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Relative Isoperimetric Inequality for Minimal Submanifolds in a Riemannian Manifold
Journal of Inequalities and Applications volume 2010, Article number: 758623 (2010)
Abstract
Let be a domain on an
-dimensional minimal submanifold in the outside of a convex set
in
or
. The modified volume
is introduced by Choe and Gulliver (1992) and we prove a sharp modified relative isoperimetric inequality for the domain
,
, where
is the volume of the unit ball of
. For any domain
on a minimal surface in the outside convex set
in an
-dimensional Riemannian manifold, we prove a weak relative isoperimetric inequality
, where
is an upper bound of sectional curvature of the Riemannian manifold.
1. Introduction
Let be the simple closed curve of a domain
in a two-dimensional space form with constant curvature
. Then the well-known sharp isoperimetric inequality is the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ1_HTML.gif)
where equality holds if and only if is a geodesic disk (see [1]).
An immediate consequence of this inequality is that if is a closed half-space of a two-dimensional space form with constant curvature
and
is a domain in
with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ2_HTML.gif)
where equality holds if and only if is a totally geodesic half-disk with the geodesic part of its boundary contained in
. This follows the original isoperimetric inequality after extending the domain by mirror symmetry with respect to
.
Motivated by this, one arises natural questions as follows.
If is a convex set in an
-dimensional space form with constant curvature
and
is a minimal surface in the outside of
with
, does
satisfy inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ3_HTML.gif)
How about an -dimensional minimal submanifold case?
Equation (1.3) is called the relative isoperimetric inequality, is called the supporting set of
, and
is called the relative area of
. A partial result is obtained by Kim [2], when part of the boundary
of the domain
is radially connected from a point
, that is,
is a connected interval. And there are some partial results on the higher-dimensional submanifold case (see [3, 4]). In case of
, the problem remains open, even in the two-dimensional case (see [5]).
In this paper, we obtain two different type relative isoperimetric inequalities. First, using the modified volume introduced by Choe and Gulliver [6], we have a modified relative isoperimetric inequality in or
without the curvature correct term:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ4_HTML.gif)
where is the volume of a unit ball of
, and
is a domain of an
-dimensional submanifold. In Theorem 2.11, (1.4) holds for
lies on a geodesic sphere of
or
. In Theorem 2.3, (1.4) holds for
,
and
is radially connected for a point
.
Second, in Section 3 we obtain an inequality on usual volume for any minimal surface of a Riemannian manifold with sectional curvature bounded above by a constant :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ5_HTML.gif)
But we cannot find a minimal surface which satisfies the equality. That is why we call (1.5) a weak relative isoperimetric inequality.
2. Modified Relative Isoperimetric Inequalities in a Space Form
We review the modified volume in and
with constant sectional curvature
and
, respectively. Let
be a point in the
-dimensional sphere
and let
be the distance from
to
in
.
Definition 2.1 (modified volume in ).
Given that is an
-dimensional submanifold in
, the modified volume of
with center at
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ6_HTML.gif)
Embed in
with
being the north pole
. For a domain, the geometric meaning of the modified volume of
is the Euclidean volume of the orthogonal projection of
into the
counting orientation. Clearly, we have in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ7_HTML.gif)
where is the usual volume of
.
Similarly, let be a point in the
-dimensional hyperbolic space
and let
be the distance from
to
in
.
Definition 2.2 (modified volume in ).
Given that is an
-dimensional submanifold in
, the modified volume of
with center at
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ8_HTML.gif)
Embed isometrically onto the hyperboloid
in
with the Minkowski metric
such that
is the point
. Then for a domain, the modified volume equals the Euclidean volume of the projection of
onto the hyperplane
. Clearly, we have in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ9_HTML.gif)
More precisely, see Choe and Gulliver's paper (see [6]).
Theorem 2.3.
Let be a closed convex set in
. Assume that
is a compact minimal surface in the outside
such that
is orthogonal to
along
. And
is the distance from
to
and
on
. If
is radially connected from the point
, that is,
is a connected interval, then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ10_HTML.gif)
Equality holds if and only if is a totally geodesic half-disk with
being a geodesic half-circle.
If is connected, then it is trivially radially connected from
. If
has two components, then using the same argument as [2, Corollary
] we obtain the following.
Corollary 2.4.
Let be a compact minimal surface satisfying the same assumptions as in Theorem 2.3 except the radially connectedness. Then the modified relative isoperimetric inequality in Theorem 2.3 holds if
is connected or
has two components that are connected by a component
of
Before giving lemmas for proving Theorem 2.3, we define a cone. Given an -dimensional submanifold
of
or
and a point
in
or
, the
-dimensional cone
with center at
is defined by the set of all minimizing geodesics from
to a point of
.
Lemma 2.5 (see [6, Lemma 4]).
