The Gagliardo-Nirenberg equation on Riemannian manifolds
Abstract
We establish geometric and analytic rigidity results for the Gagliardo-Nirenberg equation. On a complete noncompact Riemannian manifold with nonnegative Ricci curvature, the existence of a positive critical point of the Gagliardo-Nirenberg quotient forces the manifold to be Euclidean and the solution to be extremal. No minimizing assumption or volume growth hypothesis is required, thereby strengthening earlier rigidity results. In Euclidean space, we prove that all positive solutions of the normalized equation whose nonlinear mass on expanding annuli has sublinear growth are extremals. The linear threshold is optimal, as shown by explicit profiles that are constant in one direction. We also prove a sharp pointwise estimate involving the solution and its gradient, valid without assumptions at infinity, with equality at one point characterizing the extremals. The proofs use weighted divergence identities and a reflection argument controlled by annular mass. Explicit constructions reveal the geometric and variational limits of rigidity: nonflat quotient critical points exist beyond the curvature hypothesis, while the normalized equation admits positive solutions even under strictly positive sectional curvature and maximal volume growth.
Related
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- Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones
- Fractional and anisotropic Gagliardo--Nirenberg inequalities with applications
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- Normalised Hamiltonian Elliptic Systems: a Gagliardo-Nirenberg inequality for bilinear mass
BibTeX
@misc{catino2026gagliardonirenberg,
title = {{The Gagliardo-Nirenberg equation on Riemannian manifolds}},
author = {Giovanni Catino and Dario Daniele Monticelli and Alberto Roncoroni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28218},
eprint = {2610.28218},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28218},
}
RIS
TY - UNPB
TI - The Gagliardo-Nirenberg equation on Riemannian manifolds
AU - Giovanni Catino
AU - Dario Daniele Monticelli
AU - Alberto Roncoroni
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28218
UR - https://arxiv.org/abs/2610.28218
AB - We establish geometric and analytic rigidity results for the Gagliardo-Nirenberg equation. On a complete noncompact Riemannian manifold with nonnegative Ricci curvature, the existence of a positive critical point of the Gagliardo-Nirenberg quotient forces the manifold to be Euclidean and the solution to be extremal. No minimizing assumption or volume growth hypothesis is required, thereby strengthening earlier rigidity results. In Euclidean space, we prove that all positive solutions of the normalized equation whose nonlinear mass on expanding annuli has sublinear growth are extremals. The linear threshold is optimal, as shown by explicit profiles that are constant in one direction. We also prove a sharp pointwise estimate involving the solution and its gradient, valid without assumptions at infinity, with equality at one point characterizing the extremals. The proofs use weighted divergence identities and a reflection argument controlled by annular mass. Explicit constructions reveal the geometric and variational limits of rigidity: nonflat quotient critical points exist beyond the curvature hypothesis, while the normalized equation admits positive solutions even under strictly positive sectional curvature and maximal volume growth.
ER -