arXiv subjects

Reiner M. Schätzle

Publications and source records attributed to Reiner M. Schätzle.

5 recordsLinked to original sources

arXiv2022

Positivity for the clamped plate equation under high tension

In this article we consider positivity issues for the clamped plate equation with high tension $γ>0$. This equation is given by $Δ^2u - γΔu=f$ under clamped boundary conditions. Here we show, that given a positive $f$, i.e. upwards pushing, we find a $γ_0>0$ such that for all $γ\geq γ_0$ the bending $u$ is indeed positive. This $γ_0$ only depends on the domain and the ratio of the $L^1$ and $L^\infty$ norm of $f$. In contrast to a recent result by Cassani&Tarsia, our approach is valid in all dimensions.

math.AP↗

arXiv2017

The umbilic set of Willmore surfaces

It is well known that the umbilic points of minimal surfaces in spaces of constant sectional curvature consist only of isolated points unless the surface is totally umbilic on some connected component, as for example the Hopf form is holomorphic. In this note, we prove that on Willmore surfaces in codimension one the umbilic set is locally a one dimensional real-analytic manifold without boundary or an isolated point.

math.DG↗

arXiv2015

Conformal Willmore Tori in $\mathbb{R}^4$

For every two-dimensional torus $T^2$ and every $k\in \mathbb{N}$, $k\ge 3$, we construct a conformal Willmore immersion $f:T^2\to \mathbb{R}^4$ with exactly one point of density $k$ and Willmore energy $4πk$. Moreover, we show that the energy value $8π$ cannot be attained by such an immersion. Additionally, we characterize the branched double covers $T^2\to S^2 \times \{0\}$ as the only branched conformal immersions, up to Möbius transformations of $\mathbb{R}^4$, from a torus into $\mathbb{R}^4$ with at least one branch point and Willmore energy $8π$. Using a perturbation argument in order to regularize a branched double cover, we finally show that the infimum of the Willmore energy in every conformal class of tori is less than or equal to $8π$.

math.DG↗

arXiv2015

Rigidity and non-rigidity results for conformal immersions

In this paper we show a quantitative rigidity result for the minimizer of the Willmore functional among all projective planes in $\mathbb{R}^n$ with $n\ge 4$. We also construct an explicit counterexample to a corresponding rigidity result in codimension one, by showing that an Enneper surface might split-off during a blow-up process. For conformal immersions of spheres with large enough Willmore energies, we construct explicit counterexamples to a quantitative rigidity result and this complements the recently obtained rigidity results in [LaNg13].

math.DG↗

arXiv2014

Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

In [dLMu05], DeLellis and Müller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ Σ\subseteq \mathbb{R}^3\ $ with area normalized to $\ {\cal H}^2(Σ) = 4 π $, it was shown that $\ \parallel A_Σ- id \parallel_{L^2(Σ)} \leq C \parallel A^0_Σ\parallel_{L^2(Σ)}\ $, and building on this, closeness of $\ Σ $ to a round sphere in $\ W^{2,2}\ $ was established, when $\ \parallel A^0_Σ\parallel_{L^2(Σ)}\ $ is small. This was supplemented in [dLMu06] by giving a conformal parametrization $\ S^2 \stackrel{\approx}{\longrightarrow} Σ $ with small conformal factor in $\ L^\infty\ $, again when $\ \parallel A^0_Σ\parallel_{L^2(Σ)}\ $ is small. In this article, we extend these results to arbitrary codimension. In contrast to [dLMu05], our argument is not based on the equation of Mainardi-Codazzi, but instead uses the monotonicity formula for varifolds.

math.DG↗