-
(a)
If
is an
-dimensional minimal submanifold or a cone, and
is the distance in
from a fixed point, then
(2.6)
where is the Laplacian on the submanifold
and, in case
is a cone,
is the distance from the center of
.
-
(b)
Suppose that
is an
-dimensional minimal submanifold or a cone. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ12_HTML.gif)
Here again, in case of a cone ,
is the distance from the center of
.
Proposition 2.6.
Let be a closed convex set in
or
. Assume that
is an m-dimensional minimal submanifold in the outside
such that
is orthogonal to
along
and
is the distance from
to
or
. Let
. In case of
, one assumes that
for all
. Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ13_HTML.gif)
Proof.
Let and
be the unit conomals to
on
and
, respectively. By Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ14_HTML.gif)
The makes the smallest angle with
, that is, the unit normal vector to
that lies in the two-dimensional plane spanned by
and the tangent line of
such that
Clearly
where
is a unit tangent to
. Since
is a convex set,
, and
, we see that
points outward of
for every
, where
is the gradient in the
. From the orthogonality condition,
is a unit normal toward inside
along the
. So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ15_HTML.gif)
for every , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ16_HTML.gif)
The similar proof holds for
Lemma 2.7 (see [6, Lemma 6]).
Let be Green's function of
(
, resp.), whose derivative is
for
(
for
, resp.). If
is an
-dimensional minimal submanifold of
(
, resp.), then
is subharmonic on
(
, resp.).
Now we estimate the angle of viewed from a point
Recall the definition of the angle viewed from a point. For an
-dimensional rectifiable set
in
and a point
such that
for all
, the
-dimensional angle
of
viewed from
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ17_HTML.gif)
where is the geodesic sphere of radius
centered at
, and the volume is measured counting multiplicity. Clearly, the angle does not depend on
. There is obviously an analogous definition for the angle of
viewed from
.
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ18_HTML.gif)
where is the
-dimensional density of
at
.
Proposition 2.8.
Let be an m-dimensional minimal submanifold satisfying the same assumptions as in Proposition 2.6. Then for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ19_HTML.gif)
Equality holds if and only if is totally geodesic and star shaped with respect to
.
Proof.
Let be the geodesic ball centered at
, radius
,
, and
. By Lemma 2.7, we have subharmonic
, where
is distance from
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ20_HTML.gif)
where is outward unit conormal along the boundary.
Since, near ,
can be identified with totally geodesic half-sphere and
and
on
as
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ21_HTML.gif)
where is the unit conormal of the cone and the same argument holds as in Proposition 2.6.
Equality holds if and only if ,
, and
, that is,
is a star-shaped minimal cone with density at the center equal to
. Since
is the only
-dimensional minimal submanifold in
with volume
, this completes the proof for
. A similar proof holds for
Proof of Theorem 2.3.
We approach to the proof by comparison between and the cone
. Since the cone
is locally developable on a totally geodesic sphere
, we reduce the proof to the proof of Theorem
in [7] by doubling argument.
For each geodesic sphere centered at
and radius
, one has a local isometry between the curve
and a great circle on
. Hence we can develop
on a great sphere
; one can find a curve
(not necessarily closed) in
and a local isometry from
into
, where
is the north pole of
. Clearly we have
,
, and
Moreover, if we let
and
be the endpoints of
, then
.
We write , where
,
, is a connected component of
. Note that
may be empty or not.
If , then, after cutting
along an appropriate geodesic and developing it onto the great sphere
,
may be identified with the cone
in
, where
is a curve in
given in terms of the polar coordinates by
satisfying
and
Here,
is the angle parameter of the cone. Now we define the doubling
of
by the doubling parametrization as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ22_HTML.gif)
If , then, after developing it onto
,
is identified with
in
, where
is a curve in
given in terms of the polar coordinates by
defined on
, where
is the angle parameter of the cone. Choose
such that
Then we define the doubling
of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ23_HTML.gif)
In both cases, we have the following equalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ24_HTML.gif)
Now let us define . Because of doubling process, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ25_HTML.gif)
By Proposition 2.8, and then
has a self-intersection point. By the geometric meaning of the modified volume,
equals the Euclidean area of the standard projection of
onto the plane containing the equator of
. Let
be the image of
under the projection and
the origin of the plane. Then
,
, but
The last inequality arises from the fact that the projection is a length-shrinking map. Moreover, let
and
be the endpoints of
, then
. So we can apply [7, Lemma
] and conclude the following sharp isoperimetric inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ26_HTML.gif)
and equality holds if and only if is the boundary of a circle. So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ27_HTML.gif)
By Proposition 2.6, we finally get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ28_HTML.gif)
and equality holds if and only if and
, or equivalently,
is a totally geodesic half-disk centered at
.
Proposition 2.9.
