arXiv subjects

Daniel L. Stern

Publications and source records attributed to Daniel L. Stern.

3 recordsLinked to original sources

arXiv2023

Min-max harmonic maps and a new characterization of conformal eigenvalues

Given a surface $M$ and a fixed conformal class $c$ one defines $Λ_k(M,c)$ to be the supremum of the $k$-th nontrivial Laplacian eigenvalue over all metrics $g\in c$ of unit volume. It has been observed by Nadirashvili that the metrics achieving $Λ_k(M,c)$ are closely related to harmonic maps to spheres. In the present paper, we identify $Λ_1(M,c)$ and $Λ_2(M,c)$ with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing $Λ_1(M,c)$, $Λ_2(M,c)$ and, moreover, allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.

math.DG↗

arXiv2019

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 positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen-Yau minimal hypersurfaces.

math.DG↗

arXiv2019

Scalar curvature and harmonic one-forms on three-manifolds with boundary

For a homotopically energy-minimizing map $u: N^3\to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}\{θ\}$ to the scalar curvature of $N$ and the mean curvature of the boundary $\partial N$. As an application, we obtain some natural geometric estimates for the Thurston norm on $3$-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a characterization of the Thurston norm via scalar curvature and the harmonic norm for general closed, oriented three-manifolds, extending Kronheimer and Mrowka's characterization for irreducible manifolds to arbitrary topologies.

math.DG↗