arXiv subjects

Batu Güneysu

Publications and source records attributed to Batu Güneysu.

At least 19 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 Poincaré inequality. In particular, this entails the existence of a jointly Hölder 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↗

arXiv2026

Multigraphs and Time Ordered Isserlis-Wick formulae

Given a m-dimensional Gaussian process and polynomial m variables with real coefficients, we calculate the induced path odered exponenial in two different ways: one is purely algebraic in spirit and the other one is diagrammatic in spirit and uses multigraph labelings (and is inspired by the use of Feynman diagrams in quantum field theory).

math.PR↗

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↗

arXiv2025

Strongly continuous fields of operators over varying Hilbert spaces

After introducing a natural notion of continuous fields of locally convex spaces, we establish a new theory of strongly continuous families of possibly unbounded self-adjoint operators over varying Hilbert spaces. This setting allows to treat operator families defined on bundles of Hilbert spaces that are not locally trivial (such as e.g.~the tangent bundle of Wasserstein space), without referring to identification operators at all.

math.FA↗

arXiv2024

A note on the scattering theory of Kato-Ricci manifolds

In this note we prove a new $L^1$ criterion for the existence and completeness of the wave operators corresponding to the Laplace-Beltrami operators corresponding to two Riemannian metrics on a fixed noncompact manifold. Our result relies on recent estimates on the heat semigroup and its derivative, that are valid if the negative part of the Ricci curvature is in the Kato class - so called Kato-Ricci manifolds.

math.SP↗

arXiv2024

Fermionic Dyson expansions and stochastic Duistermaat-Heckman localization on loop spaces

Given a self-adjoint operator $H\geq 0$ and (appropriate) densely defined and closed operators $P_{1},\dots, P_{n}$ in a Hilbert space $\mathscr{H}$, we provide a systematic study of bounded operators given by iterated integrals \begin{align}\label{oh} \int_{\{ 0\leq s_1\leq \dots\leq s_n\leq t\}}\mathrm{e}^{-s_1H}P_{1}\mathrm{e}^{-(s_2-s_1)H}P_{2}\cdots \mathrm{e}^{-(s_n-s_{n-1})H}P_{n} \mathrm{e}^{-(t-s_n)H}\, \mathrm{d} s_{1} \ldots \mathrm{d} s_{n},\quad t>0. \end{align} These operators arise naturally in noncommutative geometry and the geometry of loop spaces. Using Fermionic calculus, we give a natural construction of an enlarged Hilbert space $\mathscr{H}^{(n)}$ and an analytic semigroup $\mathrm{e}^{-t (H^{(n)}+P^{(n)} )}$ thereon, such that $\mathrm{e}^{-t (H^{(n)}+P^{(n)} )}$ composed from the left with (essentially) a Fermionic integration gives precisely the above iterated operator integral. This formula allows to establish important regularity results for the latter, and to derive a stochastic representation for it, in case $H$ is a covariant Laplacian and the $P_{j}$'s are first-order differential operators. Finally, with $H$ given as the square of the Dirac operator on a spin manifold, this representation is used to derive a stochastic refinement of the Duistermaat-Heckman localization formula on the loop space of a spin manifold.

math.DG↗

arXiv2024

Neumann cut-offs and essential self-adjointness on complete Riemannian manifolds with boundary

We generalize some fundamental results for noncompact Riemannian manfolds without boundary, that only require completeness and no curvature assumptions, to manifolds with boundary: let $M$ be a smooth Riemannian manifold with boundary $\partial M$ and let $\hat{C}^\infty_c(M)$ denote the space of smooth compactly supported cut-off functions with vanishing normal derivative, Neumann cut-offs. We show, among other things, that under completeness: - $\hat{C}^\infty_c(M)$ is dense in $W^{1,p}(\mathring{M})$ for all $p\in (1,\infty)$; this generalizes a classical result by Aubin [2] for $\partial M=\emptyset$. - $M$ admits a sequence of first order cut-off functions in $\hat{C}^\infty_c(M)$; for $\partial M=\emptyset$ this result can be traced back to Gaffney [7]. - the Laplace-Beltrami operator with domain of definition $\hat{C}^\infty_c(M)$ is essentially self-adjoint; this is a generalization of a classical result by Strichartz [20] for $\partial M=\emptyset$.

math.DG↗

arXiv2024

A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin

Given a strongly local Dirichlet space and $λ\geq 0$, we introduce a new notion of $λ$--subharmonicity for $L^1_\loc$--functions, which we call \emph{local $λ$--shift defectivity}, and which turns out to be equivalent to distributional $λ$--subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional $L^q$-solutions of $Δf\leq f$ for complete Riemannian manifolds.

math.AP↗

arXiv2024

Asymptotic Equivalence of Identification Operators in Geometric Scattering Theory

