Searcharxiv⌕ Search

arXiv subjects

Anna Siffert

Publications and source records attributed to Anna Siffert.

At least 19 recordsLinked to original sources
arXiv2026

New Homogeneous Einstein Metrics from Detection

We use a detection method for invariant Einstein equations to construct and distinguish asymmetric homogeneous Einstein metrics on several families of compact homogeneous spaces. The method retains a geometrically meaningful symmetry-breaking parameter, eliminates the remaining Einstein equations, and then reconstructs positive metrics from the resulting detector. We prove a sharp threshold in a two-factor symplectic family, construct two asymmetric Einstein metrics on each admissible space $\SU(n)^3/\Delta\SO(n)$, obtain a sharp threshold for $\SU(2n)^3/\Delta\Sp(n)$, and construct asymmetric metrics on three exceptional three-factor families. The four-factor case exhibits a further phenomenon. On \[ M_9=\SU(18)^4/\Delta\Sp(9) \] the two symmetry types $3+1$ and $2+2$ are both realized: up to homothety and internal block permutations, the corresponding fixed families contain exactly three asymmetric Einstein metrics, two with three equal horizontal scales and one distinct scale (type $3+1$), and one with two equal pairs of horizontal scales (type $2+2$). They are pairwise non-isometric, Riemannian irreducible, and not naturally reductive. The $2+2$ metric is globally unique in the full positive $2+2$ fixed family up to block interchange.

math.DG↗
arXiv2026

Quantized Einstein Metrics on $S^7$ with $SU(3)$-Symmetry

We prove that the standard cohomogeneity-one action of \(SU(3)\) on \(S^7\), with principal orbit the Wallach flag manifold \(SU(3)/T^2\), admits infinitely many invariant Einstein metrics. The proof uses a detection approach to the Einstein boundary-value problem. Starting from one singular orbit, we follow the Einstein solutions only to a canonical hypersurface and measure there how far they are from closing smoothly. When the singular-orbit scale becomes small, this closing problem is governed by a Ricci-flat limiting solution. The linearisattion about its limiting cone has an oscillatory mode. As the scale shrinks, this oscillation repeatedly changes the sign of the closing error, producing infinitely many parameter values for which the metric closes. The closing scales satisfy an asymptotic logarithmic quantization law, with successive ratio tending to \(e^{-2\pi/\sqrt{15}}\). We also determine the geometry of the resulting sequence. Away from the two singular orbits the metrics converge to the singular sine cone over the non-normal Einstein metric on \(SU(3)/T^2\), while after rescaling by the square of the singular-orbit scale at either end they converge to the same complete Ricci-flat threshold metric. Their curvature is of order \(b_n^{-2}\), so the logarithmic phase law induces a corresponding quantization of the focal curvature scale.

math.DG↗
arXiv2026

Response Geometry for Einstein metrics

We develop a response geometry for smooth parameterized families of Einstein metrics equipped with canonical probe data. The differential of the probe observations defines a response bundle morphism; pulling back a fixed metric on the observation bundle gives a positive-semidefinite response tensor whose kernel is exactly the space of first-order invisible parameter directions. On the complete-response locus this tensor is Riemannian and yields local rigidity and quantitative reconstruction estimates whenever the response is locally realized by an observation map. We study rank-defect loci, the associated quotient geometry, Gram and conditioning operators, response volume, and covariant variation formulas for simple response eigenvalues. Under an explicit realizability hypothesis we also show that a single scalar observation can be chosen to detect every direction of a finite-dimensional parameter space. These constructions are applied to the harmonic-probe package arising from the Einstein Detection Principle. A finite-dimensional interval-matrix example illustrates the quantitative formulas, while marked unit-volume flat two-tori provide a fully explicit model: three harmonic-energy observations reconstruct the marked metric globally, the induced response metric admits an explicit positively curved hyperboloid realization, and its conditioning deteriorates toward the cusp.

math.DG↗
arXiv2026

Einstein Metrics and Equivariant Harmonic Maps: The Einstein Detection Principle

We develop a new analytical framework for the local deformation theory of compact cohomogeneity-one Einstein metrics. The framework combines an intrinsic reformulation of the Einstein boundary-value problem with an auxiliary theory of equivariant harmonic maps and leads to what we call the Einstein Detection Principle. The auxiliary harmonic maps generate a rich variational, elliptic and spectral framework while introducing no additional Einstein degrees of freedom.

math.DG↗
arXiv2026

The Nullity of a Family of Proper Biharmonic Maps via Elliptic Curves

We prove a conjecture of Montaldo, Oniciuc and Ratto concerning the nullity of a family of proper biharmonic maps from the flat two-torus to the round two-sphere. The proof reveals an unexpected connection between spectral geometry and arithmetic geometry. We show that the vanishing of a mixed Fourier eigenvalue produces a rational point on an explicitly defined affine quartic. By constructing an explicit polynomial isomorphism with an elliptic curve over $\Q$, the problem is reduced to the determination of a Mordell--Weil group. This yields a complete description of the rational points on the spectral curve and shows that none satisfies the positivity conditions required for a mixed Fourier mode. As a consequence, the mixed eigenvalues never vanish, confirming the Montaldo--Oniciuc--Ratto conjecture and proving that the nullity of every map in the family is equal to $5$.

