accretion \citep{schwa}. None of many
theories has reasonably calculated the magnetic field evolution
and how it influences dynamics. These theories have some common
pitfalls. First of all, the direction of magnetic field is usually
defined. Secondly, the magnetic field strength is prescribed by
thermal equipartition assumption. In third, dynamical effect of
magnetic field is calculated with conventional magnetic energy and
pressure. All these inaccuracies can be eliminated.

In Section 2\ref{section_method} I develop a model that abandons
equipartition prescription, calculates the magnetic field
direction and strength and employs the correct equations of
magnetized fluid dynamics. In Section 3\ref{results} I show this
accretion pattern to be in qualitative agreement with Sgr A*
spectrum models. I discuss my assumptions in Section 4
\ref{discussion}.

\section{Analytical method}\label{section_method}
 Reasonable turbulence evolution model is the key difference of my
 method. I build an averaged turbulence theory that corresponds to
numerical simulations. I start with the model of isotropic
turbulence that is consistent with simulations of collisional MHD
in three regimes. Those regimes are decaying hydrodynamic
turbulence, decaying MHD turbulence and dynamo action. I introduce
effective isotropization of magnetic field in 3D model.
Isotropization is taken to have a timescale of the order of
dissipation timescale that is a fraction $\gamma\sim1$ of the
Alfven wave crossing time $\tau_{\rm diss}=\gamma r/v_A.$

Common misconception exists about the dynamical influence of
magnetic field. Neither magnetic energy nor magnetic pressure can
represent $\vec{B}$ in dynamics. Correct averaged Euler and energy
equations were derived in \citep{scharlemann} for radial magnetic
field. Magnetic force $\vec{F}_M=[\vec{j}\times\vec{B}]$ can be
averaged over the solid angle with proper combination of
$\vec{\nabla}\cdot\vec{B}=0.$ I extend the derivation to random
magnetic field without preferred direction. Dynamical effect of
magnetic helicity \citep{biskamp03} is also investigated. I
neglect radiative and mechanical transport processes.

The derived set of equations requires some modifications and
boundary conditions to be applicable to the real astrophysical
systems. I add external energy input to turbulence to balance
dissipative processes in the outer flow. The outer turbulence is
taken to be isotropic and has magnetization $\sigma\sim1.$
Transonic smooth solution is chosen as possessing the highest
accretion rate as in \citep{bondi}.

\begin{figure}\label{fig1}
  \includegraphics[height=.5\textheight]{velocities}
  \caption{Normalized to Keplerian speed characteristic velocities of magnetized flow. Horizontal lines correspond to self-similar solution $v\sim r^{-1/2}.$}
\end{figure}

\section{Results \& Application to Sgr A*}\label{results}

\begin{figure}\label{fig2}
  \includegraphics[height=.5\textheight]{magnetization}
  \caption{Plot of magnetization $\sigma=(E_M+E_K)/E_{Th}$ with radius.}
\end{figure}
The results of my calculations confirm some known facts about
spherical magnetized accretion, agree with the results of
numerical simulations and have some previously unidentified
features.

Initially isotropic magnetic field exhibits strong anisotropy with
larger radial field $B_r.$ Perpendicular magnetic field
$B_\perp\ll B_r$ is dynamically unimportant in the inner accretion
region Fig\ref{fig1}. Because magnetic field dissipates, infall
onto the black hole can proceed \citep{schwa}.

Turbulence is supported by external driving in the outer flow
regions, but internal driving due to freezing-in amplification
takes over in the inner flow Fig\ref{fig2}. Magnetization of the
flow increases in the inner region with decreasing radius
consistently with simulations \cite{igumen06}. Density profile
appears to be $\rho\sim r^{-1.25}$ that is different from
traditional ADAF scaling $\rho\sim r^{-1.5}$ \citep{narayan}. Thus
the idea of self-similar behavior is not supported.

Compared to non-magnetized accretion, infall rate is 2-5 times
smaller depending on outer magnetization. In turn, gas density is
2-5 times smaller in the region close to the black hole, where
synchrotron radiation emerges \citep{narayan}. Sgr A* produces
relatively weak synchrotron \citep{narayan}. So, either gas
density $n$ or electron temperature $T_e$ or magnetic field $B$
are small in the inner flow or combination of factors works. Thus
low gas density in magnetized model is in qualitative agreement
with the results of modelling the spectrum.

Flow is convectively stable on average in the model of moving
blobs, where dissipation heat is released homogeneously in volume.
Moving blobs are in radial and perpendicular pressure
equilibriums. They are governed by the same equations as the
medium.

