Genericity of nondegenerate critical points and Morse geodesic functionals

Genericity of nondegenerate geodesics with general boundary conditions

Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of Biliotti, Javaloyes and Piccione for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of the product MxM that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal of MxM. Further aspects of these boundary conditions are discussed and general c…

具有柱状奇点的平均曲率流 II：稳定性与一般性

Mean curvature flows with cylindrical singularities II: stability and genericity

本文基于动力系统方法，研究非退化柱状奇点的稳定性，并证明可通过扰动将退化奇点转化为非退化奇点。

On the semi-Riemannian bumpy metric theorem

We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold $M$, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the $C^k$-topology, $k=2,...,\infty$, in the set of metrics of a given index on $M$. A higher order genericity Riemannian result of Klingenberg and Takens is extended to semi-Riemannian geometry.

Morse homology for the heat flow

We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically to nondegenerate closed geodesics of Morse index difference one.

Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2

We use the method of equivariant moving frames to revisit the problem of normal forms and equivalence of nondegenerate real hypersurfaces M \subset C^2 under the pseudo-group action of holomorphic transformations. The moving frame recurrence formulae allow us to systematically and algorithmically recover the results of Chern and Moser for hypersurfaces that are either non-umbilic at a point p \in M or umbilic in an open neighborhood of it. In the former case, the coefficients of the normal form expansion, when expressed as functions of the jet of the hypersurface at the point, provide a comple…

Morse theory on the loop space of flat tori and symplectic Floer theory

We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped with the natural symplectic structure. Both objects are isomorpic to the singular homology of the loop space. In an appendix we perturb the equations in order to eliminate degeneracies and to get to a situation with nondegenerate critical points only. The (co)homology groups turn out to be invariant under the perturbation.