Entropy for Perelman and Gravity

I tried to understand the contribution of Grigori Perelman to Mathematics some time ago. I read his papers online and felt that he must have studied Physics with some member of the Lev Landau school of Theoretical Physics in Russia. Nevertheless I recently found out the book, Perfect Rigor by Masha Gessen. To my surprise I see now that it is unlikely that Grisha, as people calls him, studied with any physicist.

How did he come up with the idea of Entropy?

I was getting excited about talking to Perelman, the man that proved Poincaré right. I have a friend that knows him; I asked him to invite him to Mexico, but he said that Perelman was not talking with anybody. Then I read Gessen book and understood.  He has cut relationships with people. I wrote a short story with that theme. I've met people a little bit like him, that is the gist of the story.

I bring this up now, because I am trying to understand the entropic character of gravity; and Grisha's insight is then even more intriguing. I guess I'll have to study his work without his help.

Why does one need to define Entropy in Differential Geometry, to prove that the only three dimensional compact object that can be contracted to a point is the three-sphere?

A three sphere is a strange object. If x_i, with i = 1, 2, 3, 4, satisfy:

x₁²+x₂²+x₃²+x₄²= 1

Then the surface one gets by that restriction is the unit three sphere. For some odd reason, one needs a function that only increases, that Perelman calls the entropy function.

Go figure.