\section{Discussion \& Conclusion}\label{discussion}
The presented accretion study self-consistently treats turbulence
in the averaged model. This model introduces many weak assumptions
instead of few strong ones.

I take dissipation rate to be that of collisional MHD simulations.
But flow in question is rather in collisionless regime.
Observations of collisionless flares in solar corona
\citep{noglik} gives dissipation rate $20$ times smaller than in
collisional simulations \citep{biskamp03}. However, flares in
solar corona may represent a large-scale reconnection event rather
than developed turbulence. It is unclear which dissipation rate is
more realistic for accretion.

Magnetic field presents another caveat. Magnetic field lines
should close, or $\vec{\nabla}\cdot\vec{B}=0$ should hold. Radial
field is much larger than perpendicular in the inner region.
Therefore, characteristic radial scale of the flow is much larger
than perpendicular. If radial turbulence scale is larger than
radius, freezing-in condition does not hold anymore. Matter can
freely slip along radial field lines into the black hole. If
matter slips already at the sonic point, the accretion rate should
be higher than calculated.

Some other assumptions are more likely to be valid. Diffusion
should be weak because of high Mach number that approaches unity
at large radius. Magnetic helicity was found to play very small
dynamical role. Only when the initial turbulence is highly
helical, magnetic helicity conservation may lead to smaller
accretion rate. Neglect of radiative cooling is justified a
posteriori. Line cooling time is about $20$ times larger that
inflow time from outer boundary.

The study is the extension of basic theory, but realistic
analytical models should include more physics. The work is
underway.
\begin{theacknowledgments}
I thank my advisor Prof. Ramesh Narayan for fruitful discussions.
\end{theacknowledgments}

\bibliographystyle{aipproc}


\section{Introduction}

Located at about 1$'$ to the NW of the Orion Trapezium, the
BN/KL region has been, as
the closest region of massive star formation, the subject of extensive studies.
Recently, Rodr\'\i guez et al. (2005) and G\'omez et al. (2005)
reported large proper motions (equivalent to velocities of the order of
a few tens of km s$^{-1}$) for the radio sources associated with the infrared sources
BN and n, as well as for the radio source I. All three objects
are located at the core of the BN/KL region and appear
to be moving away from a common point where they must all have been
located about 500 years ago.
Even when these proper motions are now available, there is no
radial velocity information for these three sources, with the
exception of the near-infrared spectroscopic study of BN
made by Scoville et al. (1983), that report an LSR radial
velocity of +21 km s$^{-1}$ for this source.
In this paper we present 7 mm continuum and H53$\alpha$
radio recombination line observations of the BN/KL region in an
attempt to obtain additional information on the radial velocities of
these sources.

\section{Observations}

The 7 mm observations were made in the B configuration
of the VLA of the NRAO\footnote{The National Radio
Astronomy Observatory is operated by Associated Universities
Inc. under cooperative agreement with the National Science Foundation.},
during 2007 December 14. The central rest frequency observed was
that of the H53$\alpha$ line, 42951.97 MHz,
and we integrated on-source for a total of
approximately 3 hours. We observed in the spectral line
mode, with 15 channels of 1.56 MHz each (10.9 km s$^{-1}$)
and both circular polarizations. The bandpass calibrator was
0319+415. A continuum channel recorded the
central 75\% of the full spectral window. The absolute amplitude
calibrator was 1331+305
(with an adopted flux density of 1.47 Jy)
and the phase calibrator was 0541$-$056 (with a bootstrapped flux density
of 1.78$\pm$0.08 Jy). The phase noise rms was about 30$^\circ$,
indicating good weather conditions. The phase center of these observations was at
$\alpha(2000) = 05^h~35^m~14\rlap.^s13;~\delta(2000) = -05^\circ~22{'}~26\rlap.^{''}6$.

The data were acquired and reduced using the recommended VLA procedures
for high frequency data, including the fast-switching mode with a
cycle of 120 seconds.
Clean maps were
obtained using the task IMAGR of AIPS with the ROBUST parameter set to 0.

