Project Slope - A study of lunar orbiter photographic evaluation secondary analysis tasks Final report

Project SLOPE /Study of Lunar Orbiter Photographic Evaluation/ techniques, implementation and accurac

WATERMAN FUND ESSAY WINNER: Splitting Clouds at the Edge of the World: How Had I Never Noticed It Before?

The 2021 winner of our annual contest for emerging writers, ecologist Jason Mazurowski, spots Mount Marcy from his Vermont apartment window at the start of the novel coronavirus pandemic and sets out to climb i

The Half-Volume Spectrum of a Manifold

We define the half-volume spectrum $\{\tilde \omega_p\}_{p\in \mathbb N}$ of a closed manifold $(M^{n+1},g)$. This is analogous to the usual volume spectrum of $M$, except that we restrict to $p$-sweepouts whose slices each enclose half the volume of $M$. We prove that the Weyl law continues to hold for the half-volume spectrum. We define an analogous half-volume spectrum $\tilde c(p)$ in the phase transition setting. Moreover, for $3 \le n+1 \le 7$, we use the Allen-Cahn min-max theory to show that each $\tilde c(p)$ is achieved by a constant mean curvature surface enclosing half the volume of $M$ plus a (possibly empty) collection of minimal surfaces with even multiplicities.Comment: 22 Pages; Comments welcom

Min-max theory for free boundary minimal hypersurfaces in locally wedge-shaped manifolds

We develop a min-max theory for the area functional in the class of locally wedge-shaped manifolds. Roughly speaking, a locally wedge-shaped manifold is a Riemannian manifold that is allowed to have both boundary and certain types of edges. Fix a dimension $3 \le n+1 \le 6$. As our main theorem, we prove that every compact locally wedge-shaped manifold $M^{n+1}$ with acute wedge angles contains a locally wedge-shaped free boundary minimal hypersurface $\Sigma^n$ which is smooth in its interior and on its faces and is $C^{2,\alpha}$ up to and including its edge. We can also handle the case of 90 degree wedge angles under an additional assumption.Comment: 47 pages, 6 figures, comments are welcome

Monotone Quantities for pp-Harmonic functions and the Sharp pp-Penrose inequality

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $\Sigma$. Fix a number $1 < p < 2$. We derive monotone quantities for $p$-harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$-capacity of $\Sigma$ in $M$, which was first proved by Xiao using weak inverse mean curvature flow.Comment: 20 pages, comments are welcome