  For any $\alpha < 1/3$, we construct weak solutions to the $3D$
incompressible Euler equations in the class $C_tC_x^\alpha$ that have nonempty,
compact support in time on ${\mathbb R} \times {\mathbb T}^3$ and therefore
fail to conserve the total kinetic energy. This result, together with the proof
of energy conservation for $\alpha > 1/3$ due to [Eyink] and [Constantin, E,
Titi], solves Onsager's conjecture that the exponent $\alpha = 1/3$ marks the
threshold for conservation of energy for weak solutions in the class
$L_t^\infty C_x^\alpha$. The previous best results were solutions in the class
$C_tC_x^\alpha$ for $\alpha < 1/5$, due to the author, and in the class $L_t^1
C_x^\alpha$ for $\alpha < 1/3$ due to Buckmaster, De Lellis and
Sz\'{e}kelyhidi, both based on the method of convex integration developed for
the incompressible Euler equations by De Lellis and Sz\'ekelyhidi. The present
proof combines the method of convex integration and a new "gluing
approximation" technique. The convex integration part of the proof relies on
the "Mikado flows" introduced by [Daneri, Sz\'ekelyhidi] and the framework of
estimates developed in the author's previous work.
