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Sebastian Boldt

Publications and source records attributed to Sebastian Boldt.

9 recordsLinked to original sources
arXiv2026

Analysis on surfaces with locally bounded integral curvature

We prove several analytic results on (possibly noncompact) complete singular surfaces having locally bounded integral curvature (in short: BIC surfaces). Regarding these as metric measure spaces with the 2-dimensional Hausdorff measure, we show that these are infinitesimally Hilbertian, locally doubling and satisfy a local Poincar\'e inequality. In particular, this entails the existence of a jointly H\"older continuous heat kernel for the Cheeger Laplacian. Assuming that the negative part of the curvature measure of a BIC surface satisfies a Dynkin-type condition, we show that the surface is bi-Lipschitz equivalent to a BIC surface with a lower bounded curvature measure, entailing global variants of the aforementioned results.

math.DG↗
arXiv2025

The speed measure and absolute continuity for curves in metric spaces

We define the speed measure $\nu$ for mappings $\gamma:I\to X$ from an interval to a metric space that are locally of bounded variation. We characterize continuity and absolute continuity of $\gamma$ in terms of $\nu$ and identify the Radon-Nikod\'ym derivative of $\nu$ with respect to Lebesgue measure as the metric speed of $\gamma$. In doing so we prove an extension of the Banach-Zaretsky theorem.

math.MG↗
arXiv2025

Lower bounds on the normal injectivity radius of hypersurfaces and bounded geometries on manifolds with boundary

We prove for the first time a pointwise lower estimate of the normal injectivity radius of an embedded hypersurface in an arbitrary Riemannian manifold. Main applications include: (i) a pointwise lower estimate of the graphing radius of a properly embedded hypersurface; (ii) the construction of metrics of bounded geometry on arbitrary manifolds with boundary; (iii) the equivalence of the classical (topological) notion of orientation with that of the geometric notion (in the sense of metric measure spaces) on arbitrary Riemannian manifolds with boundary. In addition, we prove that every manifold with boundary admits a metric with bounded geometry such that the boundary becomes convex. This result strengthens the justification of a recent notion of orientation on finite dimensional RCD spaces.

math.DG↗
arXiv2021

A Chern-Simons transgression formula for supersymmetric path integrals on spin manifolds

Earlier results show that the N = 1/2 supersymmetric path integral on a closed even dimensional Riemannian spin manifold (X,g) can be constructed in a mathematically rigorous way via Chen differential forms and techniques from non-commutative geometry, if one considers it as a current on the smooth loop space of X. This construction admits a Duistermaat-Heckman localization formula. In this note, fixing a topological spin structure on X, we prove that any smooth family of Riemannian metrics on X canonically induces a Chern-Simons current which fits into a transgression formula for the supersymmetric path integral. In particular, this result entails that the supersymmetric path integral induces a differential topological invariant on X, which essentially stems from the A-hat-genus of X.

math.DG↗
arXiv2020

Feynman-Kac formula for perturbations of order $\leq 1$ and noncommutative geometry

Let $Q$ be a differential operator of order $\leq 1$ on a complex metric vector bundle $\mathscr{E}\to \mathscr{M}$ with metric connection $\nabla$ over a possibly noncompact Riemannian manifold $\mathscr{M}$. Under very mild regularity assumptions on $Q$ that guarantee that $\nabla^{\dagger}\nabla/2+Q$ generates a holomorphic semigroup $\mathrm{e}^{-zH^{\nabla}_{Q}}$ in $\Gamma_{L^2}(\mathscr{M},\mathscr{E})$ (where $z$ runs through a complex sector which contains $[0,\infty)$), we prove an explicit Feynman-Kac type formula for $\mathrm{e}^{-tH^{\nabla}_{Q}}$, $t>0$, generalizing the standard self-adjoint theory where $Q$ is a self-adjoint zeroth order operator. For compact $\mathscr{M}$'s we combine this formula with Berezin integration to derive a Feynman-Kac type formula for an operator trace of the form $$ \mathrm{Tr}\left(\widetilde{V}\int^t_0\mathrm{e}^{-sH^{\nabla}_{V}}P\mathrm{e}^{-(t-s)H^{\nabla}_{V}}\mathrm{d} s\right), $$ where $V,\widetilde{V}$ are of zeroth order and $P$ is of order $\leq 1$. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat-Heckmann localization formula on the loop space of such a manifold.

math-ph↗
arXiv2020

Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators

Given a noncompact spin manifold $M$ with a fixed topological spin structure and two complete Riemannian metrics $g$ and $h$ on $M$ with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave operators $\mathscr{W}_{\pm}(D_h, D_g, I_{g,h})$ and $\mathscr{W}_{\pm}(D_h^2, D^2_g, I_{g,h})$, where $I_{g,h}$ is the canonically given unitary map between the underlying $L^2$-spaces of spinors. This criterion does not involve any injectivity radius assumptions and leads to a criterion for the stability of the absolutely continuous spectrum of a Dirac operator and its square under a Ricci flow.

math.DG↗
arXiv2015

Properties of the Dirac spectrum on three dimensional lens spaces

We present a spectral rigidity result for the Dirac operator on lens spaces. More specifically, we show that each homogeneous lens space and each three dimensional lens space $L(q;p)$ with $q$ prime is completely characterized by its Dirac spectrum in the class of all lens spaces.

math.DG↗
arXiv2014

An explicit formula for the Dirac multiplicities on lens spaces

We present a new description of the spectrum of the (spin-) Dirac operator $D$ on lens spaces. Viewing a spin lens space $L$ as a locally symmetric space $Γ\backslash \operatorname{Spin}(2m)/\operatorname{Spin}(2m-1)$ and exploiting the representation theory of the $\operatorname{Spin}$ groups, we obtain explicit formulas for the multiplicities of the eigenvalues of $D$ in terms of finitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions of spectral geometry, we prove with the tools developed that neither spin structures nor isometry classes of lens spaces are spectrally determined by giving infinite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer.

math.DG↗