# On scalar curvature lower bounds and scalar curvature measure

John Lott  Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840
USA<EMAIL_ADDRESS>
(Date: July 14, 2022)

###### Abstract.

We relate the (non)existence of lower scalar curvature bounds to the existence
of certain distance-decreasing maps. We also give a sufficient condition for
the existence of a limiting scalar curvature measure in the backward limit of
a Ricci flow solution.

Research partially supported by NSF grant DMS-1810700

##  1\. Introduction

In [11] we proved the following result, in which a lower bound on scalar
curvature gives a restriction on the existence of distance-nonincreasing maps
of nonzero degree. Let $R$ denote scalar curvature and let $H$ denote mean
curvature.

######  Theorem 1.1.

Let $N$ and $M$ be compact connected Riemannian manifolds-with-boundary of the
same even dimension. Let $f\>:\>(N,\partial N)\rightarrow(M,\partial M)$ be a
smooth spin map and let $\partial f\>:\>\partial N\rightarrow\partial M$
denote the restriction to the boundary. Suppose that

  * •

$f$ is $\Lambda^{2}$-nonincreasing and $\partial f$ is distance-nonincreasing,

  * •

$M$ has nonnegative curvature operator and $\partial M$ has nonnegative second
fundamental form,

  * •

$R_{N}\geq f^{*}R_{M}$ and $H_{\partial N}\geq(\partial f)^{*}H_{\partial M}$,

  * •

$M$ has nonzero Euler characteristic and

  * •

$f$ has nonzero degree.

Then $R_{N}=f^{*}R_{M}$ and $H_{\partial N}=(\partial f)^{*}H_{\partial M}$.

Furthermore,

  * •

If $0<\operatorname{Ric}_{M}<\frac{1}{2}R_{M}g_{M}$ then $f$ is a Riemannian
covering map.

  * •

If $\operatorname{Ric}_{M}>0$ and $f$ is distance-nonincreasing then $f$ is a
Riemannian covering map.

  * •

If $M$ is flat then $N$ is Ricci-flat.

In particular, the lower scalar curvature bound $R_{N}\geq f^{*}R_{M}$ means
that it is impossible for $f$ and $\partial f$ to be distance-decreasing (i.e.
have Lipschitz constant less than one), with $f$ having nonzero degree, and to
also have $H_{\partial N}>(\partial f)^{*}H_{\partial M}$. Theorem 1.1 follows
earlier work by Llarull [8] and Goette-Semmelmann [4]; we refer to [11] for
background and generalizations. The first main result of the present paper is
a converse and shows that the lack of a lower bound on the scalar curvature
implies that such distance-decreasing maps do exist.

######  Theorem 1.2.

Given $n>1$ and $K\in{\mathbb{R}}$, let $Z$ be an $n$-dimensional Riemannian
manifold and let $z\in Z$ be a point where the scalar curvature is
$R_{z}<n(n-1)K$. Then for any neighborhood $U$ of $z$, there are

  1. (1)

A codimension-zero compact submanifold-with-boundary $N\subset U$ containing
$z$ that is diffeomorphic to a ball,

  2. (2)

A codimension-zero compact submanifold-with-boundary $M$ in the
$n$-dimensional model space of constant curvature $K$, diffeomorphic to a
ball,

  3. (3)

A smooth map $f:(N,\partial N)\rightarrow(M,\partial M)$ of nonzero degree so
that $f$ and $\partial f$ are distance-decreasing, and the mean curvatures
satisfy $H_{\partial N}>(\partial f)^{*}H_{\partial M}$, and

  4. (4)

Numbers $\delta,l>0$ so that for all $n\in\partial N$ and $t\in[0,l)$, one has
$f(\exp_{n}(t\nu_{\partial N}))=\exp_{f(n)}((1-\delta)t\nu_{\partial M})$,
where $\nu_{\partial N}$ and $\nu_{\partial M}$ are the inward unit normals to
$\partial N$ and $\partial M$, respectively.

If $K\leq 0$ then we can take $M$ to be strictly convex.

Together, Theorems 1.1 and 1.2 essentially give a metric characterization of
lower scalar curvature bounds. While the geometric meaning of scalar curvature
may be hard to understand, the metric characterization is in terms of mean
curvature, which is more tractable.

The proof of Theorem 1.2 is by induction on $n$, as in the proof of a related
result by Gromov in [6]. Item (4) in the conclusion of the theorem is just for
technical convenience, in order to simplify the induction argument. In the
induction step, it is fairly easily to obtain cylindrical regions that satisfy
the conclusions of the theorem, along the lines of [6], but have codimension-
two singularities. The main technical issue is to smooth the singularities
while simultaneously maintaining the distance-decreasing property and the
inequality on mean curvatures. Given Theorem 1.2, one can somewhat simplify
Gromov’s proof of the preservation of lower scalar curvature bounds under
$C^{0}$-limits of smooth Riemannian metrics [6].

The second main result of the paper is about the existence of a limiting
scalar curvature measure, as $t\rightarrow 0$, for a Ricci flow coming out of
a metric space. If there is going to be a finite limiting measure then by
looking at the total scalar curvature along the flow, one sees that the
finiteness of
$\int_{0}^{T}\int_{M}(R^{2}-2|\operatorname{Ric}|^{2})\operatorname{dvol}_{g(t)}dt$
is a necessary condition (equation (3.25)). The next theorem essentially says
that it is also sufficient.

######  Theorem 1.3.

Let $(M,g(t))$, $t\in(0,T]$, be a Ricci flow solution on a compact
$n$-dimensional manifold $M$ satisfying

  1. (1)

$|\operatorname{Rm}_{g(t)}|<\frac{A}{t}$ for some $A<\infty$ and all $t$,

  2. (2)

$\operatorname{Ric}_{g(t)}\geq Eg(t)$ for some $E>-\infty$ and all $t$, and

  3. (3)

$R^{2}-2|\operatorname{Ric}|^{2}\in L^{1}((0,T]\times
M;dt\operatorname{dvol}_{g(t)})$.

Then there is a limit $\lim_{t\rightarrow
0}R_{g(t)}\operatorname{dvol}_{g(t)}=\mu_{0}$ in the weak-$\star$ topology.

One’s first approach to proving Theorem 1.3 might be to fix a test function
$f$ and consider the time evolution of
$\int_{M}fR_{g(t)}\operatorname{dvol}_{g(t)}$. This turns out to not be
useful. Instead we let $f$ evolve by a backward heat equation and use heat
kernel estimates from [2].

Using Theorem 1.3, we show the existence of a subsequential limiting scalar
curvature measure on a class of Ricci limit spaces. Recall that a Riemannian
manifold has $2$-nonnegative curvature operator if at each point, the sum of
the two lowest eigenvalues of the curvature operator is nonnegative.

######  Theorem 1.4.

Given $D,\mathaccent 866{A}<\infty$ and $v_{0}>0$, let
$\\{(M_{i},g_{i})\\}_{i=1}^{\infty}$ be a sequence of compact $n$-dimensional
Riemannian manifolds, $n\geq 4$, such that

  1. (1)

$\operatorname{diam}(M_{i},g_{i})\leq D$,

  2. (2)

$\operatorname{vol}(M_{i},g_{i})\geq v_{0}$,

  3. (3)

$(M_{i},g_{i})$ has $2$-nonnegative curvature operator, and

  4. (4)

$\int_{M_{i}}R_{g_{i}}\>\operatorname{dvol}_{g_{i}}\leq\mathaccent 866{A}$.

Then after passing to a subsequence, there is a Gromov-Hausdorff limit
$(X_{\infty},d_{\infty})$ with a measure $\mu_{0}$, along with continuous
Gromov-Hausdorff approximations $\eta_{i}:M_{i}\rightarrow X_{\infty}$, such
that

  1. (1)

$\lim_{i\rightarrow\infty}\left(\eta_{i}\right)_{*}\left(R_{g_{i}}\operatorname{dvol}_{g_{i}}\right)\stackrel{{\scriptstyle
weak-\star}}{{=}}\mu_{0}$, and

  2. (2)

There is a smooth $1$-parameter family of Riemannian metrics
$\\{g(t)\\}_{t\in(0,T]}$ on $X_{\infty}$, with $2$-nonnegative curvature
operator, so that $\lim_{t\rightarrow
0}(X_{\infty},g(t))\stackrel{{\scriptstyle GH}}{{=}}(X_{\infty},d_{\infty})$
and $\lim_{t\rightarrow
0}R_{g(t)}\operatorname{dvol}_{g(t)}\stackrel{{\scriptstyle
weak-\star}}{{=}}\mu_{0}$.

Condition (4) in the Theorem (for some $\mathaccent 866{A}$) may in fact
follow from the other three conditions. From the hypotheses of Theorem 1.4,
the subsequential existence of a limiting metric space (in the Gromov-
Hausdorff topology) and a limiting scalar curvature measure (in the
weak-$\star$ topology) is automatic. The content of the theorem is that the
metric space and the scalar curvature measure also arise as a continuous
limit, coming from a Ricci flow.

I thank Antoine Song for discussions and the referee for helpful comments.

##  2\. Proof of Theorem 1.2

It would be unwieldy to write out equations for all the steps in the proof of
Theorem 1.2, so we give the main ingredients. The proof is by induction on
$n$, as in [6, Section 4.9].

######  Lemma 2.1.

The theorem is true in dimension two.

###### Proof.

If $n=2$, choose normal coordinates $(r,\theta)$ around $z$ and
$(r^{\prime},\theta^{\prime})$ around a point $m\in M$. Given
$\alpha\in(0,1]$, let $N$ be given by $r\leq r_{0}$ and let $M$ be given by
$r^{\prime}\leq r_{0}^{\prime}=\alpha r_{0}$. In the normal coordinates, to
leading order,

(2.2) $\displaystyle g_{N}\sim$
$\displaystyle\>dr^{2}+r^{2}(1-\frac{1}{6}R_{z}r^{2})d\theta^{2},$
$\displaystyle g_{M}\sim$
$\displaystyle\>(dr^{\prime})^{2}+(r^{\prime})^{2}(1-\frac{1}{3}K(r^{\prime})^{2})(d\theta^{\prime})^{2},$
$\displaystyle H_{\partial N}\sim$
$\displaystyle\>\frac{1}{r_{0}}-\frac{1}{6}R_{z}r_{0},$ $\displaystyle
H_{\partial M}\sim$
$\displaystyle\>\frac{1}{r^{\prime}_{0}}-\frac{1}{3}Kr_{0}^{\prime}.$

Define a Lipschitz function $F$ by $F(r,\theta)=(\alpha r,\theta)$. Then

(2.3) $\displaystyle F^{*}g_{M}\sim$
$\displaystyle\>\alpha^{2}dr^{2}+\alpha^{2}r^{2}(1-\frac{1}{3}K\alpha^{2}r^{2})d\theta^{2},$
$\displaystyle(\partial F)^{*}H_{\partial M}\sim$
$\displaystyle\>\frac{1}{\alpha r_{0}}-\frac{1}{3}K\alpha r_{0}.$

For small $r_{0}$, if $\alpha=1$ then $F$ is distance-nonincreasing, $\partial
F$ is distance-decreasing and $H_{\partial N}>(\partial F)^{*}H_{\partial M}$.
By continuity, if $\alpha$ is slightly less than one then $F$ and $\partial F$
will be distance-decreasing and we will still have $H_{\partial N}>(\partial
F)^{*}H_{\partial M}$. If $f$ is the result of slightly smoothing $F$ near $z$
then it will satisfy the conclusion of the theorem. ∎

Now suppose that $n>2$ and the theorem is true in dimension $n-1$.

