17 problems

Rigidity conjecture. Any such solution $u$ must be a quadratic polynomial.

Let $1\leq l<k$ and consider the general Hessian quotient equation … Let $u$ be a solution on a closed Riemannian manifold $(M,g)$ endowed with a symmetric $(2,0)$ tensor $\chi$. C…

Let $M$ be a connected, closed Riemannian manifold with metric $g$, and let $\chi$ be a symmetric $(2,0)$ tensor on $M$. Suppose that there is an admissible function $v\in C^4(M)$…

Positive approximate-solution conjecture. A more appropriately chosen approximate solution should be possible, as in the cited work, and should address the failure of positivity in…

Necessary-dimension conjecture. A necessary condition for the existence of such solutions is

Higher-$\sigma_k$ gluing conjecture. Although the method is computationally restricted in the paper to $k=2$, the existence result for infinitely many solutions in the positive…

Mooney's optimal exponent conjecture. The estimate

Boundary regularity conjecture. This locally Lipschitz viscosity solution is smooth near $\partial\Omega$.

Let $X$ be a manifold, let $F\subset J^2(X)$ be a primitive subequation, and let the minimum principle mean that whenever an $F\#\mathcal P$-pseudoconvex domain…

Let $\overrightarrow{a}=(a_0,a_1,\ldots,a_p)\in(\mathbb{R}_+)^{p+1}$ and define … Here $\Gamma_n$ denotes the Gårding cone. Concavifying-function conjecture. There exists…

Let $\overrightarrow{a}=(a_0,a_1,\ldots,a_p)\in(\mathbb{R}_+)^{p+1}$ and let $F\in\mathcal{C}^2(\mathbb{R}_+^*\to\mathbb{R})$. Define … Here $\Gamma_n$ is the Gårding cone and…

Let $(X,\alpha)$ be a Kähler manifold, let $1\leq \ell<k\leq n$, and let $c_j\geq 0$ be fixed constants. Consider … where $c\geq 0$ is determined by integrating the equation over…

Let $(M^{2n},g,J_\pm)$ be a generalized Kähler structure satisfying $[J_+,J_-]=0$. Let $\mathcal H$ be the space of formally generalized Kähler classes, and let $P$ be the associat…

Convexity-relaxation conjecture. The convexity condition can be dropped, or at least relaxed, for the existence result.

CR sphere pseudohermitian k-curvature conjecture. A structure $\widetilde{\theta}\in[\hat{\theta}]_{+}$ has constant pseudohermitian $k$-curvature if and only if it has constant We…

CR $k$-Yamabe existence conjecture. If

CR sphere k-Yamabe conjecture. If the pseudohermitian structure $\widetilde{\theta}=e^{2u}\hat{\theta}\in[\hat{\theta}]_{+}$ has constant pseudohermitian $k$-curvature, then…