  I provide a new idea based on geometric analysis to obtain a positive mass
gap in pure non-abelian renormalizable Yang-Mills theory. The orbit space, that
is the space of connections of Yang-Mills theory modulo gauge transformations,
is equipped with a Riemannian metric that naturally arises from the kinetic
part of reduced classical action and admits a positive definite sectional
curvature. The corresponding regularized \textit{Bakry-\'Emery} Ricci curvature
(if positive) is shown to produce a mass gap for $2+1$ and $3+1$ dimensional
Yang-Mills theory assuming the existence of a quantized Yang-Mills theory on
$(\mathbb{R}^{1+2},\eta)$ and $(\mathbb{R}^{1+3},\eta)$, respectively. My
result on the gap calculation, described at least as a heuristic one, applies
to non-abelian Yang-Mills theory with any compact semi-simple Lie group in the
aforementioned dimensions. In $2+1$ dimensions, the square of the Yang-Mils
coupling constant $g^{2}_{YM}$ has the dimension of mass, and therefore the
spectral gap of the Hamiltonian is essentially proportional to $g^{2}_{YM}$
with proportionality constant being purely numerical as expected. Due to the
dimensional restriction on $3+1$ dimensional Yang-Mills theory, it seems one
ought to introduce a length scale to obtain an energy scale. It turns out that
a certain `trace' operation on the infinite-dimensional geometry naturally
introduces a length scale that has to be fixed by measuring the energy of the
lowest glu-ball state. However, this remains to be understood in a rigorous
way.
