Searcharxiv⌕ Search

arXiv subjects

Michael Eichmair

Publications and source records attributed to Michael Eichmair.

At least 19 recordsLinked to original sources
arXiv2026

On the Minkowski inequality near the sphere

We construct a sequence $\{Σ_\ell\}_{\ell=1}^\infty$ of closed, axially symmetric surfaces $Σ_\ell\subset \mathbb{R}^3$ that converges to the unit sphere in $W^{2,p}\cap C^1$ for every $p\in[1,\infty)$ and such that, for every $\ell$, $$ \int_{Σ_{\ell}}H_{Σ_\ell}-\sqrt{16\,π\,|Σ_{\ell}|}<0 $$ where $H_{Σ_\ell}$ is the mean curvature of $Σ_\ell$. This shows that the Minkowski inequality with optimal constant fails even for perturbations of a round sphere that are small in $W^{2,p}\cap C^1$ unless additional convexity assumptions are imposed.

math.DG↗
arXiv2026

Tangential limits of stable minimal capillary surfaces

We characterize all compact embedded stable minimal capillary surfaces with capillary angle close to either $0$ or $π$ that are supported on a complete embedded minimal surface with finite total curvature that is not an affine plane. Moreover, we characterize all compact embedded weakly stable minimal capillary surfaces with capillary angle close to either $0$ or $π$ that are supported on a closed surface whose mean curvature is positive and has no degenerate maxima. An important ingredient in our work are curvature estimates for sequences of weakly stable minimal capillary surfaces with capillary angles tending to $0$ or $π$ that enable us to analyze the tangential limits of such sequences at suitable scales.

math.DG↗
arXiv2025

Schoen's conjecture for limits of isoperimetric surfaces

Let $(M,g)$ be an $n$-dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature that admits a noncompact area-minimizing hypersurface $Σ\subset M$. In the case where $n = 3$, O. Chodosh and the first-named author have proven that $(M, g)$ is necessarily isometric to Euclidean space, confirming a conjecture of R. Schoen. In this paper, we extend this result to dimension $3 < n \leq 7$ provided that $Σ$ arises as a limit of isoperimetric surfaces. By contrast, we prove that when $3 < n \leq 7$, there is no such result for general noncompact area-minimizing $Σ\subset M$, even when additional assumptions on the stability of $Σ$ are imposed.

math.DG↗
arXiv2024

The Penrose inequality in extrinsic geometry

The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this paper, we resolve this conjecture and show that the exterior mass $m$ of an asymptotically flat support surface $S\subset\mathbb{R}^3$ with nonnegative mean curvature and outermost free boundary minimal surface $D$ is bounded in terms of $$ m\geq \sqrt{\frac{|D|}π}. $$ If equality holds, then the unbounded component of $S\setminus \partial D$ is a half-catenoid. In particular, this extrinsic Penrose inequality leads to a new characterization of the catenoid among all complete embedded minimal surfaces with finite total curvature. To prove this result, we study minimal capillary surfaces supported on $S$ that minimize the free energy and discover a quantity associated with these surfaces that is nondecreasing as the contact angle increases.

math.DG↗
arXiv2024

Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres

Let $(M,g)$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ with positive mass. We give a short proof based on Lyapunov-Schmidt reduction of the existence of an asymptotic foliation of $(M, g)$ by stable constant mean curvature spheres. Moreover, we show that the geometric center of mass of the foliation agrees with the Hamiltonian center of mass of $(M,g)$. In dimension $n = 3$, these results were shown previously by C. Nerz using a different approach. In the case where $n=3$ and the scalar curvature of $(M, g)$ is nonnegative, we prove that the leaves of the asymptotic foliation are the only large stable constant mean curvature spheres that enclose the center of $(M, g)$. This was shown previously under more restrictive decay assumptions and using a different method by S. Ma.

math.DG↗
arXiv2023

Huisken-Yau-type uniqueness for area-constrained Willmore spheres

Let $(M,g)$ be a Riemannian $3$-manifold that is asymptotic to Schwarzschild. We study the existence of large area-constrained Willmore spheres $Σ\subset M$ with non-negative Hawking mass and inner radius $ρ$ dominated by the area radius $λ$. If the scalar curvature of $(M,g)$ is non-negative, we show that no such surfaces with $\log λ\ll ρ$ exist. This answers a question of G. Huisken.