######  Lemma 2.4.

There is a (small) minimal hypersurface $V\subset U$ containing $z$ and a unit
normal vector $v\in T_{z}Z$ so that

  * •

The second fundamental form $A_{z}$ at $z$ has $|A_{z}|\ll 1$,

  * •

$\operatorname{Ric}(v,v)<(n-1)K$, and

  * •

The scalar curvature of $V$ at $z$ satisfies $R^{\prime}_{z}<(n-1)(n-2)K$.

###### Proof.

Multiplying $g_{Z}$ by a large constant $\lambda$, the geometry of a unit ball
around $z$ becomes closer and closer to Euclidean. Given a unit vector $w\in
T_{z}Z$, consider the foliation of the rescaled ball of radius two given by
hyperplanes perpendicular to $w$ with respect to the Euclidean metric in the
normal coordinates. The leaves are minimal with respect to the Euclidean
metric. Consider the part of the foliation whose height varies between $-1.5$
and $1.5$. Using stability results as in [15, 16], if $\lambda$ is large
enough then there is a small $C^{k}$-perturbation of the foliation by minimal
hypersurfaces that preserves the intersections of the leaves with the sphere
of radius two. This restricts to a minimal foliation of the unit ball with
arbitrarily small second fundamental form, if $\lambda$ is large enough. Let
$\alpha(w)$ be the choice of unit normal, to the leaf at $z$, that is close to
$w$. For sufficiently large $\lambda$, the map $\alpha:S^{n-1}\rightarrow
S^{n-1}$ is a local diffeomorphism, hence is surjective. We will take
$w=\alpha^{-1}(v)$ for an appropriately chosen $v$ that is specified below and
let $V$ be the corresponding minimal leaf through $z$.

From the Gauss-Codazzi equation, the scalar curvature $R^{\prime}_{{z}}$ of
${z}$ in $V$ is given by

(2.5)
$R^{\prime}_{{z}}=R_{{z}}-2\operatorname{Ric}(v,v)+(\operatorname{Tr}(A))^{2}-\operatorname{Tr}(A^{2}).$

Put

(2.6) $\mathaccent 866{\operatorname{Ric}}=\operatorname{Ric}-(n-1)Kg,$

with trace $\mathaccent 866{R}=R-n(n-1)K$. Let $\mathaccent
866{R}_{11}\leq\mathaccent 866{R}_{22}\leq\ldots\leq\mathaccent 866{R}_{nn}$
be the eigenvalues of $\mathaccent 866{\operatorname{Ric}}_{z}$.

If $\mathaccent 866{R}_{nn}<0$, let $v$ be a corresponding unit eigenvector.
Then $\operatorname{Ric}(v,v)<(n-1)K$ and

(2.7) $\displaystyle R_{{z}}-2\operatorname{Ric}(v,v)$
$\displaystyle=R_{11}+\ldots+R_{n-2,n-2}+(R_{n-1,n-1}-R_{nn})$
$\displaystyle\leq R_{11}+\ldots+R_{n-2,n-2}<(n-1)(n-2)K.$

If $\mathaccent 866{R}_{nn}=0$, let $v$ be a unit vector that is a slight
perturbation from $\operatorname{Ker}(\mathaccent 866{\operatorname{Ric}})$.
Then $\operatorname{Ric}(v,v)<(n-1)K$ and we still have
$R_{{z}}-2\operatorname{Ric}(v,v)<(n-1)(n-2)K$.

If $\mathaccent 866{R}_{nn}>0$ then the quadratic form $\mathaccent
866{\operatorname{Ric}}$ is indefinite and for any $\delta>0$, we can find a
unit vector $v$ so that $-\delta<\mathaccent 866{\operatorname{Ric}}(v,v)<0$.
Then

(2.8) $(n-1)K-\delta<\operatorname{Ric}(v,v)<(n-1)K,$

so $R_{{z}}-2\operatorname{Ric}(v,v)<R_{z}-2(n-1)K+2\delta$. Taking $\delta$
small enough, we can ensure that
$R_{{z}}-2\operatorname{Ric}(v,v)<(n-1)(n-2)K$.

In any case, we can achieve a negative upper bound on
$R_{{z}}-2\operatorname{Ric}(v,v)-(n-1)(n-2)K$ that is independent of
$\lambda$. Finally, taking $\lambda$ large enough to ensure that
$|(\operatorname{Tr}(A))^{2}-\operatorname{Tr}(A^{2})|$ is small, we obtain
from (2.5) that $R^{\prime}_{{z}}<(n-1)(n-2)K$. ∎

With reference to the $V$ of Lemma 2.4, let $N^{\prime}$ be an
$(n-1)$-dimensional compact submanifold-with-boundary of $V$ containing $z$
obtained by applying the induction hypothesis, with corresponding submanifold
$M^{\prime}$ of the $(n-1)$-dimensional model space of constant curvature $K$,
and with map $f^{\prime}:N^{\prime}\rightarrow M^{\prime}$ of nonzero degree
so that $f^{\prime}$ and $\partial f^{\prime}$ are distance-decreasing, and
$H_{\partial N^{\prime}}>(\partial f^{\prime})^{\star}H_{\partial
M^{\prime}}$. Taking $N^{\prime}$ small enough, we can assume that the unit
normal vector $\nu_{N^{\prime}}$ satisfies
$\operatorname{Ric}(\nu_{N^{\prime}},\nu_{N^{\prime}})<(n-1)K$ on
$N^{\prime}$.

For small $\epsilon>0$, let $N^{(2)}$ be the cylindrical region
$\\{\exp(u\nu_{N^{\prime}}):|u|\leq\epsilon\\}$ in $Z$. Similarly, put
$M^{(2)}=\\{\exp(u\nu_{M^{\prime}}):|u|\leq(1-\delta_{N^{\prime}})\epsilon\\}$
in the $n$-dimensional model space of constant curvature $K$, where
$\nu_{M^{\prime}}$ is a unit normal field to $M^{\prime}$ and
$\delta_{N^{\prime}}$ is the parameter appearing in the induction hypothesis.
See Figure 1, which illustrates the case $K=0$.

Figure 1.

In what follows we can always reduce $l_{N^{\prime}}$ and
$\delta_{N^{\prime}}$. Define $f^{(2)}:N^{(2)}\rightarrow M^{(2)}$ by
$f^{(2)}(\exp_{n^{\prime}}(u\nu_{N^{\prime}}))=\exp_{f^{\prime}(n^{\prime})}((1-\delta_{N^{\prime}})u\nu_{M^{\prime}})$.
Note that $\partial N^{(2)}$ has a top face and a bottom face, both
diffeomorphic to $N^{\prime}$, and an annular region diffeomorphic to
$[-\epsilon,\epsilon]\times\partial N^{\prime}$. The annular region meets the
top face orthogonally in a codimension-two stratum diffeomorphic to $\partial
N^{\prime}$, with a similar statement for the bottom face. The maps $f^{(2)}$
and $\partial f^{(2)}$ are distance-decreasing.

Along the geodesics in $N^{(2)}$ normal to $N^{\prime}$, we have

(2.9)
$\frac{dH}{dt}+\operatorname{Tr}(A^{2})=-\operatorname{Ric}(\gamma^{\prime},\gamma^{\prime}).$

For small $\epsilon$, if $N^{\prime}$ is taken small enough then $|A_{z}|\ll
1$ and on the top and bottom faces of $N^{(2)}$, we have
$H\sim-\epsilon\operatorname{Ric}(\nu_{N^{\prime}},\nu_{N^{\prime}})$.
Similarly, on the top and bottom faces of $M^{(2)}$, we have
$H\sim-(n-1)\epsilon\left(1-\delta_{N^{\prime}}\right)K$. Including the
annular region over $\partial N^{\prime}$, if $\epsilon$ and
$\delta_{N^{\prime}}$ are small then $H_{\partial N^{(2)}}>(\partial
f^{(2)})^{\star}H_{\partial M^{(2)}}$.

We can assume that the parameter $l_{N^{\prime}}$ in the induction assumption
is less than the focal radius of $\partial N^{\prime}\subset N^{\prime}$, and
$l_{N^{\prime}}\ll\epsilon$. Given $\epsilon^{\prime}\ll l_{N^{\prime}}$, let
$N^{(3)}$ be the points inside of $N^{(2)}$ that have distance at least
$\epsilon^{\prime}$ from $\partial N^{(2)}$. Let $N^{(4)}$ be the
$\epsilon^{\prime}$-neighborhood of $N^{(3)}$ in $Z$. Do a similar
construction for $M$, to obtain $M^{(4)}$.

The boundary $\partial N^{(4)}$ is $C^{1,1}$-regular and has a decomposition
into a top face $F_{+}^{(4)}$, a bottom face $F_{-}^{(4)}$, an annular belt
$A^{(4)}$, an upper tube $T_{+}^{(4)}$ and a lower tube $T_{-}^{(4)}$. See
Figure 2. There is a similar decomposition of $\partial M^{(4)}$.

Figure 2.

Consider the top tubular region $T_{+}^{(4)}$. It has two boundary components
$\partial T_{+,1}^{(4)}$ and $\partial T_{+,2}^{(4)}$, with $\partial
T_{+,1}^{(4)}$ also being a boundary component of $F_{+}^{(4)}$ and $\partial
T_{+,2}^{(4)}$ also being a boundary component of $A^{(4)}$.

Let $p:N^{(2)}\rightarrow N^{\prime}$ be projection onto the second factor in
the diffeomorphism $N^{(2)}\cong[-\epsilon,\epsilon]\times N^{\prime}$. Let
$\mathaccent 866{p}:M^{(2)}\rightarrow M^{\prime}$ be the analogous map on
$M^{(2)}$. Given $n^{\prime}\in\partial N^{\prime}$, put
$L_{n^{\prime}}=\\{\exp_{n^{\prime}}(t\nu_{\partial N^{\prime}}):0\leq t\leq
l_{N^{\prime}}\\}\subset N^{\prime}$ and put
$G_{n^{\prime}}=p^{-1}(L_{n^{\prime}})$. Put
$F_{\pm,n^{\prime}}^{(4)}=F_{\pm}^{(4)}\cap G_{n^{\prime}}$,
$T_{\pm,n^{\prime}}^{(4)}=T_{\pm}^{(4)}\cap G_{n^{\prime}}$ and
$A_{n^{\prime}}^{(4)}=A^{(4)}\cap G_{n^{\prime}}$. See the left-hand picture
in Figure 3, which illustrates $G_{n^{\prime}}\cap N^{(4)}$.

Figure 3.

Define $\mathaccent 866{G}_{m^{\prime}}$, $\mathaccent
866{F}_{\pm,m^{\prime}}^{(4)},\mathaccent 866{T}_{\pm,m^{\prime}}^{(4)}$ and
$\mathaccent 866{A}_{m^{\prime}}^{(4)}$ similarly for $M^{(4)}$.