\section{Continuum Analysis}

\subsection{Spectral Indices}

In Figure 1 we show the image obtained from the continuum channel.
Three sources, BN, I and n, are evident in the image. No other sources
were detected above a 5-$\sigma$ lower limit of 1.75 mJy in our $1'$
field of view.  The positions, flux
densities, and deconvolved angular sizes of these sources are
given in Table 1. The continuum flux density of the sources
has been obtained from the line-free channels.
The line emission will be discussed below.
The flux density obtained at 7 mm by us
for BN is in good agreement with the values previously reported in
the literature:
we obtain a flux density of 28.6$\pm$0.6 mJy, while
values of 31$\pm$5 and 28.0$\pm$0.6 were obtained by Menten \& Reid (1995)
and Chandler \& Wood (1997), respectively.
In the case of source I, the agreement is acceptable,
since we obtain a flux density of 14.5$\pm$0.7 mJy,
while values of
13$\pm$2 and 10.8$\pm$0.6 mJy were reported by Menten \& Reid (1995)
and Chandler \& Wood (1997), respectively.
Careful monitoring would be required
to test if the radio continuum from source I is variable in time.

The spectral indices determined from our 7 mm observations and the
3.6 cm observations of G\'omez et al. (2008) are given in the last column of Table 2.
Our spectral indices for BN and
I are in excellent agreement in this spectral range with the more detailed analysis
presented by Plambeck et al. (1995) and Beuther et al. (2004).

We have detected source n for the first time
at 7 mm and this detection allows the first estimate of the spectral index of this source
over a wide frequency range.
The value of 0.2$\pm$0.1 suggests marginally thick free-free emission, as expected in
an ionized outflow. This supports the interpretation of this source
as an ionized outflow by G\'omez et al. (2008).
The position given by us in Table 1 is consistent with the
extrapolation of the proper motions of this source discussed by G\'omez et al. (2008).

\subsection{Deconvolved Angular Sizes}

The radio source I has parameters
consistent with an optically thick free-free source (spectral
index of $1.5\pm0.1$).
Beuther et al. (2004) suggest that this spectral index is either the result of
optically thick free-free plus dust emission, or $H^-$ free-free emission
that gives rise to a power-law spectrum with an index of $\sim$1.6.

In the case of the radio source associated with the infrared source n
we only have an upper limit to its size at 7 mm. In addition,
G\'omez et al. (2008) report important morphological variations
over time in this source
that suggest that comparisons at different frequencies should be made
only from simultaneous observations.

In the case of BN,
the frequency dependences of flux density and angular size (this last
parameter taken to
be the geometric mean of the major and minor axes reported in Tables 1 and 2) can be accounted for with
a simple model of a sphere of ionized gas in which
the electron density
decreases as a power-law function of radius, $n_e \propto r^{-\alpha}$.
In this case, the flux density of the source is expected to go with
frequency as $S_\nu \propto \nu^{(6.2-4\alpha)/(1-2\alpha)}$ and the angular size is expected to go with
frequency as $\theta_\nu \propto \nu^{2.1/(1-2\alpha)}$ (Reynolds 1986).
The frequency dependences of flux density ($S_\nu \propto \nu^{1.1\pm0.1}$) and angular
size ($\theta_\nu \propto \nu^{-0.36\pm0.12}$) for
BN are consistent with a steeply declining electron density
distribution
with power law index of
$\alpha = 3.0\pm0.3$. The continuum spectrum of BN produced
by Plambeck et al. (1995) indicates that a constant
spectral index extends from 5 to 100 GHz.

\section{Analysis of the H53$\alpha$ Recombination Line Emission}

\subsection{Radial LSR Velocity}

We clearly detected the H53$\alpha$ line emission only from BN.
The spectrum is shown in Figure 2. The parameters of
the Gaussian least squares fit to the profile are given in Table 3.
We note that the radial LSR velocity determined by us, $+20.1\pm2.1$
km s$^{-1}$, agrees well with the value of $+21$ km s$^{-1}$
reported by Scoville et al. (1983) from near-IR spectroscopy.
In a single dish study of the H41$\alpha$ line made with an
angular resolution of 24$''$ toward
Orion IRc2, Jaffe \& Mart\'\i n-Pintado (1999) report emission
with $v_{LSR}$ = -3.6 km s$^{-1}$.
Most likely, this is emission from the ambient H~II region, since
its radial velocity practically coincides with the
value determined for the large H~II region (Orion A) ionized by
the Trapezium stars (e. g. Peimbert et al. 1988).
The single dish observations of the H51$\alpha$ emission
of Hasegawa \& Akabane (1984), made with an angular resolution of 33$''$,
most probably come also from the ambient ionized gas and not
from BN.