Given two measures $μ_1$ and $μ_2$ on a measurable space $X$ such that $dμ_2=ρ_{1,2} \, dμ_1$ for some bounded measurable function $ρ_{1,2}:X\to (0,\infty)$, there exist two natural identification operators $J_{1,2},\tilde{J}_{1,2}:L^2(X,μ_1)\to L^2(X,μ_2)$, namely the unitary $J_{1,2}ψ:=ψ/\sqrt{ρ_{1,2}}$ and the trivial $\tilde{J}_{1,2}ψ:=ψ$. Given self-adjoint semibounded operators $H_j$ on $L^2(X,μ_j)$, $j=1,2$, we prove a natural criterion in a topologic setting for the equality of the two-Hilbert-space wave operators $W_\pm(H_2,H_1;J_{1,2})$ and $W_\pm(H_2,H_1;\tilde{J}_{1,2})$, by showing that $J_{1,2}-\tilde{J}_{1,2}$ are asymptotically $H_1$-equivalent in the sense of Kato. It turns out that this criterion is automatically satisfied in typical situations on Riemannian manifolds and weighted infinite graphs in which one has the existence of completeness $W_\pm(H_2,H_1;\tilde{J}_{1,2})$ (and thus a-posteriori of $W_\pm(H_2,H_1;J_{1,2}))$.

math-ph↗

arXiv2023

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↗

arXiv2023

Locally convex aspects of the Kato and the Dynkin class on manifolds

We consider the Kato and the Dynkin class and their local counterparts on a smooth Riemannian manifold as Fréchet spaces. Based on recent results by Carron, Mondello and Tewodrose we show that for a Riemannian manifold $(X,g)$ of dimension $m\geq 2$ with spectral negative part $σ^-_g$ of the Ricci curvature in $L^q_{ϕ_g}(X,g)+L^\infty(X,g)$ for some $q>m/2$, the function $σ^-_g$ is in the Kato class of $(X,g)$ if and only if $(X,g)$ satisfies a Gaussian upper heat kernel bound for small times and is locally volume doubling. Here $L^q_{ϕ_g}(X,g)$ is the $L^q$-space which is weighted with the inverse volume function. By establishing a localization result for the Dynkin norm, we prove that the local Kato class and the local Dynkin class do not depend on the chosen Riemannian metric and thus can be defined as Fréchet spaces on arbitrary smooth manifolds. Moreover, we prove that smooth compactly supported functions are dense in the local Kato class and we use this result to prove that Schrödinger semigroups with Kato decomposable potentials are space-time continuous.

math.DG↗

arXiv2022

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 $Γ_{L^2}(\mathscr{M},\mathscr{E})$ (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↗

arXiv2021

Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

We prove that if the Ricci tensor $\mathrm{Ric}$ of a geodesically complete Riemannian manifold $M$, endowed with the Riemannian distance $\mathsf{d}$ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous function $k\colon M\to\mathbb{R}$ whose negative part $k^-$ satisfies, for every $t>0$, the exponential integrability condition \begin{equation*} \sup_{x\in M} \mathbb{E}\big[\mathrm{e}^{\int_0^t k^-(\mathsf{b}_r^x)/2\,\mathrm{d} r}\,1_{\{t < ζ^x\}}\big] < \infty, \end{equation*} then the lifetime $ζ^x$ of Brownian motion $\mathsf{b}^x$ on $M$ starting in any $x\in M$ is a.s. infinite. This assumption on $k$ holds if $k^-$ belongs to the Kato class of $M$. We also derive a Bismut-Elworthy-Li derivative formula for $\nabla \mathsf{P}_tf$ for every $f\in L^\infty(M)$ and $t>0$ along the heat flow $(\mathsf{P}_t)_{t\geq 0}$ with generator $Δ/2$, yielding its $L^\infty$-$\mathrm{Lip}$-regularization as a corollary. Moreover, given the stochastic completeness of $M$, but without any assumption on $k$ except continuity, we prove the equivalence of lower boundedness of $\mathrm{Ric}$ by $k$ to the existence, given any $x,y\in M$, of a coupling $(\mathsf{b}^x,\mathsf{b}^y)$ of Brownian motions on $M$ starting in $(x,y)$ such that a.s., \begin{equation*} \mathsf{d}\big(\mathsf{b}_t^x,\mathsf{b}_t^y\big) \leq \mathrm{e}^{-\int_s^t \underline{k}(\mathsf{b}_r^x,\mathsf{b}_r^y)/2\,\mathrm{d} r}\,\mathsf{d}\big(\mathsf{b}_s^x,\mathsf{b}_s^y\big) \end{equation*} holds for every $s,t\geq 0$ with $s\leq t$, involving the "average" $\underline{k}(u,v) := \inf_γ\int_0^1 k(γ_r)\,\mathrm{d} r$ of $k$ along geodesics from $u$ to $v$. Our results generalize to weighted Riemannian manifolds, where the Ricci curvature is replaced by the corresponding Bakry-Émery Ricci tensor.

