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Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds

An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the…
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Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3-dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in…
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Stability of the positive mass theorem under Ricci curvature lower bounds

We establish Gromov-Hausdorff stability of the Riemannian positive mass theorem under the assumption of a Ricci curvature lower bound. More precisely, consider a class of orientable complete…
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An explicit formula for spherical curves with constant torsion

The purpose of this article is to give an explicit formula for all curves of constant torsion $\tau$ in the unit two-sphere $S^2(1)$. These curves and their basic properties have been known since the…
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ESTIMATE FOR THE ENDANGERED SPECIES EQUATION

We prove a differential Harnack inequality for the Endangered Species Equation, which is a nonlinear parabolic equation. Our derivation relies on an idea related to the parabolic maximum principle.…

Spacetime Harmonic Functions and Applications to Mass

In the pioneering work of Stern [73], level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing…
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On positive scalar curvature cobordisms and the conformal Laplacian on end-periodic manifolds

We show that the periodic $\eta$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to the existence of cobordisms with positive scalar curvature metrics between…
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Harnack estimate for the Endangered Species Equation

We prove a differential Harnack inequality for the Endangered Species Equation, a nonlinear parabolic equation. Our derivation relies on an idea related to the parabolic maximum principle. As an…
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An intrinsic flat limit of Riemannian manifolds with no geodesics

In this paper we produce a sequence of Riemannian manifolds $$M_j^m$$ M j m , $$m \ge 2$$ m ≥ 2 , which converge in the intrinsic flat sense to the unit m -sphere with the restricted Euclidean…
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Minimal hypersurfaces and bordism of positive scalar curvature metrics

Let (Y, g) be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen–Yau, that any closed stable minimal hypersurface of Y also admits a psc-metric. We…
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