Anna Siffert

Publications


1. An infinite family of homogeneous polynomial self-maps of spheres, Manuscripta Mathematica, 2014, Volume 144, Issue 1-2, pp. 303-309.
Abstract

We provide an infinite family of homogeneous polynomial self-maps of spheres. Furthermore, we identify the gradient map of the Cartan-Münzner polynomial as a member of this infinite family and thus supply it with a geometric meaning.

2. Classification of isoparametric hypersurfaces in spheres with (g,m)=(6,1). Proc. Amer. Math. Soc. 144 (2016), 2217-2230, arXiv: 1503.04482.
Abstract

We classify the isospectral families $L(t)=\cos(t)L_0+\sin(t)L_1\in\mbox{Sym}(5,\mathbb{R})$, $t\in\mathbb{R}$, with $L_0=\mbox{diag}(\sqrt{3},1/\sqrt{3},0,-1/\sqrt{3},-\sqrt{3})$. Using this result we provide a classification of isoparametric hypersurfaces in spheres with $(g,m)=(6,1)$ and thereby give a simplified proof of the fact that any isoparametric hypersurface with $(g,m)=(6,1)$ is homogeneous. This result was first proven by Dorfmeister and Neher.

3. A new structural approach to isoparametric hypersurfaces in spheres. Annals of Global Analysis and Geometry 52 (2017), 425-456, arXiv:1410.6206.
Abstract

The classification of isoparametric hypersurfaces in spheres with four or six different principal curvatures is still not complete. In this paper we develop a structural approach that may be helpful for a classification. Instead of working with the isoparametric hypersurface family in the sphere, we consider the associated Lagrangian submanifold of the real Grassmannian of oriented 2-planes in $\mathbb{R}^{n+2}$. We obtain new geometric insights into classical invariants and identities in terms of the geometry of the Lagrangian submanifold.

4. Infinitely many new infinite families of harmonic maps between spheres. Journal of Differential Equations 260 (2016), pp. 2898-2925, arXiv: 1501.06157.
Abstract

For each of the spheres $\mathbb{S}^{n}$, $n\geq 5$, we construct a new infinite family of harmonic self-maps, and prove that their members have Brouwer degree $\pm1$ or $\pm3$. These self-maps are obtained by solving a singular boundary value problem. As an application we show that for each of the special orthogonal groups $\mathbb{SO}(4),\mathbb{SO}(5),\mathbb{SO}(6)$ and $\mathbb{SO}(7)$ there exists two infinite families of harmonic self-maps.

5. Harmonic self-maps of SU(3). J. Geom. Anal. 2018, 28, pp. 587--605, arXiv: 1512.07002.
Abstract

By constructing solutions of a singular boundary value problem we prove the existence of a countably infinite family of harmonic self-maps of $\mbox{SU}(3)$ with non-trivial, i.e. $\ne 0,±1$, Brouwer degree.

6. New equivariant harmonic maps between cohomogeneity one manifolds, joint work with Thomas Püttmann. To appear in Mathematische Annalen, arXiv: 1608.08669.
Abstract

We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups $\mathrm{SO}(4\ell+2)$, $\ell \ge 1$ with degree $-3$, of $\mathrm{SO}(8)$, $\mathrm{SO}(14)$ and $\mathrm{SO}(26)$ with degree $-5$ each, of $\mathrm{SO}(10)$ with degree $-7$, and of $\mathrm{SO}(14)$ with degree $-11$ by exhibiting linear solutions to non-linear singular boundary value problems.

7. Alexandrov Spaces with Integral Current Structure, joint work with Maree Jaramillo, Raquel Perales, Priyanka Rajan, Catherine Searle. To appear in Communications in Analysis and Geometry, arXiv: 1703.08195.
Abstract

We endow each closed, orientable Alexandrov space $(X,d)$ with an integral current structure $T$ of weight equal to 1 and $\partial T = 0$, in other words, we prove that $(X,d,T)$ is an integral current space. Combining this result with a result of Li and Perales, we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat limits agree.

