arXiv research

New extensions for homogeneous distributions on deformations to the normal cone.

This is a translation of Yves Benoist's "Actions propres sur les espaces homogènes réductifs", Ann. of Math., 144:315-347, 1996.

Develops Marsden-Meyer-Weinstein reduction for $k$-contact field theories.

We give a survey on Meyer functions, with emphasis on their application to the signatures of fibered 4-manifolds.

Three counterexamples show higher eigenvalue multiplicities than conjectured.

For each d>=2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d=4, using our Meyer function, we can define the local signature for 4-dimensional fiber spaces whose general fibers are non-hyperelliptic comp…

We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system. In particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for classical supermartingales, as well as Peng's (1999) version for right-continuous $g$-supermartingale…

Following the lines of the celebrated Riemannian result of Gromoll and Meyer, we use infinite dimensional equivariant Morse theory to establish the existence of infinitely many geometrically distinct closed geodesics in a class of globally hyperbolic stationary Lorentzian manifolds.

Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.

We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.

Quotients of torus endomorphisms have parabolic orbifolds.

Investigates Meyer risk measures and their applications in finance.

We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with $C$-positive normal curvature, if there is a closed basic 1-form $φ$ such that $Δ_Bφ=qCφ$, then the foliation is transversally isometric to the quotient of a $q$-sphere.

Gromoll and Meyer have represented a certain exotic 7-sphere $Σ^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $Σ^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\geq 1$ wh…

In this paper, we study a type of reflected BSDE with a constraint and introduce a new kind of nonlinear expectation via BSDE with a constraint and prove the Doob-Meyer decomposition with respect to the super(sub)martingale introduced by this nonlinear expectation. We then apply the results to the pricing of American o…

We give an explicit formula for the signature of handlebody bundles over the circle in terms of the homological monodromy. This gives a cobounding function of Meyer's signature cocycle on the mapping class group of a $3$-dimensional handlebody, i.e., the handlebody group. As an application, we give a topological interp…

Two reduction schemes for symplectic manifolds are shown equivalent.

We study a secondary invariant, called the Meyer function, on the fundamental group of the complement of the dual variety of a smooth projective variety. This invariant have played an important role when studying the local signatures of fibered 4-manifolds from topological point of view. As an application of our study,…

We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e. on the standard sphere and Gromoll-Meyer exotic sphere. The standard sphere $S^7_s$ can be constructed as the quotient manifold $\mathrm{Sp}(2, \mathbb{H})/S^3$ with the so-called ${\bullet}$-action of $S^3$, where…

The target of this paper is to establish the bid-ask pricing frame work for the American contingent claims against risky assets with G-asset price systems (see \cite{Chen2013b}) on the financial market under Knight uncertainty. First, we prove G-Dooby-Meyer decomposition for G-supermartingale. Furthermore, we consider …

Study geodesics on nested non-holonomic systems.

Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5…

Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotiente $\widetilde{N}^{11}=(Sp(2)\times S^4)/S^3$, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Further…

We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.

The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly t…

We describe a few abstract principles that are used in the deformation of the Gromoll-Meyer metric to positive curvature.

Werner Meyer constructed a cocycle in $H^2(Sp(2g, \mathbb{Z}); \mathbb{Z})$ which computes the signature of a closed oriented surface bundle over a surface, with fibre a surface of genus g. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this pap…

Golden age of mathematical finance in the late 20th century.

Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes whe…

We construct a co-dimension $3$ completely non-holonomic sub-bundle on the Gromoll-Meyer exotic $7$ sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7…

We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act fre…

In this paper, using functional Steiner symmetrizations, we show that Meyer and Pajor's proof of the Blaschke-Santalo inequality can be extended to the functional setting.

P. Berard and D. Meyer proved a Faber-Krahn inequality for domains in compact manifolds with positive Ricci curvature. We prove stability results for this inequality.

we introduce a generalization of the $p$-Laplace operator to act on differential forms and generalize an estimate of Gallot-Meyer for the first nonzero eigenvalue on closed Riemannian manifolds.

We classify all biquotients whose rational cohomology rings are generated by one element. As a consequence we show that the Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient.

In stochastic control problems delicate issues arise when the controlled system can jump due to both exogenous shocks and endogenous controls. Here one has to specify what the controller knows when about the exogenous shocks and how and when she can act on this information. We propose to use Meyer-$σ$-fields as a flexi…

We introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of the Kirwan convexity theorem, the Meyer-Marsden-Weinstein symplectic reduction theorem, and the Duistermaat-Heckman theorem in this context.

In this paper, we discuss relations among several invariants of 3-manifolds including Meyer's function, the eta-invariant, the von Neumann rho-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.

The central extension of mapping class groups of punctured surfaces of finite type that arises in Chekhov-Fock quantization is 12 times of the Meyer class plus the Euler classes of the punctures, which agree with the one arising in the Kashaev quantization.

Paper constructs exotic spacetimes with same physical properties.

The central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum Teichmüller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is analogous to the result obtained in \cite{FS} for the Thompson groups.

On Kahler manifolds with Ricci curvature bounded from below, we establish some theorems which are counterparts of some classical theorems in Riemannian geometry, for example, Bishop-Gromov's relative volume comparison, Bonnet-Meyers theorem, and Yau's gradient estimate for positive harmonic functions. The tool is a Boc…

Quantization and reduction for coisotropic A-branes on Hamiltonian manifolds.

Survey on smooth function and form density in Riemannian Sobolev spaces.

A physicist explores exotic 7-spheres using fibre bundles and instantons.

Extends a formula for the homomorphism defect of a signature map to coloured braids.

We construct a decomposition of the identity operator on a Riemannian manifold $M$ as a sum of smooth orthogonal projections subordinate to an open cover of $M$. This extends a decomposition of the real line by smooth orthogonal projection due to Coifman, Meyer and Auscher, Weiss, Wickerhauser, and a similar decomposit…