In 2006 he constructed counterexamples to the Compactness Conjecture for the Yamabe problem. He also gave a complete description of the asymptotic behavior of the Yamabe flow. In 2007 he proved the Differentiable Sphere Theorem (joint with Richard Schoen). In 2012 he proved the Hsiang–Lawson's conjecture and resolved a problem concerning the uniqueness of self-similar solutions to the Ricci flow which arose in the context of Grigori Perelman's work.
For his contributions to differential geometry he was awarded an EMS Prize in 2012. He delivered the 2012 Euler Lecture and the 2011 Takagi Lectures, and he was an invited speaker at the ICM 2006 in Madrid. He received an Alfred P. Sloan Fellowship in 2006. In December 2013, he was named as the recipient of the Bôcher Prize of the American Mathematical Society.