Conditional Proof of the Boltzmann-Sinai Ergodic Hypothesis

We consider the system of $N$ ($\ge 2$) elastically colliding hard balls of masses $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N$ and radius $r$ on the flat unit torus ${\mathbb{T}}^{\nu }$, $\nu \ge 2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N;r)$ of the external geometric parameters, provided that almost every singular orbit is geometrically hyperbolic (sufficient), i. e. the so called Chernov-Sinai Ansatz is true. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.