On an open question about the equivalence of monotone maximality and surjectivity condition in $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ spaces: Negative and partially positive answers

In a Hilbert space, it is known that a monotone operator is maximal if and only if the \emph{surjectivity condition}, \ie its resolvent has a full domain, holds. This equivalence is an open question in a $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ space, knowing only that the surjectivity condition implies the monotone maximality. In fact, there have been three approaches to generalize the monotonicity from a Hilbert space into a $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ space $X$ — $\mathbf{\text{A:}}$ the ones of dual space $X^{\ast }$, $\mathbf{\text{B:}}$ of linear dual space $X^{\diamond }$, and $\mathbf{\text{C:}}$ of tangent bundle $TX$. In this paper, we answer \emph{negatively} on the first two approaches by constructing counterexamples on an $\mathbb{R}$-tree. We then go on proving that the equivalence is \emph{affirmative} on the same $\mathbb{R}$-tree, without any assumption on the domain. Moreover, we show that the answer is still \emph{positive} in some other cases, through a flat slice principle that transfers Minty’s principle from Hilbert spaces to flat convex subsets, and through Minty’s principle on proper Hilbert-Hadamard spaces satisfying a convex half-space condition. The question of equivalence \emph{remains open} under the general setting with the tangential approach.