We define a map $\partial f^{(4)}:\partial N^{(4)}\rightarrow\partial M^{(4)}$
as follows. Given $n^{\prime}\in\partial N^{\prime}$, we first send the curve
$T_{+,n^{\prime}}^{(4)}\cup A_{n^{\prime}}^{(4)}\cup T_{-,n^{\prime}}^{(4)}$
to $\mathaccent 866{T}_{+,(\partial
f^{\prime})(n^{\prime})}^{(4)}\cup\mathaccent 866{A}_{(\partial
f^{\prime})(n^{\prime})}^{(4)}\cup\mathaccent 866{T}_{-,(\partial
f^{\prime})(n^{\prime})}^{(4)}$ piecewise linearly with respect to arc length.
Next, given the point $x=\partial T_{+,1}^{(4)}\cap G_{n^{\prime}}$, write
$p(x)$ as $\exp_{n^{\prime}}(\tau_{n^{\prime}}\nu_{\partial N^{\prime}})$ for
some $\tau_{n^{\prime}}\in(0,l_{N^{\prime}})$. The parameter
$\tau_{n^{\prime}}$ is comparable to $\epsilon^{\prime}$. Define $\mathaccent
866{\tau}_{(\partial f^{\prime})(n^{\prime})}$ similarly for $M^{\prime}$. Let
$\lambda_{n^{\prime}}$ be the increasing linear bijection from
$[\tau_{n^{\prime}},l_{N^{\prime}}]$ to $[\mathaccent 866{\tau}_{(\partial
f^{\prime})(n^{\prime})},(1-\delta_{N^{\prime}})l_{N^{\prime}}]$. Given
$t\in[\tau_{n^{\prime}},l_{N^{\prime}}]$, let $y_{t}$ be the point in
$F_{+,n^{\prime}}^{(4)}$ with $p(y_{t})=\exp_{n^{\prime}}(t\nu_{\partial
N^{\prime}})$ and let $(\partial f^{(4)})(y_{t})$ be the point in $\mathaccent
866{F}_{+,(\partial f^{\prime})(n^{\prime})}^{(4)}$ whose image under
$\mathaccent 866{p}$ is $\exp_{(\partial
f^{\prime})(n^{\prime})}(\lambda_{n^{\prime}}(t)\nu_{\partial M^{\prime}})$.
Define $\partial f^{(4)}$ on the remaining points of $F_{+,n^{\prime}}^{(4)}$
to be the same as $\partial f^{(2)}$. Finally, define $\partial f^{(4)}$ on
$F_{-,n^{\prime}}^{(4)}$ by a similar construction.

The lengths of $T_{\pm,n^{\prime}}^{(4)}$ and $\mathaccent
866{T}_{\pm,m^{\prime}}^{(4)}$ are
$\frac{\pi}{2\epsilon^{\prime}}+O((\epsilon^{\prime})^{0})$ as
$\epsilon^{\prime}\rightarrow\infty$. The Lipschitz constant of $\partial
f^{(4)}$ is $1+O(\epsilon^{\prime})$. We can extend $\partial f^{(4)}$ to a
map $f^{(4)}:N^{(4)}\rightarrow M^{(4)}$, sending $G_{n^{\prime}}\cap N^{(4)}$
to $\mathaccent 866{G}_{(\partial f^{\prime})(n^{\prime})}\cap M^{(4)}$, whose
Lipschitz constant is $1+O(\epsilon^{\prime})$.

By tube formulas, the mean curvature on $T_{\pm}^{(4)}$ is
$\frac{1}{\epsilon^{\prime}}+O(\epsilon^{\prime})$, and similarly for the tube
regions of $\partial M^{(4)}$ [5, Theorem 9.23]. We now perturb $N^{(4)}$ to
increase the mean curvature on $T_{\pm}^{(4)}$. To do so, we effectively
borrow some of the mean curvature from $F_{\pm}^{(4)}$ and $A^{(4)}$

We do some preliminary calculations. Let
$\phi:[0,\infty)\rightarrow{\mathbb{R}}$ be a smooth nonnegative function such
that $\phi(0)=0$, $\phi^{\prime}(0)=1$ and $\phi(x)$ vanishes when $x\geq 2$.
Given constants $c,c^{\prime}>0$ so that $cl_{N^{\prime}}\gg 1$ and
$c\epsilon^{\prime}\ll 1$, and $L<\infty$, define $u_{L}\in
C^{1,1}({\mathbb{R}})$ by

(2.10) $u_{L}=\begin{cases}\frac{c^{\prime}L}{2c}\phi(-cx)&x<0\\\
\frac{c^{\prime}}{2}x\left(x-L\right)&0\leq x\leq L\\\
\frac{c^{\prime}L}{2c}\phi(c(x-L))&x>L.\end{cases}$

Then

(2.11)
$u_{L}^{\prime\prime}=\begin{cases}\frac{c^{\prime}cL}{2}\phi^{\prime\prime}(-cx)&x<0\\\
c^{\prime}&0<x<L\\\
\frac{c^{\prime}cL}{2}\phi^{\prime\prime}(c(x-L))&x>L.\end{cases}$

Let $d_{1}$ be the intrinsic distance function on $\partial N^{(4)}$ from
$\partial T_{+,1}^{(4)}$ and let $d_{2}$ be the intrinsic distance function on
$\partial N^{(4)}$ from $\partial T_{+,2}^{(4)}$. Let $\pi_{1}:\partial
N^{(4)}\rightarrow\partial T_{+,1}^{(4)}$ be nearest point projection with
respect to the intrinsic distance on $\partial N^{(4)}$, and similarly for
$\pi_{2}$. (In the application, the nearest point will be unique.) Define a
function $V_{+}\in C^{1,1}(\partial N^{(4)})$ by

(2.12) $V_{+}=\begin{cases}-\>\frac{c^{\prime}}{2}d_{1}d_{2}&\text{on
}T_{+}^{(4)}\\\
\frac{c^{\prime}}{2c}(d_{2}\circ\pi_{1})\>\phi(cd_{1})&\text{on
}F_{+}^{(4)}\\\
\frac{c^{\prime}}{2c}(d_{1}\circ\pi_{2})\>\phi(cd_{2})&\text{on
}A_{+}^{(4)}\\\ 0&\text{on }T_{-}^{(4)}\cup F_{-}^{(4)}.\end{cases}$

Define $V_{-}\in C^{1,1}(\partial N^{(4)})$ similarly, replacing $T_{+}^{(4)}$
by $T_{-}^{(4)}$. Put $V=V_{+}+V_{-}$. Deform $\partial N^{(4)}$ by distance
$V$ in the inward normal direction. See the right-hand picture in Figure 3,
where the deformation is indicated by dashed lines. Let $N^{(5)}$ be the
region bounded by the ensuing hypersurface, i.e. $\partial N^{(5)}$ is the
image of the $C^{1,1}$-diffeomorphism $D:\partial N^{(4)}\rightarrow\partial
N^{(5)}$ given by $D(x)=\exp_{x}(V(x)\nu_{\partial N^{(4)}})$. Note that the
deformation is outward on $T^{(4)}_{\pm}$, of magnitude comparable to
$c^{\prime}(\epsilon^{\prime})^{2}$, and inward on the rest of $\partial
N^{(4)}$, of magnitude comparable to $c^{\prime}c^{-1}\epsilon^{\prime}$.

The first variation formula for mean curvature is

(2.13) $H^{\prime}=\triangle V+(|A|^{2}+\operatorname{Ric}(\mu,\mu)))V.$

If $x$ denotes the length variable on a minimal arc in $T_{+}^{(4)}$ between
$q\in\partial T_{+,2}^{(4)}$ and $\pi_{1}(q)\in\partial T_{+,1}^{(4)}$ then on
$T_{+}^{(4)}$, to leading order $\triangle V\sim\frac{d^{2}}{dx^{2}}V$ and
$|A|^{2}\sim(\epsilon^{\prime})^{-2}$. From (2.11), we deduce that on
$T_{\pm}^{(4)}$ the change in $H$ roughly ranges between
$c^{\prime}\left(1-\frac{\pi^{2}}{32}\right)$ and $c^{\prime}$. On the rest of
$\partial N^{(4)}$, the change in $H$ is bounded in magnitude by
$\operatorname{const.}c^{\prime}\epsilon^{\prime}(c+c^{-1})$. Put
$M^{(5)}=M^{(4)}$.

Define subsets of $\partial N^{(5)}$ by $\mathaccent
869{F}_{\pm}^{(5)}=D({F}_{\pm}^{(4)})$, $\mathaccent
869{T}_{\pm}^{(5)}=D({T}_{\pm}^{(4)})$ and $\mathaccent
869{A}^{(5)}=D(A^{(4)})$. Define $\partial f^{(5)}:\partial
N^{(5)}\rightarrow\partial M^{(5)}$ by $\partial f^{(5)}=(\partial
f^{(4)})\circ D^{-1}$. On $\mathaccent 869{T}_{\pm}^{(5)}$, the map $\partial
f^{(5)}$ has a Lipschitz bound comparable to that of $\partial f^{(4)}$,
namely $1+O(\epsilon^{\prime})$, using the fact that the perturbation on
${T}_{\pm}^{(4)}$ is outward. (It may seem paradoxical that $\mathaccent
869{T}_{\pm}^{(5)}$ lies outside of ${T}_{\pm}^{(4)}$ but has a higher mean
curvature. One way to understand this is by looking at Figure 3 and comparing
the total turning angle of ${T}_{+,n^{\prime}}^{(4)}$ with the total turning
angle of the corresponding dotted segment on the right.) If the Lipschitz
bound of $\partial f^{(2)}$ is $1-\sigma$, where $\sigma>0$, then the
Lipschitz bound of $\partial f^{(5)}$ on the rest of $\partial N^{(5)}$ is
$1-\sigma+\operatorname{const.}c^{\prime}\epsilon^{\prime}$. In sum, $\partial
f^{(5)}$ has a Lipschitz bound that is $1+O(\epsilon^{\prime})$. We can extend
$\partial f^{(5)}$ to a map $f^{(5)}:N^{(5)}\rightarrow M^{(5)}$ which also
has a Lipschitz bound that is $1+O(\epsilon^{\prime})$. We can assume that
$f^{(5)}$ maps normal geodesics to normal geodesics, in a small neighborhood
of $\partial N^{(5)}$.

On $\mathaccent 869{T}_{\pm,n^{\prime}}^{(5)}$, the ratio $\frac{H_{\partial
N^{(5)}}}{(\partial f^{(5)})^{\star}H_{\partial M^{(5)}}}$ is bounded below by
$(1+\operatorname{const.}\epsilon^{\prime}c^{\prime})$. If
$c^{\prime}\epsilon^{\prime}(c+c^{-1})\ll 1$ then on the rest of $\partial
N^{(6)}$, there is a uniform lower bound for the ratio that is greater than
one, coming from $\partial N^{(2)}$. Hence by taking $c^{\prime}$ sufficiently
large and then $\epsilon^{\prime}$ sufficiently small, we can ensure that the
Lipschitz constant of $f^{(5)}$ is strictly less than the minimum of
$\frac{H_{\partial N^{(5)}}}{(\partial f^{(5)})^{\star}H_{\partial M^{(5)}}}$.

Put $N^{(6)}=N^{(5)}$. Taking normal coordinates around $f^{(5)}(z)\in
M^{(5)}$, let $M^{(6)}$ be the result of a slight radial shrinking of
$M^{(5)}$. If $f^{(6)}:N^{(6)}\rightarrow M^{(6)}$ is the composite map then
we can ensure that $f^{(6)}$ and $\partial f^{(6)}$ are distance-decreasing,
while $H_{\partial N^{(6)}}>(\partial f^{(6)})^{\star}H_{\partial M^{(6)}}$.

