Let
N
N
be a compact convex
n
n
-dimensional Riemannian manifold with a boundary
∂
N
\partial N
having normal curvatures
⩾
κ
>
0
\geqslant \kappa > 0
. Suppose the sectional curvature
>
−
κ
2
> - {\kappa ^2}
in
N
N
. Let
H
H
be the integral mean curvature of
∂
N
\partial N
,
V
V
be the volume of
N
N
,
k
s
c
{k_{sc}}
be the scalar curvature and
k
¯
R
(
p
)
{\bar k_R}(p)
,
p
∈
N
p \in N
, be the maximum Ricci curvature at
p
p
. Then
\[
H
⩾
n
−
2
2
κ
2
V
−
1
2
(
n
−
1
)
∫
N
k
s
c
d
V
,
H
⩾
(
n
−
2
)
κ
2
V
−
1
n
−
1
∫
N
k
¯
R
d
V
.
H \geqslant \frac {{n - 2}} {2}{\kappa ^2}V - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;dV} ,\quad H \geqslant (n - 2){\kappa ^2}V - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV.}
\]
Let
N
−
{N_ - }
denote
N
N
with nonpositive sectional curvature. Let
G
G
be the integral Gauss curvature of
∂
N
−
\partial {N_ - }
. Then
G
⩾
−
κ
n
−
2
∫
N
−
k
¯
R
d
V
G \geqslant - {\kappa ^{n - 2}}\int _{N - } {{{\bar k}_R}\;dV}
. These three estimates are sharp. For a ball in
3
3
-dimensional hyperbolic space, the ratio of the right-hand part of each estimate to its left-hand part (i.e.
V
(
κ
2
+
3
)
/
2
H
V({\kappa ^2} + 3)/2H
,
V
(
κ
2
+
1
)
/
H
V({\kappa ^2} + 1)/H
and
2
κ
V
/
G
2\kappa V/G
respectively) approaches 1 as the
radius
→
∞
{\operatorname {radius}} \to \infty
. The same ratios for the estimates
\[
H
⩾
−
1
2
(
n
−
1
)
∫
N
k
s
c
d
V
and
H
⩾
−
1
n
−
1
∫
N
k
¯
R
d
V
H \geqslant - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;} dV\quad {\text {and}}\quad H \geqslant - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV}
\]
(rougher ones but without
κ
\kappa
) approach
3
4
\tfrac {3} {4}
and
1
2
\tfrac {1} {2}
respectively.