arXiv subjects
Publications and source records attributed to Jan Nienhaus.
We classify fixed point homogeneous Riemannian orbifolds with positive sectional curvature up to equivariant diffeomorphism. As a corollary we obtain a classification of positively curved orbifolds with maximal symmetry rank.
We prove a Tachibana-type result for K\"ahler-Einstein manifolds isolating complex projective space based on its Calabi curvature operator. The proof uses a new Bochner formula expressing the Lichnerowicz curvature term via the Calabi curvature operator. This resolves one of the problems posed at the AIM workshop "The Bochner Technique'' (May 2026).
Hopf conjectured that even-dimensional closed Riemannian manifolds with positive sectional curvature have positive Euler characteristic. The conclusion of the conjecture is known to fail if the positive sectional curvature assumption is relaxed in any number of ways, including to positive second intermediate Ricci curvature. Here we prove that if a manifold with positive second intermediate Ricci curvature has dimension divisible by four and torus symmetry of rank at least ten, then it has positive Euler characteristic. A crucial new tool is a non-trivial extension of the first author's Four Periodicity Theorem to situations where the periodicity of the cohomology does not extend all the way down to degree zero.
It is shown that a compact $n$-dimensional K\"ahler manifold with $\frac{n}{2}$-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric with its symmetric metric has $\frac{n}{2}$-nonnegative Calabi curvature operator, yet $b_n =2.$ Furthermore, the compact K\"ahler manifolds with an $\frac{n}{2}$-nonnegative Calabi curvature operator are classified. In addition, the previously known results for the K\"ahler curvature operator are improved when the metric is K\"ahler--Einstein.
We prove the existence of three non-round, non-isometric Einstein metrics with positive scalar curvature on the sphere $S^{10}.$ Previously, the only even-dimensional spheres known to admit non-round Einstein metrics were $S^6$ and $S^8.$
We prove that there exists a gradient expanding Ricci soliton asymptotic to any given cone over the product of a round sphere and a Ricci flat manifold. In particular we obtain asymptotically conical expanding Ricci solitons with positive scalar curvature on $\mathbb{R}^3 \times S^1.$ More generally we construct continuous families of gradient expanding Ricci solitons on trivial vector bundles over products of Einstein manifolds with arbitrary Einstein constants.
We show that an $n$-dimensional Riemannian manifold with $n$-nonnegative or $n$-nonpositive curvature operator of the second kind has restricted holonomy $SO(n)$ or is flat. The result does not depend on completeness and can be improved provided the space is Einstein or Kähler. In particular, if a locally symmetric space has $n$-nonnegative or $n$-nonpositive curvature operator of the second kind, then it has constant curvature. When the locally symmetric space is irreducible this can be improved to $\frac{3n}{2}\frac{n+2}{n+4}$-nonnegative or $\frac{3n}{2}\frac{n+2}{n+4}$-nonpositive curvature operator of the second kind.
In the 1930s, H. Hopf conjectured that a closed, even-dimensional manifold of positive sectional curvature has positive Euler characteristic. We show this under the additional assumption of an isometric $T^4$-action on the manifold, improving from previous theorems of Kennard, Wiemeler and Wilking assuming a $T^5$-action. More specifically, this is achieved by giving a rational cohomology classification of possible fixed point components. The main new tool is an improvement on the four-periodicity theorem originally developed by Kennard through the use of characteristic class theory. As a second application we give a rational cohomology classification of closed positively curved even-dimensional manifolds without odd rational cohomology that admit an isometric $T^6$-action.
We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number provided that the curvature operator of the second kind is $C(p,n)$-positive. Our curvature conditions become weaker as $p$ increases. For $p=\frac{n}{2}$ we have $C(p,n)= \frac{3n}{2} \frac{n+2}{n+4} $, and for $5 \leq p \leq \frac{n}{2}$ we exhibit a $C(p,n)$-positive algebraic curvature operator of the second kind with negative Ricci curvatures.