We run the mean curvature flow on $\partial N^{(6)}$ and $\partial M^{(6)}$
for a time $\tau\ll(\epsilon^{\prime})^{2}$ (c.f. [3]) to obtain smooth
hypersurfaces $\partial N$ and $\partial M$, and hence $N$ and $M$. As the
mean curvature obeys a diffusion-type equation under mean curvature flow, the
main effect on the mean curvature at a point will be to average the mean
curvature with respect to a Gaussian centered at the point with scale on the
order of $\sqrt{\tau}$. In particular, if $\tau$ is small enough then for the
map $\partial f:\partial N\rightarrow\partial M$, obtained from $f^{(6)}$ by
following the flows, the inequalities will be preserved. We can then extend it
to a smooth distance-decreasing map $f:N\rightarrow M$. By construction, after
choosing orientations on $N$ and $M$ (which are diffeomorphic to balls), the
degree is nonzero. Finally, after a small perturbation of $f$, we can assume
that there are numbers $\delta,l>0$ so that for all $n\in\partial N$ and
$t\in[0,l)$, one has $f(\exp_{n}t\nu_{\partial
N})=\exp_{f(n)}((1-\delta)t\nu_{\partial M})$.

If $K<0$ then for sufficiently small $\tau$, the preceding steps preserve the
strict convexity of $M$. If $K=0$ then they preserve the convexity of $M$ and
we can slightly perturb $M$ at the end, for example by the mean curvature
flow, to make it strictly convex.

##  3\. Proof of Theorem 1.3

######  Lemma 3.1.

There is a constant $C^{\prime}=C^{\prime}(n,A)<\infty$ so that if $0<s\leq t$
then

(3.2) $d_{s}-C^{\prime}\left(\sqrt{t}-\sqrt{s}\right)\leq d_{t}\leq
e^{E(s-t)}d_{s}.$

###### Proof.

This follows from distance distortion estimates for Ricci flow, as in [7,
Remark 27.5 and Corollary 27.16]. ∎

######  Corollary 3.3.

The diameter of $(M,g(t))$ is uniformly bounded above in $t$.

######  Lemma 3.4.

There are some $v_{0},A^{\prime}>0$ so that for all $(x,t)\in M\times(0,T]$,

  1. (1)

$\operatorname{vol}_{g(t)}\left(B_{g(t)}(x,1)\right)\geq v_{0}$, and

  2. (2)

$\operatorname{vol}_{g(t)}\left(B_{g(t)}(x,\sqrt{t})\right)>A^{\prime}t^{n/2}$

###### Proof.

From the evolution of volume under Ricci flow,

(3.5)
$\frac{d}{dt}\operatorname{Vol}(M,g(t))=-\int_{M}R_{g(t)}\>\operatorname{dvol}_{g(t)}\leq-
nE\operatorname{Vol}(M,g(t)).$

It follows that $\operatorname{Vol}(M,g(t))\geq
e^{nE(T-t)}\operatorname{Vol}(M,g(T))$. The lower Ricci curvature bound, the
diameter bound from Corollary 3.3 and Bishop-Gromov comparison now give
numbers $v_{0},A^{\prime}$ as in the statement of the lemma. ∎

######  Lemma 3.6.

Let $F$ be a solution of the backward heat equation

(3.7) $\partial_{t}F=-\triangle F.$

Then $\max|F|$ and $e^{-2Et}\max|\nabla F|$ are nondecreasing in $t$.

###### Proof.

From the maximum principle, $\max|F|$ is nondecreasing in $t$. Next, we have

(3.8) $\partial_{t}|\nabla F|^{2}=2\operatorname{Ric}(\nabla F,\nabla
F)-2\langle\nabla F,\nabla\triangle F\rangle$

and

(3.9) $\triangle|\nabla F|^{2}=2\langle\nabla F,\nabla\triangle
F\rangle+2|\operatorname{Hess}F|^{2}+2\operatorname{Ric}(\nabla F,\nabla F).$

Hence

(3.10) $(\partial_{t}+\triangle)|\nabla F|^{2}\geq 4\operatorname{Ric}(\nabla
F,\nabla F)\geq 4E|\nabla F|^{2},$

or

(3.11) $(\partial_{t}+\triangle)\left(e^{-4Et}|\nabla F|^{2}\right)\geq 0.$

By the maximum principle, $e^{-4Et}\max|\nabla F|^{2}$ is nondecreasing in
$t$. This proves the lemma. ∎

######  Lemma 3.12.

Given $f\in C^{\infty}(M)$, there is a function
$\alpha:(0,T]\rightarrow{\mathbb{R}}^{+}$ with $\lim_{t\rightarrow
0}\alpha(t)=0$ having the following property. Given $\mathaccent
866{t}\in(0,T]$, let $F$ be the solution to (3.7) on $(0,\mathaccent 866{t}]$
with $F(\mathaccent 866{t})=f$. If $s\in(0,\mathaccent 866{t}/2]$ then
$\|F(s)-f\|_{\infty}\leq\alpha(\mathaccent 866{t})$.

###### Proof.

Let $G(x,t;y,s)$, $0<s<t$, be the Green’s function for (3.7), meaning that for
fixed $(x,t)$, the function $G(x,t;\cdot,\cdot)$ satisfies

(3.13) $(\partial_{s}+\triangle_{y,s})G(x,t;\cdot,\cdot)=0,$

and $\lim_{s\rightarrow t}G(x,t;y,s)=\delta_{x}(y)$. Then $G$ is positive and
for given $(y,s)$, one has

(3.14) $\int_{M}G(x,t;y,s)\>\operatorname{dvol}_{g(t)}(x)=1.$

Also,

(3.15) $F(y,s)=\int_{M}G(x,\mathaccent
866{t};y,s)\>f(x)\>\operatorname{dvol}_{g(\mathaccent 866{t})}(x).$

By [2, Proposition 3.1], there is a constant $C=C(n,A)<\infty$ so that

(3.16) $G(x,t;y,s)<Ct^{-\>\frac{n}{2}}e^{-\>\frac{d_{s}^{2}(x,y)}{Ct}}$

whenever $s\leq\frac{t}{2}$.

Given $L<\infty$, we have

(3.17) $\displaystyle F(y,s)-f(y)=$ $\displaystyle\int_{M}G(x,\mathaccent
866{t};y,s)\>(f(x)-f(y))\>\operatorname{dvol}_{g(\mathaccent 866{t})}(x)$
$\displaystyle=$ $\displaystyle\int_{B_{g(\mathaccent
866{t})}(y,L\sqrt{t})}G(x,\mathaccent
866{t};y,s)\>(f(x)-f(y))\>\operatorname{dvol}_{g(\mathaccent 866{t})}(x)+$
$\displaystyle\int_{M-B_{g(\mathaccent 866{t})}(y,L\sqrt{t})}G(x,\mathaccent
866{t};y,s)\>(f(x)-f(y))\>\operatorname{dvol}_{g(\mathaccent 866{t})}(x),$

so

(3.18) $\displaystyle|F(y,s)-f(y)|\leq$ $\displaystyle\max_{x\in
B_{g(\mathaccent 866{t})}(y,L\sqrt{t})}|f(x)-f(y)|+$ $\displaystyle
2(\max|f|)\int_{M-B_{g(\mathaccent 866{t})}(y,L\sqrt{t})}C\mathaccent
866{t}^{-\>\frac{n}{2}}e^{-\>\frac{d_{s}^{2}(x,y)}{C\mathaccent
866{t}}}\>\operatorname{dvol}_{g(\mathaccent 866{t})}(x).$

From Lemma 3.6,

(3.19) $\max_{x\in B_{g(\mathaccent 866{t})}(y,L\sqrt{\mathaccent
866{t}})}|f(x)-f(y)|\leq L\sqrt{\mathaccent 866{t}}\max|\nabla
f|_{g(\mathaccent 866{t})}\leq L\sqrt{\mathaccent 866{t}}e^{2E(\mathaccent
866{t}-T)}\max|\nabla f|_{g(T)}.$

From (3.2), we have $d_{s}^{2}(x,y)\geq e^{2E(\mathaccent
866{t}-s)}d_{\mathaccent 866{t}}^{2}(x,y)\geq e^{-2|E|\mathaccent
866{t}}d_{\mathaccent 866{t}}^{2}(x,y)$. Then Bishop-Gromov comparison gives

(3.20) $\displaystyle\int_{M-B_{g(\mathaccent
866{t})}(y,L\sqrt{t})}C\mathaccent
866{t}^{-\>\frac{n}{2}}e^{-\>\frac{d_{s}^{2}(x,y)}{C\mathaccent
866{t}}}\>\operatorname{dvol}_{g(\mathaccent 866{t})}(x)\leq$ $\displaystyle
C\mathaccent
866{t}^{-\>\frac{n}{2}}\operatorname{vol}(S^{n-1})\int_{L\sqrt{t}}^{\infty}e^{-\>\frac{e^{-2|E|\mathaccent
866{t}}r^{2}}{C\mathaccent
866{t}}}\left(\frac{1}{\sqrt{|E|}}\sinh(r\sqrt{|E|})\right)^{n-1}\>dr=$
$\displaystyle
C\operatorname{vol}(S^{n-1})\int_{L}^{\infty}e^{-\>\frac{e^{-2|E|\mathaccent
866{t}}u^{2}}{C}}\left(\frac{1}{\sqrt{|E|\mathaccent
866{t}}}\sinh(u\sqrt{|E|\mathaccent 866{t}}\right)^{n-1}\>du\leq$
$\displaystyle
C\operatorname{vol}(S^{n-1})\int_{L}^{\infty}e^{-\>\frac{u^{2}}{2C}}\left(\sinh(u)\right)^{n-1}\>du$

for $\mathaccent 866{t}$ small. Taking $L=\mathaccent
866{t}^{-\>\frac{1}{4}}$, the lemma follows from (3.18), (3.19) and (3.20). ∎

######  Lemma 3.21.

If $F$ is a solution of (3.7) then

(3.22)
$\frac{d}{dt}\int_{M}FR\>\operatorname{dvol}=-\int_{M}F\left(R^{2}-2|\operatorname{Ric}|^{2}\right)\>\operatorname{dvol}.$

###### Proof.

We have

(3.23) $\displaystyle\frac{d}{dt}\int_{M}FR\>\operatorname{dvol}=$
$\displaystyle\int_{M}\left(\frac{\partial F}{\partial
t}R\operatorname{dvol}+F\frac{\partial R}{\partial
t}\operatorname{dvol}+FR\frac{d\operatorname{dvol}}{dt}\right)$
$\displaystyle=$ $\displaystyle\int_{M}\left(-(\triangle
F)R\operatorname{dvol}+F(\triangle
R+2|\operatorname{Ric}|^{2})\operatorname{dvol}-FR^{2}\operatorname{dvol}\right)$
$\displaystyle=$
$\displaystyle-\int_{M}F\left(R^{2}-2|\operatorname{Ric}|^{2}\right)\>\operatorname{dvol}.$

This proves the lemma. ∎

######  Lemma 3.24.

There is a $C^{\prime\prime}<\infty$ such that $\|R_{g(t)}\|_{L^{1}}\leq
C^{\prime\prime}$ for all $t\in(0,T]$.

