Search results for "Submanifold"
showing 10 items of 23 documents
Partial Stabilization of Input-Output Contact Systems on a Legendre Submanifold
2017
This technical note addresses the structure preserving stabilization by output feedback of conservative input-output contact systems, a class of input-output Hamiltonian systems defined on contact manifolds. In the first instance, achievable contact forms in closed-loop and the associated Legendre submanifolds are analysed. In the second instance the stability properties of a hyperbolic equilibrium point of a strict contact vector field are analysed and it is shown that the stable and unstable manifolds are Legendre submanifolds. In the third instance the consequences for the design of stable structure preserving output feedback are derived: in closed-loop one may achieve stability only rel…
Regular 1-harmonic flow
2017
We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some …
Flots de Smale en dimension 3: présentations finies de voisinages invariants d'ensembles selles
2002
Abstract Given a vector field X on a compact 3-manifold, and a hyperbolic saddle-like set K of that vector field, we consider all the filtering neighbourhood of K: by such, we mean any submanifold which boundary is tranverse to X, the maximal invariant of which is equal to K and which intersection with every orbit of X is connected. Up to topological equivalence, there is only a finite number of such neighbourhoods. We give a finite combinatorial presentation of the global dynamics on any such neighbourhood. A key step is the construction of a unique model of the germ of X along K; this model is, roughly speaking, the simplest three-dimensional manifold and the simplest Smale flow exhibitin…
Topology guaranteeing manifold reconstruction using distance function to noisy data
2006
Given a smooth compact codimension one submanifold S of Rk and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisy-approximation that generalize sampling conditions introduced by Amenta & al. and Dey & al. Our results are based upon critical point theory for distance functions. For the two approximation conditions, we prove that the connected components of the boundary of unions of balls centered on K are isotopic to S. Our results allow to consider balls of different radii. For the first approximation condition, we also prove th…
Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3
1987
AbstractA cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate toWe define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent toOur main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to
On the points realizing the distance to a definable set
2011
Abstract We prove a definable/subanalytic version of a useful lemma, presumably due to John Nash, concerning the points realizing the Euclidean distance to an analytic submanifold of R n . We present a parameter version of the main result and we discuss the properties of the multifunction obtained.
Critical points of higher order for the normal map of immersions in Rd
2012
We study the critical points of the normal map v : NM -> Rk+n, where M is an immersed k-dimensional submanifold of Rk+n, NM is the normal bundle of M and v(m, u) = m + u if u is an element of NmM. Usually, the image of these critical points is called the focal set. However, in that set there is a subset where the focusing is highest, as happens in the case of curves in R-3 with the curve of the centers of spheres with contact of third order with the curve. We give a definition of r-critical points of a smooth map between manifolds, and apply it to study the 2 and 3-critical points of the normal map in general and the 2-critical points for the case k = n = 2 in detail. In the later case we a…
A comparison theorem for the first Dirichlet eigenvalue of a domain in a Kaehler submanifold
1994
AbstractWe give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.
Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF
2015
Let $(\bar{M},,e^\psi)$ be a Riemannian manifold with a density, and let $M$ be a closed $n$-dimensional submanifold of $\bar{M}$ with the induced metric and density. We give an upper bound on the first eigenvalue $\lambda_1$ of the closed eigenvalue problem for $\Delta_\psi$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector ${\vec{H}}_{{\psi}} + {\bar \nabla}$ with respect to the volume form induced by the density, where ${\vec{H}}_{{\psi}}$ is the mean curvature of $M$ associated to the density $e^\psi$. When $\bar{M}=\Bbb R^{n+k}$ or $\bar{M}=S^{n+k-1}$, the equality between $\lambda_1$ and its bound implies that $e^\psi$ is a Gaussian den…
Area of intrinsic graphs and coarea formula in Carnot Groups
2020
AbstractWe consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ C 1 regularity ($$C^1_H$$ C H 1 ). Our first main result is an area formula for $$C^1_H$$ C H 1 intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ C H 1 submanifolds into level sets of a $$C^1_H$$ C H 1 function.