An arXiv preprint presents a conditional proof that a conformal class on a smooth, closed five-dimensional Riemannian manifold can contain a metric with both positive scalar curvature and positive Q-curvature. The result requires two mathematical quantities attached to that class, called the Yamabe invariant and the Yamabe-type Q-curvature invariant, to be positive. It also requires an additional condition, holding at every point, on an initial metric in the class.
A path through possible metrics
Here, a conformal class is the family of metrics linked by the rescaling used in the proof. This is an existence result from analysis, not a finding drawn from a dataset. The work treats a general class of such manifolds and follows positive solutions along a path of conformal metrics as a continuity parameter changes. Its central question is whether the solution can be carried through the full path.
To set up that path, the paper rewrites the problem as a relation between Q-curvature and sigma2-curvature, a second elementary symmetric curvature polynomial. The conformal metric is formed by rescaling a reference metric with a power of a positive factor. The path's auxiliary quantities are defined from a parameter t and a smooth function f that does not depend on t. This formulation gives the proof a single parameter to track.
Keeping the path open
One part of the continuity argument is openness, the step that allows the solution set to persist as the parameter moves. In dimension five, the relevant linearized operator is stated to be positive definite throughout the allowed parameter interval. That positivity supports openness of the set of parameters admitting a suitable positive solution.
The other major step is control of the solutions. For the dimension and path range covered by the lemma, the a priori estimates give every positive solution a uniform bound in the norm used by the proof and a positive lower bound. The constant is independent of both the solution and the path parameter. The supplied analysis does not give its numerical value.
An integral inequality and a bootstrap, meaning a sequence of regularity estimates, then supply a bound on the solution's maximum size, two-derivative estimates for every finite integrability exponent greater than one, and controls on derivatives of all orders. These estimates establish closedness of the solution set, the second condition needed to complete the continuity argument.
What the proof establishes
With openness and closedness established, the argument reaches the zero value of the path parameter. The solution set fills the full interval used in the proof, including zero, and the resulting conformal metric has positive Q-curvature and positive scalar curvature. No statistical uncertainty is reported. The result is a theoretical existence statement, not an empirical estimate.
The condition that remains
The qualification is central. Positivity of the two conformal invariants by itself is not enough for the theorem as presented. The initial metric must also pass a pointwise test, expressed equivalently as positivity of a specified combination of its Q-curvature and sigma2-curvature together with positivity of its J-quantity. The paper does not establish that this extra requirement can be removed.
A related corollary
A related corollary starts from positivity of the sigma2-related conformal invariant. In dimension five, the paper uses an inequality between that invariant and the Yamabe-type Q-curvature invariant to conclude the latter is positive. That implication lets the main theorem be applied when its other conditions, including the initial-metric test, are satisfied.
What the preprint leaves open
The author describes the work as a continuity-method proof of the dimension-five Gursky-Hang-Lin conjecture under the added initial-metric condition. A different continuity path may eventually remove that condition, the paper says, but that remains future work rather than an established result. The document is identified as arXiv version 1, dated 28 Aug 2026.
The declarations say that no datasets were generated or analyzed, so data sharing is not applicable. The work was supported in part by an Institute Post Doctoral Fellowship from the Indian Institute of Technology Bombay, and the author reports no conflict of interest.
Paper data and sources
Original title: A note on the positivity of $Q$-curvature via the continuity method
Authors: Ramesh Mete
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text