###### Proof.

Taking $F=1$ in (3.22) gives

(3.25) $\int_{M}R_{g(T)}\operatorname{dvol}_{g(T)}-\int_{M}R_{g(\mathaccent
869{t})}\operatorname{dvol}_{g(\mathaccent 869{t})}=-\int_{\mathaccent
869{t}}^{T}\int_{M}\left(R^{2}-2|\operatorname{Ric}|^{2}\right)(x,t)\operatorname{dvol}_{g(t)}(x)dt.$

As $\operatorname{Ric}_{g(\mathaccent 869{t})}\geq Eg(\mathaccent 869{t})$, it
follows that $R_{g(\mathaccent 869{t})}\geq nE$. The lemma now follows from
(3.25). ∎

With the hypotheses of Lemma 3.12, from (3.22) we obtain

(3.26) $\int_{M}fR_{g(\mathaccent 866{t})}\operatorname{dvol}_{g(\mathaccent
866{t})}-\int_{M}F(s)R_{g(s)}\operatorname{dvol}_{g(s)}=-\int_{s}^{\mathaccent
866{t}}F(t)\left(R^{2}-2|\operatorname{Ric}|^{2}\right)\operatorname{dvol}_{g(t)}dt.$

Now

(3.27) $\displaystyle\left|\int_{M}fR_{g(\mathaccent
866{t})}\operatorname{dvol}_{g(\mathaccent
866{t})}-\int_{M}fR_{g(s)}\operatorname{dvol}_{g(s)}\right|\leq$
$\displaystyle\left|\int_{M}fR_{g(\mathaccent
866{t})}\operatorname{dvol}_{g(\mathaccent
866{t})}-\int_{M}F(s)R_{g(s)}\operatorname{dvol}_{g(s)}\right|+$
$\displaystyle\left|\int_{M}(F(s)-f)R_{g(s)}\operatorname{dvol}_{g(s)}\right|.$

From Lemma 3.6 and (3.26),

(3.28) $\left|\int_{M}fR_{g(\mathaccent
866{t})}\operatorname{dvol}_{g(\mathaccent
866{t})}-\int_{M}F(s)R_{g(s)}\operatorname{dvol}_{g(s)}\right|\leq(\max|f|)\int_{0}^{\mathaccent
866{t}}\left|R^{2}-2|\operatorname{Ric}|^{2}\right|\operatorname{dvol}_{g(t)}dt.$

From Lemmas 3.12 and 3.24,

(3.29) $\left|\int_{M}(F(s)-f)R_{g(s)}\operatorname{dvol}_{g(s)}\right|\leq
C^{\prime\prime}\alpha(\mathaccent 866{t}).$

Hence

(3.30) $\displaystyle\left|\int_{M}fR_{g(\mathaccent
866{t})}\operatorname{dvol}_{g(\mathaccent
866{t})}-\int_{M}fR_{g(s)}\operatorname{dvol}_{g(s)}\right|\leq$
$\displaystyle(\max|f|)\int_{0}^{\mathaccent
866{t}}\left|R^{2}-2|\operatorname{Ric}|^{2}\right|\operatorname{dvol}_{g(t)}dt+C^{\prime\prime}\alpha(\mathaccent
866{t}).$

From (3.30), the sequence
$\\{\int_{M}fR_{g(2^{-j}T)}\operatorname{dvol}_{g(2^{-j}T)}\\}_{j=0}^{\infty}$
is a Cauchy sequence and so has a limit $M_{f}\in{\mathbb{R}}$. Then from
(3.30), $\lim_{t\rightarrow
0}\int_{M}fR_{g(t)}\operatorname{dvol}_{g(t)}=M_{f}$.

Given $f,f^{\prime}\in C^{\infty}(M)$, we have

(3.31)
$\left|\int_{M}fR_{g(t)}\operatorname{dvol}_{g(t)}-\int_{M}f^{\prime}R_{g(t)}\operatorname{dvol}_{g(t)}\right|\leq
C^{\prime\prime}\|f-f^{\prime}\|_{\infty}.$

It follows that the map $f\rightarrow M_{f}$ extends to a bounded linear
function on $C(M)$, and so defines a Borel measure $\mu_{0}$ on $M$.

This proves Theorem 1.3.

######  Example 3.32.

In dimension two, $R^{2}-2|\operatorname{Ric}|^{2}=0$. Let $\Sigma$ be a
compact boundaryless two dimensional Alexandrov space. It is known that there
is a Ricci flow solution $(M,g(t))$, defined for an interval $(0,T]$, that
satisfies the assumptions of Theorem 1.3 and for which $\lim_{t\rightarrow
0}(M,g(t))\stackrel{{\scriptstyle GH}}{{=}}\Sigma$ [14]. Theorem 1.3
reproduces the canonical curvature measure on $\Sigma$, as defined in [13].

######  Remark 3.33.

The scalar curvature measure $\mu_{0}$ is defined using the Ricci flow. We do
not know if it just depends on the Gromov-Hausdorff limit $\lim_{t\rightarrow
0}(M,g(t))$. There are examples of distinct Ricci flows coming out of a cone
[1]; however those Ricci flows do not have a lower bound on the Ricci
curvature.

If the time slices $(M,g(t))$ have nonnegative curvature operator then by the
uniqueness result of [9], the scalar curvature measure $\mu_{0}$ agrees with
the measure constructed there, and hence only depends on the limit space.

##  4\. Proof of Theorem 1.4

Note that if $(M,g)$ has $2$-nonnegative curvature operator then it has
nonnegative Ricci curvature.

The next lemma is probably well known; we give the direct proof.

######  Lemma 4.1.

Let $H$ be a finite dimensional real inner product space and let $S$ be a
symmetric operator on $H$. Let $\lambda_{1}\leq\lambda_{2}\leq\ldots$ be the
eigenvalues of $S$, listed with multiplicity. If $J$ is a $j$-dimensional
subspace of $H$, let $P_{J}$ be orthogonal projection onto $J$. Then

(4.2)
$\operatorname{Tr}\left(P_{J}SP_{J}\right)\geq\sum_{i=1}^{j}\lambda_{i}.$

###### Proof.

By continuity and the compactness of the Grassmannian of $j$-planes in $H$,
there is some $J$ that minimizes the left-hand side of (4.2). Suppose that $J$
is a minimizer. If $U\in O(H)$ then $P_{UJ}=UP_{J}U^{-1}$, so
$\operatorname{Tr}\left(SP_{J}\right)\leq\operatorname{Tr}\left(SUP_{J}U^{-1}\right)$.
Considering one-parameter subgroups of $O(H)$, it follows that

(4.3)
$0=\operatorname{Tr}\left(S[\eta,P_{J}]\right)=\operatorname{Tr}\left(\eta[P_{J},S]\right)$

for all skew-symmetric $\eta$. As $[P_{J},S]$ is skew-symmetric, it must
vanish. If $H=\bigoplus_{k}H_{k}$ is the spectral decomposition of $S$ into
eigenspaces of distinct eigenvalue then we must have $J=\bigoplus_{k}W_{k}$,
where $W_{k}\subset H_{k}$. Considering $J$’s just of this form and minimizing
$\operatorname{Tr}\left(P_{J}SP_{J}\right)$, the lemma follows. ∎

######  Lemma 4.4.

Suppose that

  1. (1)

$n=3$ and $(M,g)$ has nonnegative sectional curvature, or

  2. (2)

$n\geq 4$ and $(M,g)$ has 2-nonnegative curvature operator.

Then $R^{2}-2|\operatorname{Ric}|^{2}\geq 0$

###### Proof.

We have

(4.5)
$R^{2}-2|\operatorname{Ric}|^{2}=\operatorname{Tr}\left(\operatorname{Ric}(Rg-2\operatorname{Ric})\right).$

Let $\\{e_{i}\\}$ be an orthonormal basis of $T_{m}M$ that diagonalizes
$\operatorname{Ric}$. Then

(4.6)
$R^{2}-2|\operatorname{Ric}|^{2}=\sum_{i}R_{ii}(Rg-2\operatorname{Ric})_{ii}.$

Now

(4.7)
$(Rg-2\operatorname{Ric})_{ii}=R-2\operatorname{Ric}_{ii}=\sum_{j,k}R_{jkjk}-2\sum_{j}R_{ijij}=\sum_{j,k\neq
i}R_{jkjk}.$

If $(M,g)$ has nonnegative sectional curvature then the last term in (4.7) is
clearly nonnegative. Suppose that $(M,g)$ has $2$-nonnegative curvature
operator. The last term in (4.7) is

(4.8) $\sum_{j,k\neq i}R_{jkjk}=2\sum_{\begin{subarray}{c}j,k\neq i\\\
j<k\end{subarray}}\langle e_{j}\wedge e_{k},{\mathcal{R}}(e_{j}\wedge
e_{k})\rangle.$

We apply Lemma 4.1 with $H=\Lambda^{2}(T_{m}M)$ and
$J=\operatorname{span}\\{e_{j}\wedge e_{k}\\}_{\begin{subarray}{c}j,k\neq i\\\
j<k\end{subarray}}$. We conclude that

(4.9) $\sum_{\begin{subarray}{c}j,k\neq i\\\ j<k\end{subarray}}\langle
e_{j}\wedge e_{k},{\mathcal{R}}(e_{j}\wedge
e_{k})\rangle\geq\sum_{l=1}^{\binom{n-1}{2}}\lambda_{l}.$

If $n\geq 4$ then $\binom{n-1}{2}\geq 2$ and
$\sum_{l=1}^{\binom{n-1}{2}}\lambda_{l}\geq\lambda_{1}+\lambda_{2}\geq 0$.
This proves the lemma. ∎

######  Corollary 4.10.

Let $(M,g(t))$, $t\in(0,T]$, be a Ricci flow solution on a compact
$n$-dimensional manifold $M$. Suppose that

  1. (1)

$|\operatorname{Rm}_{g(t)}|<\frac{A}{t}$,

  2. (2)

$n=3$ and each $(M,g(t))$ has nonnegative sectional curvature, or $n\geq 4$
and each $(M,g(t))$ has $2$-nonnegative curvature operator, and

  3. (3)

There is a uniform upper bound on
$\int_{M}R_{g(t)}\operatorname{dvol}_{g(t)}$.

Then there is a limit $\lim_{t\rightarrow
0}R_{g(t)}\operatorname{dvol}_{g(t)}=\mu_{0}$ in the weak-$\star$ topology.

