I am currently trying to understand Wilking's paper "A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities""A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities" (DOI: 10.1515/crelle.2012.018, arXiv:1011.3561). If I may, I wish to clarify a few points where I am getting stuck:
I am not sure how to prove Lemma 2.3. Wilking says that one can use estimates similar to Shi for linearized equations built from the PDE satisfied by $S_k - Hk$, where $Hk$ is a Harnack operator, and $S_k$ is an approximation as $k \to \infty$. Is this straightforward? Also, it is not at all clear to me why the sequence $S_k$ needs to be introduced at all for proving Theorem 2.1.
On page 234, Wilking takes a closed convex set $C$ of maximal dimension inside the vector space $S^2_B(\mathfrak{g})$, and defines the cone at infinity as $$\partial_\infty C := \lim_{\lambda \to \infty}\frac{C}{\lambda}.$$ I am having trouble understanding exactly what this notation means.
Any help will be highly appreaciated. Thanks!