arXiv subjects

Michael B. Law

Publications and source records attributed to Michael B. Law.

5 recordsLinked to original sources

arXiv2026

An analytical characterization of Eguchi-Hanson space and its higher-dimensional analogs

Let $(M,g)$ be a complete 4-dimensional Ricci-flat ALE orbifold with finitely many orbifold points and group at infinity $\mathbb{Z}_2$. We prove that if the $L^2$ kernel of its Lichnerowicz Laplacian has dimension at most 3, then $(M,g)$ is either the Eguchi-Hanson space or the flat orbifold $\mathbb{R}^4/\mathbb{Z}_2$. A similar uniqueness result is proved for Calabi's higher-dimensional analogs of the Eguchi-Hanson space among Ricci-flat Kähler ALE orbifolds with group at infinity $\mathbb{Z}_m$.

math.DG↗

arXiv2026

On steady and expanding Ricci solitons with asymptotic symmetries

We establish a symmetry principle for asymptotically cylindrical steady gradient Ricci solitons (GRSs) and asymptotically conical expanding GRSs with homogeneous links. Using this, we show that the Bryant steady soliton is the unique asymptotically cylindrical steady GRS that has a round spherical link and satisfies a particular quantitative rigidity condition. A similar characterization is proved for Bryant's expanding solitons. Finally, we establish a global symmetry result for GRSs which exhibit the aforementioned asymptotics with quotient-Berger sphere asymptotic links.

math.DG↗

arXiv2025

Concavity for elliptic and parabolic equations in locally symmetric spaces with nonnegative curvature

We establish a concavity principle for solutions to elliptic and parabolic equations on locally symmetric spaces with nonnegative sectional curvature, extending the results of Langford and Scheuer. To the best of our knowledge, this is the first general concavity principle established on spaces with non-constant sectional curvature.

math.AP↗

arXiv2025

Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds

We construct and classify all polynomial growth solutions to certain drift-harmonic equations on complete manifolds with paraboloidal asymptotics. These encompass the natural drift-harmonic equations on certain steady gradient Ricci solitons. Specifically, we show that all drift-harmonic functions with polynomial growth asymptotically separate variables, and compute the dimensions of spaces of drift-harmonic functions with a given polynomial growth rate. The proof uses an inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.

math.DG↗

arXiv2024

Positive mass and Dirac operators on weighted manifolds and smooth metric measure spaces

We establish a weighted positive mass theorem which unifies and generalizes results of Baldauf--Ozuch and Chu--Zhu. Our result is in fact equivalent to the usual positive mass theorem, and can be regarded as a positive mass theorem for smooth metric measure spaces. We also study Dirac operators on certain warped product manifolds associated to smooth metric measure spaces. Applications of this include, among others, an alternative proof for a special case of our positive mass theorem, eigenvalue bounds for the Dirac operator on closed spin manifolds, and a new way to understand the weighted Dirac operator using warped products.

math.DG↗