###### Proof.

From Lemma 4.4 and (3.25), it follows that $R^{2}-2|\operatorname{Ric}|^{2}\in
L^{1}((0,T]\times M;dt\operatorname{dvol}_{g(t)})$. The corollary now follows
from Theorem 1.3. ∎

We now prove Theorem 1.4. There is a uniform existence time $[0,T]$ for the
Ricci flow solutions $(M_{i},g_{i}(t))$ with initial condition
$g_{i}(0)=g_{i}$ [2]. The flows have $2$-nonnegative curvature operator and
satisfy $|\operatorname{Rm}_{g_{i}(t)}|<\frac{A}{t}$ for some $A<\infty$. By
(3.25) and Lemma 4.4,
$\int_{M_{i}}R_{g_{i}(t)}\>\operatorname{dvol}_{g_{i}(t)}\leq\mathaccent
866{A}$. By Cheeger-Hamilton compactness, after passing to a subsequence there
are

  1. (1)

A smooth manifold $X_{\infty}$,

  2. (2)

A Ricci flow solution $(X_{\infty},g_{\infty}(t))$ defined for $t\in(0,T]$ and

  3. (3)

Diffeomorphisms $\phi_{i}:X_{\infty}\rightarrow M_{i}$

so that for any $[a,b]\subset(0,T)$,
$\lim_{i\rightarrow}\phi_{i}^{*}g_{i}=g_{\infty}$ smoothly on $[a,b]\times
X_{\infty}$. Then $g_{\infty}(t)$ has $2$-nonnegative curvature operator, and
satisfies $|\operatorname{Rm}_{g_{\infty}(t)}|<\frac{A}{t}$ and
$\int_{X_{\infty}}R_{g_{\infty}(t)}\>\operatorname{dvol}_{g_{\infty}(t)}\leq\mathaccent
866{A}$, for all $t\in(0,T]$. From Lemma 3.1, there is a Gromov-Hausdorff
limit $\lim_{t\rightarrow
0}(X_{\infty},g_{\infty}(t))=(X_{\infty},d_{\infty})$. By Corollary 4.10,
there is a limit $\lim_{t\rightarrow
0}R_{g_{\infty}(t)}\operatorname{dvol}_{g_{\infty}(t)}=\mu_{0}$ in the
weak-$\star$ topology.

The proof of Theorem 1.3 can be made uniform in the underlying geometry. Hence
given $f\in C(X_{\infty})$, it follows that

(4.11)
$\displaystyle\lim_{i\rightarrow\infty}\int_{X_{\infty}}f\>\left(\phi_{i}^{-1}\right)_{*}\left(R_{g_{i}(t)}\operatorname{dvol}_{g_{i}(t)}\right)=$
$\displaystyle\lim_{i\rightarrow\infty}\int_{X_{\infty}}f\>R_{\phi_{i}^{*}g_{i}(t)}\operatorname{dvol}_{\phi_{i}^{*}g_{i}(t)}=\int_{X_{\infty}}f\>R_{g_{\infty}(t)}\operatorname{dvol}_{g_{\infty}(t)},$

uniformly in $i$ and $t\in(0,T]$. Thus

(4.12)
$\lim_{i\rightarrow\infty}\int_{X_{\infty}}f\>\left(\phi_{i}^{-1}\right)_{*}\left(R_{g_{i}}\operatorname{dvol}_{g_{i}}\right)=\int_{X_{\infty}}f\>d\mu_{0},$

which means that
$\lim_{i\rightarrow\infty}\left(\phi_{i}^{-1}\right)_{*}\left(R_{g_{i}}\operatorname{dvol}_{g_{i}}\right)\stackrel{{\scriptstyle
weak-\star}}{{=}}\mu_{0}$.

######  Remark 4.13.

There is a conjecture that for any $n\in{\mathbb{Z}}^{+}$ and $v>0$, there is
some $\mathaccent 866{A}=\mathaccent 866{A}(n,v)<\infty$ so that if $(M,g)$ is
a complete $n$-dimensional Riemannian manifold with
$\operatorname{Ric}\geq-(n-1)g$, and $B(m,1)$ is a unit ball in $M$ with
$\operatorname{vol}(B(m,1))\geq v$, then
$\int_{B(m,1)}R\>\operatorname{dvol}_{g}\leq\mathaccent 866{A}$. This
conjecture is known to be true if $M$ is a polarized Kähler manifold [10,
Proposition 1.7] or if $M$ has sectional curvature bounded below by $-1$ [12].
If the conjecture holds then condition (4) in Theorem 1.4 follows from the
first three conditions.

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Addendum to “On scalar curvature lower bounds and scalar curvature measure”
about (3.19)

From Lemma 3.1, if $d_{\mathaccent 866{t}}(x,y)<L\sqrt{\mathaccent 866{t}}$
then $d_{T}(x,y)<L\sqrt{\mathaccent 866{t}}e^{E(\mathaccent 866{t}-T)}$. As

$|f(x)-f(y)|\leq d_{T}(x,y)\max|\nabla f|_{g(T)},$

we obtain

$\max_{x\in B_{g(\mathaccent 866{t})}(y,L\sqrt{\mathaccent
866{t}})}|f(x)-f(y)|\leq L\sqrt{\mathaccent 866{t}}e^{E(\mathaccent
866{t}-T)}\max|\nabla f|_{g(T)}.$

The rest of the proof follows as in the paper.


figure table

# High-angular resolution and high-contrast VLTI observations from Y to L band
with the Asgard instrumental suite

###### Abstract

The Very Large Telescope Interferometer is one of the most proficient
observatories in the world for high angular resolution. Since its first
observations, it has hosted several interferometric instruments operating in
various bandwidths in the infrared. As a result, the VLTI has yielded
countless discoveries and technological breakthroughs. Here, we introduce a
new concept for the VLTI, Asgard: an instrumental suite comprised of four
natively collaborating instruments: BIFROST, a combiner whose main science
case is studying the formation processes and properties of stellar and
planetary systems; NOTT, a nulling interferometer dedicated to imaging young
nearby planetary systems in the L band; HEIMDALLR, an all-in-one instrument
performing both fringe tracking and stellar interferometry with the same
optics; Baldr, a Strehl optimiser. These instruments share common goals and
technologies. The goals are diverse astrophysical cases such as the study of
the formation and evolution processes of binary systems, exoplanetary systems
and protoplanetary disks, the characterization of orbital parameters and spin-
orbit alignment of multiple systems, the characterization of the exoplanets,
and the study of exozodiacal disks. Thus, the idea of this suite is to make
the instruments interoperable and complementary to deliver unprecedented
sensitivity and accuracy from the J to M bands to meet these goals. The
interoperability of the Asgard instruments and their integration in the VLTI
are major challenges for this project.

Marc-Antoine<EMAIL_ADDRESS>Denis Defrèrea,
Michael Irelandb, Stefan Krausc, Frantz Martinached,
Peter Tuthille, Azzurra Bigiolia, Julia Bryante, Sorabh Chhabrac, Benjamin
Courtney-Barrerb,
Fred Crouse, Nick Cvetojevicd, Colin Dandumontf, Germain Garreaua, Tiphaine
Lagadecb,
Romain Laugiera, Daniel Mortimerc, Barnaby Norrise, Gordon Robertsone, Adam
Tarase

aInstitute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium

bThe Australian National University, Canberra, Australia

cUniversity of Exeter, School of Physics and Astronomy, Stocker Road, Exeter,
United Kingdom

dLaboratoire Lagrange, Observatoire de la Côte d’Azur, Nice, France

eSydney Institute for Astronomy, School of Physics, University of Sydney, NSW
2006, Australia

fSpace sciences, Technologies & Astrophysics Research (STAR) Institute,
University of Liège, Liège, Belgium

_Keywords_ integrated-optics, wavefront control, infrared, high contrast
imaging, high angular resolution, optical fibers, long baseline
interferometry, exoplanets, AGN, protoplanetary disk, spectroscopy

##  1 Introduction

The emphatic triumph of the Very Large Telescope Interferometer (VLTI) and its
second-generation instruments in delivering unique science has set European
astronomy apart, motivating the ongoing facility upgrade within the Gravity+
framework (Eisenhauer, 2019) promising still further ground-breaking
scientific discoveries. In parallel, major technology and scientific
milestones have been achieved on other interferometric facilities, such as the
Center for High Angular Resolution Astronomy Array (CHARA) and the Large
Binocular Telescope Interferometer (LBTI). New ideas and laboratory
demonstrations have also emerged in recent years opening and enabling the path
to novel scientific capabilities for optical interferometry. Leveraging these
recent developments, the Asgard instrument suite (Martinod et al., 2022) will
extend the scientific capabilities of the VLTI with a set of four instrument
modules: BIFROST (Kraus et al., 2022; Chhabra et al., 2022; Mortimer et al.,
2022) (Beam-combination Instrument for studying the Formation and fundamental
paRameters of Stars and planeTary systems) which is a Y/J/H-band combiner
optimized for high spectral resolution, HEIMDALLR (Ireland et al., 2018)
(High-Efficiency Multiaxial Do-it ALL Recombiner) which is a high-sensitivity
K-band fringe tracker, NOTT (Defrère et al., 2022; Dandumont et al., 2022a, b;
Garreau et al., 2022; Laugier et al., 2022b, a; Sanny et al., 2022) (Nulling
Observations of dusT and planeTs), formerly known as Hi-5 (Defrère et al.,
2018), which is an L-band nuller optimized for high-contrast observations, and
Baldr which is an H-band injection optimizer for BIFROST and NOTT.

This paper presents the instruments, capabilities and key scientific
objectives of Asgard. The first step in bringing the Asgard suite to the VLTI
is to propose it as a visitor instrument, serving as a critical new platform
for test and launch of future innovations to deliver benefit to a broad
community for years to come.

##  2 Science cases for Asgard

Asgard science cases mainly revolve around stellar physics, multiple systems
and exoplanetary science (Kraus et al., 2022; Defrère et al., 2022).

#### Formation process of binary systems:

BIFROST and HEIMDALLR will aim to spatially resolve and completely
characterize up to 6000 binaries among the 28 million non-single stars in the
GAIA DR3 catalog (Kraus et al., 2022). Both instruments will measure
visibilities and closure phases. They will derive different information about
multiple systems from these observables. They will find the statistics of
orbital parameters by solving the degeneracy of the GAIA measurement of the
photocenter. Combined with statistics coming from radial velocity observations
and direct imaging, this data will help to constrain the formation process of
binary systems. These instruments will also allow the derivation of the
dynamical masses, providing an input for the refinement of evolutionary models
that are the cornerstone of modern astrophysics, as these models currently
face uncertainties, for instance the treatment of mass-loss in massive stars
(Feiden and Chaboyer, 2012; Mann et al., 2015). The uncertainties are
particularly notable in the pre-main-sequence phase, where predicted and
measured masses currently differ by $\geq 10\%$ (Stassun et al., 2014).
HEIMDALLR and BIFROST will measure the binary flux ratio in the K-band and
Y/J-band respectively, where BIFROST’s operating wavelengths are close to the
GAIA band (0.4-0.9 $\mu$m). This will allow deriving dynamical masses with a
precision of $\sim 3\%$. The mass constraints will also give access to the
ages of the stars, which is important for Galactic Archaeology studies.
Finally, BIFROST will give information about the processes involved in the
formation and dynamical evolution of binaries and planetary systems by
measuring the spin-orbit alignment of the binaries and planet-host stars. This
is enabled through BIFROST’s very high-spectral resolution (VHR) mode with
R=25,000 ($\Delta v\sim 12$ km/s), which allows measuring the stellar spin
orientation in moderate/slow rotating stars. The statistics obtained from
these observations will enable tests of theories concerning the origin of this
obliquity, such as multiple modes of formation, the Kozai-Lidov mechanism or
the flyby of neighboring stars.

#### Mass accretion and ejection:

BIFROST will also have a particular interest in studying the process of mass
accretion and ejection from Young Stellar Objects (YSO) and Active Galactic
Nuclei (AGN). To do so, the instrument will perform differential visibility
measurements in the Paschen lines and the He I 1.083 $\mu$m line; these are
stronger markers than Br$\gamma$ lines in the K band (Nicholls et al., 2017).
Thus, it will gather critical information on accretion disks, wind-wind
collisions, disk winds and relativistic jets from micro-quasars. In addition,
by spatially and spectrally resolving multiple transitions of hydrogen
recombination lines, BIFROST can constrain the physical conditions (gas
density, temperature, excitation) and the velocity field at different regions
in the circumstellar environment. This will enable new approaches for
reconstruction of the 3-dimensional velocity field in the inner regions of
protoplanetary disks, where accretion occurs and winds are launched from the
star, the disk, or the interaction region between the stellar and disk
magnetic fields.