\subsection{LTE Interpretation}

If we assume that the line emission is optically thin and in LTE,
the electron temperature, $T_e^*$, is given by
(Mezger \& H\"oglund 1967; Gordon 1969; Quireza et al. 2006):

\begin{equation}\Biggl[{{T_e^*} \over {K}}\Biggr] = \Biggl[7100 \biggl({{\nu_L} \over {GHz}} \biggr)^{1.1}
\biggl({{S_C} \over {S_L}} \biggr) \biggl({{\Delta v} \over {km~s^{-1}}}\biggr)^{-1}
(1 + y^+)^{-1} \Biggr]^{0.87}, \end{equation}

\noindent where $\nu_L$ is the line frequency, $S_C$ is the continuum flux density,
$S_L$ is the peak line flux density, $\Delta v$ is the FWHM line width, and
$y^+$ is the ionized helium to ionized hydrogen abundance ratio.
In the case of BN, we can adopt $y^+ \simeq 0$ given that the
source is not of very high luminosity, and using the values given in Tables 1 and 3,
we obtain $T_e^* \simeq 8,200$ K. This value is similar to that
determined for the nearby Orion A from radio recombination lines (e. g. Lichten, Rodr\'\i guez, \&
Chaisson 1979).

It is somewhat
surprising that we get a very reasonable estimate for $T_e^*$ when our previous discussion
seemed to imply that BN is partially optically thick at 7 mm.
One possibility is that we have two effects fortuitously canceling each other. For example, the
optical thickness of the source will diminish the
line emission, while maser effects (such as those observed
in MWC 349; Mart\'\i n-Pintado et al. 1989) will amplify the line.
However, in an attempt to understand this result in LTE conditions, we will discuss the expected
LTE radio recombination line emission from
a sphere of ionized gas in which the electron density
decreases as a power-law function of radius, $n_e \propto r^{-\alpha}$.
As noted before, the  modeling of the continuum emission from such a source
was presented in detail by Panagia \& Felli (1975) and Reynolds (1986). The radio recombination line emission
for the case $\alpha = 2$ has been discussed by Altenhoff, Strittmatter, \&
Wendker (1981) and Rodr\'\i guez (1982).
Here we generalize the derivation of the recombination line emission
to the case of $\alpha > 1.5$. This lower limit is
adopted to avoid the total emission from the source to diverge.

For a sphere of ionized gas, the free-free continuum emission will be given by
(Panagia \& Felli 1975):

\begin{equation}S_C = 2 \pi {{r_0^2} \over {d^2}} B_\nu \int_0^\infty
\biggl(1 - exp[-\tau_C(\xi)]\biggr)~ \xi~ d\xi, \end{equation}

\noindent where $r_0$ is a reference radius, $d$ is the distance to the source,
$B_\nu$ is Planck's function, $\xi$ is the projected radius in units of $r_0$,
and $\tau_C(\xi)$ is the continuum optical depth along the line of sight with
projected radius $\xi$. On the other hand, the free-free continuum plus
radio recombination line emission will be given by an equation similar to eqn. (2), but with the
continuum opacity substituted by the continuum plus line opacity (Rodr\'\i guez 1982):

\begin{equation}S_{L+C} = 2 \pi {{r_0^2} \over {d^2}} B_\nu \int_0^\infty \biggl(1 - exp[-\tau_{L+C}(\xi)]
\biggr) \xi d\xi, \end{equation}

\noindent where $\tau_{L+C}(\xi)$ is the line plus continuum optical depth along the line of sight with
projected radius $\xi$.

The line-to-continuum ratio will be given by:

\begin{equation}{{S_L} \over {S_C}} = {{S_{L+C} - S_C} \over {S_C}}. \end{equation}

The opacity of these emission processes depends on projected radius as (Panagia \& Felli 1975):

\begin{equation}\tau(\xi) \propto \xi^{-(2 \alpha -1)}. \end{equation}

We now introduce the definite integral (Gradshteyn \& Ryzhik 1994)

\begin{equation}\int_0^\infty [1- exp(-\mu x^{-p})]~x~ dx =
- {{1} \over {p}}~ \mu^{{2} \over{p}}~ \Gamma(-{{2} \over{p}}), \end{equation}

\noindent valid for $\mu > 0$ and $p > 0$ and with $\Gamma$ being the Gamma function.
Substituting eqns. (2) and (3) in eqn. (4), and using the integral
defined in eqn. (7), it can be shown that

\begin{equation}{{S_L} \over {S_C}} = \Biggl[{{\kappa_L + \kappa_C}
\over {\kappa_C}} \Biggr]^{1/(\alpha -0.5)} - 1, \end{equation}

