The Alexandrov-Bakel'man-Pucci (ABP) estimate is one of
the most beautiful applications of geometric ideas in PDE and it is the
backbone of the regularity theory of fully nonlinear elliptic PDE. I will
start from the classical ABP estimate and then talk about its general-
ization on Riemannian manifolds, obtained in joint work with Yu Wang.
As applications, I will present results about the Harnack inequalities for
non-divergent PDE on manifolds and also an ABP approach to the clas-
sical Minkowski and Heintze-Karcher inequalities. In the second part of
the talk, I will give a brief overview of the classical Minkowski integral
formulae which are related to the divergence structure of some elliptic
operators. I will present the spacetime analogue of this type formula
I obtained with co-authors. Motivated by the problems from general
relativity, we consider the co-dimension two submanifolds in Lorentzian
spacetimes and establish some new Minkowski formulae in this setting.