Let be a Riemannian manifold. We prove two versions of volume comparison results. The first one compares the volume of a geodesic ball with a geodesic ball of the same radius in another Riemannian manifold (not necessarily a space form), under certain assumption on the Ricci curvatures. This is similar to the classical result of Bishop-Gromov in spirit. The second one compares the “exterior region” of a closed hypersurface with the “exterior region” of a corresponding hypersurface . Roughly speaking, when the hypersurface “shrinks” to a point, then this reduces to the first result.

**1. Comparing volumes of geodesic balls **

Let be a complete Riemannian manifold and . The spherical normal coordinates is given by

for and . Here . Let be the distance from . Let be the geodesic ball of radius centered at , , the injectivity radius of .

Of course, within the cut locus of , is just the coordinate function , but we prefer to distinguish these two notions and regard as the distance function and as a parameter for comparison.

Let be another complete Riemannian manifold and . The distance from is denoted by and the geodesic ball of radius centered at is denoted by . We use the spherical normal coordinates to identify the coordinate neighborhood of with that of . More precisely, we identify with by identifying two orthonormal bases of and of , then for , we have a diffeomorphism from to by

Under this identification, we can make sense of the following

**Ricci curvature condition**:

To state the comparison theorem, we also need to impose another condition on . Let and denote the geodesic sphere of radius centered at and that centered at respectively. We impose the following

**Umbilical condition: **For all , the sphere is umbilical at each point. i.e. the second fundamental form of is pointwisely proportional to its induced metric at each point (the proportional constant can depend on and ):

This is for example satisfied if is a warped product metric: , where is the round metric on the standard unit sphere. In particular, any space form satisfies this condition.

We can now state the main result:

Theorem 1 (Volume comparison)Let and be two complete Riemannian manifolds such that the curvature condition (2) and the “umbilical condition” (3) are satisfied. Then for all ,

The equality holds if and only if is isometric to .

Let us first focus on the distance function on . Let and denote the connection and Laplacian on respectively. We first state some classic results.

Lemma 2 (Gauss lemma)Within the cut locus of ,

By the Gauss lemma, we immediately get that on and , the mean curvature of . (We use the convention that the second fundamental form of the standard unit sphere w.r.t. the outward normal is positive and its mean curvature w.r.t. is . Also, on .)

Lemma 3 (Riccati equation)Within the cut locus of ,

Here . (More intrinsically, .)

**Proof**: See the proof of Lemma 12. See also [CLN] p.63 Equation 1.136 or [Petersen] p.265 Equation 2.

By taking the trace of the Riccati equation and applying the Cauchy-Schwarz inequality , we have

Corollary 4 (Riccati inequality)Within the cut locus of ,

The equality holds at the point if and only if is umbilical at this point.

Obviously Corollary 4 is also true on , with “” replaced by “” if the “umbilical condition” (3) holds.

The following maximum principle should be well-known:

Lemma 5 (Maximum principle)Let be a compact manifold, be a smooth function () and is a locally bounded function on . Suppose and

whenver . Then

**Proof**: Let and suppose on . Let and define , then . Suppose there is a first time such that attains at the point , then clearly is the spatial maximum of at time . Thus,

* as . This contradiction shows that and thus on by first taking and then . *

We say a function defined on a metric space (e.g. a Riemannian manifold) to be locally Lipschitz if its restriction on any compact set is Lipschitz. For our purpose, we restate the previous result in a form which is more useful for our argument.

Lemma 6 (Comparison principle)Let be a compact Riemannian manifold. Suppose and are smooth functions such that and for some locally Lipschitz function on . Assume that and as , then

**Proof**: It suffices to prove the result on where . We can assume on , and that on for some constant . Let , then and

* where , . We can now take and apply Proposition 5 to conclude that on . *

Let and denote the connection and Laplacian on respectively.

Lemma 7 (Laplacian comparison)If the umbilical condition (3) and hold, then .

**Proof**: Let and . Then by Corollary 4,

* Note that be defined by is locally Lipschitz. We also have , where is the mean curvature of . It is well-known that both and are of the order as . So , and in particular , so we can apply Lemma 6 to conclude that . *

Let the volume element of and be and respectively. Then within , there is a function such that

where is the area element of . Similarly . Note also that is the area element on .

As , and , we have

Lemma 8With the same assumption as in Lemma 7, the function is non-increasing and . In particular, .

**Proof**: Since , by Lemma 7, we have

* It is easy to see that . *

We can now prove Theorem 1.

**Proof of Theorem 1**: By Lemma 8,

If the equality holds for some , then for , which implies . From the proof of Lemma 7, the equality case of the Riccati inequality (Corollary 4) holds for , which implies that for . In other words, for any tangential to .

