"Taking the long view: The life of Shiing-Shen Chern" http://takingthelongviewfilm.com/ This very nice documentary was produced recently on the occasion of the Chern Centennial Conference.

Let me clarify a couple of issues from the previous answers/comments: 1) The linearization of $Rc-\tfrac{1}{2}Rg$ has mixed signs, and for this reason the equation $\partial_t g=-(Rc-\tfrac{1}{2}Rg)$ ...

There is a nice interpretation of Perelman's monotonicity formulas in terms of optimal transportation, see e.g. these lecture notes by Peter Topping http://homepages.warwick.ac.uk/~maseq/...

I think that in a borderline case like this, the math department is indeed the better choice. At least according to my experience, the math community seems to be more flexible, give you a nicer ...

The Sobolev-inequality holds for general metric measure spaces satisfying CD(K,n), in particular for your smooth ones. See e.g. Theorem 21.15 in Villani's book http://math.univ-lyon1.fr/~villani/...

Let me add some uniqueness theorems for CMC and minimal surfaces: 1) A classical theorem of Hopf says that any immersed CMC sphere in $\mathbb{R}^3$ is the round sphere. 2) A classical theorem of ...

Here is a toy model: Consider a function $u=u(t,x,y,z)$. Then the standard heat equation $\partial_tu=(\partial_x^2+\partial_y^2+\partial_z^2)u$ is strongly parabolic, while e.g. the equation $\...

It's very different! Convex (in the Huisken sense) means that all the principal curvatures $\lambda_1,\ldots,\lambda_n$ are positive, i.e. $\lambda_1>0,\ldots,\lambda_n>0$. Mean convex only ...

First of all, a canonical reference for special geometric structures is the book "Compact manifolds with special holonomy" by Dominic Joyce. A1: As you observed, specifying an $O(n)$-structure is the ...

The implication holds e.g. on compact manifolds. To see this, you just have to convince yourself that in this case the flow of a vector field without zeros cannot have fixed points for very small ...

This is the Sobolev space of $L^2$ functions with one derivative in $L^2$. A more common notation for that space is $W^{1,2}$. Let me give an example first: On $S^3$ the min-max is attained by the ...

EDIT: As Misha and Deane pointed out, the question is not terribly well posed. So I will interpret it somewhat broadly to better reflect our actual understanding of canonical metrics on 3-manifolds ...