Let be a closed convex set in
(
, resp.) and let
be an
-dimensional compact minimal submanifold of
(
, resp.) satisfying that
is orthogonal to
along
. Define
for
. Then
(
, resp.) is a monotonically nondecreasing function of
for
(
, resp.), where
is the
-dimensional geodesic ball of radius
centered
in
(
, resp.).
Proof.
Let .
is a convex set,
, and the same argument holds as in Proposition 2.6; we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ29_HTML.gif)
Denote the volume forms on and
by
and
, respectively. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ30_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ31_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ32_HTML.gif)
In the above inequality we used the fact that and
on
Therefore we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ33_HTML.gif)
This completes the proof for
The similar argument applies to
Remark 2.10.
The classical monotonicity of a minimal submanifold in the Euclidean or hyperbolic space can be found in [6, 8, 9].
In case lies on a geodesic sphere,
automatically satisfies the radially connectivity. In this case, the relative isoperimetric inequality can be extended to the hyperbolic space, and to a minimal submanifold case (not necessarily a minimal surface case). More precisely, we have the following theorem.
Theorem 2.11.
Let be a closed convex set in
or
and let
be an
-dimensional compact minimal submanifold of
or
satisfying that
is orthogonal to
along
. Assume that
lies on a geodesic sphere centered at a point
and that
is the distance in
or
from
. Furthermore, in case of
, assume that
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ34_HTML.gif)
where equality holds if and only if is an
-dimensional totally geodesic half-ball centered at
.
Proof.
Assume that is a minimal submanifold in
with
lying on a geodesic sphere. Let
be the radius of the geodesic sphere. Since
is a convex set and the same argument holds as in Proposition 2.6, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ35_HTML.gif)
By Proposition 2.9, Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ36_HTML.gif)
and so the desired inequality follows. Equality holds if and only if is a cone with density at the center equal to
, or equivalently,
is a totally geodesic half-ball in
. A similar proof holds for
3. Weak Relative Isoperimetric Inequalities in a Riemannian Manifold
The results in Section 2 are sharp but those require some extra assumptions on their boundary. And the results are concerned with the modified volume. In this section, by contrast, we obtain weak relative isoperimetric inequality which holds for any minimal surface and holds for the usual volume.
From now on, we denote as
and
as
for simplicity.
Theorem 3.1.
Let be a closed convex set in a complete simply connected Riemannian manifold
of sectional curvature bounded above by a constant
. Assume that
is a compact minimal surface in the outside of
such that
is orthogonal to
along
. In case of
, one assumes that
. Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ37_HTML.gif)
To prove the above theorem, we begin with the following lemmas on the Laplacian on functions of distance.
Lemma 3.2.
Let be an m-dimensional compact minimal submanifold in a simply connected Riemannian manifold
of sectional curvature bounded above by a constant
. Define
for a fixed point
.
(a)If , then one has
on
.
(b)If , then one has
on
.
(c)If , then one has
on
.
Proof.
Lemma 3.3.
Let be a compact minimal surface satisfying the same assumptions as in Theorem 3.1. Define
for any
.
(a)If , then one has on
(1)
(b)If , then one has on
(2),
(3).
(c)If , then one has on
(4) if
,
(5) if
.
Proof.
For , using Lemma 3.2(c), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ38_HTML.gif)
Near ,
can be identified with totally geodesic half-sphere with constant sectional curvature
. Since
is a fundamental solution of Laplacian on
,
. Hence we obtain (4).
Next for (5) we compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ39_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ40_HTML.gif)
which proves (5) by (4).
For , using Lemma 3.2(a) and
being a fundamental solution of Laplacian on
, we get (1).
For , using Lemma 3.2(b) and
being a fundamental solution of Laplacian on
, we get (2). By similar argument as before, we get (3).
Proof of Theorem 3.1.
-
(i)
. Let
be the distance from
to
. Integrating Lemma 3.3(3) over
for the point
, we get
(3.5)
Since the same argument holds as in Proposition 2.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ42_HTML.gif)
This inequality holds for all , and we can integrate it over
and apply Fubini's theorem to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ43_HTML.gif)
By Lemma 3.2(b) and convexity of , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F758623/MediaObjects/13660_2009_Article_2241_Equ44_HTML.gif)
-
(ii)
. Integrate Lemma 3.3(1) twice and apply Lemma 3.2(a) as in (i).
-
(iii)
. Integrate Lemma 3.3(5) twice and apply Lemma 3.2(c) as in (i).
This completes the proof.
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Acknowledgment
This work was supported in part by KRF-2007-313-C00057.
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Pyo, J. Relative Isoperimetric Inequality for Minimal Submanifolds in a Riemannian Manifold. J Inequal Appl 2010, 758623 (2010). https://doi.org/10.1155/2010/758623
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DOI: https://doi.org/10.1155/2010/758623