\noindent where $\kappa_L$ and $\kappa_C$ are the line and continuum absorption coefficients
at the frequency of observation, respectively.
In this last step we have also
assumed that the opacity of the line and continuum processes are proportional to
the line and continuum absorption coefficients, respectively, that is, that the
physical depths producing the line and continuum emissions are the
same. Under the LTE assumption, we have
that

\begin{equation}{{\kappa_L} \over {\kappa_C}} = 7100 \biggl({{\nu_L} \over {GHz}} \biggr)^{1.1}
\biggl({{T_e^*} \over {K}} \biggr)^{-1.1} \biggl({{\Delta v} \over {km~s^{-1}}}\biggr)^{-1}
(1 + y^+)^{-1}. \end{equation}

For $\nu \leq$ 43 GHz and typical parameters of an H II region, we
can see from eqn. (8) that $\kappa_L<\kappa_C$, and
eqn. (7) can be approximated by:

\begin{equation}{{S_L} \over {S_C}} \simeq {{1} \over
{(\alpha -0.5)}} \Biggl[{{\kappa_L} \over {\kappa_C}} \Biggr]. \end{equation}

That is, the expected optically-thin, LTE line-to-continuum ratio:

\begin{equation}{{S_L} \over {S_C}} \simeq \Biggl[{{\kappa_L} \over {\kappa_C}} \Biggr], \end{equation}

\noindent becomes attenuated by a factor $1/(\alpha -0.5)$. In the case of $\alpha = 2$,
the factor is 2/3, and we reproduce the result of Altenhoff, Strittmatter, \&
Wendker (1981) and Rodr\'\i guez (1982). In the case of BN, we have that $\alpha \simeq 3$, and
we expect the attenuation factor to be 2/5. If BN can be modeled this way, we would have expected
to derive electron temperatures under the LTE assumption (see eqn. 1) of order

\begin{equation}T_e^*(\alpha = 3) \simeq 2.2~ T_e^*(thin). \end{equation}

However, from the discussion in the first paragraph of this section observationally
we determine that

\begin{equation}T_e^*(\alpha = 3) \simeq T_e^*(thin). \end{equation}

Summarizing: i) BN seems to have significant optical depth in the continuum at
7 mm, ii) this significant optical depth should attenuate the observed recombination
line emission with respect to the optically-thin case, but iii) the line emission seems
to be as strong as in the optically-thin case.

As possible explanations for the ``normal'' (apparently optically-thin and in LTE)
radio recombination line emission
observed from BN we can think of two options.
The first is that, as noted before, there is a non-LTE line-amplifying
mechanism that approximately compensates for the optical depth attenuation.
The second possibility is that the free-free emission from BN at 7 mm is already optically thin.
However, this last possibility seems to be in contradiction with the results
of Plambeck et al. (1995) that suggest a single spectral index
from 5 to 100 GHz. Observations of radio recombination lines around
100 GHz are needed to solve this problem.

A comparison with the H53$\alpha$ emission from the hypercompact H~II
region G28.20-0.04N is also of interest.
The continuum flux densities from this source at
21, 6, 3.6, and 2 cm are 49, 135, 297, and 543 mJy, respectively
(Sewilo et al. 2004). At 7 mm the continuum flux density is 641 mJy
(Sewilo et al. 2008), indicating
that the source has become optically thin at this wavelength.
Using the H53$\alpha$ line parameters given by (Sewilo et al. 2008)
we derive an LTE electron temperature of $T_e^* \simeq 7,600$ K,
similar to the value for BN and in this case consistent with
the optically-thin nature of G28.20-0.04N.

The non detection of H53$\alpha$ emission from radio source I is consistent
with its expected large optical depth. The formulation above implies $\alpha \simeq 5$, and an
attenuation factor of 2/9.
This confirms the notion that BN and radio source I are two sources
intrinsically very different in nature.
This difference is also evident in the brightness temperature of both sources.
At 7 mm, the brightness temperature of a source is

\begin{equation}\Biggl[{{T_B} \over {K}} \Biggr] \simeq 0.96 \Biggl[{{S_\nu} \over {mJy}}
\Biggr] \Biggl[{{\theta_{maj} \times
\theta_{min}} \over {arcsec^2}} \Biggr]^{-2}. \end{equation}

Using the values of Table 1, we get $T_B \simeq$ 7,800 K for BN, confirming
its nature as photoionized gas. However, for the radio source I we get
$T_B \simeq$ 2,600 K. So, even when source I seems to be optically thick, its
brightness temperature is substantially lower than that expected for
a photoionized region. Reid et al. (2007) have discussed as possible
explanations for this low brightness temperature $H^-$ free-free opacity or
a photoionized disk.

