arXiv subjects

Oskar Riedler

Publications and source records attributed to Oskar Riedler.

6 recordsLinked to original sources

arXiv2026

Harmonic morphisms and minimal submanifolds

Harmonic morphisms, maps which preserve Laplace's equation, are intimately connected to the topic of minimal submanifolds. In this article we first characterise harmonic morphisms between Riemannian manifolds as the weakly horizontally conformal maps that preserve the equation for minimal submanifolds of co-dimension $2$. We further derive additional reduction properties of harmonic morphisms for minimal submanifolds of other co-dimensions. These theorems are then applied in an example case, yielding a novel family of degree $4$ area-minimising hypercones in $\mathbb R^m$, $m\geq32$.

math.DG↗

arXiv2026

Scalar-rigid submersions are Riemannian products

Scalar-rigid maps are Riemannian submersions by works of Llarull, Goette--Semmelmann, and the second named author. In this article we show that they are essentially Riemannian products of the base manifold with a Ricci-flat fiber. As an application we obtain a Llarull-type theorem for non-zero degree maps onto products of manifolds of non-negative curvature operator and positive Ricci curvature with some enlargeable manifold. The proof is based on spin geometry for Dirac operators and an analysis connecting Clifford multiplication with the representation theory of the curvature operator.

math.DG↗

arXiv2025

Polynomial harmonic morphisms and eigenfamilies on spheres

The eigenfamilies of Gudmundsson and Sakovich can be used to generate harmonic morphisms, proper $r$-harmonic maps, and minimal co-dimension $2$ submanifolds. This article begins by characterising the globally defined eigenfamilies of the sphere $S^m$; they correspond to orthogonal families of homogeneous polynomial harmonic morphisms from $\Bbb{R}^{m+1}$ to $\Bbb C$, all of the same degree. We investigate and construct such families, paying special attention to those that are not congruent to families of holomorphic polynomials. Strong restrictions for families of such polynomials are found in low dimensions, and the pairs of degree $2$ maps that induce an eigenfamily are classified.

math.DG↗

arXiv2025

Global eigenfamilies on closed manifolds

We study globally defined $(λ,μ)$-eigenfamilies on closed Riemannian manifolds. Among others, we provide (non-)existence results for such eigenfamilies, examine their topological properties and classify $(λ,μ)$-eigenfamilies on flat tori. It is further shown that for $f=f_1+i f_2$ being an eigenfunction decomposed into its real and its imaginary part, the powers $\{f_1^a f_2^b\mid a,b\in\mathbb N\}$ satisfy highly rigid orthogonality relations in $L^2(M)$. In establishing these orthogonality relations one is led to combinatorial identities involving determinants of products of binomials, which we view as being of independent interest.

math.DG↗

arXiv2024

$(λ,λ)$-eigenfunctions on compact manifolds

In this note we study $(λ,μ)$-eigenfamilies on compact Riemannian manifolds when $λ= μ$. We show that any compact manifold admitting a $(λ,λ)$-eigenfunction is a mapping torus and that any $(λ,λ)$-eigenfamily is one dimensional. Additionally, we consider generalised eigenfamilies, which can have higher dimension, and relate these to harmonic Riemannian submersions to a torus.

math.DG↗

arXiv2023

Closed embedded self-shrinkers of mean curvature flow

In this article we show the existence of closed embedded self-shrinkers in $\Bbb{R}^{n+1}$ that are topologically of type $S^1\times M$, where $M\subset S^n$ is any isoparametric hypersurface in $S^n$ for which the multiplicities of the principle curvatures agree. This yields new examples of closed self-shrinkers, for example self-shrinkers of topological type $S^1\times S^k\times S^k\subset \Bbb R^{2k+2}$ for any $k$. If the number of distinct principle curvatures of $M$ is one the resulting self-shrinker is topologically $S^1\times S^{n-1}$ and the construction recovers Angenent's shrinking doughnut.

math.DG↗