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BMS conjecture on positivity preservation of geodesically complete Riemannian manifolds

Let (M,g)(M,g) be a Riemannian manifold. It is geodesically complete if every geodesic can be extended indefinitely. The manifold is L2L^{2}-positivity preserving if every distributional function u∈L2(M)u\in L^{2}(M) satisfying

(−Δ+1)u≥0(-\Delta+1)u\geq 0

obeys u≥0u\geq 0 almost everywhere on MM.

BMS conjecture. Assume that (M,g)(M,g) is geodesically complete. Then (M,g)(M,g) is L2L^{2}-positivity preserving.

The conjecture concerns the LpL^{p}-positivity preserving property of Riemannian manifolds and was formulated in connection with self-adjointness questions for covariant Schrödinger operators. The supplied text does not establish whether the conjecture is resolved.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. BMS conjecture on positivity preservation of geodesically complete Riemannian manifolds

    Let (M,g)(M,g) be a geodesically complete Riemannian manifold. It is called L2L^{2}-positivity preserving if every u∈L2(M)u\in L^{2}(M) satisfying

    (−Δ+1)u≥0(-\Delta+1)u\geq 0

    distributionally on MM also satisfies u≥0u\geq 0 almost everywhere on MM. BMS conjecture. Every geodesically complete Riemannian manifold (M,g)(M,g) is L2L^{2}-positivity preserving. This conjecture concerns positivity preservation for distributional subsolutions of the shifted Laplace operator and was formulated in connection with essential self-adjointness of covariant Schrödinger operators. Its resolution is not specified in the supplied text.

    source: Stefano Pigola and Giona Veronelli, “L^p Positivity Preserving and a conjecture by M. Braverman, O. Milatovic and M. Shubin”, arXiv:2105.14847 (2023).

Sources & referencesView supporting material

Primary source

Stefano Pigola, Daniele Valtorta and Giona Veronelli, “Approximation, regularity and positivity preservation on Riemannian manifolds”, arXiv:2301.05159 (2023).

elliptic-operatorselliptic-pdepositivity-preservationriemannian-geometry

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