Riemannian manifold where volume of a unit ball is larger than surface area

Question

Is there any Riemannian manifold where where volume of a unit ball is larger than surface area? In Euclidean space it is not possible but by imposing some curvature restriction is it possible? Thank you

Answer 1

This is not really well-defined as a question because volume and surface are measures in different dimensions. So you could just take $\mathbb{R}^3$ as your manifold and use the standard metric times a constant factor. This would be the same as instead of looking at the unit ball looking at a ball with a fixed radius. If you change the radius of the ball, the relation between volume and surface area becomes meaningless.

With proper scaling you can make a well-defined question out of this. The keyword to look at is 'isoperimetric'. One can study all kinds of questions related to how much volume you can get with a given surface and the answer does depend on the curvature of the underlying space. In general, high positive curvature will give you a lot of volume per surface, negative curvature low volume per surface.