math.PR↗

arXiv2021

Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms

With $\vecΔ_j\geq 0$ is the uniquely determined self-adjoint realization of the Laplace operator acting on $j$-forms on a geodesically complete Riemannian manifold $M$ and $\nabla$ the Levi-Civita covariant derivative, we prove amongst other things a Li-Yau type heat kernel bound for $\nabla \mathrm{e}^{ -t\vecΔ_j }$, if the curvature tensor of $M$ and its covariant derivative are bounded, an exponentially weighted $L^p$ bound for the heat kernel of $\nabla \mathrm{e}^{ -t\vecΔ_j }$, if the curvature tensor of $M$ and its covariant derivative are bounded, that $\nabla \mathrm{e}^{ -t\vecΔ_j }$ is bounded in $L^p$ for all $1\leq p<\infty$, if the curvature tensor of $M$ and its covariant derivative are bounded, and a second order Davies-Gaffney estimate (in terms of $\nabla$ and $\vecΔ_j$) for $\mathrm{e}^{ -t\vecΔ_j }$ for small times, if the $j$-th degree Bochner-Lichnerowicz potential $V_j=\vecΔ_j-\nabla^{\dagger}\nabla$ of $M$ is bounded from below (where $V_1=\mathrm{Ric}$), which is shown to fail for large times if $V_j$ is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform $\nabla (\vecΔ_j+κ)^{-1/2}$ in $L^p$ for all $1\leq p<\infty$ (which we prove for $1\leq p\leq 2$), and explain its implications to geometric analysis, such as the $L^p$-Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for $\nabla \mathrm{e}^{ -t\vecΔ_j }$.

math.AP↗

arXiv2021

The Chern Character of θ-summable Fredholm Modules over dg Algebras and Localization on Loop Space

We introduce the notion of a {\vartheta}-summable Fredholm module over a locally convex dg algebra Ω and construct its Chern character as a cocycle on the entire cyclic complex of Ω, extending the construction of Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this Chern character, we prove an index theorem involving an abstract version of a Bismut-Chern character constructed by Getzler, Jones and Petrack in the context of loop spaces. Our theory leads to a rigorous construction of the path integral for N=1/2 supersymmetry which satisfies a Duistermaat-Heckman type localization formula on loop space.

math.KT↗

arXiv2021

Hölder estimates for magnetic Schrödinger semigroups in $\mathbb{R}^{d}$ from mirror coupling

We use the mirror coupling of Brownian motion to show that under a $β\in (0,1)$-dependent Kato type assumption (which is satisfied under a suitable $L^q$-assumption on the electro-magnetic potential, where $q$ depends on $β$ and the dimension $d$) on the possibly nonsmooth electro-magnetic potential, the corresponding magnetic Schrödinger semigroup in $\mathbb{R}$ has a global $L^{p}$-to-$C^{0,β}$ Hölder smoothing property for all $p\in [1,\infty]$, in particular all eigenfunctions are uniformly $β$-Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly $β$-Hölder continuous under weak $L^q$-assumptions on the magnetic potential.

math-ph↗

arXiv2020

$\mathrm{RCD}^*(K,N)$ spaces and the geometry of multi-particle Schrödinger semigroups

With $(X,\mathfrak{d},\mathfrak{m})$ an $\mathrm{RCD}^*(K,N)$ space for some $K\in\mathbf{R}$, $N\in [1,\infty)$, let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given $α\in [0,1]$ we introduce the $α$-Kato class of potentials on $(X,\mathfrak{d},\mathfrak{m})$, and given a potential $V:X\to \mathbf{R}$ in this class, with $H_V$ the natural self-adjoint realization of the Schrödinger operator $H+V$ in $L^2(X,\mathfrak{m})$, we use Brownian coupling methods and perturbation theory to prove that for all $t>0$ there exists an explicitly given constant $A(V,K,α,t)<\infty$, such that for all $Ψ\in L^{\infty}(X,\mathfrak{m})$, $x,y\in X$ one has \begin{align*} \big|e^{-tH_V}Ψ(x)-e^{-tH_V}Ψ(y)\big|\leq A(V,K,α,t) \|Ψ\|_{L^{\infty}}\mathfrak{d}(x,y)^α. \end{align*} In particular, all $L^{\infty}$-eigenfunctions of $H_V$ are globally $α$-Hölder continuous. This result applies to multi-particle Schrödinger semigroups and, by the explicitness of the Hölder constants, sheds some light into the geometry of such operators.

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↗