Michael Eichmair
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The Penrose inequality in extrinsic geometry Open
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…
The development of mathematics expectancy-value profiles during the secondary–tertiary transition into STEM fields Open
Background To master the secondary–tertiary transition into fields of science, technology, engineering, and mathematics (STEM), academic self-beliefs play a pivotal role, especially those related to learning mathematics. The framework of e…
The isoperimetric inequality Open
We discuss several classical and recent proofs of the isoperimetric inequality and the Sobolev inequality.
View article: Enhancing user-centred educational design: Developing personas of mathematics school students
Enhancing user-centred educational design: Developing personas of mathematics school students Open
Persona development techniques are a well-established method to create relatable descriptions of representatives of target users of digital systems. In the field of education, research on learner characteristics has yielded comprehensive r…
Impact of a Mathematics Bridging Course on the Motivation and Learning Skills of University Students Open
The transition from secondary to tertiary education is an exciting and yet challenging event in the educational biography of students. During this transition, students often meet with unexpected challenges, which may cause them to drop out…
Proof of the Michael–Simon–Sobolev inequality using optimal transport Open
We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2.
Trends in expectancy for success and value beliefs at the secondary-tertiary transition into STEM fields Open
International audience
On the Minkowski inequality near the sphere Open
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 $\ell$, $$ \…
Schoen's conjecture for limits of isoperimetric surfaces Open
Let $(M,g)$ be an $n$-dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature that admits a noncompact area-minimizing hypersurface $Σ\subset M$. In the case where $n = 3$, O. Chodosh and the first-named autho…
Doubling of Asymptotically Flat Half-spaces and the Riemannian Penrose Inequality Open
Building on previous works of Bray, of Miao, and of Almaraz, Barbosa, and de Lima, we develop a doubling procedure for asymptotically flat half-spaces ( M , g ) with horizon boundary $$\Sigma \subset M$$ and mass $$m\in {\mathbb {R}}$$ . I…
View article: Mathematics student personas for the design of technology-enhanced learning environments
Mathematics student personas for the design of technology-enhanced learning environments Open
To benefit from the quickly expanding range of new possibilities of technology-enhanced education, school systems, schools, and teachers need to adapt quickly. Conversely, the needs of students and teachers in a technology-enhanced classro…
Personas Characterising Secondary School Mathematics Students: Development and Applications to Educational Technology Open
Information technology plays an increasingly prominent role in our personal and professional lives. It also plays an important role in schools, and especially in mathematics education. To realise their full potential in education, technolo…
Huisken-Yau-type uniqueness for area-constrained Willmore spheres Open
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 $…
Developing personas to support professional practices of mathematics teacher educators Open
International audience
Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres Open
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 …
Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds Open
Let ( M , g ) be an asymptotically flat Riemannian 3 ‐manifold with nonnegative scalar curvature and positive mass. We show that each leaf of the canonical foliation of the end of ( M , g ) through stable constant mean curvature spheres en…
Large area-constrained Willmore surfaces in asymptotically Schwarzschild 3-manifolds Open
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…
On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds Open
We extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifo…
The Mathematics of Richard Schoen Open
describes his work on harmonic maps into NPC spaces, and
On far-outlying CMC spheres in asymptotically flat Riemannian $3$-manifolds Open
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…
Global uniqueness of large stable CMC spheres in asymptotically flat Riemannian three-manifolds Open
Let $(M, g)$ be a complete Riemannian $3$-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We \textsl{unconditionally} characterize the large, embedded stable constant mean curvature sphe…
Global uniqueness of large stable CMC surfaces in asymptotically flat 3-manifolds Open
Let $(M, g)$ be a complete Riemannian 3-manifold that is asymptotic to Schwarzschild with positive mass and which has non-negative scalar curvature. We show that there are no outlying closed embedded stable constant mean curvature surfaces…
Isoperimetry, scalar curvature, and mass in asymptotically flat\n Riemannian $3$-manifolds Open
Let $(M, g)$ be an asymptotically flat Riemannian $3$-manifold with\nnon-negative scalar curvature and positive mass. We show that each leaf of the\ncanonical foliation through stable constant mean curvature surfaces of the end\nof $(M, g)…
Jenkins–Serrin-type results for the Jang equation Open
Let (M, g, k) be an initial data set for the Einstein equations of general relativity. We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This ans…