Following the discussion of Reid et al. (2007), we consider
it unlikely that dust emission could be a dominant contributor to the 7 mm emission of BN or
Orion I. A dense, warm, dusty disk would be expected to show many molecular lines at
millimeter/submillimeter wavelengths. While Beuther et al. (2006) and Friedel
\& Snyder(2008) find numerous, strong,
molecular lines toward the nearby "hot core", they find no strong lines toward the position of
Orion I (with the exception of
the strong SiO masers slightly offset from Orion I) or BN.
Also, the brightness temperatures derived by us at 7 mm (7,800 K for BN and
2,600 K for source I) are
high enough to sublimate dust and suggest that free-free emission from
ionized gas dominates the continuum emission.
Finally, the continuum spectra of BN and of source I measured by Plambeck et al.(1995)
and Beuther et al. (2006), respectively, suggest that the dust
emission becomes dominant only above $\sim$300 GHz.

In the case of source n, no detection was expected given its
weakness even in the continuum.

\subsection{Spatial Distribution of the H53$\alpha$ Line Emission}

The H53$\alpha$ line emission in the individual velocity
channels shows evidence of structure but unfortunately the signal-to-noise
ratio is not large enough to reach reliable conclusions from the
analysis of these individual channels. However, an image
with good signal-to-noise ratio can be obtained averaging over the velocity
range of -21.2 to +66.1 km s$^{-1}$, using the task MOMNT in
AIPS. This line image is compared
in Figure 3 with a continuum image
made from the line-free channels.
The larger apparent size of the continuum image is simply the
result of its much better signal-to-noise ratio.
For the total line emission we obtain an upper limit of
$0\rlap.{''}12$ for its size, that is consistent with the
size of the continuum emission given in Table 1.
We also show images of the blueshifted (-21.2 to +22.5 km s$^{-1}$)
and redshifted (+22.5 to 66.1 km s$^{-1}$) line emission in Figure 3.
The cross in the figure indicates the centroid of the total line
emission. The centroid of the line emission does not appear to
coincide with the centroid of the continuum emission and
we attribute this to opacity effects.

An interesting conclusion comes from comparing the total
line emission, with the blueshifted and redshifted components.
The blueshifted emission seems slightly shifted to the SW, while the
redshifted emission seems slightly shifted to the NE, suggesting a
velocity gradient. This result supports the suggestion of
Jiang et al. (2005) of the presence of an outflow in BN along a
position angle of 36$^\circ$. Given the modest signal-to-noise ratio
of the data, it is difficult to estimate the magnitude
of the velocity shift and we crudely assume it is of order one
channel ($\sim$10 km s$^{-1}$), since most of the line
emission is concentrated in the central two channels
of the spectrum (see Figure 2). The position shift between the blueshifted and
the redshifted emissions is $0\rlap.{''}028 \pm 0\rlap.{''}007$
($12 \pm 3$ AU at the distance of 414 pc given by Menten et al. 2007), significant to the
4-$\sigma$ level. Unfortunately, the data of Jiang et al. (2005) does not
include line observations and there is no kinematic information in their paper to
compare with our results.

The small velocity gradient observed by us in BN is consistent with a
slow bipolar outflow but also with Keplerian rotation around a central mass
of only 0.2 $M_\odot$.

\section{Conclusions}

We presented observations of the H53$\alpha$ recombination line
and adjacent continuum toward the Orion BN/KL region.
In the continuum we detect the BN object, the radio source
I (GMR I) and the radio counterpart of the infrared source n
(Orion-n) and discuss its parameters.
In the H53$\alpha$ line we only detect the BN object,
the first time that radio recombination lines have been detected from this source.
The LSR radial velocity of BN from the H53$\alpha$ line, $v_{LSR} = 20.1 \pm 2.1$
km s$^{-1}$,
is consistent with that found from previous studies in near-infared lines,
$v_{LSR} = 21$ km s$^{-1}$.
We discuss the line-to-continuum ratio from BN and present evidence
for a possible velocity gradient across this source.

\acknowledgments

LFR and LAZ acknowledge the support
of CONACyT, M\'exico and DGAPA, UNAM.

{\it Facilities:} \facility{VLA}