8. A Dirichlet problem on balls and harmonic maps into spheres. Submitted, arXiv: 1709.02155.
Abstract

We provide infinitely many solutions of a Dirichlet problem on balls. Furthermore, we show the existence of infinitely many new smooth harmonic maps between spheres with nontrivial (i.e. not $0$ or $\pm 1$) degree.

9. A note on Kuttler-Sigillito's inequalities, joint work with Asma Hassannezhad. To appear in Annales mathématiques du Québec, arXiv:1709.09841.
Abstract

We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with $C^2$ boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kutller and Sigillito from subsets of $\mathbb{R}^2$ to the manifold setting.

10. The systole of large genus minimal surfaces in positive Ricci curvature, joint work with Henrik Matthiesen. Submitted, arXiv:1808.01157.
Abstract

We use Colding--Minicozzi lamination theory to study the systole of large genus minimal surfaces in ambient three manifold of positive Ricci curvature.

11. New Biharmonic functions on the compact Lie groups $\mbox{SO}(n)$, $\mbox{SU}(n)$, $\mbox{Sp}(n)$, joint work with Sigmundur Gudmundsson. To appear in Journal of Geometric Analysis, arXiv:1812.02777.
Abstract

We develop a new scheme for the construction of explicit complex-valued proper biharmonic functions on Riemannian Lie groups. We exploit this and manufacture many infinite series of uncountable families of new solutions on the special unitary group $\mathbb{SU}(n)$. We then show that the special orthogonal group $\mathbb{SO}(n)$ and the quaternionic unitary group $\mathbb{Sp}(n)$ fall into the scheme. As a by-product we obtain new harmonic morphisms on these groups. All the constructed maps are defined on open and dense subsets of the corresponding spaces.

12. Escobar constants of planar domains, joint work with Asma Hassannezhad. Submitted, arXiv:1905.07634
Abstract

We study the higher order Escobar constants on planar domains with non-empty boundary. For a domain in $\mathbb{R}^2$ with Lipschitz and piecewise smooth boundary, we introduce bounds which depend on the corner angles of this domain. Moreover, we conjecture that the $k$-th Escobar constant of such domains is always bounded above by the $k$-th Escobar constant of the disk.

13. Existence of metrics maximizing the first eigenvalue on non-orientable surfaces, joint work with Henrik Matthiesen. To appear in Journal of Spectral Theory, arXiv:1703.01264.
Abstract

We prove the existence of metrics maximizing the first eigenvalue normalized by area on closed, non-orientable surfaces assuming two spectral gap conditions. These spectral gap conditions are proved by the authors in their paper "Handle attachment and the normalized first eigenvalue".

14. Sharp asymptotics of the first eigenvalue on some degenerating surfaces, joint work with Henrik Matthiesen. Submitted, arXiv:1909.02974.
Abstract

We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial surface has some symmetries. If these symmetries are not present we show that the first eigenvalue normalized by area strictly decreases for the same range of parameters. These results are the main motivation for the construction in the authors paper "Handle attachment and the normalized first eigenvalue", where we show a monotonicity result for the normalized first eigenvalue without any symmetry assumptions.

15. Handle attachment and the normalized first eigenvalue, joint work with Henrik Matthiesen. Submitted, arXiv:1909.03105.
Abstract

We show that the first eigenvalue of a closed Riemannian surface normalized by the area can be strictly increased by attaching a cylinder or a cross cap. As a consequence we obtain the existence of maximizing metrics for the normalized first eigenvalue on any closed surface of fixed topological type. Since these metrics are induced by (possibly branched) minimal immersions into spheres, we find new examples of immersed minimal surfaces in spheres.

16. Proper $r$-harmonic functions on the Thurston geometries, joint work with Sigmundur Gudmundsson. Submitted, arXiv:1910.13477.
Abstract

For any positive natural number $r\in\mathbb{N}$, we construct new explicit proper $r$-harmonic functions on the well-known $3$-dimensional Thurston geometries $\mathbb{Sol}$, $\mathbb{Nil}$, $\mathbb{SL}_2$, $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$.