Now, recall that the diffeomorphism in (1) is defined by . To prove that is an isometry, it suffices to show it is an isometry along for all . Without loss of generality, assume . (From now on, we ignore the dependence on for any function.) By the canonical identification for , we define the normal Jacobi field (), and similarly define along . It is easy to see that . Consider

Similarly, . Therefore for ,

where , and . By multiplying the integrating factor and integrate,

* It is known that , , and as , so as . Hence by taking . This implies is an isometry. *

If is a warped product metric of the form

around the point , where is the round metric on the standard unit sphere, then we can get a better result (Theorem 10).

Lemma 9Let be two continuous functions which are positive for and is non-increasing for , then is non-increasing for .

**Proof**: For ,

* *

Theorem 10Let and be two complete Riemannian manifolds which satisfy the curvature condition 2 and the “umbilical condition: (3). Suppose is a warped product metric of the form (5) around . Then Then for , the following function is non-increasing:

**Proof**: Since , we have . By Lemma 8, is non-increasing, so

* is also non-increasing. By Lemma 9, is also non-increasing. *

Example 1Let around , then it can be computed that . So if , then when ,

**2. Comparing volumes of exterior regions **

A geodesic ball can be regarded as the “exterior region” of a point, and so Theorem 1 can be regarded as comparing the volumes of two exterior regions of two different points. We now consider the volume of the “exterior region of radius ” of a hypersurface. The method is the same as in Section 1 with some suitable modification.

Let be a two-sided hypersurface embedded in with a unit normal field (chosen to be outward whenever this makes sense). The local coordinates of adapted to is then defined by

By the inverse function theorem, this defines a local coordinates of around any point in . Let to be the distance from , i.e. . Let be the cut function in the outward direction of , i.e. . Then is continuous and is smooth on ; we say a point is within the cut locus of if it lies in this region.

Lemma 11 (Gauss lemma)On ,

**Proof**: Let be a local coordinates of , and extend it by using the coordinates in (6) (such that ). As , it suffices to prove that . To see this, consider

* Therefore is constant and at , . The claim is proved. *

We define . If , then is also a smooth embedded hypersurface for . By the above lemma, for , is the second fundamental form of and is the mean curvature of at . Since the second fundamental form varies continuously along , is the second fundamental form of when restricted to and , the mean curvature of .

Lemma 12 (Riccati equation)On ,

Here . (More intrinsically, .)

**Proof**: We have

* We have used the fact that for any in the last line. *

Corollary 13 (Riccati inequality)On ,

The equality holds at the point if and only if is umbilical at this point.

Now, recall that is another complete Riemannian manifold with connection . Suppose can be isometrically embedded in by with image , which we assume to be two-sided with a unit (outward) normal field . Let be the region bounded between and , i.e. and . We can also define a coordinates adapted to as in (6).

For , we have a diffeomorphism from to by

Again, under this identification, we can make sense of the following

**Ricci curvature condition**:

As before, we define the

**Umbilical condition: **For all , the sphere is umbilical at each point. i.e. the second fundamental form of is pointwisely proportional to its induced metric at each point (the proportional constant can depend on and ):

We identify with and let and be the mean curvature of and respectively. We also write and .

Lemma 14 (Laplacian comparison)Suppose the umbilical condition (9) and hold. Assume that , then .

**Proof**: The proof is exactly the same as that of Lemma 7 except that the initial condition is replaced by , and apply Lemma 6.

Let the volume element of and be and respectively. Then within , there is a function such that

where is the area element of . Similarly . As in (4), we have

By the same proof as in Lemma 8, we have

Lemma 15With the same assumption as in Lemma 14, the function is non-increasing and . In particular, .

We have the following analogue of Theorem 1:

Theorem 16 (Volume comparison)Let and be two complete Riemannian manifolds. be a closed hypersurface in which can be isometrically embedded into such that (8) and the “umbilical condition” (9) are satisfied. Assume further that . Then for all ,

The equality holds if and only if is isometric to .

**Proof**: The proof of the inequality is the same as Theorem 1. If the equality holds for some , then for , which implies . From the proof of Lemma 7, the equality case of the Riccati inequality (Corollary 13) holds for , which implies that for . Let be a local coordinates around a point in (and hence by identification), we use the coordinates in (6) to extend it to a local coordinates in (and ). Then for ,

Similarly,

* Since , this implies , and hence in (7) is an isometry. *

The following is the analogue of Theorem (10):

Theorem 17With the same assumptions as in Theorem 16. Suppose is a warped product metric of the formoutside , where is the area element of . Then for , the following function is non-increasing:

It is clear that for Theorem 16 and Theorem 17 to hold, it is not really necessary that and to be complete, instead we can assume they to be geodesically complete for geodesics pointing outside and respectively.