#### Formation and evolution of exoplanetary systems and exoplanet
atmospheres:

Although the GAIA mission provides useful information about the masses of
giant gas exoplanets and the ages of the systems, it only brings a partial
knowledge about the formation and evolution of exoplanetary systems. NOTT will
fill the gap by measuring the size and the effective temperatures of the
exoplanets in order to better constrain their planet-formation models (Wallace
and Ireland, 2019).

NOTT’s spectroscopic capability will enable us to trace the thermal emission
from these exoplanets, opacities of absorbers (molecular bands, dust, clouds)
and BIFROST will be able to probe the the exoplanet atmospheres using its off-
axis mode, in particular those of Jupiter-like exoplanets. They will give
access to critical information about the non-equilibrium chemistry of their
atmosphere, their vertical temperature profiles, atmospheric dynamics,
rotation rates and formation processes.

#### Protoplanetary and circumplanetary disks:

Protoplanetary disks are structured with the presence of rings, gaps, spirals
and large cavities that provide clues about the formation process. The
observed cavities are often large ($>20$ AU) and are observed in 10% of the
disks (van der Marel et al., 2018) and many of them show asymmetric features
like spirals or vortices (van der Marel et al., 2021) that are linked to the
presence of companions carving the cavities in the disk (Norfolk et al.,
2021). NOTT observations will provide unprecendented inputs for hydrodynamical
models to understand planet-disk interaction and constrain planet formation
and evolution in the very early stages. The high-contrast nulling mode of NOTT
combined with the long VLTI baselines will provide the dynamic range and
angular resolution required to observe the cavities of transition disks imaged
at shorter wavelengths by current direct imaging instruments such as SPHERE,
and currently inaccessible to other L-band (single-dish) exoplanet imagers.

BIFROST’s high-resolution spectroscopic capability will probe the
circumplanetary disks where exoplanets are formed. The instrument will
investigate the Pa$\gamma$ and Pa$\beta$ lines in the circumplanetary disk,
thereby revealing the kinematics in the circumplanetary disk and the accretion
onto the forming planets. BIFROST offers up to 6 times higher spectral
resolution than GRAVITY and opens a spectral window of particular interest for
line studies.

#### Exozodiacal dust

Exozodiacal dust is a warm and hot dust in the inner regions of main-sequence
planetary systems, including the Habitable Zone (HZ). It presents both a
scientific interest and an obstacle for imaging a future exo-Earth. Indeed,
the light from HZ dust adds photon noise and causes confusion in the
discrimination of disk structure (Defrère et al., 2012). Exozodiacal dust has
so far only been observed in thermal emission due to the lower dust-to-star
contrast compared to scattered light, but the dust’s excess emission is still
too faint to detect photometrically. Thus, it needs to be spatially resolved
from the star, and the small angular scales and high contrasts involved (1 AU
at 10 parsec corresponds to 0.1”) require the use of precision interferometry.
Particularly, nulling interferometry on the VLTI with NOTT will provide
tenfold better sensitivity to extended circumstellar emission than the stellar
optical long-baseline interferometers like PIONIER (Le Bouquin et al., 2011)
and FLUOR (Coudé du Foresto et al., 2003; Scott et al., 2013). NOTT will
enable image reconstruction and thus detailed study of the origin of the dust
through its distribution, in addition to the previously mentioned structures
showing planet-disk interaction.

##  3 Instrument Capabilities

Asgard will extend the scientific capabilities of the VLTI in many ways:
shorter wavelengths combined with high-spectral resolution, high-sensitivity
and high-performance fringe tracking, and high-contrast direct imaging with
the first VLTI nulling instrument. To do so, Asgard relies on the following
four instruments:

  * •

BIFROST will offer dramatic benefits compared to the 2nd-generation VLTI
instruments, namely a better angular resolution (due to shorter wavelengths),
a higher continuum flux for blue sources (as we observe closer to the peak
wavelength of the photosphere), around 6 times higher spectral resolution
(R=25,000), and access to new line tracers, including Pa$\beta$ 1.282 $\mu$m,
Pa$\gamma$ 1.094 $\mu$m, the He I 1.083 $\mu$m accretion-tracing line (Fischer
et al., 2008), and forbidden lines (e.g. [Fe II] 1.257 $\mu$m). To do so,
BIFROST will have two arms: one with a low resolution (LR) spectrograph and
the other one with a high resolution (HR) spectrograph (Kraus et al., 2022).

  * •

HEIMDALLR is a dual-band and multiaxial beam combiner, where beams are placed
in a two-dimensional array in a non-redundant layout, then combined in the
manner analogous to an aperture-masking interferometer. This innovation
delivers: low order wavefront control by the retention of the full two-
dimensional information contained in the interferogram, high sensitivity by
the use of bulk optics, and accurate closure-phase using the proven concept of
aperture masking interferometry.

  * •

Baldr uses the same detector as HEIMDALLR, and provides a Zernike Wavefront
Sensor or a photonic lantern to augment NAOMI (in number of modes and speed)
or GPAO (Gravity+ AO, in speed for low-order modes). This system determines
the practical magnitude limit for BIFROST as a secondary adaptive optics
system is needed to increase short-wavelength Strehl. It can operate in either
J or H bands, leaving the rest of Y,J,H for BIFROST science.

  * •

NOTT will be the first nulling instrument at the VLTI, the first long-baseline
nuller in the Southern hemisphere, and the first operating at L band, where
young giant planets are the most luminous. Leveraging the long-baseline of the
VLTI and the ongoing Gravity+ facility upgrade, it will be able to directly
image the snow line where most giant exoplanets are located (Fulton et al.,
2021). It relies on integrated optics for beam combination in order to provide
a compact and stable design (Sanny et al., 2022). In addition, NOTT will be
able to compensate the fringe drift due to water vapor and CO2 dispersion.

The main instrumental features of the VLTI Asgard science modules are given in
Table 1 and the preliminary performances are given in Table 2.

[b]  Feature BIFROST HEIMDALLR Baldr NOTT Photometric band Y, J, H K H L
Central wavelength ($\mu$m) 1.35 2.18 1.6 3.75 Bandwidth ($\mu$m) 0.6 0.45 0.3
0.5 Spectral resolutions LR arm: R=50 HR arm MR: R=1000 (1.05-1.65 $\mu$m) HR:
R=5000 (1.05-1.65 $\mu$m) VHR: R=25,000 (around He I+Pa$\beta$ and Pa$\gamma$)
R=5 (2 channels) None R = 20 R = 400 R = 2000∗ Polarization split YES
(optional) NO - YES∗ Off-axis mode YES∗ - - - Inner Working Angle @3.75 $\mu$m
- - - 2 mas (AT), 3 mas (UT) Diameter of the field of view 155 mas (AT), 34
mas (UT) @1.35 $\mu$m 222 mas (AT), 49 mas (UT) @ 1.94 $\mu$m 183 mas (AT), 41
mas (UT) @ 1.6 $\mu$m 430 mas (AT), 94 mas (UT) @3.75 $\mu$m Interferometric
observable V2, closure phase, differential phases V2, closure phase - Null
depth, differential null depth Magnitude limit Limited by Baldr K=11.5 (ATs)
or 13.5 (UTs) H=9.6 (ATs) or 12.9 (UTs) L’=11 (ATs) or 13 (UTs)

Table 1: Main instrumental features of the VLTI Asgard modules

  * ∗

Contingent on additional funding.

[b]  Feature BIFROST HEIMDALLR Baldr NOTT Precision on $V^{2}$ 3% (LR arm) 5%
- - Precision on closure phase $0.5^{\circ}$ (LR arm) $1^{\circ}$ - -
Precision on differential phase $0.1^{\circ}$ (HR arm) - - - Residual piston
RMS (nm) - $<100$ (UT) $<50$ (AT) - $<100$ (UT) $<50$ (AT) Strehl (in J band)
- - $>50$% (AT) $>40$% (UT) - RMS intensity fluctuations (in L band) - - $<2$%
- Precision on null depth (L=4 & R=18) - - - $5\times 10^{-5}$ (UT) $3\times
10^{-4}$ (AT)

Table 2: Preliminary performance of the VLTI Asgard modules

##  4 Instrumental overview

###  4.1 Instrumental setup and components

Asgard is a suite of four instruments that will be located on the Visitor 2
table (formerly the AMBER table). It will operate from Y to L bands (1 to 4
microns). The conceptual layout of Asgard is given in Figure 1. The four 18-mm
beams from the VLTI (bottom right) are first reflected on the beam
compressors, which reduce the beam size to 12 mm using off-axis paraboloids.
The deformable mirrors are located in the pupil plane between the two sets of
mirrors of the beam compressors. The L band light is then sent to NOTT with a
dichroic while the Y-K band light is sent to the other modules. More
information on the sub-systems of each module is given below.

Figure 1: Conceptual layout of Asgard. This diagram is not to scale and the
positions of the modules are subject to change but the order of the optical
components should remain.

Asgard has two common modules for all the instruments. The first is the Asgard
Calibration and Alignment Unit (ACAU), which contains a light source for
spectral calibration and a 4-beam coherent light source for internal alignment
and cophasing. It also provides pickoffs that are flippable mirrors to select
between light from ACAU and the VLTI.

The second common module is the Asgard Common Optics (ACO) which includes the
following subsystems:

  * •

Beam compressors (ASG-BC): consist of 4 pairs of optics to reduce the beams’
size from 18 mm to 12 mm;

  * •

Deformable mirrors (ASG-DM): comprise 4 deformable mirrors (DM) actuated by
MEMS technology. They perform fringe tracking and injection control as well as
the correction of higher aberration modes to maximise injection into the
optical fibers of BIFROST and NOTT;

  * •

Dichroic (ASG-DIC): transmitting the L band to NOTT and reflecting the shorter
wavelengths to the other modules.

HEIMDALLR has the following sub-systems:

  * •

Dichroic and periscope (HEI-DIPER): the dichroic separates the Y-J-H and K
bands. The periscope sends the Y-J-H band to Baldr/BIF-INJ at the upper level.
The K band is transmitted toward HEIMDALLR’s optics;

  * •

Focus and alignment mirrors (HEI-FA): comprise 4 pairs of mirrors. They align
and cophase HEIMDALLR. The mirrors are moved by stepper motor actuators;

  * •

Pupil reconfiguration (HEI-COMB): comprising knife-edge mirrors to reconfigure
the beams into a two-dimensional pattern. The K band-beams are reconfigured
according to a 2-dimensional pattern providing 6 non-redundant baselines. The
fringe pattern is used to perform wavefront sensing and science;

  * •

Camera (HEI-DET): comprises the cooling system, its electronics and the
detector. It collects and records the fringe pattern of HEIMDALLR and the
signal from Baldr.

