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Sharp gradient comparison and rigidity for semilinear equations on manifolds
Abstract
We prove sharp amplitude and gradient comparisons for positive solutions of a class of semilinear elliptic equations on complete manifolds under a Ricci lower bound. Equality at one point characterizes hyperbolic space and the corresponding radial solution. The local estimates are uniform in the mixing coefficients and number of terms for positive mixtures of subcritical powers. For a class of conformal compactifications defined by a source equation, we obtain sharp boundary scalar-curvature bounds and hyperbolic rigidity. We also construct non-Einstein examples whose normalization persists under small compactly supported metric perturbations.
Related
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
- Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient
- Sub-Laplacian Comparison Theorem and Gradient Estimates on Complete Noncompact Contact Riemannian Manifolds
- Stability and gradient estimates for fully nonlinear elliptic equations on hermitian manifolds
- Interior gradient estimates for mean curvature equations on Riemannian manifolds
BibTeX
@misc{sun2026sharp,
title = {{Sharp gradient comparison and rigidity for semilinear equations on manifolds}},
author = {Linlin Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34207},
eprint = {2610.34207},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34207},
}
RIS
TY - UNPB
TI - Sharp gradient comparison and rigidity for semilinear equations on manifolds
AU - Linlin Sun
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34207
UR - https://arxiv.org/abs/2610.34207
AB - We prove sharp amplitude and gradient comparisons for positive solutions of a class of semilinear elliptic equations on complete manifolds under a Ricci lower bound. Equality at one point characterizes hyperbolic space and the corresponding radial solution. The local estimates are uniform in the mixing coefficients and number of terms for positive mixtures of subcritical powers. For a class of conformal compactifications defined by a source equation, we obtain sharp boundary scalar-curvature bounds and hyperbolic rigidity. We also construct non-Einstein examples whose normalization persists under small compactly supported metric perturbations.
ER -