12 citations · 38 across the 16 of their papers we have counts for
19 papers
Unique continuation estimates on manifolds with Ricci curvature bounded below
Christian Rose, Martin Tautenhahn
We prove quantitative unique continuation estimates for relatively dense sets and spectral subspaces associated to small energies of Schrödinger operators on Riemannian manifolds w…
Anchored heat kernel upper bounds on graphs with unbounded geometry and anti-trees
Matthias Keller, Christian Rose
We derive Gaussian heat kernel bounds on graphs with respect to a fixed origin for large times under the assumption of a Sobolev inequality and volume doubling on large balls. The…
Gaussian upper bounds for heat kernels on graphs with unbounded geometry
Matthias Keller, Christian Rose
We prove large-time Gaussian upper bounds for continuous-time heat kernels of Laplacians on graphs with unbounded geometry. Our estimates hold for centers of large balls satisfying…
Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains
Xavier Ramos Olivé, Christian Rose, Lili Wang +1
We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of th…
Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
Alexander Dicke, Christian Rose, Albrecht Seelmann +1
Recent (scale-free) quantitative unique continuation estimates for spectral subspaces of Schrödinger operators are extended to allow singular potentials such as certain -funct…
Quantitative Sobolev extensions and the Neumann heat kernel for integral Ricci curvature conditions
Olaf Post, Xavier Ramos Olivé, Christian Rose
We prove the existence of Sobolev extension operators for certain uniform classes of domains in a Riemannian manifold with an explicit uniform bound on the norm depending only on t…
Eigenvalue estimates for Kato-type Ricci curvature conditions
Christian Rose, Guofang Wei
We prove that optimal lower eigenvalue estimates of Zhong-Yang type as well as a Cheng-type upper bound for the first eigenvalue hold on closed manifolds assuming only a Kato condi…
Spectrally positive Bakry-Émery Ricci curvature on graphs
Florentin Münch, Christian Rose
We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assum…
Almost positive Ricci curvature in Kato sense -- an extension of Myers' theorem
Christian Rose
It is shown that if the Kato constant of the negative part of the Ricci curvature below a positive level is small, then the volume of the corresponding manifold can be bounded abov…
Liouville property and non-negative Ollivier curvature on graphs
Jürgen Jost, Florentin Münch, Christian Rose
For graphs with non-negative Ollivier curvature, we prove the Liouville property, i.e., every bounded harmonic function is constant. Moreover, we improve Ollivier's results on conc…
Bounds on the first Betti number - an approach via Schatten norm estimates on semigroup differences
Marcel Hansmann, Christian Rose, Peter Stollmann
We derive new estimates for the first Betti number of compact Riemannian manifolds. Our approach relies on the Birman-Schwinger principle and Schatten norm estimates for semigroup…
Geometric and spectral estimates based on spectral Ricci curvature assumptions
Gilles Carron, Christian Rose
We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold for which the lowest eigenvalue of the Ricci tensor is such that the Schrödi…
Manifolds with Ricci curvature in the Kato class: heat kernel bounds and applications
Christian Rose, Peter Stollmann
We review recent results about heat kernel estimates based on Kato conditions on the negative part of the Ricci curvature.
Distance bounds for graphs with some negative Bakry-Émery curvature
Shiping Liu, Florentin Münch, Norbert Peyerimhoff +1
We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-po…
Li-Yau gradient estimate for compact manifolds with negative part of Ricci curvature in the Kato class
Christian Rose
This article shows that if the negative part of Ricci curvature lies in the Kato class, the heat kernel satisfies a Li-Yau type estimate. Additionally, using the resulting heat ker…
The Kato class on compact manifolds with integral bounds on the negative part of Ricci curvature
Christian Rose, Peter Stollmann
We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci…
Heat kernel upper bound on Riemannian manifolds with locally uniform Ricci curvature integral bounds
Christian Rose
This article shows that under locally uniformly integral bounds of the negative part of Ricci curvature the heat kernel admits a Gaussian upper bound for small times. This provides…
A quantitative Carleman estimate for second order elliptic operators
Ivica Nakić, Christian Rose, Martin Tautenhahn
We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued…
Multiscale unique continuation properties of eigenfunctions
Denis Borisov, Ivica Nakić, Christian Rose +2
Quantitative unique continuation principles for multiscale structures are an important ingredient in a number applications, e.g. random Schrödinger operators and control theory. We…