math.DG↗
arXiv2026

A framework for producing harmonic maps

We present a general framework for producing families of harmonic maps from the unit ball into the Euclidean sphere starting from a fixed harmonic map. In particular, our analysis provides a theoretical background for a generalized radial projection which was recently introduced by Nakauchi. Our approach is based on work of Toth who established a machinery that generates eigenmaps between spheres from a given eigenmap. Finally, we point out how these families of harmonic maps can be used to construct solutions to associated variational problems such as intrinsic biharmonic and triharmonic maps, $p$-harmonic maps as well as extrinsic polyharmonic maps.

math.DG↗
arXiv2025

Stable proper biharmonic maps in Euclidean spheres

We construct an explicit family of stable proper weak biharmonic maps from the unit ball $B^m$, $m\geq 5$, to Euclidean spheres. To the best of the authors knowledge this is the first example of a stable proper weak biharmonic map from at compact domain. To achieve our result we first establish the second variation formula of the bienergy for maps from the unit ball into a Euclidean sphere. Employing this result, we examine the stability of the proper weak biharmonic maps $q:B^m\to\mathbb{S}^{m^{\ell}}$, $m,\ell\in\mathbb{N}$ with $\ell\leq m$, which we recently constructed in \cite{BS25} and thus deduce the existence of an explicit family of stable proper biharmonic maps to Euclidean spheres.

math.DG↗
arXiv2025

On the stability of the generalized equator map

The energy, the $p$-energy ($p\in\mathbb{R}$ with $p\geq 2$) and the extrinsic $k$-energy ($k\in\mathbb{N}$) for maps between Riemannian manifolds are central objects in the geometric calculus of variations. The equator map from the unit ball to the Euclidean sphere provides an explicit critical point of all aforementioned energy functionals. During the last four decades many researchers studied the stability of this particular map when considered as a critical point of one of these energy functionals, see e.g. \cite{MR4436204}, \cite{MR705882}. Recently, Nakauchi \cite{MR4593065} introduced a generalized radial projection map and proved that this map is both a critical point of the energy and a critical point of the $p$-energy. This generalized radial projection map gives rise to a generalized equator map which is also both a critical point of the energy and a critical point of the $p$-energy. In this manuscript we first of all show that the generalized equator map is also a critical point of the extrinsic $k$-energy. Then, the main focus is a detailed stability analysis of this map, considered as a critical point of both the extrinsic $k$-energy and the $p$-energy. We thus establish a number of interesting generalizations of the classical (in)stability results of J\"ager and Kaul \cite{MR705882}.

math.DG↗
arXiv2025

A family of triharmonic maps to spheres in all dimensions greater than two

We present a construction method for triharmonic maps to spheres. In particular, we show that for any $m\in\mathbb{N}$ with $m\geq 3$ there exists a triharmonic map from $\mathbb{R}^m\setminus\{0\}$ into a round sphere. In addition, we provide a construction method for proper $r$-harmonic maps between spheres based on a suitable deformation of eigenmaps.

math.DG↗
arXiv2024

Magnetic Steklov problem on surfaces

The magnetic Dirichlet-to-Neumann map encodes the voltage-to-current measurements under the influence of a magnetic field. In the case of surfaces, we provide precise spectral asymptotics expansion (up to arbitrary polynomial power) for the eigenvalues of this map. Moreover, we consider the inverse spectral problem and from the expansion we show that the spectrum of the magnetic Dirichlet-to-Neumann map, in favourable situations, uniquely determines the number and the length of boundary components, the parallel transport and the magnetic flux along boundary components. In general, we show that the situation complicates compared to the case when there is no magnetic field. For instance, there are plenty of examples where the expansion does $\textit{not}$ detect the number of boundary components, and this phenomenon is thoroughly studied in the paper.

math.AP↗
arXiv2024

Construction of $r$-harmonic submanifolds

We provide a construction method for biharmonic submanifolds in cohomogeneity one manifolds. In particular, we give new examples of biharmonic submanifolds and study the normal index of these submanifolds. We use this strategy to construct metrics on the sphere admitting biharmonic non-minimal hypersurfaces with three distinct principal curvatures. Finally, we perform a similar study of $r$-harmonic submanifolds of cohomogeneity one manifolds.

math.DG↗
arXiv2024

Global eigenfamilies on closed manifolds

We study globally defined $(\lambda,\mu)$-eigenfamilies on closed Riemannian manifolds. Among others, we provide (non-)existence results for such eigenfamilies, examine their topological properties and classify $(\lambda,\mu)$-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↗
arXiv2022

Infinite families of harmonic self-maps of ellipsoids in all dimensions

We prove that for given $k\in\mathbb{N}$, $k\geq 3$, $d\in\mathbb{N}$ and each $a\in\mathbb{R}^{*}$ with \begin{align*} a^2<4d (d+k-2)(k-2)^{-2} \end{align*} the ellipsoid $E_a:=\{x\in\mathbb{R}^k\,\lvert\,a^{-2}x_1^2+x_2^2+\ldots+x_k^2=1\}$ admits infinitely many harmonic self-maps.

math.DG↗
arXiv2022

On p-harmonic self-maps of spheres

In this manuscript we study rotationally $p$-harmonic maps between spheres. We prove that for $p\in\mathbb{N}$ given, there exist infinitely many $p$-harmonic self-maps of $\mathbb{S}^m$ for each $m\in\mathbb{N}$ with $p m$, the identity map of $\mathbb{S}^m$ is stable when interpreted as a $p$-harmonic self-map of $\mathbb{S}^m$.

math.DG↗