Monotonicity - analytic and geometric implications
  In this expository article, we discuss various monotonicity formulas for parabolic and elliptic operators and explain how the analysis of the function spaces and the geometry of the underlining spaces are intertwined.   After briefly discussing some of the well-known analytical applications of monotonicity for parabolic operators, we turn to their elliptic counterparts, their geometric meaning, and some geometric consequences. 
Tobias Holck Colding and William P. Minicozzi II
math.DG math.AP