Let be a complete Riemannian manifold with non-negative Ricci curvature. Suppose we have a geodesic ball in whose boundary has the same surface area as a Euclidean ball (not necessarily of the same radius), we want to compare the volumes of these two balls.
Note that the usual comparison theorem says that the volume of a geodesic ball is bounded from above (often by the volume of a ball in a space form with the same radius) given that is bounded from below. In this note, we investigate a dual problem: instead of comparing the volumes of balls with the same radius, we compare the balls with the same surface area. Naturally, the main ingredient is a comparison theorem of Bishop.
We use the following notations. Denote the geodesic ball and geodesic sphere in centered at with radius by and respectively. Let be fixed and define and . Similarly, we define and , where and . Let , we define
so that the Euclidean ball of radius has surface area .
Theorem 1 Let be a complete Riemannian manifold with . Suppose is a geodesic ball such that its boundary has area , then . The equality holds if and only if is isometric to a Euclidean ball.
From now on, we fix a geodesic ball in and let be the area of . Let . Note that for . Obviously, to prove Theorem 1, it suffices to show .
For , we define
It is easy to see that is increasing (and is upper semi-continuous). Indeed, if , then by the continuity of , there exists such that . Therefore .
We also have the simple properties that
Lemma 2 There exists a non-decreasing function such that for ,
Proof: By dividing and by the same constant, we can assume , and regard and as functions of (i.e. ). In the same way we normalize and so that . By a comparison theorem of Bishop (cf. [BC] p.256 Cor. 3 or [SY] Theorem 1.2), there exists a non-increasing function such that
Let be fixed and substitute in the above equation,
Clearly is non-decreasing (as is increasing). The result follows.
As a consequence, we have:
By the equality case of the comparison theorem of Bishop, if and only if is isometric to a Euclidean ball.
Question: Can Theorem 1 be generalized to a Riemannian manifold with smooth boundary?