Sharp Schrödinger inequalities for embedded hypersurfaces
Abstract
Let Δ = −div ∇. For a closed connected embedded minimal hypersurface Mⁿ ⊂ Sⁿ⁺¹, n ≥ 2, which is not totally geodesic, we prove the quadratic-form inequality Δ + |h|² ≥ n. Taking the test function equal to one gives Perdomo's average-curvature inequality, with equality precisely for minimal Clifford hypersurfaces. Let Q be a positive-definite self-adjoint endomorphism of ℝⁿ⁺¹ and let F(X) = ½⟨QX, X⟩. For a connected complete properly embedded hypersurface without boundary satisfying H = ⟨QX, ν⟩, where H = tr h, and which is not a hyperplane, we prove Δ_F + |h|² + ⟨Qν, ν⟩ ≥ 2λ_min(Q). Properness implies finite weighted volume. Equality in the corresponding average inequality characterizes generalized cylinders whose spherical factor lies in the least eigenspace of Q. In dimensions n ≥ 2, the isotropic case gives the sharp Gaussian average inequalities conjectured by Li and Zhao. The proofs construct a positive supersolution from the sum of two tangent-ball curvature functions. The singular parts of their distributional Hessians are retained in the weak inequality. Consequences include compact spectral comparisons, the embedded low-index classification, and a Bernstein theorem in specified eigendirections.
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BibTeX
@misc{nifa2026sharp,
title = {{Sharp Schrödinger inequalities for embedded hypersurfaces}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22999024},
url = {https://hal.science/hal-05769842v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Schrödinger inequalities for embedded hypersurfaces
AU - Anass Nifa
PY - 2026
DA - 2026/09/08
PB - HAL
DO - 10.5281/zenodo.22999024
UR - https://hal.science/hal-05769842v1
AB - Let Δ = −div ∇. For a closed connected embedded minimal hypersurface Mⁿ ⊂ Sⁿ⁺¹, n ≥ 2, which is not totally geodesic, we prove the quadratic-form inequality Δ + |h|² ≥ n. Taking the test function equal to one gives Perdomo's average-curvature inequality, with equality precisely for minimal Clifford hypersurfaces. Let Q be a positive-definite self-adjoint endomorphism of ℝⁿ⁺¹ and let F(X) = ½⟨QX, X⟩. For a connected complete properly embedded hypersurface without boundary satisfying H = ⟨QX, ν⟩, where H = tr h, and which is not a hyperplane, we prove Δ_F + |h|² + ⟨Qν, ν⟩ ≥ 2λ_min(Q). Properness implies finite weighted volume. Equality in the corresponding average inequality characterizes generalized cylinders whose spherical factor lies in the least eigenspace of Q. In dimensions n ≥ 2, the isotropic case gives the sharp Gaussian average inequalities conjectured by Li and Zhao. The proofs construct a positive supersolution from the sum of two tangent-ball curvature functions. The singular parts of their distributional Hessians are retained in the weak inequality. Consequences include compact spectral comparisons, the embedded low-index classification, and a Bernstein theorem in specified eigendirections.
ER -