Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)

Question

There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs,
where $P_k$ is given by \begin{equation} \begin{aligned} P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < i_{k}\leq n} (\lambda_{i_{1}}+\cdots+ \lambda_{i_{k}}). \end{aligned} \end{equation} In addition, $\lambda_{i_{1}}+\cdots+ \lambda_{i_{k}}>0$ for any $1\leq i_{1}<\cdots < i_{k}\leq n$.

The operator $P_1$ is related to Calabi conjecture in K\"ahler geometry, while $P_{n-1}$ is related to Gauduchon conjecture in Hermitian geometry. These two geometric problems can be reduced to solving fully nonlinear elliptic equations on the given compact complex manifold.

My question: Are there books or literatures could present the detailed discussion about these functions $P_k$ or $\log P_k$ ($1\leq k <n$)，especially for $P_{n-1}$?

Thanks and best.

Answer 1

For operator f(\lambda)=\log P_k(\lambda), in Guan Bo's several papers, say arXiv:1409.3633v1, Page 5 below Theorem 1.9, they mention some basic properties about it, such as it satisfies (1.2) to (1.6) in suitable cones.

Answer 2

I'm not sure if these are too similar to Bo Guan's work to be right for your application, but Gabor Szekelyhidi also has some work on these equations. For instance, see the following paper. https://arxiv.org/abs/1501.02762v2

In fact, there are quite a few people working on these sort of problems. The papers which cite the above work gives some overview. For instance, there is the paper of Szekelyhidi, Tosatti, and Weinkove on the problem of finding Gauduchon metrics (https://arxiv.org/abs/1503.04491). There is also the work of Phong, Picard and Zhang on fully non-linear equations (https://arxiv.org/abs/1508.03254).