Two problems in geometric analysis

In this thesis, we prove two results concerning the global geometry of manifolds. The first part of this thesis studies asymptotically at spaces (AF), which play a vital role in mathematical gravitational theory. For graph AF manifolds, we establish the Riemannian Positive Mass Theorem (RPMT) and the Riemannian Penrose Inequality (RPI) with respect to the Gauss-Bonnet-Chern (GBC) masses. Due to the coordinate-invariance required for any relativistic gravity model, the notion of global energy or mass of a gravitating system at a given time cannot be defined as the space integral of the energy density. The global mass is instead defined as an asymptotic boundary integral, making its positivity in theory far from obvious even with a positive energy density. The RPI is robust evidence for the Weak Cosmic Censorship Hypothesis and, hence, the causal structure in any relativistic model of gravity. In the second part of this thesis, we study a sharp form of the Isoperimetric inequality. Given a fixed quantitative distortion of the standard round sphere, we obtain a lower bound for the indeterminate optimal dimensional constant in the Bonnesen-type inequality in Fuglede's work for convex domains of dimension larger than three. To obtain precisely the lower bound, we use a variational approach to obtain the domains minimizing the isoperimetric quotient within a suitable family of domains. These Bonnesen-type inequalities allow us to use the isoperimetric quotient to bound the Gromov-Hausdor distance between a domain's boundary and the round sphere of the same dimension. The three-dimensional case remains open.