Baldr’s sub-systems are:

  * •

Atmospheric Dispersion Compensation (BAL-ADC): comprises 4 pairs of prisms.
These devices will compensate for the chromatic dispersion effect between the
Y and H bands, which degrades the injection efficiency and induces
differential phases. They are actuated by stepper-motorised rotary stages;

  * •

Beam splitters (BAL-DIC): reflecting the H-band to Baldr and transmitting the
J-Y band to BIFROST. An alternative beam splitter can instead be translated
into place that reflects the J band to Baldr and transmits the H band to
BIFROST;

  * •

Zernike optics (BAL-WFS): is the phase sensor. It will either be a wavefront
sensor and comprise phase masks and shutters to decompose the wavefront on a
Zernike aberration base, or photonic lanterns (Leon-Saval et al., 2013; Norris
et al., 2020, 2022);

  * •

Periscope (BAL-PER): sends the signals from BAL-ZER to the HEIMDALLR’s camera.

BIFROST’s sub-systems are:

  * •

Longitudinal dispersion compensator (BIF-LDC): each beam contains an LDC
mounted on a piezo-electric linear stage to compensate for atmospheric
longitudinal dispersion. Its location is yet to be determined: either in the
common path with HEIMDALLR or in the BIFROST path only. The choice will depend
on manufacturing quality (it needs an anti-reflection coating surface quality
at better than 1%), and the available space, as we have to be able to remove
the subsystem entirely when NOTT and HEIMDALLR are used.

  * •

Shutters (BIF-SHUT): comprises four shutters. They can individually close each
beam or all of them to perform the calibration tasks or protect the detectors
from bright light while BIFROST is not operating;

  * •

Birefringence correction (BIF-BIR): comprises LiNbO3 plates that compensate
for the birefringence of the beam trains and the guided optics. The LiNbO3
plates are set on stepper-motorised rotary stages and are adjusted to maximise
contrast on BIFROST;

  * •

OPD & TT alignment mirrors (BIF-OTT): to cophase the input beams between
BIFROST and HEIMDALLR and to optimise the light injection into BIF-INJ. A
Differential Delay Line mirror mounted on a linear stage is used to cophase
BIFROST with Asgard. A Tip-Tilt fast steering mirror is used to actively
optimise the coupling into the optical fibers;

  * •

Injection module (BIF-INJ): comprises off-axis paraboloids (OAPs) and the
fiber tips. The fiber tips receive the light to transport it to the combiner
of BIFROST through single-mode fibers. The fiber tips are fixed at the focus
of the OAPs;

  * •

Combiner (BIF-COMB): comprises the combiner and a 90/10 beam splitter that can
be moved in to record data simultaneously with BIFROST’s LR arm and HR arm.
The combiner interferes the four beams of starlight;

  * •

LR arm (BIF-LR): comprises imaging optics, a prism for low-spectral dispersion
at a fixed resolving power (R=50 to 80, precise value TBD) and a camera. Also,
it contains a Wollaston prism on a translation stage that can be moved in to
measure separate polarisation states to improve the visibility calibration.
The LR arm measures OPD drift, dispersion and fringe jumps to feed a feedback
loop on the LDC and the BIFROST DLLs. The camera is a SAPHIRA-based APD
detector system;

  * •

LR arm detector (BIF-DET1): comprises the detector, its electronics and
cooling system. It collects the low-dispersion fringe pattern and the
dispersed photometric channels;

  * •

HR arm (BIF-HR): comprises imaging optics, a filter wheel with grisms for high
spectral dispersion of the fringe pattern and a camera. It is used for
measuring wavelength-differential quantities and closure phases;

  * •

HR arm detector (BIF-DET2): comprises the detector, its electronics and
cooling system. It collects the high-dispersion fringe pattern. The camera is
an APD detector system.

NOTT’s sub-systems are:

  * •

Beam conditioner (NOTT-SHUT): comprises shutters and diaphragms. It shuts the
beams individually for calibration purposes and equalises the beam intensities
to optimise the null depth;

  * •

OPD & TT alignment mirrors (NOTT-OTT): to cophase the input beams between NOTT
and Asgard and to optimise the light injection into the photonic chip. They
are actuated by stepper motors;

  * •

Warm optics (NOTT-WO): comprises LDCs, polarisation control optics, the slicer
and alignment camera. The LDC compensates for the chromatic phase across the L
band and the water vapour effects via a feedback loop. The polarisation
control compensates for polarisation effects which reduce the instrumental
null depth. The slicer overlaps the four telescope pupils on the cold stop at
the entrance of the cryostat. The alignment camera is an Infratec ImageIR
5300;

  * •

Cryostat (NOTT-CRYO): comprises a window, a cold stop, the photonic chip
(Sanny et al., 2022) which combines the beams and creates photometric outputs,
a wheel and imaging optics, the camera, the cryocooler and the vacuum system.
The wheel contains grisms for different spectral resolutions. The chip is
mounted on a 3-axis mount for alignment and focusing with NOTT-WO;

  * •

Camera (NOTT-DET): comprises the detector and its electronics. It collects the
spectrally-dispersed signal delivered by the photonic chip. It consists of
Teledyne’s 5-micron Hawaii-2RG with SIDECAR cold electronics and Astroblank
MACIE warm electronics;

###  4.2 Opto-mechanical design

The conceptual layout (Fig. 1) has been developed into a preliminary opto-
mechanical design to verify that Asgard can fit on the VLTI optical table
(Fig. 2 and 3). The beams of the VLTI enter on the left hand side of the
Visitor 2 table and come across ACO and HEIMDALLR’s paths on the lower level.
A dichroic splits the light to send its L-band component to NOTT (gray beams)
where they encounter dispersion correctors and the injection systems that are
part of NOTT’s warm optics and are then sent to the cryostat. The remaining
light is split again to send the K-band to HEIMDALLR’s combiner and
HEIMDALLR’s detector while a periscope brings the YJH-band components to the
upper level to be dispatched between Baldr (green beam) and BIFROST (red
beams). The red beams go along the visitor table, parallel to NOTT’s beams.
They meet the delay lines before being injected into optical fibers (not
shown) that carry them to the integrated-optical chip where they are
coherently combined. The beams of the coherent calibration source ACAU (blue
beams) are created on the upper level and sent to the entrance of the table
via periscopes. This preliminary design demonstrates the possibility to fit 4
full scale instruments into a single optical table.

Figure 2: Preliminary version of the full opto-mechanical design of Asgard.
The optical paths are colored for each module: HEIMDALLR (Purple), BIFROST
(red), NOTT (gray), Baldr (green) and ACAU (Blue). Figure 3: Preliminary
version of the opto-mechanical design of Asgard (optical table only). The
optical paths are colored for each module: HEIMDALLR (Purple), BIFROST (red),
NOTT (gray), Baldr (green) and ACAU (Blue).

###  4.3 Control architecture

Figure 4: Preliminary Control architecture of Asgard.

The overall Asgard control architecture is given in Figure 4. The hardware
architecture is designed so that two terminals can operate all the modules.
This centralisation allows the observer to run all the instruments
simultaneously and to communicate with the VLTI systems. These terminals can
connect to the Observing Workstation (Observing WS) on Asgard through the ESO
network while not necessarily being ESO compliant.

Asgard IT hardware and software are non-ESO compliant, hence must be kept
separate. However, Asgard modules need to interact with ESO systems such as
the Reflective Memory Network (RMN), the delay lines and the calibration
sources. The suggested solution is to create a local network on which the IT
components of each module are connected: the _Asgard Internal Network_ called
_Yggdrasil_ 222Sacred tree which supports all the worlds in the Nordic
mythology. Asgard systems are run by the users through the Observing WS that
is accessed either directly (during in-laboratory operations) or via the
terminals (during the observations). An _ESO-link WS_ , called _Hermod_
333Messenger of the gods in the Nordic mythology, connected to _Yggdrasil_ ,
will interact with ESO systems, the RMN and transmit data to and from Asgard
modules as required. It is also connected to the _Asgard Archive_ to store
Asgard data acquired over several weeks. In a future extension, the Asgard
archive could also serve as the hub for feeding Asgard data into the ESO
archive. _Hermod_ will respect ESO’s standards for hardware and software. From
the ESO network point of view, Asgard is a unique computer represented by
_Hermod_.

Each module has one or two Local Control Units (LCU) or Programmable Logic
Controllers (PLC) to drive Asgard’s sub-systems and components including
monitor cameras, science detectors and the piezo-actuators). The LCUs are
connected to the Asgard Internal Network and are driven from the terminals.
The PLCs directly control the opto-mechanical components and are controlled by
the terminals similarly to the LCUs. Each module (except ACAU) has its own
data storage embedded in its respective LCU.

The LCUs managing the detectors have direct access to the archive to write the
frames and the user can access the storage via the _Observing WS_. The
connections between the WS and LCUs are made with Ethernet cables and the TTL
protocol.

The Instrument Software Package will be subdivided into the following software
modules:

  * •

The ICS (Instrument Control Software): to control all devices connected to the
LCU on which it is installed, except the detector. There will be one ICS per
module.

  * •

The DCS (Detector Control Software): to control the detector. There will be
one DCS per LCU connected to a detector.

  * •

The OS (Observing Software): to coordinate the acquisitions of the instrument,
interfaces with the __ESO-link WS__ , with the local storage diskspace of the
modules, and the Asgard archive.

  * •

The LS (Link Software): exchanges information and instructions between ESO and
Asgard networks.

###  4.4 Data flow

The data flow is shown in Figure 5. Asgard will use its own observing blocks.
After each observing night, the data (in FITS format) and the observing logs
are copied to the local Asgard database. The data and the logs are copied to
different places depending on the instrument (BIFROST data sent to the
University of Exeter, HEIMDALLR/Baldr sent to ANU and NOTT data sent to KUL).
The transfer would be done by physically moving the storage drives. The data
are reduced and quality controls check the quality of the final data. Reduced
data are formatted in OIFITS files in order to be published and put on the
JMMC OiDB with their observing logs.

Figure 5: Preliminary data flow plan of Asgard.

##  5 Summary

Asgard is a new instrument suite proposed to ESO (P110, March 2022) to open
new unique scientific capabilities at the VLTI (YJH high-spectral resolution
and L-band high-contrast nulling interferometry) and to enable exoplanet
imaging at milli-arcsecond angular separation. It consists of 4 different
modules: BIFROST (YJH high-spectral resolution (R=50, 1000, 5000, 25,000)),
Baldr (Zernike Wavefront Sensor in H band), HEIMDALLR (high-sensitivity fringe
tracker (dual K band)), and NOTT (L-band (3.5-4 microns) nulling
interferometer based on a photonic chip and self-calibration data reduction
techniques). The suite is currently in the design and preparation phases. The
design aims at making the instruments interoperable and making its data
integrated to the existing data flows.

## acknowledgments

M-A.M. has received funding from the European Union’s Horizon 2020 research
and innovation programme under grant agreement No. 101004719.
S.K. and S.C. acknowledge support from an ERC Consolidator Grant (“GAIA-
BIFROST”, grant agreement No. 101003096).
S.K. and D.J.M. acknowledge support from STFC Consolidated Grant
(ST/V000721/1).
SCIFY (A.B., C.D., D.D., G.G., and R.L.) has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (grant agreement CoG - 866070). We are
grateful for the kind support and constructive interactions with colleagues at
ESO, in particular Frédéric Gonte, Xavier Haubois, Antoine Mérand, Nicolas
Schuhler, and Julien Woillez.
This research was partially funded by the Australian Government through the
Australian Research Council. (LE220100126).

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