It is known how $S^2 \subset \mathbb{R}^3$ minimises surface area for a fixed volume, i.e. fixing the surface area, it maximises the volume within
In Other words:
Constant-volume, surfaceArea minimisation requires constant (+1) sectional curvature of the outcome structure
BUT here is the question:
- How to show a Constant-volume, surfaceArea MAXIMIZATION requires constant (-1) sectional curvature?
one intuitive argument follows, that fixing a volume; the surface should DEFLATE (possibly create cusps) or INFLATE depending on whether the surfaceArea has to be respectively, MAXIMISED or MINIMISED. Where deflation would cause a negative sectional curvature [or atleast the sectional curvature should be bounded by some negative number], and positive, for inflation. Also,
- Is there a possibility to impose a metric on arbitrary compact manifold of FIXED volume in $\mathbb{R}^3$, and inflate or deflate (Probably the technical way to say it, is in terms of Ricci Flows) and depending on whether the surface area is minimising or maximising, it is to check the outcome metric which would indeed give atleast mean-curvature to be consecutively +ve or -ve?
I've found in a lecture by Prof. Mahan MJ, that surface area maximisation for fixed volume leads to hyperbolic structure 3D as an outcome and is the source of this question.
I am in longstanding doubt on this, since trying to learn them by myself. Therefore any clarification would be extremely helpful.