arXiv subjects

Dominik Schlagenhauf

Publications and source records attributed to Dominik Schlagenhauf.

4 recordsLinked to original sources

arXiv2026

Improved Morse Index Stability for Sequences of Harmonic Maps from Degenerating Riemann Surfaces

We study the stability of the extended Morse index, defined as the number of negative and zero eigenvalues of the Jacobi operator, for sequences of harmonic maps on degenerating Riemann surfaces. As the conformal structure approaches the boundary of moduli space, collar collapse creates major analytical challenges. We analyze the second variation of the energy under these degenerations and identify conditions ensuring upper semicontinuity of the extended Morse index. Refining earlier results of the first and second authors in [7], we obtain sharper control of the spectrum of the Jacobi operator on degenerating domains. A key new aspect is the explicit contribution of geodesics arising as limits of the images of degenerating collars. We show that these neck regions converge to geodesic segments whose Morse index contributes nontrivially to the limiting extended index.

math.DG↗

arXiv2026

A Discontinuous Solution of the Critical n-Laplace System with Antisymmetric Potential

Let $n>2$. We construct a map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $B^n \setminus \{0\}$, together with an antisymmetric potential $Ω\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)$ such that $-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=Ω\cdot |\nabla U|^{n-2}\nabla U$ in $D'(B^n)$. This gives a negative answer to a regularity question posed by Rivière. Our potential admits the Lorentz-space regularity $Ω\in \bigcap_{q>2}L^{(n,q)} \setminus L^{(n,2)}$. In addition for given $1<p<\infty$ we can enforce $\nabla U \in L^{(n,p)}$ but $\nabla U \notin L^{(n,1)}$. The construction does not give a counterexample to regularity for weakly $n$-harmonic maps or for higher-dimensional $H$-systems. The example was generated by ChatGPT 5.6 Sol on August 5, 2026. The work itself was written by the author and thoroughly reviewed to ensure its correctness.

math.AP↗

arXiv2025

Stability of the Morse Index for the $p$-harmonic Approximation of Harmonic Maps into Homogeneous Spaces

In the joint work of the author with Da Lio and Rivière (Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as $p\searrow2$. These are critical points of the energy $E_p(u) := \int_Σ\left( 1+|\nabla u|^2\right)^{p/2} \ dvol_Σ,$ where $u:Σ\rightarrow S^n$ is a map from a closed Riemannian surface $Σ$ into a sphere $ S^n$. In this paper we extend the results found in our previous work to the case of Sacks-Uhlenbeck sequences into homogeneous spaces, by incorporating the strategy introduced by Bayer and Roberts (Energy identity and no neck property for $ε$-harmonic and $α$-harmonic maps into homogeneous target manifolds). In the spirit of the work of Da Lio, Gianocca and Rivière (Morse Index Stability for Critical Points to Conformally invariant Lagrangians), we show in this setting the upper semicontinuity of the Morse index plus nullity and an improved pointwise estimate of the gradient in the neck regions around blow up points.

math.AP↗

arXiv2025

Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere

In this paper we consider sequences of $p$-harmonic maps, $p>2$, from a closed Riemann surface $Σ$ into the $n$-dimensional sphere $\mathbb{S}^n$ with uniform bounded energy. These are critical points of the energy $E_p(u) :=\int_Σ\left( 1+|{\nabla u}|^2\right)^{p/2} \ dvol_Σ.$ Our two main results are an improved pointwise estimate of the gradient in the neck regions around blow up points and the proof that the necks are asymptotically not contributing to the negativity of the second variation of the energy $E_p.$ This allows us, in the spirit of the paper of the first and second authors in collaboration with M. Gianocca {\em Morse index stability for critical points to conformally invariant Lagrangians}, to show the upper semicontinuity of the Morse index plus nullity for sequences of $p$-harmonic maps into a sphere.

math.AP↗