On an open question about the equivalence of monotone maximality and surjectivity condition in CAT(0) spaces: Negative and partially positive answers
Abstract
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 CAT(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 CAT(0) space X --- \textbf{A:} the ones of dual space X∗, \textbf{B:} of linear dual space Xdiamond, and \textbf{C:} of tangent bundle TX. In this paper, we answer \emph{negatively} on the first two approaches by constructing counterexamples on an ℝ-tree. We then go on proving that the equivalence is \emph{affirmative} on the same 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.
Related
- Interface condition for the Poisson equation in a partially porous medium
- Equivalence of the minimality conditions for the root functions of Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter
- When surjectivity demands pathology
- On the surjectivity conjectures of Dupont and Monod
- Maximality and configuration spaces on real algebraic curves
BibTeX
@misc{chaipunya2026open,
title = {{On an open question about the equivalence of monotone maximality and surjectivity condition in $\mathrmCAT(0)$ spaces: Negative and partially positive answers}},
author = {Parin Chaipunya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25080},
eprint = {2610.25080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25080},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On an open question about the equivalence of monotone maximality and surjectivity condition in CAT(0) spaces: Negative and partially positive answers
AU - Parin Chaipunya
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25080
UR - https://arxiv.org/abs/2610.25080
AB - 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 CAT(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 CAT(0) space X --- \textbf{A:} the ones of dual space X^∗, \textbf{B:} of linear dual space X^diamond, and \textbf{C:} of tangent bundle TX. In this paper, we answer \emph{negatively} on the first two approaches by constructing counterexamples on an ℝ-tree. We then go on proving that the equivalence is \emph{affirmative} on the same 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.
ER -