Search results for "Mathematics::Symplectic Geometry"
showing 10 items of 184 documents
Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds
2017
We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.
Elementary deformations and the hyperK\"ahler-quaternionic K\"ahler correspondence
2014
The hyperK\"ahler-quaternionic K\"ahler correspondence constructs quaternionic K\"ahler metrics from hyperK\"ahler metrics with a rotating circle symmetry. We discuss how this may be interpreted as a combination of the twist construction with the concept of elementary deformation, surveying results of our forthcoming paper. We outline how this leads to a uniqueness statement for the above correspondence and indicate how basic examples of c-map constructions may be realised in this context.
Quillen superconnections and connections on supermanifolds
2013
Given a supervector bundle $E = E_0\oplus E_1 \to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan-Koszul supermanifold $(M;\Omega (M))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections.
Boundary rigidity for Randers metrics
2021
If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium.
The linearized Calder\'on problem on complex manifolds
2018
In this note we show that on any compact subdomain of a K\"ahler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calder\'on problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of K\"ahler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot by treated by standard methods for the Calder\'on problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends resu…
Bounded geometry, growth and topology
2010
We characterize functions which are growth types of Riemannian manifolds of bounded geometry.
Evolution by mean curvature flow of Lagrangian spherical surfaces in complex Euclidean plane
2016
We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.
Universal infinitesimal Hilbertianity of sub-Riemannian manifolds
2019
We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.
L2-torsion of hyperbolic manifolds
1998
The L^2-torsion is an invariant defined for compact L^2-acyclic manifolds of determinant class, for example odd dimensional hyperbolic manifolds. It was introduced by John Lott and Varghese Mathai and computed for hyperbolic manifolds in low dimensions. In this paper we show that the L^2-torsion of hyperbolic manifolds of arbitrary odd dimension does not vanish. This was conjectured by J. Lott and W. Lueck. Some concrete values are computed and an estimate of their growth with the dimension is given.
On the stability of flat complex vector bundles over parallelizable manifolds
2017
We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G / \Gamma$, where $G$ is a complex connected Lie group and $\Gamma$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_\rho$ associated to any irreducible representation $\rho : \Gamma \rightarrow \text{GL}(r,{\mathbb C})$. More precisely, we prove that $E_{\rho}$ is holomorphically isomorphic to a vector bundle of the form $E^{\oplus n}$, where $E$ is a stable vector bundle. All the rational Chern classes of $E$ vanish, in particular, its degree is zero. We deduce a stability result for flat holomorphic vector bundles $E_{\r…