math.DG↗
arXiv2023

Initial data rigidity results

We present several rigidity results for initial data sets motivated by the positive mass theorem. An important step in our proofs here is to establish conditions that ensure that a marginally outer trapped surface is "weakly outermost". A rigidity result for Riemannian manifolds with a lower bound on their scalar curvature is included as a special case.

gr-qc↗
arXiv2023

Doubling of asymptotically flat half-spaces and the Riemannian Penrose inequality

Building on previous works of H. L. Bray, of P. Miao, and of S. Almaraz, E. Barbosa, and L. L. de Lima, we develop a doubling procedure for asymptotically flat half-spaces $(M,g)$ with horizon boundary $Σ\subset M$ and mass $m\in\mathbb{R}$. If $3\leq \dim(M)\leq 7$, $(M,g)$ has non-negative scalar curvature, and the boundary $\partial M$ is mean-convex, we obtain the Riemannian Penrose-type inequality $$ m\geq\left(\frac{1}{2}\right)^{\frac{n}{n-1}}\,\left(\frac{|Σ|}{ω_{n-1}}\right)^{\frac{n-2}{n-1}} $$ as a corollary. Moreover, in the case where $\partial M$ is not totally geodesic, we show how to construct local perturbations of $(M,g)$ that increase the scalar curvature. As a consequence, we show that equality holds in the above inequality if and only if the exterior region of $(M,g)$ is isometric to a Schwarzschild half-space. Previously, these results were only known in the case where $\dim(M)=3$ and $Σ$ is a connected free boundary hypersurface.

math.DG↗
arXiv2022

Large area-constrained Willmore surfaces in asymptotically Schwarzschild 3-manifolds

We apply the method of Lyapunov-Schmidt reduction to study large area-constrained Willmore surfaces in Riemannian 3-manifolds asymptotic to Schwarzschild. In particular, we prove that the end of such a manifold is foliated by distinguished area-constrained Willmore spheres. The leaves are the unique area-constrained Willmore spheres with large area, non-negative Hawking mass, and distance to the center of the manifold at least a small multiple of the area radius. Unlike previous related work, we only require that the scalar curvature satisfies mild asymptotic conditions. We also give explicit examples to show that these conditions on the scalar curvature are necessary.

math.DG↗
arXiv2022

The Willmore center of mass of initial data sets

We refine the Lyapunov-Schmidt analysis developed in our recent paper arxiv:2101.12665 to study the geometric center of mass of the asymptotic foliation by area-constrained Willmore surfaces of initial data for the Einstein field equations. If the scalar curvature of the initial data vanishes at infinity, we show that this geometric center of mass agrees with the Hamiltonian center of mass. By contrast, we show that the position of large area-constrained Willmore surfaces is sensitive to the distribution of the energy density. In particular, the geometric center of mass may differ from the Hamiltonian center of mass if the scalar curvature does not satisfy asymptotic symmetry assumptions.

math.DG↗
arXiv2018

A splitting theorem for scalar curvature

We show that a Riemannian $3$-manifold with non-negative scalar curvature is flat if it contains an area-minimizing cylinder. This scalar-curvature analogue of the classical splitting theorem of J.~Cheeger and D.~Gromollhas been conjectured by D.~Fischer-Colbrie and R.~Schoen and by M.~Cai and G.~Galloway.

math.DG↗
arXiv2018

Characterization of large isoperimetric regions in asymptotically hyperbolic initial data

Let $(M,g)$ be a complete Riemannian $3$-manifold asymptotic to Schwarzschild-anti-deSitter and with scalar curvature $R \geq - 6$. Building on work of A.~Neves and G.~Tian and of the first-named author, we show that the leaves of the canonical foliation of $(M, g)$ are the unique solutions of the isoperimetric problem for their area. The assumption $R \geq -6$ is necessary. This is the first characterization result for large isoperimetric regions in the asymptotically hyperbolic setting that does not assume exact rotational symmetry at infinity.

math.DG↗
arXiv2017

On far-outlying CMC spheres in asymptotically flat Riemannian $3$-manifolds

We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the work of S. Brendle and the second-named author to the "far-off-center" regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable CMC spheres that depend very delicately on the behavior of scalar curvature at infinity.

math.DG↗
arXiv2016

Effective versions of the positive mass theorem

The study of stable minimal surfaces in Riemannian $3$-manifolds $(M, g)$ with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when $(M, g)$ is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of R. Schoen: An asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat $\mathbb{R}^3$.

math.DG↗